Hyperspace: A Scientific Odyssey through Parallel Universes, Time Warps, and the Tenth Dimension

HYPERSPACE

This page intentionally left blank

HYPERSPACE

A Scientific Odyssey Through Parallel Universes, Time Warps, and The Tenth Dimension Michio Kak u Illustrations by Robert O'Keefe

New Yor k Oxfor d OXFORD UNIVERSIT Y PRESS 1994

Oxford Universit y Press Oxiford Ne w York Toront o Delhi Bomba y Calcutt a Madra s Karach i Kuala Lumpur Singapor e Hon g Kong Toky o Nairobi Da r es Salaam Cap e Tow n Melbourne Aucklan d Madri d and associate d companie s in Berlin Ibada n

Copyright © 199 4 b y Oxford Universit y Press, Inc . Published b y Oxford Universit y Press, Inc. , 200 Madison Avenue , New York, New York 1001 6 Oxford i s a registere d trademar k o f Oxford Universit y Press All rights reserved. N o part o f this publication ma y be reproduced, stored i n a retrieva l system, or transmitted , in an y form o r b y any means, electronic, mechanical, photocopying, recording , o r otherwise, without th e prio r permissio n of Oxford Universit y Press . Library of Congress Cataloging-in-Publicatio n Dat a Kaku, Michio. Hypcrspacc : a scientifi c odyssc y through paralle l universes, time warps, and the tent h dimensio n / Miehio Kaku. p. cm . Includes bibliographica l references an d index . ISBN 0-19-508514- 0 1. Katu/a-Klei n theories . 2 . Supcrstring theories. 3. Hyperspace. I . Title. QC793.3.F5K35 199 4 530. 1 '42—dc20 93-791 0 "Cosmic Gall. " From Telephone. Poles and Other Poems by John Updike . Copyright © 196 0 byjoh n Updike . Reprinted b y permission o f Alfred A . Knopf, Inc. Originally appeared i n Th e New Yorker. Excerpt fro m "Fir e an d Ice. " Fro m Th e Poetry o f Robert Frost, edited b y Edward Conner y Lathem . Copyright 195 1 b y Robert Frost . Copyright 1923 , © 196 9 b y Henry Holt , avid Company , inc . Reprinted b y permission o f Henr y Holt an d Company, Inc .

24 G8 97 531

Printed i n th e Unite d States of America on acid-fre e pape r

This boo k i s dedicated to m y parents

This page intentionally left blank

Preface

Scientific revolutions , almost by definition, def y commo n sense . If al l ou r common-sens e notion s abou t th e univers e wer e correct , then scienc e would have solved the secret s o f the univers e thousands of years ago. The purpos e o f science is to peel back the layer of the appear ance o f objects to reveal their underlyin g nature. I n fact , i f appearanc e and essenc e were th e same thing , ther e woul d be no nee d for science. Perhaps th e mos t deepl y entrenche d common-sens e notio n abou t our worl d i s tha t i t i s thre e dimensional . I t goe s withou t saying tha t length, width, and breadt h suffic e t o describ e al l objects in ou r visibl e universe. Experiments with babies an d animal s have shown that we are born wit h a n innat e sens e tha t ou r worl d i s thre e dimensional . I f we include time as another dimension , then fou r dimension s are sufficien t to recor d al l events in th e universe . No matte r wher e ou r instrument s have probed , fro m dee p within the ato m t o the farthes t reaches o f the galactic cluster, we have onl y found evidenc e of these fou r dimensions . To claim otherwis e publicly , tha t other dimensions might exist o r that our univers e ma y coexist with others, is to invite certain scorn . Yet this deeply ingraine d prejudic e abou t ou r world , firs t speculate d o n b y ancient Gree k philosophers 2 millennia ago, is about to succumb to th e progress o f science. This book is about a scientific revolution created by the theory o f hyperspace,1 whic h states that dimensions exist beyond the commonly accepted four o f spac e an d time . Ther e i s a growin g acknowledgmen t amon g physicists worldwide, including several Nobel laureates, that the universe may actuall y exist in higher-dimensiona l space . I f this theor y i s proved correct, i t will creat e a profound conceptua l an d philosophica l revolu tion in our understandin g o f the universe. Scientifically, th e hyperspac e theory goe s b y the name s of Kaluza-Klein theor y and supergravity . But

viii Preface its mos t advanced formulatio n is called superstrin g theory , which even predicts th e precise number of dimensions: ten. The usual three dimensions o f spac e (length , width, and breadth ) an d on e o f tim e ar e no w extended b y six more spatia l dimensions. We caution tha t th e theor y o f hyperspac e ha s no t ye t been experi mentally confirmed and would , in fact, be exceedingly difficult t o prove in the laboratory. However, the theory has already swept across the majo r physics researc h laboratorie s o f th e worl d an d ha s irrevocabl y altered the scientific landscape of modern physics, generating a staggering number o f researc h paper s i n th e scientifi c literatur e (ove r 5,00 0 b y on e count). However, almost nothing ha s been writte n for th e la y audience to explai n th e fascinatin g propertie s o f higher-dimensiona l space . Therefore, th e genera l publi c is only dimly aware, if at all , of this revolution. In fact , th e glib references to other dimension s an d paralle l universes i n th e popula r cultur e ar e ofte n misleading . Thi s i s regrettabl e because th e theory' s importanc e lie s i n it s powe r t o unif y al l known physical phenomen a i n a n astonishingl y simple framework . This boo k makes available, for th e first time, a scientifically authoritative but accessible account o f the curren t fascinatin g research o n hyperspace . To explain why the hyperspac e theory has generated s o much excitement within the world of theoretical physics , I have developed fou r fundamental theme s tha t run throug h thi s book lik e a thread. Thes e fou r themes divide the boo k int o fou r parts. In Par t I, I develop th e earl y history of hyperspace, emphasizing the theme tha t th e law s of nature becom e simple r an d mor e elegan t when expressed i n higher dimensions . To understan d ho w adding highe r dimension s can simplif y physical problems, conside r th e followin g example : T o th e ancien t Egyptians, the weathe r was a complete mystery . What caused th e seasons ? Why did it get warmer as they traveled south? Why did th e wind s generally blow in one direction? The weather was impossible to explain from the limited vantage point of the ancient Egyptians, to whom the earth appeared flat , like a two-dimensional plane. But now imagine sending the Egyptians in a rocke t int o oute r space , where the y can se e th e eart h a s simple and whole in its orbit aroun d th e sun . Suddenly, the answer s to thes e ques tions become obvious . From oute r space , i t i s clear tha t th e earth' s axi s i s tilted abou t 2 3 degrees from th e vertical (th e 'vertical " bein g th e perpendicular t o the plane of the earth's orbit around th e sun). Because of this tilt, the north ern hemispher e receive s much less sunlight during on e par t of its orbit than during another part. Hence we have winter and summer. And since

Preface i

x

the equator receive s more sunligh t then th e northern o r southern pola r regions, it becomes warmer as we approach th e equator . Similarly , since the eart h spin s counterclockwise to someone sittin g on th e nort h pole , the cold , pola r ai r swerve s as it move s south towar d th e equator . Th e motion o f hot an d col d masse s of air, set in motion b y the earth' s spin, thus help s t o explai n wh y the wind s generall y blo w in on e direction , depending o n where you are on th e earth . In summary, the rather obscure laws of the weather are easy to understand onc e we view the earth from space. Thus th e solution t o the prob lem is to go u p into space, into the third dimension. Facts that were impossible t o understan d i n a fla t worl d suddenl y becom e obviou s whe n viewing a three-dimensional earth . Similarly, th e law s o f gravit y and ligh t seem totall y dissimilar. They obey differen t physica l assumption s an d differen t mathematics . Attempts t o splic e thes e tw o force s hav e alway s failed . However, if we add one mor e dimension , a fifth dimension, to the previous four dimensions of space an d time , then th e equation s governin g light and gravity appear t o merg e togethe r lik e tw o pieces o f a jigsaw puzzle . Light , i n fact, ca n b e explaine d a s vibrations i n th e fift h dimension . I n thi s way, we see tha t th e law s of light and gravit y become simple r in fiv e dimen sions. Consequently, many physicists are now convinced that a conventional four-dimensional theor y is "too small " to describe adequately the forces that describe ou r universe . In a four-dimensional theory , physicists have to squeeze together th e forces of nature i n a clumsy, unnatural fashion . Furthermore, this hybrid theory is incorrect. Whe n expressed i n dimen sions beyon d four , however , w e hav e "enoug h room " t o explai n th e fundamental force s in an elegant , self-containe d fashion. In Part II, we further elaborate o n thi s simple idea, emphasizing tha t the hyperspac e theor y may be able to unify all known laws of nature int o one theory . Thu s th e hyperspac e theor y ma y be th e crownin g achieve ment o f 2 millenni a o f scientifi c investigation : th e unificatio n o f al l known physical forces. It may give us the Hol y Grail of physics, the "the ory of everything" tha t elude d Einstei n for s o many decades . For th e pas t half :century, scientists have been puzzle d as to why the basic force s tha t hol d togethe r th e cosmos—gravity , electromagnetism, and th e stron g an d wea k nuclear forces—diffe r s o greatly. Attempts by the greatest mind s of the twentieth century to provide a unifying pictur e of al l th e know n force s hav e failed . However , th e hyperspac e theor y allows th e possibilit y of explainin g th e fou r force s o f natur e a s well a s the seemingly random collectio n of subatomic particles in a truly elegant

X Preface

fashion. I n th e hyperspac e theory , "matter " ca n b e als o viewed as th e vibrations that ripple through th e fabric of space and time. Thus follow s the fascinatin g possibilit y that everythin g we see aroun d us , fro m th e trees an d mountain s to th e star s themselves, are nothin g but vibrations in hyperspace. I f thi s i s true, the n thi s give s u s a n elegant , simple , an d geometric mean s o f providing a coherent an d compellin g descriptio n of the entir e universe. In Par t III , we explore th e possibilit y that, unde r extrem e circum stances, spac e ma y be stretche d unti l i t rip s o r tears . I n othe r words , hyperspace ma y provide a mean s t o tunne l throug h spac e an d time . Although we stress that thi s is still highly speculative, physicists are seriously analyzing the propertie s o f "wormholes, " o f tunnels that link distant parts of space and time . Physicists at the California Institute of Technology, for example, hav e seriously proposed the possibilit y of building a tim e machine , consistin g o f a wormhole tha t connect s th e pas t wit h the future . Time machine s hav e now left th e real m o f speculation an d fantasy an d hav e become legitimat e fields of scientific research . Cosmologists hav e eve n propose d th e startlin g possibilit y tha t ou r universe i s just on e amon g a n infinit e numbe r o f paralle l universes . These universes might be compared to a vast collection of soap bubbles suspended i n air . Normally , contact betwee n thes e bubbl e universe s is impossible, but , b y analyzin g Einstein' s equations , cosmologist s hav e shown that there might exist a web of wormholes, or tubes, that connec t these paralle l universes . On eac h bubble , we can defin e our own distinctive space and time , which have meaning onl y on its surface; outsid e these bubbles , space and tim e have no meaning . Although man y consequences o f this discussion ar e purel y theoretical, hyperspac e trave l may eventually provide th e mos t practica l application of all: to sav e intelligent life , includin g ours , from the deat h of the universe . Scientists universally believe tha t th e univers e must eventually die, an d wit h i t all life tha t ha s evolve d over billions of years. For example, accordin g to the prevailin g theory, called the Big Bang, a cosmic explosio n 1 5 to 2 0 billio n year s ag o se t th e univers e expanding , hurling star s and galaxie s awa y from us at grea t velocities. However, if the univers e on e da y stop s expandin g an d begin s t o contract , i t wil l eventually collapse into a fiery cataclysm called the Big Crunch, in which all intelligent life wil l be vaporized by fantastic heat. Nevertheless, some physicists hav e speculate d tha t th e hyperspac e theor y ma y provide th e one an d onl y hope of a refuge for intelligent life. I n th e las t seconds of the deat h o f ou r universe , intelligen t lif e ma y escape th e collaps e by fleeing into hyperspace.

Preface x

i

In Part IV , we conclude wit h a final, practical question: If the theor y is proved correct, then when will we be able t o harness th e powe r o f the hyperspace theory? This is notjust a n academic question, because i n the past, th e harnessin g o f just one o f the fou r fundamenta l forces irrevo cably changed th e course of human history, lifting us from th e ignoranc e and squalo r o f ancient, preindustrial societie s to modern civilization . In some sense , eve n th e vas t sweep o f huma n histor y can b e viewe d i n a new light, in terms of the progressiv e mastery of each of the fou r forces. The histor y of civilization has undergone a profound chang e a s each of these force s was discovered an d mastered . For example , whe n Isaa c Newto n wrot e dow n th e classica l laws o f gravity, h e develope d th e theor y o f mechanics, which gave us th e law s governing machines. This, in turn, greatly accelerated th e Industrial Revolution, whic h unleashed politica l force s that eventually overthrew th e feudal dynasties of Europe. I n the mid-1860s, when James Clerk Maxwell wrote down the fundamenta l law s of the electromagneti c force , he ushered i n th e Electri c Age, which gave u s th e dynamo , radio, television , radar, household appliances , the telephone, microwaves , consumer electronics, the electronic computer , lasers, and many other electroni c marvels. Without th e understandin g an d utilizatio n o f the electromagneti c force, civilizatio n would have stagnated, froze n i n a time befor e th e discovery of the ligh t bulb an d th e electri c motor. I n th e mid-1940s , when the nuclea r forc e wa s harnessed, th e worl d wa s again turne d upsid e down wit h th e developmen t o f th e atomi c an d hydroge n bombs , th e most destructive weapons on the planet. Because we are riot on the verge of a unifie d understandin g o f all the cosmi c force s governing th e uni verse, one migh t expect tha t any civilization that masters the hyperspac e theory will become lor d o f the universe . Since th e hyperspac e theor y is a well-defined body of mathematical equations, w e can calculat e th e precis e energ y necessar y to twis t spac e and tim e int o a pretze l or t o creat e wormhole s linking distant parts of our universe . Unfortunately , the result s ar e disappointing . Th e energ y required fa r exceed s anythin g tha t ou r plane t ca n muster . In fact , th e energy i s a quadrillion times larger tha n th e energ y of our larges t ato m smashers. We must wait centuries or even millennia until our civilization develops th e technica l capability of manipulating space—time , o r hop e for contac t wit h a n advance d civilizatio n tha t ha s alread y mastere d hyperspace. Th e boo k therefor e end s b y exploring th e intriguin g bu t speculative scientifi c question o f wha t leve l o f technolog y i s necessary for u s to become masters o f hyperspace . Because the hyperspace theory takes us far beyond normal, common-

xii Preface sense conception s o f spac e an d time , I hav e scattere d throughou t th e text a few purely hypothetical stories . I was inspired t o utiliz e thi s pedagogical techniqu e b y th e stor y o f Nobe l Priz e winne r Isidor e I . Rab i addressing a n audienc e o f physicists. He lamente d th e abysma l state of science educatio n i n th e Unite d State s an d scolde d th e physic s com munity for neglectin g it s duty in popularizin g th e adventur e o f science for th e genera l publi c and especiall y for the young . In fact , h e admon ished, science-fictio n writer s ha d don e mor e t o communicat e th e romance o f science than al l physicists combined. In a previou s book, Beyond Einstein: Th e Cosmic Quest for th e Theory o f the Universe (coauthore d wit h Jennifer Trainer) , I investigate d super string theory, described th e nature o f subatomic particles, and discusse d at length the visible universe and ho w all the complexitie s of matter might be explaine d b y tiny, vibrating strings. In thi s book, I have expanded o n a differen t them e an d explore d th e invisible universe—that is, the worl d of geometry and space-time . The focu s o f this book i s not th e natur e o f subatomic particles , bu t th e higher-dimensiona l worl d i n whic h the y probably live . I n th e process , reader s wil l se e tha t higher-dimensiona l space, instead of being an empty , passive backdrop agains t which quarks play ou t thei r eterna l roles , actuall y becomes th e centra l acto r i n th e drama o f nature. In discussing the fascinatin g history of the hyperspac e theory, we will see tha t th e searc h fo r th e ultimat e natur e o f matter , begu n b y th e Greeks 2 millennia ago , ha s bee n a lon g an d tortuou s one . Whe n th e final chapte r i n thi s long sag a is written b y future historians o f science, they ma y well recor d tha t th e crucia l breakthrough wa s the defea t o f common-sense theorie s o f three o r fou r dimension s an d th e victor y of the theor y o f hyperspace. New York M.K

May 1993

.

Acknowledgments

In writin g this book, I hav e been fortunat e t o hav e Jeffrey Robbin s as my editor. He was the editor who skillfully guided the progress of three of my previous textbooks i n theoretica l physic s written for th e scientific community, concerning the unified field theory, superstring theory, and quantum fiel d theory . This book, however , marks th e first popular science boo k aime d a t a general audienc e tha t I have written for him . It has always been a rare privileg e to work closely with him. I would als o like to than k Jennife r Trainer, wh o has been m y coauthor o n tw o previous popular books . Onc e again , sh e ha s applied he r considerable skill s to make the presentatio n a s smooth an d coheren t as possible. I a m als o gratefu l t o numerou s othe r individual s who hav e helpe d to strengthe n an d criticiz e earlier draft s o f thi s book : Bur t Solomon , Leslie Meredith, Eugene Mallove , and m y agent, Stuart Krichevsky. Finally, I woul d lik e t o than k th e Institut e fo r Advance d Stud y at Princeton, where muc h o f this book wa s written, for it s hospitality. The Institute, where Einstein spent th e las t decades o f his life, was an appro priate plac e t o writ e abou t th e revolutionar y development s tha t hav e extended an d embellishe d much o f his pioneering work.

This page intentionally left blank

Contents

Part I Enterin g the Fifth Dimensio n 1. World s Beyond Spac e an d Time , 3 2. Mathematician s and Mystics , 30 3. Th e Ma n Who "Saw " th e Fourt h Dimension, 55 4. Th e Secre t o f Light: Vibrations i n th e Fift h Dimension , 8 0

Part I I Unificatio n in Ten Dimensions 5. Quantu m Heresy , 111 6. Einstein' s Revenge , 136 7. Superstrings , 151 8. Signal s from th e Tent h Dimension, 178 9. Befor e Creation , 19 1

xvi Contents Part II I Wormholes : Gateway s t o Anothe r Universe ? 10. Blac k Holes and Paralle l Universes , 21 7 11. T o Buil d a Time Machine , 232 12. Collidin g Universes, 252

Part I V Master s o f Hyperspac e 13. Beyon d th e Future , 273 14. Th e Fat e of th e Universe , 301 15. Conclusion , 31 3 Notes, 33 5 References an d Suggested Reading , 353 Index, 35 5

PART I Entering the Fift h Dimensio n

But the creative principle resides in mathematics. In a certain sense, therefore , I hol d i t tru e tha t pur e though t ca n gras p reality, as the ancient s dreamed . Albert Einstein

1 Worlds Beyon d Space and Time I want to know how God created this world. I am not intereste d in thi s or that phenomenon. I want to know His thoughts, th e rest are details . Albert Einstein

The Educatio n of a Physicist

T

WO incidents from m y childhood greatl y enriched m y understanding o f th e worl d an d sen t m e o n cours e t o becom e a theoretica l physicist. I remembe r tha t m y parents woul d sometimes take m e t o visi t th e famous Japanese Te a Garde n i n Sa n Francisco . On e o f m y happies t childhood memorie s i s of crouching nex t to th e pond , mesmerize d by the brilliantl y colored car p swimming slowly beneath th e water lilies. In thes e quie t moments, I felt fre e t o le t m y imagination wander ; I would ask myself silly questions that a only child might ask, such as how the car p i n tha t pon d woul d view th e worl d aroun d them . I thought , What a strange world theirs must be! Living their entir e live s in th e shallo w pond, th e car p would believe that their "universe" consiste d of the murky water and th e lilies. Spending most of their tim e foraging on th e botto m o f the pond , the y would be onl y dimly aware that a n alie n worl d could exis t above th e surface.

3

4 ENTERIN

G TH E FIFTH DIMENSIO N

The natur e o f m y worl d wa s beyon d thei r comprehension . I wa s intrigued that I could sit only a few inches from th e carp, yet be separated from the m b y an immense chasm . The car p an d I spent our lives in two distinct universes, never enterin g each other' s world, yet were separated by only the thinnes t barrier, th e water' s surface. I onc e imagine d tha t ther e ma y be car p "scientists " livin g among the fish . The y would , I thought , scof f a t an y fish who propose d tha t a parallel world could exist just above the lilies . To a carp "scientist, " th e only thing s tha t were rea l wer e what the fis h coul d se e o r touch . Th e pond wa s everything. An unseen worl d beyon d th e pon d mad e n o scientific sense . Once I was caught in a rainstorm. I noticed that the pond's surfac e was bombarde d b y thousand s o f tin y raindrops . Th e pond' s surfac e became turbulent , and th e wate r lilies were being pushe d i n al l directions by water waves. Taking shelter from th e wind and th e rain , I wondered ho w all this appeared t o the carp . To them, the water lilies would appear t o b e movin g aroun d b y themselves, without anythin g pushing them. Sinc e th e wate r the y lived i n woul d appea r invisible , much lik e the ai r and spac e around us , they would be baffle d tha t th e wate r lilies could mov e around b y themselves. Their "scientists, " I imagined , woul d concoc t a cleve r inventio n called a "force " i n orde r t o hid e thei r ignorance . Unabl e t o compre hend tha t ther e coul d be waves on th e unsee n surface , they would conclude that lilies could move without being touched because a mysterious, invisible entity called a force acted betwee n them . They might give this illusion impressive, lofty name s (suc h as action-at-a-distance, or th e ability of the lilie s to move without anything touching them) . Once I imagined wha t would happe n i f I reache d dow n an d lifte d one o f the car p "scientists " ou t o f the pond . Befor e I threw him back into the water, he migh t wiggle furiously as I examined him . I wondered how thi s would appea r t o th e res t o f the carp . T o them , i t would be a truly unsettling event. They would first notice that on e o f their "scien tists" ha d disappeare d fro m thei r universe . Simpl y vanished, without leaving a trace. Whereve r they would look, there woul d be no evidence of the missin g carp i n their universe. Then, second s later, when I threw him back into th e pond, th e "scientist " would abruptly reappear ou t of nowhere. T o th e othe r carp , i t would appea r tha t a miracl e ha d hap pened. After collectin g his wits , th e "scientist " woul d tel l a trul y amazin g story. "Withou t warning, " h e woul d say, "I wa s somehow lifted ou t o f the universe (the pond) an d hurled into a mysterious nether world, with

Worlds Beyond Space and Time 5 blinding lights and strangely shaped object s that I had never seen before . The stranges t o f all was the creatur e wh o held m e prisoner, wh o did no t resemble a fish in th e slightest . I was shocked t o see that i t had n o fins whatsoever, but nevertheless coul d mov e without them. It struck me that the familiar laws of nature n o longer applie d i n this nether world . Then , just a s suddenly, I found mysel f thrown bac k int o ou r universe. " (Thi s story, o f course, o f a journey beyond th e univers e would be s o fantastic that mos t o f the car p woul d dismiss it as utter poppycock. ) I often thin k that we are like the car p swimmin g contentedly i n tha t pond. We live out ou r live s in our ow n "pond," confident that our universe consist s of onl y thos e thing s w e can se e o r touch . Lik e th e carp , our univers e consist s o f onl y th e familia r an d th e visible . W e smugly refuse t o admi t tha t paralle l universe s o r dimension s ca n exis t nex t t o ours, just beyond ou r grasp . If our scientist s invent concepts like forces, it i s only because the y canno t visualiz e the invisibl e vibrations tha t fill the empt y spac e aroun d us . Som e scientist s snee r a t th e mentio n o f higher dimension s becaus e the y canno t b e convenientl y measured i n the laboratory . Ever since that time, I have been fascinated by the possibility of other dimensions. Lik e mos t children , I devoured adventur e storie s i n which time traveler s entere d othe r dimension s an d explore d unsee n paralle l universes, wher e th e usua l law s o f physic s coul d b e convenientl y suspended. I grew up wondering i f ships that wandered int o th e Bermud a Triangle mysteriousl y vanished int o a hole i n space; I marveled a t Isaa c Asimov's Foundation Series , i n which the discover y of hyperspace travel led t o the ris e o f a Galactic Empire . A secon d inciden t fro m m y childhoo d als o mad e a deep , lastin g impression o n me . Whe n I was 8 years old, I heard a stor y that woul d stay with me for the res t of my life. I remember m y schoolteachers tellin g the clas s abou t a grea t scientis t who ha d just died . The y talke d abou t him with great reverence , callin g him one o f the greatest scientist s in all history. They said tha t ver y few people coul d understan d hi s ideas, bu t that his discoveries changed th e entir e world and everythin g around us . I didn' t understan d muc h o f what they were tryin g to tel l us , but wha t most intrigue d m e abou t thi s ma n wa s that h e die d befor e h e coul d complete hi s greatest discovery . They said he spent years on this theory , but h e die d with his unfinished papers stil l sitting on hi s desk. I wa s fascinated b y th e story . T o a child , thi s wa s a grea t mystery . What wa s his unfinishe d work ? What wa s in thos e paper s o n hi s desk? What proble m coul d possibl y be so difficult an d s o important tha t suc h a great scientis t would dedicate year s of his life t o its pursuit? Curious, I

6 ENTERIN

G T H E F I F T H D I M E N S I ON

decided t o lear n al l I coul d abou t Alber t Einstei n and hi s unfinished theory. I still have warm memories of spending many quiet hours reading every book I could fin d abou t thi s grea t ma n an d hi s theories. Whe n I exhausted th e book s i n our loca l library, I began t o scour libraries and bookstores acros s th e city , eagerl y searchin g fo r mor e clues . I soo n learned tha t thi s story was far mor e excitin g tha n an y murder mystery and mor e importan t tha n anythin g I could ever imagine. I decided tha t I would try to get t o th e roo t of this mystery, even if I had t o become a theoretical physicis t to do it. I soon learne d tha t the unfinished papers on Einstein's desk were an attempt t o construc t wha t he calle d th e unifie d fiel d theory , a theor y that coul d explai n al l the law s o f nature , fro m th e tinies t atom t o th e largest galaxy. However, being a child, I didn't understand tha t perhaps there was a link between the car p swimming in the Tea Garden an d th e unfinished paper s lyin g on Einstein' s desk . I didn' t understan d tha t higher dimension s might be th e ke y to solving the unifie d field theory. Later, in high school, I exhausted mos t of the local libraries and often visited th e Stanfor d Universit y physics library. There, I came acros s th e fact tha t Einstein' s work made possibl e a new substance calle d antimatter, whic h would ac t lik e ordinar y matte r bu t woul d annihilat e upo n contact wit h matter i n a burs t of energy. I also read tha t scientist s had built larg e machines , o r "ato m smashers, " tha t coul d produc e micro scopic quantities of this exotic substance in th e laboratory . One advantag e of youth is that it is undaunted b y worldly constraints that woul d ordinarily seem insurmountabl e t o most adults . Not appre ciating the obstacle s involved, I set out t o build m y own atom smasher . 1 studied th e scientifi c literatur e until I was convinced that I could build what was called a betatron, whic h could boos t electron s t o million s of electron volts . ( A million electro n volt s i s the energ y attaine d b y electrons accelerated b y a field of a million volts.) First, I purchased a small quantity of sodium-22, which is radioactive and naturall y emits positrons (th e antimatter counterpar t of electrons). Then 1 built wha t is called a clou d chamber , whic h makes visible the tracks left by subatomic particles. I was able to take hundreds of beautiful photographs o f the track s left behin d b y antimatter. Next , I scavenge d around larg e electronic warehouses in the area, assembled the necessary hardware, including hundreds of pounds o f scrap transformer steel, and built a 2.3-million-electron-volt betatron i n my garage that would be powerful enough to produce a beam of antielectrons. To construct the monstrous magnet s necessar y fo r th e betatron , I convince d m y parents t o help me wind 22 miles of cooper wire on th e high-schoo l football field.

Worlds Beyond Space and Time 7

We spen t Christma s vacation o n th e 50-yar d line , winding and assem bling th e massiv e coil s tha t woul d ben d th e path s o f th e high-energ y electrons. When finall y constructed , th e 300-pound , 6-kilowat t betatro n con sumed every ounce o f energy my house produced . Whe n I turned i t on, I would usually blow every fuse, an d th e hous e woul d suddenl y becam e dark. Wit h th e hous e plunge d periodicall y int o darkness , m y mothe r would ofte n shak e he r head . ( I imagined tha t sh e probabl y wondere d why she couldn't hav e a child who played baseball or basketball, instead of building these hug e electrica l machines in the garage. ) I was gratified that th e machin e successfull y produce d a magneti c fiel d 20,00 0 time s more powerfu l tha n th e earth' s magneti c field , whic h i s necessary t o accelerate a beam o f electrons. Confronting th e Fift h Dimensio n Because my family was poor, m y parents wer e concerned tha t I wouldn't be able to continue m y experiments and m y education. Fortunately, the awards tha t I won for m y various scienc e project s caught th e attentio n of the atomi c scientist Edward Teller. Hi s wife generousl y arranged fo r me t o receiv e a 4-year scholarship t o Harvard , allowin g me t o fulfill m y dream. Ironically, although a t Harvar d I began m y formal training in theo retical physics, it was also where my interest i n higher dimension s grad ually die d out . Lik e other physicists , I began a rigorou s an d thoroug h program o f studyin g the highe r mathematic s o f eac h o f th e force s of nature separately , in complete isolation from on e another. I still remember solvin g a proble m i n electrodynamic s for m y instructor, and the n asking him wha t the solutio n migh t look like if space were curved i n a higher dimension . H e looke d a t m e i n a strange way , as if I were a bi t cracked. Lik e others befor e me , I soon learne d t o put asid e m y earlier, childish notions about higher-dimensional space . Hyperspace, I was told, was not a suitable subject of serious study. I was never satisfied with this disjointed approach t o physics, and m y thoughts would often drift back to the the carp living in the Tea Garden . Although th e equation s we used fo r electricit y and magnetism , discovered by Maxwell in the nineteenth century , worked surprisingly well, the equations seeme d rathe r arbitrary . I fel t tha t physicist s (like th e carp ) invented thes e "forces " t o hide our ignoranc e o f how objects can move each othe r withou t touching .

8 ENTERIN

G TH E FIFTH DIMENSIO N

In m y studies, 1 learned tha t on e o f th e grea t debate s o f th e nine teenth centur y ha d bee n abou t ho w ligh t travel s throug h a vacuum . (Light from the stars, in fact, can effortlessly travel trillions upon trillions of miles through th e vacuu m o f outer space.) Experiment s als o showed beyond questio n tha t ligh t i s a wave . But i f ligh t wer e a wave , then i t would require something t o be "waving." Soundwave s require air, water waves require water , but sinc e there i s nothing to wave in a vacuum, we have a paradox. Ho w can ligh t be a wave if there is nothing t o wave? So physicists conjure d u p a substanc e calle d th e aether , whic h fille d th e vacuum and acte d a s the mediu m fo r light. However, experiments con clusively showed tha t th e "aether " does no t exist. * Finally, when I became a graduate studen t in physics at the University of California at Berkeley , I learned quit e b y accident tha t ther e was an alternative, albei t controversial , explanatio n o f ho w ligh t ca n trave l through a vacuum . Thi s alternativ e theor y wa s s o outlandis h tha t I received quit e a jolt whe n I stumbled acros s it . That shock was similar to th e on e experience d b y many Americans when the y first heard tha t President John Kennedy had bee n shot . They can invariably remember the precise moment whe n they heard th e shocking news, what they were doing, and t o whom they were talking at that instant. We physicists, too, receive quit e a shock when we first stumbl e across Kalu/a-Klein theor y for th e firs t time . Sinc e the theor y was considered t o be a wild specula tion, i t was never taugh t i n graduat e school ; s o young physicists are lef t to discover i t quite b y accident i n thei r casua l readings . This alternativ e theor y gave the simples t explanation o f light: that i t was really a vibration o f the fift h dimension , o r what used t o called th e fourth dimensio n b y the mystics . If light could trave l through a vacuum, it was because th e vacuu m itsel f was vibrating, becaus e th e "vacuum " really existe d i n fou r dimension s o f spac e an d on e o f time . B y adding the fift h dimension , th e forc e of gravity and ligh t coul d b e unifie d in a startlingly simple way. Looking bac k at my childhood experience s a t th e Tea Garden , I suddenly realize d tha t thi s was the mathematica l theor y for which I had bee n looking . The ol d Kaluza-Klein theory, however, had man y difficult, technica l problems tha t rendere d i t useless for ove r hal f a century. All this, however, ha s change d i n th e pas t decade . Mor e advance d version s o f th e theory, lik e supergravity theory an d especiall y superstring theory , have *Surprisingly, eve n toda y physicists stil l d o no t hav e a rea l answe r to thi s puzzle, bu t over th e decade s w e have simply gotte n use d t o th e ide a tha t ligh t can trave l through a vacuum eve n if there i s nothing to wave.

Worlds Beyond Space and Time 9

finally eliminated the inconsistencies of the theory . Rather abruptly, the theory of higher dimension s is now being champione d i n researc h laboratories aroun d th e globe . Man y of the world' s leading physicists now believe that dimensions beyond th e usua l four of space and tim e might exist. This idea, in fact, ha s become th e foca l point of intense scientifi c investigation. Indeed , man y theoretica l physicist s no w believ e tha t higher dimensions may be the decisive step in creating a comprehensiv e theory that unites the law s of nature—a theory of hyperspace. If it proves to be correct , the n futur e historian s o f science may well record that one of the great conceptual revolutions in twentieth-century science was the realization that hyperspace may be the key to unlock the deepest secret s of nature an d Creatio n itself . This seminal concept has sparked an avalanche of scientific research: Several thousan d paper s writte n b y theoretical physicist s i n th e majo r research laboratorie s around th e world have been devote d t o exploring the propertie s of hyperspace . The page s of Nuclear Physics and Physics Letters, tw o leading scientifi c journals, hav e bee n floode d wit h article s analyzing the theory . Mor e than 20 0 international physics conferences have been sponsored to explore the consequences of higher dimensions. Unfortunately, we are still far from experimentall y verifying tha t ou r universe exist s in highe r dimensions . (Precisel y what it woul d tak e t o prove th e correctnes s o f the theor y an d possibl y harness th e powe r of hyperspace wil l b e discusse d late r i n thi s book. ) However , this theor y has no w becom e firml y establishe d a s a legitimat e branc h o f moder n theoretical physics . The Institut e fo r Advance d Study at Princeton, fo r example, wher e Einstei n spen t th e las t decades o f his lif e (an d wher e this book was written), is now one of the activ e centers of research on higher-dimensional space-time . Steven Weinberg, who won the Nobe l Prize in physics in 1979 , summarized thi s conceptual revolutio n when h e commente d recentl y that theoretical physic s seems t o be becomin g mor e an d mor e lik e science fiction. Why Can't W e Se e Highe r Dimensions ? These revolutionar y idea s see m strang e a t firs t becaus e w e tak e fo r granted that our everyda y world has three dimensions . As the late physicist Heinz Pagels noted, "One featur e of our physical world is so obvious that most people are not even puzzled by it—the fact that space is threedimensional."1 Almost by instinct alone, we know that any object can be

10 E N T E R I N

G TH E F I F T H D I M E N S I O N

described b y giving it s height, width , and depth . B y giving three num bers, we can locat e an y position i n space . I f we want to mee t someon e for lunc h i n Ne w York, we say, "Meet me o n th e twenty-fourt h floor o f the building at the corner of Forty-second Street and First Avenue." Two numbers provid e us the street corner ; an d th e third , th e height off the ground. Airplane pilots , too, kno w exactly where the y are wit h thre e num bers—their altitude and tw o coordinates tha t locate thei r position o n a grid o r map . I n fact , specifyin g thes e thre e number s ca n pinpoin t any location in our world, from th e ti p of our nos e t o the end s of the visible universe. Even babies understand this : Tests with infants have shown that they will crawl to the edg e o f a cliff, pee r over the edge , and craw l back. In additio n t o understandin g "left " an d "right " an d "forward " an d "backward" instinctively , babie s instinctivel y understan d "up " an d "down." Thus the intuitive concept of three dimensions is firmly embedded i n our brain s fro m a n earl y age. Einstein extended thi s concept t o include time as the fourth dimension. For example, to meet that someone for lunch, we must specify tha t we shoul d mee t at , say , 12:30 P.M . i n Manhattan ; tha t is , t o specif y a n event, we also need t o describ e it s fourth dimension , the time at which the even t takes place. Scientists today are interested in going beyond Einstein's conception of the fourt h dimension. Current scientific interes t centers on th e fifth dimension (th e spatia l dimensio n beyon d tim e an d th e thre e dimen sions of space) and beyond . (T o avoid confusion, throughout thi s book I hav e bowe d t o custo m an d calle d th e fourt h dimensio n th e spatial dimension beyon d length , breadth , an d width . Physicists actually refer to thi s as the fift h dimension , but I will follo w historica l precedent. We will call time the fourt h temporal dimension.) How do we see the fourt h spatial dimension? The proble m is , we can't. Higher-dimensiona l space s are impossible to visualize ; s o it is futile eve n t o try . The prominen t Germa n physicist Hermann vo n Helmholt z compare d th e inabilit y to "see " th e fourt h dimension with th e inabilit y of a blind ma n t o conceive of the concep t of color. No matter how eloquently we describe "red" t o a blind person, words fail to impart the meaning of anything as rich in meaning as color. Even experience d mathematician s and theoretica l physicist s who have worked with higher-dimensional space s for years admit that they cannot visualize them. Instead, they retreat into the world of mathematical equations. Bu t whil e mathematicians , physicists , an d computer s hav e n o problem solvin g equations i n multidimensiona l space , human s fin d i t impossible to visualize universes beyond thei r own.

Worlds Beyond Space an d Time 1

1

At best, we can use a variety of mathematical tricks , devised by mathematician an d mysti c Charles Hinton a t the tur n o f the century, to visualize shadows of higher-dimensional objects . Other mathematicians, like Thomas Banchoff , chairman o f the mathematic s departmen t a t Brown University, have written computer programs tha t allow us to manipulat e higher-dimensional object s by projecting thei r shadow s ont o flat , twodimensional compute r screens . Like the Gree k philosopher Plato, who said that we are like cave dwellers condemned t o see only the dim , gray shadows of th e ric h lif e outsid e ou r caves , Banchoff s computers allow only a glimpse of the shadow s of higher-dimensional objects. (Actually, we cannot visualize higher dimension s because o f an acciden t of evolution. Ou r brain s hav e evolve d t o handl e myria d emergencies i n thre e dimensions. Instantly , without stopping t o think, we can recognize an d react t o a leapin g lio n o r a chargin g elephant . I n fact , thos e human s who coul d bette r visualiz e ho w object s move, turn , an d twis t i n thre e dimensions had a distinct survival advantage ove r those who could not . Unfortunately, ther e wa s no selectio n pressur e place d o n human s t o master motio n i n fou r spatia l dimensions. Being able t o see the fourth spatial dimensio n certainl y did no t hel p someon e fen d of f a chargin g saber-toothed tiger . Lion s an d tiger s d o no t lung e a t u s throug h th e fourth dimension. ) The Law s of Natur e Are Simple r in Highe r Dimension s One physicis t who delights in teasing audiences about the properties of higher-dimensional universes is Peter Freund, a professor of theoretical physics at the Universit y of Chicago's renowned Enrico Fermi Institute. Freund wa s one o f the earl y pioneers workin g on hyperspac e theorie s when it was considered to o outlandish fo r mainstream physics. For years, Freund an d a small group o f scientists dabbled in the science of higher dimensions in isolation; now, however, it has finally become fashionable and a legitimate branch o f scientific research . To his delight, he is finding tha t his early interest is at last paying off. Freund doe s not fit the traditiona l imag e of a narrow, crusty, disheveled scientist . Instead, h e is urbane, articulate , an d cultured , an d ha s a sly, impis h gri n tha t captivate s nonscientist s with fascinating stories o f fast-breaking scientifi c discoveries . He i s equally at ease scribbling on a blackboard littere d with dense equations o r exchanging light banter a t a cocktail party. Speaking with a thick, distinguished Romanian accent, Freund ha s a rar e knac k for explainin g th e mos t arcane , convolute d concepts o f physics in a lively, engaging style .

!2 E N T E R I N G T H E F I F T H D I M E N S I O

N

Traditionally, Freun d remind s us , scientist s hav e viewe d highe r dimensions with skepticism because they could not be measured and did not hav e an y particula r use . However , th e growin g realizatio n amon g scientists toda y i s tha t an y three-dimensiona l theor y i s "to o small " t o describe th e force s that govern ou r universe . As Freund emphasizes, one fundamental theme running through th e past decade o f physic s ha s bee n tha t th e laws o f nature become simpler an d elegant when expressed i n higher dimensions, whic h i s thei r natura l home . The law s of light and gravit y fin d a natural expressio n when expressed in higher-dimensiona l space-time. The ke y step in unifyin g th e law s of nature is to increase the number o f dimensions of space-time unti l more and mor e force s can be accommodated . I n higher dimensions , we have enough "room " to unif y al l known physical forces. Freund, in explaining why higher dimension s are exciting the imagination o f the scientifi c world, uses the followin g analogy: "Think, for a moment, o f a cheetah , a sleek, beautiful animal, on e o f th e fastes t o n earth, which roams freely on the savannas of Africa. I n its natural habitat, it is a magnificent animal, almost a work of art, unsurpassed i n speed o r grace by any other animal . Now," h e continues , think o f a cheeta h tha t ha s been capture d an d throw n int o a miserabl e cage in a zoo. It has lost its original grace and beauty , and i s put o n display for ou r amusement . We see onl y th e broke n spiri t o f the cheeta h i n th e cage, not it s original power and elegance . The cheeta h can be compared to th e law s o f physics , whic h ar c beautifu l i n thei r natura l setting . Th e natural habita t o f th e law s o f physic s i s higher-dimensional space-time . However, we ca n onl y measure th e law s o f physics when the y hav e bee n broken an d place d o n displa y i n a cage , whic h i s our three-dimensiona l laboratory. We only see the cheeta h when it s grace and beaut y have been stripped away. 2 For decades, physicist s have wondered wh y the fou r force s o f nature appear t o b e s o fragmented—wh y th e "cheetah " look s so pitifu l an d broken i n his cage. The fundamental reason wh y these four forces seem so dissimilar, notes Freund , i s that we have been observin g the "cage d cheetah." Ou r three-dimensiona l laboratorie s ar e steril e zoo cages for the law s o f physics . Bu t whe n w e formulate th e law s i n higher-dimen sional space-time, their natural habitat, we see their tru e brillianc e and power; the laws become simple and powerful. The revolutio n now sweeping over physics is the realizatio n that th e natura l hom e for the cheeta h may be hyperspace.

Worlds Beyond Space an d Time 1

3

To illustrat e how adding a higher dimensio n ca n mak e thing s simpler, imagin e ho w major wars were fought b y ancient Rome. The grea t Roman wars , ofte n involvin g many smaller battlefields , were invariably fought wit h great confusion, wit h rumors and misinformatio n pourin g in o n bot h side s from man y different directions. With battles raging on several fronts , Roman general s wer e often operatin g blind . Rom e won its battles more from brute strengt h tha n fro m the eleganc e o f its strategies. Tha t i s why one o f th e firs t principle s o f warfare is to seiz e th e high ground—tha t is, to go u p into the thir d dimension , above the twodimensional battlefield . From th e vantag e poin t o f a larg e hil l wit h a panoramic vie w o f th e battlefield , the chao s o f war suddenly become s vastly reduced. I n othe r words , viewed from th e thir d dimensio n (tha t is, fro m th e to p o f th e hill) , th e confusio n o f th e smalle r battlefields becomes integrate d int o a coherent singl e picture . Another applicatio n o f this principle—that nature become s simple r when expresse d i n highe r dimensions—i s the centra l ide a behin d Ein stein's special theory of relativity. Einstein revealed time to be the fourth dimension, and h e showe d tha t spac e an d tim e coul d convenientl y be unified i n a four-dimensional theory. This, in turn, inevitabl y led t o th e unification o f all physical quantities measure d b y space an d time , such as matter an d energy . He the n foun d th e precis e mathematica l expres sion for this unity between matter and energy : E = m
14 E N T E R I N G T H E F I F T H D I M E N S I O

N

microwaves, tracks our plane s and spac e probes with radar , and electrifies ou r powe r plants. More recently, the powe r of the electromagneti c force has been used in electronic computers (whic h have revolutionized the office , home , school, and military ) and i n laser s (whic h hav e introduced ne w vistas in communications, surgery , compact disks , advanced Pentagon weaponry , and eve n the check-ou t stands in groceries). More than half the gross national product of the earth, representing th e accumulated wealth of our planet , depends in some way on th e electromag netic force. The Strong Nuclear Forc e

The strong nuclear force provides the energy that fuels the stars; it makes the star s shine and create s the brilliant , life-giving rays of the sun . If the strong forc e suddenl y vanished, th e su n woul d darken, endin g al l lif e on earth . In fact , som e scientist s believe that th e dinosaur s were driven to extinction 65 million years ago when debris from a comet impact was blown hig h int o th e atmosphere , darkenin g th e eart h an d causin g the temperature aroun d th e plane t t o plummet . Ironically , it i s als o th e strong nuclea r forc e tha t ma y on e da y tak e bac k th e gif t o f life . Unleashed i n th e hydroge n bomb , th e stron g nuclea r forc e could on e day end al l life o n earth . The Wea k Nuclea r Forc e

The wea k nuclea r forc e govern s certai n form s o f radioactiv e decay . Because radioactive materials emit heat when they decay or break apart, the weak nuclear force contributes to heating the radioactive rock deep within th e earth' s interior . Thi s heat , i n turn , contribute s t o th e hea t that drives the volcanoes, the rare but powerful eruptions of molten rock that reach th e earth's surface . Th e weak and electromagneti c forces are also exploited t o treat seriou s diseases: Radioactive iodine is used to kil l tumors of the thyroid gland and fight certain forms of cancer. The force of radioactive deca y can als o be deadly : It wreaked havoc at Three Mile Island and Chernobyl ; it also creates radioactive waste, the inevitable byproduct of nuclear weapons production an d commercia l nuclear power plants, which may remain harmfu l for million s of years. The Gravitationa l Forc e

The gravitationa l force keep s th e eart h an d th e planet s i n thei r orbit s and bind s th e galaxy . Without the gravitationa l force o f th e earth , we

Worlds Beyond Space an d Time 1

5

would be flung into space like rag dolls by the spi n of the earth . The air we breathe would be quickly diffused int o space, causing us to asphyxiate and makin g life o n eart h impossible . Without the gravitationa l force of the sun , al l the planets , includin g th e earth , woul d b e flun g fro m th e solar syste m into the col d reache s o f deep space , where sunligh t is too dim t o support life. I n fact, without the gravitational force, the su n itself would explode. The su n is the result of a delicate balancing act between the force of gravity, which tends to crush the star, and th e nuclear force, which tend s t o blast th e su n apart . Withou t gravity, the su n would detonate like trillions upon trillion s of hydrogen bombs . The centra l challeng e of theoretical physics today is to unif y thes e fou r forces int o a single force. Beginning with Einstein, the giant s of twentieth-century physics have tried and faile d to find such a unifying scheme . However, the answe r that eluded Einstei n for th e las t 30 years of his lif e may lie in hyperspace . The Quest fo r Unificatio n

Einstein onc e said , "Natur e show s us only the tai l of the lion . But I d o not doub t tha t th e lio n belong s t o i t eve n thoug h h e canno t a t onc e reveal himself because of his enormous si/,e." 3 If Einstein is correct, then perhaps these four forces are the "tail of the lion," and the "lion" itself is higher-dimensiona l space-time . Thi s ide a ha s fuele d th e hop e tha t the physica l laws of the universe , whose consequences fil l entir e library walls with books densely packed with tables and graphs , may one da y be explained b y a single equation . Central to thi s revolutionary perspective on th e univers e is the realization tha t higher-dimensiona l geometry ma y be th e ultimat e source of unity i n th e universe . Simpl y put , th e matte r i n th e univers e an d th e forces tha t hold i t together, whic h appear i n a bewildering, infinite variety of complex forms, may be nothin g bu t differen t vibration s of hyperspace. Thi s concept , however , goe s agains t th e traditiona l thinkin g among scientists, who have viewed space and tim e as a passive stage on which th e star s an d th e atom s pla y th e leadin g role. T o scientists , th e visible universe of matter seemed infinitel y richer and more diverse than the empty , unmovin g aren a o f th e invisibl e univers e o f space—time . Almost all the intens e scientific effort an d massiv e government fundin g in particl e physic s has historicall y gone t o catalogin g th e propertie s of subatomic particles , such a s "quarks" an d "gluons, " rathe r tha n fath -

16 E N T E R I N

G T H E F I F T H D I M E N S I ON

oming the nature of geometry. Now, scientists are realizing that the "use less" concept s o f space an d tim e may be th e ultimat e source o f beauty arid simplicity in nature . The firs t theor y of higher dimension s was called Kaluza-Klein theory, after tw o scientists who proposed a new theory of gravity in which light could b e explained as vibrations in the fifth dimension. When extende d to ^-dimensional space (wher e jVcan stand for any whole number), th e clumsy-looking theorie s o f subatomi c particle s dramaticall y take o n a startling symmetry . The ol d Kaluza—Klei n theory , however , could no t determine th e correct value of N, and ther e were technical problems in describing al l the subatomi c particles. A more advance d version of this theory, called supergravity theory, als o had problems . Th e recen t interest in th e theor y was sparked i n 198 4 by physicists Michael Green an d John Schwarz, wh o prove d th e consistenc y of th e mos t advance d versio n of Kaluza-Klein theory , calle d superstring theory, whic h postulate s tha t al l matter consist s of tiny vibrating strings. Surprisingly, the superstring theory predicts a precise numbe r o f dimensions for space an d time : ten.* The advantag e o f ten-dimensiona l spac e i s tha t w e hav e "enoug h room" in which to accommodate al l four fundamental forces. Furthermore, we have a simple physical picture i n which to explain th e confus ing jumble of subatomic particles produced by our powerful atom smashers. Over th e pas t 30 years, hundreds o f subatomic particles have bee n carefully cataloge d an d studie d b y physicists amon g th e debri s create d by smashin g togethe r proton s an d electron s wit h atoms. Lik e bug col lectors patientl y giving names t o a vas t collectio n o f insects , physicist s have at times been overwhelmed by the diversity and complexit y of these subatomic particles. Today, this bewildering collection of subatomic particles can be explaine d a s mere vibrations of the hyperspac e theory. Traveling Through Space and Time

The hyperspace theory has also reopened th e question of whether hyperspace can be used t o travel through spac e and time . To understand thi s *Freund chuckles whe n aske d when we will be able to see these higher dimensions. W e cannot see these higher dimensions because they have "curled up" int o a tiny ball so small that the y ca n n o longe r be detected . According t o Kaluza-Klei n theory , th e siz e o f these curled up dimension s i s called the Planck length, 4 which i s 100 billion billio n time s smalle r than the proton , too small to be probed by even b y our larges t atom smasher . High-energ y physicists had hope d that the $1 1 billion superconductin g supercollider (SSC ) (which was canceled b y Congres s i n Octobe r 1993 ) migh t hav e bee n abl e t o revea l som e indirec t glimmers o f hyperspace.

Worlds Beyond Space an d Time 1

7

concept, imagine a race of tiny flatworms living on the surfac e o f a large apple. It's obvious to these worms that their world, which they call Appleworld, is flat and tw o dimensional, like themselves. One worm , however, named Columbus, is obsessed by the notion that Appleworld is somehow finite an d curve d i n somethin g h e call s the thir d dimension . He eve n invents two new words, up and down, to describe motion in this invisible third dimension . His friends, however, call hi m a fool fo r believing that Appleworld coul d b e ben t i n some unsee n dimensio n tha t n o on e can see or feel. One day , Columbus sets out o n a long and arduou s journey and disappear s ove r th e horizon . Eventuall y he return s t o hi s starting point, provin g tha t th e worl d i s actuall y curved i n th e unsee n thir d dimension. Hi s journey prove s tha t Appleworl d is curved i n a highe r unseen dimension , th e thir d dimension . Although weary from hi s travels, Columbu s discovers that ther e i s yet another wa y to trave l between distant point s on th e apple : By burrowing into th e apple , he ca n carve a tunnel, creating a convenient shortcut to distant lands. These tunnels, which considerabl y reduce th e tim e and discomfor t of a long journey, he call s wormholes. They demonstrate tha t th e shortest path between two points is not necessaril y a straight line, as he's been taught , but a wormhole. One strang e effec t discovere d b y Columbus i s that whe n h e enter s one o f these tunnel s an d exit s at th e othe r end , h e find s himsel f back in th e past . Apparently , thes e wormhole s connec t part s o f th e apple wher e tim e beat s a t differen t rates . Som e o f th e worm s eve n claim tha t thes e wormhole s ca n b e molde d int o a workabl e tim e machine. Later, Columbu s make s a n eve n mor e momentou s discovery—hi s Appleworld i s actually not th e onl y one i n th e universe . It i s but on e apple i n a larg e appl e orchard . Hi s apple, h e find s out , coexist s with hundreds o f others , som e wit h worm s lik e themselves , an d som e without worms . Unde r certai n rar e circumstances , he conjectures , i t may eve n b e possibl e t o journey betwee n th e differen t apple s i n th e orchard. We human beings are like the flatworms. Common sense tells us that our world , lik e thei r apple , i s fla t an d thre e dimensional . N o matte r where we go with our rocke t ships, the universe seems flat. However, the fact that our universe, like Appleworld, is curved in an unseen dimension beyond ou r spatia l comprehension ha s been experimentall y verified by a numbe r o f rigorous experiments . Thes e experiments , performe d o n the pat h o f light beams, show that starlight is bent as it moves across the universe.

18 E N T E R I N

G T H E F I F TH D I M E NS I O N

Multiply Connecte d Universe s

When w e wake up i n th e mornin g an d ope n th e window to let in som e fresh air , we expect to see the fron t yard. We do not expect to fac e the towering pyramids of Egypt. Similarly, when we open the fron t door, we expect t o see the car s on th e street , no t th e crater s an d dea d volcanoes of a bleak, lunar landscape . Withou t even thinkin g abou t it , we assume that w e can safel y ope n window s or door s without being scare d ou t o f our wits . Our world , fortunately, is not a Steven Spielberg movie . We act on a deepl y ingraine d prejudic e (whic h is invariably correct ) tha t ou r world i s simply connected, tha t ou r window s an d doorway s ar e no t entrances t o wormholes connecting ou r home to a far-away universe. (In ordinary space , a lasso of rope can always be shrunk to a point. I f this is possible, then the spac e is called simply connected. However , if the lasso is placed around the entrance o f the wormhole, then it cannot be shrunk to a point. Th e lasso , in fact , enter s th e wormhole . Suc h spaces , where lassos ar e no t contractible , ar e calle d multiply connected. Althoug h th e bending o f our univers e i n a n unsee n dimensio n ha s been experimen tally measured, th e existenc e o f wormholes and whethe r our univers e is multiply connected o r no t i s still a topic of scientific controversy.) Mathematicians dating bac k to Georg Bernhard Riemann have studied th e propertie s o f multipl y connecte d space s i n whic h differen t regions o f space and tim e are spliced together . An d physicists, who once thought thi s was merely an intellectua l exercise, are now seriously studying multipl y connecte d world s a s a practica l mode l o f ou r universe . These models ar e th e scientifi c analogue o f Alice's looking glass. When Lewis Carroll' s Whit e Rabbi t fall s dow n th e rabbi t hol e t o ente r Won derland, h e actuall y falls dow n a wormhole. Wormholes ca n b e visualize d with a shee t o f pape r an d a pai r o f scissors: Take a piece o f paper, cu t tw o holes i n it , and the n reconnec t the tw o holes with a long tube (Figur e 1.1) . As long as you avoid walking into the wormhole, ou r worl d seems perfectly normal. Th e usua l laws of geometry taugh t i n schoo l ar e obeyed . However , i f yo u fal l int o th e wormhole, yo u ar e instantl y transported t o a different region o f spac e and time . Onl y by retracing you r step s and fallin g bac k int o th e wormhole ca n you return t o your familiar world. Time Trave l an d Bab y Universe s

Although wormholes provide a fascinating area of research, perhap s th e most intriguin g concept t o emerg e fro m thi s discussio n o f hyperspac e

Worlds Beyond Space and Time

19

Figure 1.1. Parallel universes may be graphically represented by two parallel planes. Normally, they never interact with each other. However, at times wormholes or tubes may open up between them, perhaps making communication and travel possible between them,. This is now the subject of intense interest among theoretical physicists.

is th e questio n o f tim e travel . I n th e fil m Back t o the Future, Michae l J. Foxjourneys back in time and meet s his parents a s teenagers befor e the y were married. Unfortunately, his mother falls in love with him and spurns his father , raisin g th e ticklis h questio n o f ho w h e wil l b e bor n i f hi s parents neve r marr y an d hav e children . Traditionally, scientist s hav e hel d a di m opinio n o f anyon e wh o raised the questio n o f time travel . Causality (the notion tha t ever y effect is preceded, no t followed , by a cause ) i s firmly enshrined i n th e foun -

20 E N T E R I N G T H E F I F T H D I M E N S I O

N

dations of modern science. However, in the physics of wormholes, "acausal" effect s sho w up repeatedly . In fact, we have to make strong assumptions i n orde r t o preven t tim e trave l fro m takin g place . Th e mai n problem i s that wormholes may connect no t onl y two distant points in space, but als o the futur e wit h th e past. In 1988 , physicis t Kip Thorne of the Californi a Institute of Technology and hi s collaborator s mad e th e astonishin g (an d risky ) clai m tha t time trave l is indeed no t onl y possible, but probabl e unde r certai n conditions. They published thei r claim not i n an obscure "fringe " journal , but i n th e prestigiou s Physical Review Letters. This marked th e firs t tim e that reputable physicists, and not crackpots, were scientifically advancing a claim about changin g th e cours e o f time itself. Their announcemen t was base d o n th e simpl e observatio n tha t a wormhol e connect s tw o regions tha t exis t i n differen t tim e periods . Thu s th e wormhol e ma y connect th e presen t t o th e past . Sinc e trave l through th e wormhol e is nearly instantaneous , on e coul d us e th e wormhol e t o g o backwar d in time. Unlike the machin e portrayed i n H . G . Wells's Th e Time Machine, however, whic h could hur l th e protagonis t hundred s o f thousand s of years into England's distant future with the simple twist of a dial, a wormhole ma y require vast amounts o f energy for it s creation, beyon d what will be technicall y possible for centurie s to come . Another bizarre consequence o f wormhole physics is the creatio n of "baby universes" in the laboratory. We are, of course, unable to re-create the Big Bang and witnes s the birth of our universe . However, Alan Guth of th e Massachusett s Institut e o f Technology , wh o ha s mad e man y important contribution s i n cosmology , shocked man y physicist s a few years ag o when he claime d tha t th e physic s of wormholes may make i t possible t o create a baby universe of our ow n in th e laboratory . By concentrating intense heat and energ y in a chamber, a wormhole may eventually open up, servin g as an umbilical cord connectin g our univers e to another, muc h smalle r universe. If possible, it would give a scientist an unprecedented view of a universe as it is created i n th e laboratory . Mystics an d Hyperspac e Some of tiSese concepts are not new . For the past several centuries, mystics and philosopher s have speculated about the existenc e of other universes an d tunnel s between them . The y hav e lon g bee n fascinate d by the possibl e existenc e of other worlds , undetectable by sight or sound , yet coexistin g with ou r universe . They hav e been intrigue d b y the pos -

Worlds Beyond Space and Time 2

1

sibility tha t thes e unexplored , nethe r world s may even b e tantalizingl y close, i n fac t surroundin g u s an d permeatin g u s everywher e we move, yetjust beyond our physical grasp and eludin g our senses. Such idle talk, however, wa s ultimately useless because ther e wa s no practica l wa y in which t o mathematically express and eventuall y test these ideas. Gateways betwee n ou r univers e an d othe r dimension s ar e als o a favorite literary device. Science-fiction writer s find higher dimensions to be an indispensable tool , using them as a medium for interstellar travel. Because o f the astronomica l distance s separatin g th e star s i n the heav ens, science-fictio n writers use highe r dimension s a s a cleve r shortcu t between th e stars . Instead o f taking the long , direct rout e t o other gal axies, rocket s merel y zi p alon g i n hyperspac e b y warpin g th e spac e around them. For instance, in th e film Star Wars, hyperspace is a refug e where Luk e Skywalke r ca n safel y evad e th e Imperia l Starship s o f th e Empire. In the televisio n series "Star Trek: Dee p Spac e Nine," a wormhole open s u p nea r a remote spac e station , makin g it possible t o span enormous distance s across the galax y within seconds. Th e spac e station suddenly become s th e cente r o f intens e intergalacti c rivalr y ove r who should contro l suc h a vital link to other part s of the galaxy. Ever since Fligh t 19, a group o f U.S. military torpedo bombers, vanished in the Caribbean 3 0 years ago, mystery writers too have used higher dimensions a s a convenient solutio n t o the puzzl e of the Bermud a Tri angle, o r Devil' s Triangle. Som e hav e conjecture d tha t airplane s an d ships disappearing i n the Bermud a Triangl e actuall y entered som e sor t of passageway to another world . The existenc e of these elusiv e parallel worlds has also produced end less religious speculation over the centuries. Spiritualist s have wondered whether th e soul s of departed love d one s drifte d int o anothe r dimen sion. The seventeenth-centur y British philosopher Henr y More argue d that ghost s and spirit s did indeed exis t and claime d that they inhabited the fourth dimension. In Enchiridion Metaphysicum (1671) , he argued fo r the existenc e of a nether real m beyon d ou r tangibl e senses that served as a home fo r ghost s and spirits. Nineteenth-century theologians, at a loss to locate heaven an d hell , pondered whethe r the y might b e foun d i n a higher dimension . Som e wrote abou t a univers e consistin g o f thre e paralle l planes : th e earth , heaven, and hell . God himself, according t o the theologia n Arthu r Willink, foun d hi s home i n a world far remove d fro m thes e thre e planes ; he live d in infinite-dimensional space. Interest i n highe r dimension s reache d it s peak betwee n 187 0 an d 1920, when the "fourt h dimension " ( a spatial dimension, different from

22 E N T E R I N G T H E F I F T H D I M E N S I O

N

what we know as the fourt h dimensio n o f time) seize d th e publi c imagination an d graduall y cross-fertilize d ever y branch o f th e art s an d sci ences, becoming a metaphor for the strange an d mysterious. The fourth dimension appeare d i n th e literar y work s of Oscar Wilde, Fyodo r Dos toyevsky, Marce l Proust , H . G . Wells , and Josep h Conrad ; i t inspire d some o f th e musica l works of Alexander Scriabin , Edgar d Varese , an d George Antheil . I t fascinate d such divers e personalitie s a s psychologist William James, literary figure Gertrude Stein , and revolutionar y socialist Vladimir Lenin . The fourt h dimensio n als o inspire d th e work s o f Pablo Picass o an d Marcel Ducham p an d heavil y influenced th e developmen t o f Cubis m and Expressionism , tw o of th e mos t influentia l art movement s i n thi s century. Art historian Linda Dalrymple Henderson writes, "Like a Black Hole, 'th e fourt h dimension ' possesse d mysteriou s qualities that coul d not be completely understood, even by the scientists themselves. Yet, the impact of 'the fourth dimension ' was far more comprehensiv e than tha t of Blac k Hole s o r an y othe r mor e recen t scientifi c hypothesi s except Relativity Theory afte r 1919." 5 Similarly, mathematician s hav e lon g bee n intrigue d b y alternative forms of logic and bizarr e geometries that defy every convention of common sense . Fo r example , th e mathematicia n Charle s L. Dodgson, who taught a t Oxford University , delighted generation s o f schoolchildren by writing books—a s Lewi s Carroll—that incorporat e thes e strang e math ematical ideas. When Alice falls dow n a rabbit hole or steps through th e looking glass , she enter s Wonderland , a strang e plac e wher e Cheshir e cats disappea r (leavin g only their smile) , magic mushroom s tur n chil dren into giants, arid Mad Hatters celebrate "unbirthdays. " The looking glass someho w connects Alice's world with a strang e lan d wher e everyone speak s in riddles an d commo n sens e isn' t so common. Some o f th e inspiratio n fo r Lewi s Carroll' s idea s mos t likel y cam e from th e great nineteenth-centur y German mathematicia n Georg Bernhard Riemann , who was the first t o lay the mathematica l foundation of geometries i n higher-dimensiona l space. Rieman n change d th e cours e of mathematic s fo r th e nex t centur y b y demonstrating tha t thes e uni verses, a s strange a s they ma y appear t o th e layperson , ar e completel y self-consistent and obe y their own inner logic. To illustrate some of these ideas, thin k o f stacking many sheet s o f paper, on e o n to p o f another . Now imagine tha t eac h shee t represent s a n entir e worl d and tha t eac h world obey s its own physica l laws, differen t fro m thos e o f al l the othe r worlds. Ou r universe , then , woul d no t b e alone , bu t woul d b e on e o f

Worlds Beyond Space an d Time 2

3

many possible parallel worlds. Intelligen t beings might inhabi t some of these planes , completely unaware of the existence of the others. On on e sheet o f paper, w e might hav e Alice's bucolic Englis h countryside . O n another shee t might be a strange world populated b y mythical creature s in th e worl d of Wonderland . Normally, life proceeds o n each of these parallel planes independen t of the others. On rare occasions, however, th e planes may intersect and , for a brie f moment , tea r th e fabri c o f spac e itself , whic h open s u p a hole—or gateway—betwee n thes e tw o universes . Lik e th e wormhol e appearing i n "Star Trek: Dee p Space Nine," these gateways make travel possible between these worlds, like a cosmic bridge linking two different universes o r tw o points i n th e sam e univers e (Figur e 1.2) . Not surpris ingly, Carroll found children much more open to these possibilities tha n adults, whose prejudices about space and logi c become mor e rigi d ove r time. I n fact , Riemann' s theory of higher dimensions , a s interpreted b y Lewis Carroll, has become a permanent par t of children's literatur e an d folklore, givin g birth t o other children' s classic s over th e decades , suc h as Dorothy's Land o f Oz and Pete r Pan' s Never Never Land. Without any experimental confirmatio n or compelling physical motivation, however, these theories of parallel worlds languished a s a branc h of science. Ove r 2 millennia, scientists have occasionall y picked u p th e notion o f highe r dimensions , onl y t o discar d i t a s a n untestabl e an d therefore sill y idea . Althoug h Riemann' s theor y o f highe r geometrie s was mathematically intriguing, it was dismissed as clever but useless . Scientists willing to risk their reputations o n higher dimension s soon found themselves ridicule d b y th e scientifi c community . Higher-dimensiona l space became th e las t refuge fo r mystics, cranks, and charlatans . In thi s book , w e wil l stud y th e wor k o f dies e pioneerin g mystics , mainly becaus e the y devise d ingeniou s way s i n whic h a nonspecialis t could "visualize " wha t higher-dimensiona l object s migh t loo k like . These trick s will prov e usefu l t o understan d ho w thes e higher-dimen sional theorie s may be graspe d b y the genera l public . By studying the wor k of these earl y mystics, we also see mor e clearly what wa s missing fro m thei r research . W e se e tha t thei r speculation s lacked two important concepts : a physical and a mathematical principle . From the perspective of modern physics , we now realize that the missing physical principle is that hyperspace simplifies th e law s of nature, provid ing the possibility of unifying all the forces of nature by purely geometric arguments. The missin g mathematical principle is called field theory, which is the universa l mathematical languag e o f theoretical physics.

Figure 1.2. Wormholes may connect a universe with itself, perhaps providing a means of interstellar travel. Since wormholes may connect two different time eras, they may also provide a means for time travel. Wormkoles may also connect an infinite series of parallel universes. The hope is that the hyperspace theory will be able to determine whether wormholes are physically possible or merely a mathematical curiosity.

24

Worlds Beyond Space an d Time 2

5

Field Theory : Th e Languag e o f Physic s Fields were first introduced b y the grea t nineteenth-centur y British scientist Michael Faraday. The son of a poor blacksmith, Farada y was a selftaught geniu s who conducted elaborat e experiment s on electricit y and magnetism. He visualized "lines of force" that , like long vines spreading from a plant, emanate d fro m magnets and electri c charges in al l directions an d fille d u p al l o f space . Wit h hi s instruments , Farada y coul d measure the strengt h o f these lines of force from a magnetic or an electric charge a t any point in his laboratory. Thu s h e coul d assig n a series of numbers (th e strength and directio n o f the force) t o that point (an d any point i n space) . He christene d th e totalit y of these number s a t any point in space, treated a s a single entity, a field. (There is a famous story concerning Michael Faraday. Because his fame had sprea d fa r and wide , he wa s often visite d by curiou s bystanders . When on e aske d wha t his work was good for, he answered , "What i s the us e of a child? It grows to be a man." On e day , William Gladstone, the n Chancellor o f the Exchequer, visited Faraday in his laboratory. Knowing nothing about science, Gladstone sarcasticall y asked Faraday what use the hug e electrica l contraptions i n hi s laborator y coul d possibl y hav e fo r England . Farada y replied, "Sir , I know not wha t these machines will be used for, but I am sure that one da y you will tax them." Today, a large portion o f the tota l wealth of England i s invested in th e frui t o f Faraday's labors.) Simply put, a field is a collectio n o f numbers define d a t ever y point in spac e tha t completel y describe s a forc e a t tha t point . Fo r example , three number s a t eac h poin t i n spac e ca n describ e th e intensit y and direction o f the magneti c line s of force. Another thre e number s every where in spac e ca n describ e th e electri c field. Faraday got this concep t when he though t o f a "field" plowe d by a farmer. A farmer's field occupies a two-dimensiona l regio n o f space. A t each poin t i n th e farmer' s field, one ca n assig n a series of numbers (whic h describe, for example , how many seeds ther e are at that point). Faraday's field, however, occupies a three-dimensional regio n o f space. At each point , there is a series of si x numbers tha t describe s bot h th e magneti c an d electri c lines of force. What makes Faraday's fiel d concep t s o powerful is that al l forces of nature ca n b e expresse d a s a field. However, we need on e mor e ingre dient befor e w e can understan d th e natur e o f an y force: We mus t b e able to write down the equation s tha t thes e fields obey. The progress of the pas t hundre d year s in theoretica l physic s can be succinctl y summarized a s the searc h for th e field equations of the force s of nature.

26 E N T E R I N G T H E F I F T H D I M E N S I O

N

For example , i n th e 1860s , Scottis h physicis t James Cler k Maxwell wrote down th e fiel d equation s fo r electricit y and magnetism . In 1915 , Einstein discovere d th e fiel d equation s fo r gravity . After innumerabl e false starts , the field equations for the subatomi c forces were finally written dow n in th e 1970s , utilizin g the earlie r work of C. N. Yang and hi s student R . L . Mills . Thes e fields , whic h gover n th e interactio n o f al l subatomic particles, are now called Yang-Mills fields. However , the puzzle that has stumped physicists within this century is why the subatomic field equations look s o vastly different from the fiel d equation s o f Einstein— that is , why the nuclea r forc e seems s o different fro m gravity . Som e o f the greates t mind s in physic s have tackled thi s problem, onl y to fail . Perhaps the reaso n for their failure is that they were trapped b y common sense . Confine d t o thre e o r fou r dimensions , th e fiel d equation s of the subatomi c world and gravitatio n are difficul t t o unify. The advantage of the hyperspace theory is that the Yang-Mills field, Maxwell's field , and Einstein' s field can all be placed comfortabl y within the hyperspace field. We see that these fields fit together precisel y within the hyperspac e field like pieces i n a jigsaw puzzle. The othe r advantag e o f field theory is that it allows us to calculate the precise energies at which we can expec t space an d tim e t o form wormholes. Unlik e the ancients , therefore , we have th e mathematica l tool s to guid e u s in buildin g th e machine s tha t may one da y bend spac e an d tim e t o our whims.

The Secret of Creatio n

Does thi s mean tha t big-gam e hunter s ca n no w start organizing safaris to th e Mesozoi c er a t o ba g larg e dinosaurs ? No . Thorne , Guth , an d Freund will all tell you that the energy scale necessary to investigate thes e anomalies i n spac e i s far beyon d anythin g availabl e o n earth . Freun d reminds us that the energ y necessar y t o probe the tent h dimensio n is a quadrillion time s larger tha n th e energ y tha t ca n b e produce d b y our largest atom smasher . Twisting space—tim e int o knot s require s energ y o n a scale tha t wil l not b e availabl e within the nex t several centuries o r eve n millennia—i f ever. Even if all the nation s o f the world were to band togethe r t o buil d a machine tha t could probe hyperspace, they would ultimately fail. And, as Guth points out, the temperature s necessar y to create a baby universe in th e laborator y i s 1,00 0 trillio n trillio n degrees , fa r i n exces s o f anything availabl e t o us . I n fact , tha t temperatur e i s muc h greate r tha n anything foun d i n the interio r of a star. So, although i t is possible tha t

Worlds Beyond Space and Time 27

Einstein's law s an d th e law s o f quantu m theor y migh t allo w fo r tim e travel, thi s i s not withi n th e capabilitie s o f earthling s lik e us , who ca n barely escape th e feebl e gravitational fiel d o f our ow n planet. Whil e we can marvel a t the implications of wormhole research, realizing it s potential i s strictly reserved fo r advance d extraterrestria l civilizations . There wa s only one perio d o f time whe n energ y on thi s enormou s scale wa s readily available, and tha t wa s at th e instan t o f Creation . I n fact, th e hyperspac e theor y canno t b e tested b y our larges t ato m smash ers because th e theor y i s really a theory of Creation. Onl y at the instan t of the Big Bang do we see the ful l powe r of the hyperspac e theor y coming into play. This raises the excitin g possibility that the hyperspace the ory may unlock the secre t o f the origi n o f the universe . Introducing highe r dimension s ma y be essential for prying loose th e secrets o f Creation. Accordin g t o thi s theory , befor e th e Bi g Bang, ou r cosmos wa s actually a perfect ten-dimensiona l universe , a world wher e interdimensional trave l wa s possible . However , thi s ten-dimensiona l world wa s unstable, an d eventuall y i t "cracked " i n two , creatin g tw o separate universes : a four- and a six-dimensional universe. The univers e in which we live was born in that cosmic cataclysm. Our four-dimensiona l universe expande d explosively, while our twi n six-dimensiona l universe contracted violently , unti l i t shran k t o almos t infinitesima l size . Thi s would explai n th e origi n o f th e Bi g Bang. I f correct, thi s theor y dem onstrates tha t the rapid expansio n o f the universe was just a rather mino r aftershock o f a muc h greate r cataclysmi c event , th e crackin g o f spac e and tim e itself . Th e energ y tha t drive s th e observe d expansio n o f th e universe is then found in the collapse often-dimensional space and time. According t o the theory, the distant stars and galaxies are receding fro m us at astronomical speed s becaus e of the origina l collaps e of ten-dimensional spac e an d time . This theor y predict s tha t ou r univers e stil l ha s a dwarf twin , a com panion univers e that has curled u p into a small six-dimensional ball tha t is too small to be observed. This six-dimensional universe, far from bein g a useless appendage to our world , may ultimately be our salvation . Evading th e Death o f th e Univers e

It i s often sai d tha t th e onl y constants o f human societ y are deat h an d taxes. For the cosmologist , the only certainty is that the universe will on e day die. Som e believ e tha t th e ultimat e death o f the univers e will com e in the form of the Big Crunch. Gravitatio n wil l reverse the cosmic expan -

28 E N T E R I N G T H E F I F T H D I M E N S I O

N

sion generate d b y the Bi g Bang an d pul l th e star s an d galaxie s back , once again , int o a primordial mass . As the star s contract, temperature s will rise dramatically until all matter an d energ y in the univers e are con centrated int o a colossal fireball that will destroy the universe as we know it. All life form s wil l b e crushe d beyon d recognition . Ther e wil l b e n o escape. Scientist s and philosophers , lik e Charle s Darwi n and Bertran d Russell, hav e writte n mournfull y about th e futilit y o f ou r pitifu l exis tence, knowin g that ou r civilizatio n will inexorabl y di e when ou r worl d ends. Th e law s of physics, apparently, hav e issue d th e final , irrevocabl e death warran t for al l intelligent life i n th e universe . According to the late Columbia University physicist Gerald Feinberg, there i s one, an d perhap s onl y one, hop e of avoiding the final calamity. He speculate d tha t intelligent life, eventually mastering the mysterie s of higher-dimensional space over billions of years, will use the other dimensions as an escap e hatc h fro m th e Bi g Crunch. I n th e fina l moment s of the collapse of our universe, our siste r universe will open up onc e again , and interdimensiona l trave l wil l becom e possible . A s al l matte r i s crushed i n th e fina l moment s befor e doomsday , intelligen t lif e form s may b e abl e t o tunne l int o higher-dimensiona l spac e o r a n alternativ e universe, avoiding the seemingl y inevitable death o f our universe. Then, from thei r sanctuar y in higher-dimensiona l space , thes e intelligen t lif e forms ma y be abl e t o witness the deat h o f th e collapsin g univers e i n a fiery cataclysm. A s our hom e univers e i s crushed beyon d recognition , temperatures wil l ris e violently , creating ye t anothe r Bi g Bang. Fro m their vantag e poin t i n hyperspace , thes e intelligen t lif e form s will hav e front-row seat s to the rares t o f all scientific phenomena , th e creatio n o f another univers e and o f their ne w home .

Masters o f Hyperspac e Although fiel d theor y show s that th e energ y necessar y t o creat e thes e marvelous distortions of space and tim e is far beyond anything that modern civilizatio n can muster, this raises two important questions : How long will it take for ou r civilization , which is growing exponentially in knowledge and power, to reach th e point of harnessing the hyperspace theory? And wha t abou t othe r intelligen t lif e form s i n th e universe , wh o may already have reache d tha t point ? What makes thi s discussion interestin g i s that seriou s scientists have tried t o quantif y th e progres s o f civilizations far int o th e future , whe n space trave l wil l have becom e commonplace an d neighborin g sta r sys -

Worlds Beyond Space an d Time 2

9

terns o r eve n galaxie s wil l hav e bee n colonized . Althoug h th e energ y scale necessar y to manipulate hyperspac e i s astronomically large, thes e scientists point ou t tha t scientifi c growt h will probabl y continue t o rise exponentially ove r th e nex t centuries , exceedin g th e capabilitie s o f human mind s to grasp it . Since World War II, the su m total of scientific knowledge ha s double d ever y 1 0 to 2 0 o r s o years, so th e progres s o f science an d technolog y int o th e twenty-firs t centur y ma y surpas s ou r wildest expectations . Technologie s tha t ca n onl y be dreame d o f toda y may become commonplac e i n th e nex t century . Perhaps the n on e ca n discuss the questio n o f when we might become master s of hyperspace. Time travel . Parallel universes . Dimensional windows. By themselves, these concepts stand at the edge of our understandin g of th e physica l universe. However , becaus e th e hyperspac e theor y i s a genuine fiel d theory , w e eventuall y expec t i t t o produc e numerica l answers determinin g whether thes e intriguing concept s are possible. I f the theor y produce s nonsensica l answer s tha t disagre e wit h physica l data, the n i t must be discarded, n o matter ho w elegant its mathematics. In th e fina l analysis , we are physicists , not philosophers . Bu t if it proves to be correct and explain s the symmetries of modern physics , then i t will usher i n a revolutio n perhap s equa l t o th e Copernica n o r Newtonia n revolutions. To have an intuitiv e understanding o f these concepts , however , it is important t o start at the beginning. Befor e we can feel comfortabl e with ten dimensions , w e must learn ho w t o manipulat e fou r spatia l dimen sions. Using historical examples, we will explore th e ingeniou s attempt s made b y scientists over th e decade s t o giv e a tangible , visua l represen tation of higher-dimensional space. The first part of the book, therefore, will stress the histor y behind th e discover y of higher-dimensional space , beginning wit h th e mathematicia n wh o starte d i t all , Geor g Bernhar d Riemann. Anticipating the nex t century of scientific progress , Riemarm was the first to state tha t natur e finds its natural hom e i n the geometr y of higher-dimensional space .

2 Mathematicians and Mystic s Magic i s any sufficientl y advanced technology .

O

Arthur C. Clarke

N June 10 , 1854, a new geometry was born. The theor y of higher dimension s was introduced whe n Geor g Bernhard Riemann gave his celebrated lectur e before th e facult y o f the University of Gottingen in Germany. In one masterfu l stroke, like opening u p a musty , darkened roo m t o th e brillianc e o f a warm summer' s sun, Riemann's lecture expose d th e world to the dazzlin g properties of higher-dimensional space . His profoundly important an d exceptionall y elegant essay , "On th e Hypotheses Whic h Li e a t th e Foundatio n o f Geometry, " topple d th e pillars of classical Greek geometry, which had successfull y weathered all assaults by skeptics for 2 millennia. The ol d geometry of Euclid, in which all geometric figure s are two or three dimensional, came tumbling down as a new Riemannian geometry emerged fro m its ruins. The Riemannian revolution woul d hav e vast implications fo r th e futur e o f th e art s an d sciences. Withi n 3 decade s o f hi s talk , th e "mysteriou s fourt h dimen sion" woul d influenc e the evolutio n o f art, philosophy , an d literatur e in Europe . Withi n 6 decades o f Riemann's lecture , Einstei n would use four-dimensional Riemannia n geometr y t o explai n th e creatio n o f th e universe an d it s evolution . And 13 0 years afte r hi s lecture , physicist s

30

Mathematicians an d Mystics 3

1

would us e ten-dimensiona l geometr y to attemp t t o unite al l the law s of the physica l universe. The cor e o f Riemann's work was the realizatio n that physica l laws simplif y i n higher-dimensiona l space , th e ver y them e of this book.

Brilliance Amid Povert y

Ironically, Riemann was the leas t likely person t o usher i n suc h a deep and thorough-goin g revolutio n i n mathematica l an d physica l thought. He wa s excruciatingly, almost pathologically, shy and suffere d repeate d nervous breakdowns. He als o suffered fro m th e twi n ailments that have ruined th e live s o f s o man y of th e world' s great scientist s throughou t history: abject poverty and consumptio n (tuberculosis) . His personality and temperamen t showe d nothing of the breath-taking boldness, sweep, and suprem e confidenc e typica l o f his work. Riemann was born in 182 6 in Hanover, Germany , the son of a poor Lutheran pastor , th e secon d o f six children. His father, who fought in the Napoleoni c Wars , struggled a s a country pastor t o fee d and cloth e his larg e family . A s biographer E . T. Bel l notes, "th e frai l healt h an d early deaths o f most of the Rieman n childre n wer e the resul t of under nourishment i n thei r yout h an d wer e no t du e t o poo r stamina . Th e mother als o died befor e he r childre n wer e grown." 1 At a ver y earl y age, Rieman n exhibite d hi s famou s traits : fantastic calculational ability, coupled wit h timidity , and a life-long horror of any public speaking . Painfull y shy , he wa s the but t o f crue l jokes b y othe r boys, causin g him t o retrea t furthe r into th e intensel y private world of mathematics. He als o was fiercely loya l to his family, strainin g his poor health an d constitution to buy presents for his parents and especiall y for his beloved sisters. T o pleas e hi s father, Riemann se t ou t t o becom e a studen t o f theology. Hi s goal was to ge t a paying position a s a pastor a s quickly as possible to help with his family's abysmal finances. (It is difficult t o imagine a more improbabl e scenari o than tha t of a tongue-tied, timi d young boy imaginin g tha t h e coul d delive r fiery , passionat e sermon s railin g against sin and drivin g out th e devil.) In high school, he studied the Bible intensely, but his thoughts always drifted bac k t o mathematics ; he eve n trie d t o provid e a mathematical proof o f the correctnes s o f Genesis. He als o learned s o quickly that h e kept outstripping th e knowledg e of his instructors, who found it impossible t o kee p u p wit h th e boy . Finally, the principa l o f hi s school gav e

32 ENTERIN

G TH E FIFT H DIMENSIO N

Riemann a ponderou s boo k t o kee p hi m occupied . Th e boo k wa s Adrien-Marie Legendre' s Theory o f Numbers, a hug e 859-pag e master piece, the world's most advanced treatis e on the difficul t subjec t of number theory . Riemann devoure d th e boo k i n 6 days. When hi s principal asked , "Ho w fa r di d you read? " th e youn g Riernann replied , "Tha t is certainly a wonderful book. I have mastered it. " Not reall y believing the bravad o o f this youngster, the principa l several months late r aske d obscur e question s fro m th e book , whic h Riemann answered perfectly. 2 Beset by the daily struggle to put food on the table, Riemarm's father might have sent the boy to do menial labor. Instead, he scraped togethe r enough funds t o send his 19-year-old so n to the renowne d Universit y of Gottingen, wher e h e firs t me t Car l Friedric h Gauss , th e acclaime d "Prince o f Mathematicians," on e o f the greates t mathematician s of all time. Even today, i f you as k any mathematician t o ran k th e thre e mos t famous mathematicians i n history, the name s of Archimedes, Isaac Newton, an d Car l Gauss will invariably appear. Life fo r Riemann , however , wa s an endles s serie s o f setback s an d hardships, overcome onl y with the greatest difficulty an d b y straining his frail health . Each triumph was followed by tragedy and defeat . For example, just a s his fortunes began t o improve an d h e undertoo k hi s formal studies under Gauss, a full-scale revolutio n swept Germany. The working class, lon g sufferin g unde r inhuma n livin g conditions, ros e u p agains t the government , wit h workers in score s o f citie s throughou t German y taking up arms. The demonstrations and uprisings in early 1848 inspired the writing s of anothe r German , Kar l Marx , an d deepl y affecte d th e course o f revolutionary movements throughout Europ e for th e nex t 50 years. With al l o f German y swep t up i n turmoil , Riemann' s studie s were interrupted. H e was inducted int o th e studen t corps , where he ha d th e dubious hono r o f spendin g 1 6 weary hours protectin g someon e eve n more terrifie d tha n he : th e king , wh o wa s quivering wit h fea r i n hi s royal palac e i n Berlin , tryin g t o hid e fro m th e wrat h o f th e workin g class. Beyond Euclidea n Geometr y Not only in Germany, but in mathematics, too, fierce revolutionary winds were blowing . Th e proble m tha t rivete d Riemann' s interes t wa s th e impending collapse of yet another bastion of authority, Euclidean geom-

Mathematicians an d Mystics 3

3

etry, whic h hold s tha t spac e i s thre e dimensional . Furthermore , thi s three-dimensional spac e i s "flat " (i n fla t space , th e shortes t distanc e between tw o points is a straight line; this omits the possibilit y that space can b e curved , as on a sphere). In fact , afte r th e Bible , Euclid's Elements was probably the mos t influential boo k o f all time . Fo r 2 millennia, th e keenes t mind s o f Western civilization have marveled a t its elegance and th e beauty of its geometry. Thousands o f the fines t cathedral s in Europe were erected accordin g to its principles. I n retrospect , perhap s i t was too successful . Ove r th e cen turies, it became somethin g o f a religion; anyon e who dared t o propose curved spac e o r highe r dimension s was relegated t o th e rank s of crackpots o r heretics . For untol d generations , schoolchildre n hav e wrestled with th e theorem s o f Euclid' s geometry: tha t th e circumferenc e o f a circle is IT time s the diameter , an d tha t th e angle s within a triangle ad d up t o 18 0 degrees. However , try as they might, th e fines t mathematica l minds fo r severa l centurie s coul d no t prov e thes e deceptivel y simple propositions. In fact, the mathematicians of Europe began t o realize that even Euclid' s Elements, whic h ha d bee n revere d fo r 2,30 0 years , was incomplete. Euclid' s geometry wa s still viabl e i f on e staye d withi n th e confines o f fla t surfaces , bu t i f on e straye d int o th e worl d o f curve d surfaces, i t was actually incorrect. To Riemann , Euclid' s geometr y wa s particularly sterile whe n com pared with the ric h diversit y of the world. Nowhere in th e natural world do w e se e th e flat , idealize d geometri c figure s o f Euclid . Mountain ranges, ocea n waves , clouds , an d whirlpool s are no t perfec t circles, triangles, and squares, but are curved objects that bend and twis t in infinite diversity. The tim e was ripe fo r a revolution, but wh o would lea d i t an d what would replac e th e ol d geometry?

The Ris e of Riemannia n Geometry Riemann rebelled agains t the apparent mathematical precision of Greek geometry, whose foundation, he discovered, ultimately was based o n th e shifting san d o f commo n sens e an d intuition , no t th e fir m groun d o f logic. It is obvious, said Euclid, that a point ha s no dimensio n a t all. A line has on e dimension : length . A plan e ha s tw o dimensions: lengt h an d breadth. A solid has three dimensions : length, breadth, and height. And there it stops. Nothing has four dimensions. These sentiment s were ech-

34 E N T E R I N G T H E F I F T H D I M E N S I O

N

oed by the philosophe r Aristotle , who apparently was the first person t o state categorically that the fourt h spatial dimension i s impossible. In O n Heaven, he wrote, "The lin e has magnitude i n one way, the plane in two ways, an d th e soli d i n thre e ways , an d beyon d thes e ther e i s no othe r magnitude becaus e th e thre e ar e all. " Furthermore , i n A.D . 150, th e astronomer Ptolemy from Alexandria went beyond Aristotle and offered , in hi s boo k O n Distance, th e firs t ingeniou s "proo f tha t th e fourt h dimension i s impossible. First, he said, draw three mutually perpendicular lines . For example, the corne r o f a cub e consist s o f thre e mutuall y perpendicula r lines . Then, h e argued , tr y to draw a fourth lin e tha t is perpendicular t o th e other three lines . No matter ho w one tries , he reasoned , fou r mutually perpendicular line s ar e impossibl e t o draw . Ptolem y claime d tha t a fourth perpendicula r lin e is "entirely without measure and withou t definition." Thus th e fourt h dimension i s impossible. What Ptolemy actually proved was that it is impossible to visualize th e fourth dimensio n with our three-dimensiona l brains . (I n fact , toda y we know that many objects in mathematics cannot b e visualized but can be shown to exist.) Ptolemy may go down in history as the man who opposed two great idea s in science: the sun-centere d sola r system and th e fourth dimension. Over th e centuries , i n fact , som e mathematician s wen t out o f their way t o denounc e th e fourt h dimension . I n 1685 , th e mathematicia n John Walli s polemicized agains t th e concept , callin g it a "Monste r i n Nature, less possible tha n a Chimera or Centaure. . . . Length, Breadth, and Thickness, take up the whole of Space. Nor can Fansie imagine how there should b e a Fourth Loca l Dimensio n beyond thes e Three." 3 For several thousan d years , mathematician s woul d repea t thi s simpl e bu t fatal mistake , that th e fourth dimension cannot exist because we cannot picture i t in our minds . The Unit y of All Physica l La w

The decisiv e break with Euclidean geometry came when Gauss asked his student Rieman n t o prepare a n ora l presentatio n o n th e "foundatio n of geometry." Gaus s was keenly interested i n seeing if his student could develop a n alternativ e to Euclidean geometry . (Decade s before, Gauss had privately expressed deep and extensive reservations about Euclidean geometry. He even spoke to his colleagues of hypothetical "bookworms " that migh t live entirel y on a two-dimensional surface. He spok e o f gen-

Mathematicians an d Mystics 3

5

eralizing thi s t o th e geometr y o f higher-dimensiona l space . However , being a deeply conservative man, he never published an y of his work on higher dimension s becaus e o f th e outrag e i t would creat e amon g th e narrow-minded, conservativ e ol d guard . H e derisivel y calle d the m "Boeotians" after a mentally retarded Gree k tribe. 4) Riemann, however, was terrified. This timid man, terrified of public speaking, was being asked by his mentor to prepare a lecture before th e entire faculty on the most difficult mathematica l problem o f the century. Over th e nex t severa l months, Rieman n bega n painfull y developin g the theor y o f higher dimensions , straining his health t o th e poin t o f a nervous breakdown. His stamina further deteriorated becaus e of his dismal financia l situation. H e wa s forced t o tak e low-payin g tutoring jobs to provid e fo r hi s family . Furthermore , h e wa s becoming sidetracke d trying t o explai n problem s o f physics . I n particular , h e wa s helpin g another professor , Wilhelm Weber, conduc t experiment s i n a fascinating ne w field of research, electricity. Electricity, of course, had been known to the ancient s in the form of lightning and sparks . But in th e earl y nineteenth century , this phenomenon becam e th e centra l focu s o f physic s research. I n particular , th e discovery tha t passin g a curren t o f wir e acros s a compas s needl e ca n make th e needl e spi n rivete d th e attentio n o f the physic s community. Conversely, movin g a bar magne t acros s a wire can induc e a n electri c current i n th e wire . (This is called Faraday' s Law, and toda y all electric generators an d transformers—an d henc e muc h o f th e foundatio n o f modern technology—ar e based o n thi s principle.) To Riemann , thi s phenomenon indicate d tha t electricit y and mag netism ar e someho w manifestation s o f th e sam e force . Rieman n was excited b y the ne w discoveries and wa s convinced tha t h e coul d giv e a mathematical explanatio n tha t woul d unif y electricit y and magnetism . He immerse d himsel f i n Weber' s laboratory , convince d tha t th e ne w mathematics woul d yiel d a comprehensiv e understandin g o f thes e forces. Now, burdened wit h having to prepare a major public lecture on th e "foundation o f geometry, " t o suppor t hi s family , an d t o conduc t sci entific experiments , hi s health finally collapsed an d h e suffere d a ner vous breakdow n i n 1854 . Later , h e wrot e t o hi s father , " I becam e s o absorbed i n m y investigation of the unit y of all physical laws tha t when the subjec t of the tria l lecture was given me, I could not tea r myself away from m y research. Then, partly as a result of brooding o n it, partly from staying indoor s to o muc h i n thi s vile weather , I fel l ill." 5 This lette r i s significant, fo r i t clearl y show s that , eve n durin g month s o f illness ,

36 E N T E R I N G T H E F I F T H D I M E N S I O

N

Riemann firmly believed that he would discover the "unit y of all physical laws" an d tha t mathematic s would eventually pave th e wa y for thi s unification.

Force = Geometr y

Eventually, despite hi s frequent illnesses, Riemann developed a startling new picture of the meanin g of a "force." Eve r since Newton, scientist s had considere d a force to be a n instantaneou s interactio n betwee n two distant bodies. Physicist s called it action-at-a-distance, which meant tha t a bod y could influenc e th e motion s o f distant bodie s instantaneously. Newtonian mechanic s undoubtedl y coul d describ e th e motion s of th e planets. However, over th e centuries , critic s argued tha t action-at-a-distance wa s unnatural, becaus e i t mean t tha t one bod y could chang e th e direction o f another withou t even touching it. Riemann develope d a radicall y new physica l picture . Lik e Gauss' s "bookworms," Rieman n imagine d a race o f two-dimensional creatures living o n a sheet o f paper. Bu t th e decisiv e break h e mad e wa s to pu t these bookworms on a crumpled sheet of paper.6 What would these bookworms think about their world? Riemann realize d tha t the y would conclude that their world was still perfectly flat. Becaus e their bodies would also be crumpled, these bookworms would never notice that their world was distorted. However , Riemann argue d tha t if these bookworm s trie d to mov e across the crumple d shee t of paper, the y would feel a mysterious, unseen "force " tha t prevented the m from moving in a straight line. They would be pushed lef t and righ t every time their bodies moved over a wrinkle on th e sheet . Thus Rieman n mad e th e first momentous brea k with Newton in 200 years, banishing th e action-at-a-distanc e principle . T o Riemann , "force" was a consequence of geometry. Riemann the n replace d th e two-dimensiona l shee t wit h ou r three dimensional worl d crumple d i n th e fourt h dimension . I t would not b e obvious t o u s that ou r univers e was warped. However , we would immediately realiz e tha t somethin g wa s amis s whe n w e trie d t o wal k i n a straight line. We would walk like a drunkard, as though a n unseen forc e were tuggin g at us, pushing us left an d right . Riemann conclude d tha t electricity , magnetism , an d gravit y ar e caused b y th e crumplin g o f ou r three-dimensiona l univers e i n th e unseen fourt h dimension . Thus a "force" ha s no independent lif e o f its own; it is only the apparen t effec t cause d b y the distortio n o f geometry.

Mathematicians an d Mystics 3

7

By introducing the fourth spatial dimension, Riemann accidentally stumbled o n wha t would becom e on e o f the dominan t theme s i n moder n theoretical physics, that the laws of nature appear simple when expressed in higher-dimensiona l space . H e the n se t abou t developin g a mathe matical language i n which this idea could be expressed .

Riemann's Metri c Tensor: A Ne w Pythagorean Theore m Riemann spent several months recovering from his nervous breakdown. Finally, when he delivere d hi s oral presentatio n i n 1854 , th e receptio n was enthusiastic . In retrospect , thi s was, without question , on e o f th e most importan t publi c lecture s i n th e histor y o f mathematics . Word spread quickly throughout Europ e tha t Riemann had decisivel y broken out o f the confine s of Euclidean geometry tha t ha d rule d mathematics for 2 millennia . New s o f th e lectur e soo n sprea d throughou t al l th e centers o f learnin g i n Europe , an d hi s contribution s t o mathematic s were bein g haile d throughou t th e academi c world . His tal k wa s translated int o several languages and create d quit e a sensation in mathematics. Ther e was no turnin g back to the work of Euclid. Like man y o f th e greates t work s in physic s and mathematics , th e essential kerne l underlyin g Riemann's grea t pape r i s simple to under stand. Rieman n bega n wit h th e famou s Pythagorean Theorem, on e o f the Greeks' greatest discoveries in mathematics. The theorem establishe s the relationship between the lengths of the three sides of a right triangle: It states that the sum of the squares of the smaller sides equals the square of the longes t side, the hypotenuse ; that is, if a and b are th e length s of the tw o short sides , and c is the lengt h o f the hypotenuse , the n a 2 + V= c 2. (Th e Pythagorea n Theorem , o f course , i s th e foundatio n o f all architecture; every structure buil t on thi s planet is based o n it.) For three-dimensional space, the theorum can easily be generalized. It states that th e su m o f the square s of three adjacent sides of a cube is equal to the squar e of the diagonal ; so if a, b, and c represent the side s of a cube, and d is its diagonal length , the n a 2 + b 2 + c 2 = d 2 (Figur e 2.1). It is now simple to generalize this to the cas e of /^dimensions. Imagine a n ^-dimensiona l cube. I f a,b,c, . . . are th e length s of the side s of a "hypercube," an d z is the lengt h o f the diagonal , the n « 2 + 6 2 + c 2 + d~ + . . . = z 2. Remarkably, even thoug h ou r brain s cannot visualiz e a n Aklimensional cube , i t i s easy to writ e down th e formul a for it s sides . (This i s a commo n featur e o f workin g in hyperspace . Mathematically

38

ENTERING TH E FIFT H DIMENSIO N

a 2 +b 2 + c2 =d 2 Figure 2.1. The length of a diagonal of a cube is given by a three-dimensional version o f the Pythagorean Theorem: a 2 - f b 2 + cz = d2. B y simply adding more terms to the Pythagorean Theorem, this equation easily generalizes to the diagonal of a hypercube in N dimensions. Thus although higher dimensions cannot be visualized, it is easy to represent N dimensions mathematically.

manipulating TV-dimensiona l space i s no mor e difficul t tha n manipulating three-dimensiona l space . I t i s nothing shor t o f amazin g tha t o n a plain sheet o f paper, yo u can mathematically describe th e propertie s o f higher-dimensional object s that cannot b e visualized by our brains.) Riemann the n generalize d thes e equation s fo r space s o f arbitrar y dimension. Thes e space s ca n b e eithe r fla t o r curved . If flat , the n th e usual axiom s of Euclid apply: The shortes t distanc e between two points is a straight line, parallel line s neve r meet , an d th e su m o f the interio r angles o f a triangl e ad d t o 18 0 degrees. Bu t Rieman n als o foun d tha t surfaces ca n hav e "positiv e curvature, " a s i n th e surfac e o f a sphere , where paralle l line s always mee t an d wher e th e su m o f the angle s o f a triangle ca n excee d 18 0 degrees. Surface s can als o hav e "negativ e cur -

Mathematicians an d Mystics 3

9

vature," a s in a saddle-shape d o r a trumpet-shape d surface . On thes e surfaces, th e su m of the interio r angles of a triangle add t o less than 180 degrees. Give n a lin e an d a poin t of f tha t line , ther e ar e a n infinit e number o f parallel lines one ca n draw through tha t point (Figur e 2.2). Riemann's ai m was to introduc e a ne w object in mathematic s tha t would enable hi m t o describe al l surfaces, n o matte r ho w complicated. This inevitably led hi m t o reintroduce Faraday's concept o f the field. Faraday's field, we recall, was like a farmer's field, which occupies a region o f two-dimensiona l space . Faraday' s fiel d occupie s a regio n o f three-dimensional space; at any point i n space, we assign a collection of numbers tha t describes the magnetic or electric force at that point. Riemann's ide a was to introduce a collection of numbers a t ever y point i n space tha t would describe how much i t was bent or curved. For example , fo r a n ordinar y two-dimensiona l surface , Rieman n introduced a collection of three numbers at every point that completely describe th e bending of that surface. Riemann found that in four spatial dimensions, on e need s a collectio n o f te n number s a t eac h poin t t o describe its properties. No matter how crumpled o r distorted th e space, this collectio n o f te n number s a t eac h poin t i s sufficient t o encod e al l the information about that space. Let us label these ten number s by the symbols g,,, g l2, g]3, . . .. (Whe n analyzing a four-dimensional space, the lower index ca n range from on e t o four.) The n Riemann' s collection of ten numbers can be symmetrically arranged as in Figure 2.3.7 (It appears as thoug h ther e ar e 1 6 components. However , g-, 2 = g,,, g l?i = g ?>l, an d so on, s o there are actuall y only ten independen t components. ) Today , this collectio n o f number s i s called th e Rieman n metric tensor. Roughly speaking, th e greate r th e valu e o f th e metri c tensor , th e greate r th e crumpling o f th e sheet . N o matte r ho w crumpled th e shee t o f paper , the metri c tensor give s us a simple means of measuring its curvature at any point. If we flattened the crumple d sheet completely, then we would retrieve th e formul a of Pythagoras. Riemann's metric tensor allowe d him t o erect a powerful apparatu s for describin g spaces of any dimension with arbitrar y curvature. To hi s surprise, h e foun d that all these spaces are wel l defined and self-consis tent. Previously , it was thought tha t terribl e contradiction s would arise when investigatin g the forbidde n worl d o f highe r dimensions . T o hi s surprise, Riemann found none. In fact, i t was almost trivial to extend his work t o Aklimensiona l space . Th e metri c tenso r woul d no w resembl e the square s o f a checke r boar d tha t wa s N X N i n size . Thi s wil l hav e profound physica l implications when w e discus s the unificatio n o f al l forces i n th e nex t several chapters.

Zero curvature

Positive curvature

Negative curvature

Figure 2.2. A plane has zero curvature. In Euclidean geometry, the interior angles of a triangle sum to 180 degrees, and parallel lines never meet. In non-Euclidean geometry, a sphere has positive curvature. A triangle's interior angles sum to greater than 180 degrees and parallel lines always meet. (Parallel lines include arcs whose centers coincide with the center of the sphere. This rules out latitudinal lines.) A saddle has negative curvature. The interior angles sum to less than 180 degrees. There are an infinite number of lines parallel to a given line that go through a fixed point.

40

Mathematicians and Mystics

41

Figure 2.3. Riemann's metric tensor contains all the information necessary to describe mathematically a curved space in N dimensions. It takes 16 numbers to describe the metric tensor for each point in four-dimensional space. These numbers can be arranged in a square array (six of these numbers are actually redundant; so the metric tensor has ten independent numbers).

(The secre t o f unification, w e will see, lies in expandin g Riemann's metric to N- dimensional space and then chopping it up into rectangular pieces. Each rectangular piec e correspond s t o a different force . I n this way, we can describ e th e variou s forces of nature b y slotting them int o the metric tensor like pieces of a puzzle. This is the mathematical expression o f th e principl e tha t higher-dimensiona l spac e unifie s th e law s of nature, tha t ther e i s "enough room " t o unite the m i n TV-dimensiona l space. More precisely, there is "enough room" in Riemann's metric to unite th e force s of nature.) Riemann anticipated anothe r developmen t in physics; he was one of the first to discuss multiply connected spaces, or wormholes. To visualize this concept, take two sheets of paper and place one o n top of the other . Make a short cu t o n eac h shee t wit h scissors . Then glu e the tw o sheets together alon g th e tw o cuts (Figure 2.4). (Thi s is topologically the same as Figure 1.1 , except that th e nec k of the wormhol e has length zero.) If a bug lives on th e to p sheet, he may one da y accidentally walk into the cut and find himself on the bottom sheet. He will be puzzled because everything is in th e wron g place. After muc h experimentation , th e bu g

42

ENTERING TH E FIFT H DIMENSIO N

Figure 2.4. Riemann 's cut, with two sheets are connected together along a line. If we walk around the cut, we stay within the same space. But if we walk through the cut, we pass from one sheet to the next. This is a multiply connected surface.

will find that h e ca n re-emerge i n his usual world by re-entering th e cut . If h e walk s aroun d th e cut , the n hi s world look s normal ; bu t whe n h e tries to take a short-cut throug h th e cut , he ha s a problem . Riemann's cut s ar e a n exampl e o f a wormhol e (excep t tha t i t ha s zero length) connectin g tw o spaces. Riemann's cuts were used with great effect b y the mathematicia n Lewi s Carroll i n hi s boo k Through th e Looking-Glass. Riemann's cut , connectin g Englan d wit h Wonderland, i s th e looking glass. Today, Riemann's cuts survive in two forms. First, they are cited i n ever y graduate mathematic s cours e i n th e worl d whe n applie d to the theory of electrostatics or conforrnal mapping. Second, Riemann's cuts ca n b e foun d i n episode s o f "Th e Twiligh t Zone." (I t shoul d b e stressed tha t Rieman n himsel f di d no t vie w his cuts as a mode o f travel between universes.)

Riemann's Legac y

Riemann persisted wit h his work in physics. In 1858, h e even announce d that h e ha d finally succeeded i n a unified description o f light and elec tricity. He wrote, "I am fully convinced that my theory is the correct one , and tha t i n a fe w years i t wil l b e recognize d a s such." H Althoug h hi s metric tenso r gav e hi m a powerful way to describ e an y curved spac e i n any dimension , h e di d no t kno w the precis e equation s tha t th e metri c tensor obeyed ; that is , he di d not kno w what made th e shee t crumple .

Mathematicians an d Mystics 4

3

Unfortunately, Riemann' s efforts t o solve this problem wer e continually thwarte d b y grinding poverty . His successes did no t translat e int o money. H e suffere d anothe r nervou s breakdow n i n 1857 . Afte r man y years, he was finally appointed t o Gauss's coveted position at Gottingen, but it was too late. A life of poverty had broke n hi s health, and like many of the greatest mathematician s throughout history , he died prematurely of consumption a t the age of 39, before he could complete his geometric theory of gravity arid electricit y and magnetism . In summary , Riemann di d muc h mor e tha n la y the foundatio n o f the mathematics of hyperspace. In retrospect, we see that Riemann anticipated som e o f the majo r themes i n moder n physics . Specifically, 1. H e used higher-dimensional space to simplify the laws of nature; that is, to him, electricity and magnetis m a s well as gravity were just effect s cause d b y the crumplin g or warping of hyperspace. 2. H e anticipate d th e concep t o f wormholes. Riemann' s cuts are the simples t examples o f multiply connected spaces . 3. H e expresse d gravit y a s a field . Th e metri c tensor , becaus e i t describes th e forc e o f gravity (via curvature) a t ever y point i n space, i s precisel y Faraday' s fiel d concep t whe n applie d t o gravity. Riemann was unable t o complete his work on force fields because he lacked th e fiel d equation s tha t electricit y and magnetis m an d gravit y obey. In other words, he did not kno w precisely how the universe would be crumple d i n order t o yield the forc e of gravity. He trie d t o discover the field equations for electricity and magnetism, but he died before he could finish that project. At his death, h e stil l had n o way of calculating how much crumpling would be necessary to describe th e forces . These crucial developments would be lef t t o Maxwel l and Einstein . Living i n a Space Warp

The spel l was finally broken. Riemann, in hi s short life , lifte d th e spel l cast by Euclid more tha n 2,000 years before. Riemann's metri c tensor was the weapon with which young mathematician s coul d def y th e Boeotians , wh o howle d a t an y mention o f higher dimensions . Those who followed in Riemann's footsteps found it easier to speak of unseen worlds. Soon, research bloomed al l over Europe. Prominent scientists began

44

ENTERING TH E FIFT H DIMENSIO N

Figure 2.5. A two-dimensional being cannot eat. Its digestive tract necessarily divides it into two distinct pieces, and the being falls apart.

popularizing th e ide a fo r the genera l public . Hermann vo n Helmholtz, perhaps the most famous German physicis t of his generation, was deeply affected b y Riemann's work and wrote and spok e extensivel y to the gen eral publi c abou t th e mathematic s o f intelligent beings livin g on a ball or sphere . According to Helmholtz, these creatures , with reasoning powers similar t o ou r own , would independentl y discove r tha t al l of Euclid's pos-

Mathematicians an d Mystics 4

5

tulates and theorem s were useless. On a sphere, for example , the sums of the interio r angle s of a triangl e do not add up to 180 degrees. The "bookworms" first talked about by Gauss now found themselve s inhabiting Helmholtz's two-dimensional spheres. Helmholt z wrote that "geo metrical axiom s mus t vary according t o th e kin d o f space inhabite d by beings whose powers of reasoning ar e quit e i n conformit y with ours." 9 However, i n hi s Popular Lectures o f Scientific Subjects (1881) , Helmholtz warned hi s reader s tha t i t i s impossible fo r u s t o visualiz e the fourt h dimension. I n fact , h e sai d "suc h a 'representation ' i s as impossible as the 'representation ' o f colours would be to one born blind." 10 Some scientists , marveling at th e eleganc e o f Riemann's work, tried to find physical applications for such a powerful apparatus.'' While some scientists wer e explorin g th e application s o f highe r dimension , othe r scientists asked more practical , mundane questions , such as : How does a two-dimensional being eat? In order for Gauss's two-dimensional people t o eat , thei r mouth s would have t o fac e t o th e side . Bu t i f we now draw thei r digestiv e tract , w e notic e tha t thi s passagewa y completely bisects thei r bodies (Figur e 2.5). Thus i f they eat, thei r bodie s wil l spli t into tw o pieces. I n fact , an y tub e tha t connect s tw o openings i n thei r bodies wil l separat e the m int o two unattached pieces . This present s us with a difficult choice . Either these people ea t like we do and thei r bodies break apart , o r the y obey different law s of biology. Unfortunately, th e advance d mathematic s of Rieman n outstrippe d the relativel y backward understanding o f physics in the nineteent h cen tury. Ther e wa s no physica l principl e t o guid e furthe r research . W e would hav e t o wai t anothe r centur y for th e physicist s t o catc h u p with the mathematicians . Bu t this did no t sto p nineteenth-centur y scientists from speculatin g endlessly about what beings from th e fourth dimension would loo k like . Soon , the y realize d tha t suc h a fourth-dimensional being would have almost God-like powers.

To B e a Go d Imagine being abl e t o walk through walls . You wouldn't have to bother with opening doors; you could pass right through them . You wouldn't hav e t o g o aroun d buildings ; you coul d enter the m throug h thei r wall s an d pillar s an d ou t throug h th e bac k wall. Yo u wouldn't have t o detou r aroun d mountains ; you coul d ste p right int o them . Whe n hungry , you coul d simpl y reac h throug h th e

46 E N T E R I N G T H E F I F T H D I M E N S 1 0

N

refrigerator doo r withou t openin g it . You could neve r b e accidentally locked outsid e your car; you could simply step throug h th e ca r door . Imagine bein g able t o disappear o r reappea r a t will. Instea d of driving t o school o r work , you would just vanis h an d rematerializ e i n you r classroom o r office . Yo u wouldn' t nee d a n airplan e t o visi t far-awa y places, you coul d just vanis h and rematerializ e where you wanted. You would never be stuck in cit y traffic durin g rus h hours ; you and your car would simply disappear an d rematerializ e at your destination. Imagine havin g x-ray eyes. You would b e abl e t o se e accident s happening fro m a distance. Afte r vanishin g and rematerializin g at th e sit e of an y accident , yo u coul d se e exactl y where th e victim s were, even if they were buried unde r debris . Imagine being abl e t o reach int o an objec t without opening it . You could extrac t th e section s from a n orange without peeling o r cutting it. You would b e haile d a s a master surgeon , wit h the abilit y t o repair th e internal organs o f patients without ever cutting the skin, thereby greatly reducing pain an d th e risk of infection. You would simply reach into the person's body, passing directl y through th e skin , and perfor m th e delicate operation. Imagine what a criminal could do with these powers. He could ente r the most heavily guarded bank . He could see through th e massive doors of the vaul t for th e valuables and cas h an d reac h insid e and pul l the m out. H e coul d the n strol l outside a s the bullet s from th e guard s passe d right throug h him . With these powers, no prison coul d hold a criminal. No secrets could be kept from us. No treasures could be hidden from us. N o obstruction s could sto p us . We would trul y be miracl e workers, performing feat s beyon d th e comprehensio n o f mortals. We would also be omnipotent . What being coul d possess such God-like power ? The answer : a being from a higher-dimensional world. Of course, these feats ar e beyon d th e capability o f any three-dimensiona l person . Fo r us , walls ar e soli d an d prison bar s are unbreakable . Attempting to walk through wall s will only give u s a painful, bloody nose . Bu t for a four-dimensional being, thes e feats would be child' s play. To understan d ho w these miraculou s feats ca n b e performed , con sider agai n Gauss' s mythica l two-dimensional beings , livin g o n a twodimensional tabl e top . T o jai l a criminal, the Flatlander s simply draw a circle around him . No matter which way the crimina l moves, he hits the impenetrable circle . However, it is a trivial tas k for us to spring the prisoner fro m jail. W e just reac h down , gra b th e Flatlander , pee l hi m off the two-dimensional world, and redeposit him elsewhere on his world (Fig-

Mathematicians and Mystics

47

Figure 2.6. In Flatland, a "jail" is a circle drawn around a person. Escape from this circle is impossible in two dimensions. However, a three-dimensional person can yank a Flatlander out of jail into the third dimension. To a jailer, it appears as though the prisoner has mysteriously vanished into thin air. ure 2.6) . This feat, which is quite ordinary in three dimensions , appears fantastic i n tw o dimensions. To his jailer, th e prisoner ha s suddenly disappeared fro m a n escapeproof prison, vanishing into thin air. Then just as suddenly, he reappear s somewhere else. If you explain to the jailer tha t the prisone r was moved "up" an d of f Flatland, h e would not understan d wha t you were saying. The wor d u p does no t exis t i n th e Flatlander' s vocabulary , nor ca n h e visualize th e concept . The othe r feat s can be similarly explained. For example, notice tha t the interna l organs (lik e the stomac h or heart) o f a Flatlander are completely visible to us, in the same way that we can see the internal structure of cells on a microscope slide . It's now trivial to reach inside a Flatlander and perfor m surgery without cutting the skin. We can also peel the Flat-

48

ENTERING TH E FIFT H DIMENSIO N

Figure 2. 7. If we peel a Flatlander from his world and flip him over in three dimensions, his heart, now appears on the right-hand side. All his internal organs have been reversed. This transformation is a medical impossibility to someone who lives strictly in Flatland.

lander of f his world, flip hi m around , an d pu t hi m bac k down . Notice that hi s lef t an d righ t organ s ar e no w reversed, s o that hi s heart i s on the righ t side (Figur e 2.7). Viewing Flatland , notic e als o tha t w e ar e omnipotent . Eve n i f th e Flatlander hide s insid e a house o r unde r th e ground , w e can se e him perfectly. H e woul d regard ou r power s as magical; we, however, would know that no t magic , but simpl y a more advantageou s perspective, is at work. (Althoug h such feats o f "magic" are, in principle, possible within the realm of hyperspace physics , we should caution , once again, that th e technology necessar y t o manipulat e space-tim e fa r exceed s anythin g possible on the earth, at least for hundreds of years. The abilit y to manipulate space-time may be within the domain of only some extraterrestrial life i n th e univers e far in advanc e o f anything found o n th e earth , with the technolog y t o maste r energ y o n a scal e a quadrillio n time s larger than ou r mos t powerfu l machines. ) Although Riemann's famous lecture was popularized b y the work of Helmholtz an d man y others , th e la y public coul d mak e littl e sense o f this o r th e eatin g habit s o f two-dimensional creatures. Fo r th e averag e

Mathematicians an d Mystics 4

9

person, th e questio n wa s more direct : Wha t kin d o f being s ca n wal k through walls , se e throug h steel , an d perfor m miracles ? What kin d o f beings are omnipoten t an d obe y a set of laws different fro m ours? Why ghosts, of course! In the absenc e o f any physical principle motivating the introductio n of higher dimensions, the theor y of the fourth dimension suddenly took an unexpected turn . We will now begin a strange but importan t detou r in th e histor y of hyperspace , examinin g it s unexpected bu t profoun d impact o n th e art s an d philosophy . Thi s tou r throug h popula r cultur e will sho w ho w th e mystic s gave u s cleve r ways i n whic h t o "visualize " higher-dimensional space .

Ghosts fro m the Fourt h Dimension

The fourt h dimensio n penetrate d th e public' s consciousnes s in 1877 , when a scandalous tria l in London gav e it an internationa l notoriety. The Londo n newspaper s widel y publicize d th e sensationa l claim s and bizarr e trial of psychic Henry Slade. The raucou s proceedings drew in som e o f th e mos t prominen t physicist s of th e day . As a resul t of all the publicity , tal k o f th e fourt h dimensio n lef t th e blackboard s o f abstract mathematicians and burst into polite society, turning up in dinner-table conversation s throughou t London . Th e "notoriou s fourt h dimension" wa s now the tal k of the town . It al l began , innocentl y enough , whe n Slade , a psychi c from th e United States , visited London an d hel d seance s with prominen t townspeople. H e was subsequently arrested fo r fraud an d charge d wit h "using subtle craft s an d devices , by palmistr y and otherwise, " t o deceiv e hi s clients.12 Normally , this trial migh t hav e gon e unnoticed . Bu t Londo n society was scandalized and amuse d when eminent physicists came to his defense, claimin g tha t hi s psychi c feats actuall y proved tha t h e coul d summon spirit s living in th e fourt h dimension. This scandal was fueled by th e fac t tha t Slade' s defender s were no t ordinar y Britis h scientists, but rathe r some of the greates t physicists in the world . Many went on t o win th e Nobe l Prize in physics. Playing a leading role in stirring up thi s scandal was Johann Zollner, a professor o f physics and astronom y at the Universit y of Leipzig. It was Zollner who marshaled a galaxy of leading physicists to come t o Slade' s defense. That mystic s coul d perfor m parlo r trick s fo r th e roya l cour t an d proper society , o f course , wa s nothin g new . Fo r centuries , the y ha d

50 E N T E R I N G T H E F I F T H D I M E N S 1 O

N

claimed tha t they could summo n spirit s to read th e writing within closed envelopes, pul l objects from close d bottles , resea l broke n matc h sticks, and intertwin e rings. The strang e twis t t o thi s trial was that leading scientists claimed thes e feat s wer e possible b y manipulating object s in th e fourth dimension . I n th e process, the y gav e th e publi c it s firs t under standing of how to perform thes e miraculous feats via the fourth dimension. Zollner enlisted the help of internationally prominent physicist s who participated i n the Society for Psychical Research an d wh o even rose t o lead th e organization , including some o f the mos t distinguishe d names of nineteenth-century physics : William Crookes, inventor of the cathod e ray tube, which today is used i n ever y television set and compute r mon itor in the world;13 Wilhelm Weber, Gauss's collaborator an d the mento r of Rieman n (today , th e internationa l uni t o f magnetis m i s officiall y named th e "weber " afte r him) ; J. J. Thompson , wh o won the Nobe l Prize in 190 6 fo r th e discover y of the electron ; an d Lor d Rayleigh , recognized by historians as one o f the greates t classical physicists of the lat e nineteenth centur y and winne r of the Nobe l Prize in physics in 1904 . Crookes, Weber, and Zollner , in particular, took a special interest in the wor k of Slade , who was eventually convicted o f fraud b y the court . However, he insiste d tha t h e coul d prov e hi s innocence b y duplicating his feat s befor e a scientifi c body . Intrigued , Zollne r too k u p th e chal lenge. A number o f controlled experiment s were conducted i n 187 7 to test Slade's ability to send object s through th e fourth dimension. Several distinguished scientist s were invited by Zollner to evaluate Slade' s abilities. First, Slad e wa s given two separate, unbroke n woode n rings . Could he pus h on e woode n rin g past the other , so that the y were intertwined without breaking? If Slade succeeded, Zollnerwrote, itwould "represent a miracle , that is , a phenomenon whic h our conception s heretofor e o f physical an d organi c processe s woul d b e absolutel y incompeten t t o explain."14 Second, h e wa s given the shel l of a sea snail, which twiste d either t o the right or t o the left . Coul d Slad e transfor m a right-handed shel l into a left-hande d shell and vic e versa? Third, h e was given a closed loo p o f rope made o f dried anima l gut. Could h e mak e a knot i n th e circula r rope without cutting it? Slade was also given variations of these tests. For example, a rope was tied int o a right-hande d kno t an d it s end s wer e seale d wit h wa x an d impressed with Zollner's personal seal. Slade was asked to untie the knot, without breaking th e wax seal, and rede the rope in a left-handed knot .

Mathematicians an d Mystics 5

1

Since knot s ca n alway s b e untie d i n th e fourt h dimension , thi s fea t should b e eas y for a fourth-dimensional person . Slad e was also asked to remove th e content s o f a sealed bottl e withou t breaking th e bottle . Could Slad e demonstrat e thi s astounding ability ? Magic i n the Fourth Dimensio n Today we realize tha t the manipulatio n o f higher-dimensional space, as claimed by Slade, would require a technology far in advance of anything possible on this planet for the conceivable future. However , what is interesting about thi s notorious cas e is that Zollne r correctly concluded tha t Slade's feats of wizardry could be explained i f one coul d somehow move objects throug h th e fourt h dimension . Thu s fo r pedagogica l reasons , the experiment s o f Zollner ar e compellin g and wort h discussing. For example, in thre e dimensions , separat e ring s cannot b e pushe d through eac h othe r unti l the y intertwine without breaking them . Simi larly, closed, circular pieces of rope cannot b e twisted into knots without cutting them . An y Boy or Gir l Scou t wh o ha s struggle d wit h knots fo r his o r he r meri t badge s know s tha t knot s i n a circula r loo p o f rop e cannot b e removed . However , i n highe r dimensions , knot s ar e easil y unraveled an d ring s can be intertwined . Thi s is because ther e i s "more room" in which to move ropes past each other and rings into each other . If the fourt h dimensio n existed , ropes and ring s could b e lifte d of f our universe, intertwined , an d the n returne d t o ou r world . In fact , i n th e fourth dimension , knot s ca n neve r remai n tied . The y ca n alway s b e unraveled withou t cuttin g th e rope . Thi s fea t i s impossibl e i n thre e dimensions, bu t trivia l in th e fourth . The thir d dimension , a s it turn s out, i s the onl y dimension i n whic h knots sta y knotted. (Th e proo f o f this rather unexpecte d resul t is given in th e notes. 15) Similarly, in thre e dimension s it is impossible t o conver t a rigid left handed objec t into a right-hande d one . Human s ar e bor n wit h heart s on thei r lef t side , an d n o surgeon , n o matte r no w skilled, can revers e human interna l organs . Thi s i s possible (a s first pointed ou t b y mathematician Augus t Mobiu s i n 1827 ) onl y i f w e lif t th e bod y ou t o f ou r universe, rotate i t in the fourth dimension, and then reinsert it back into our universe . Two of these tricks are depicted i n Figure 2.8; they can b e performed onl y if objects can b e move d i n th e fourt h dimension . Polarizing the Scientific Community Zollner sparke d a stor m o f controvers y when, publishing i n bot h th e Quarterly Journal o f Science an d Transcendental Physics, h e claime d tha t

Figure 2.8. The mystic Henry Slade clamed to be able to change right-handed snail shells into left-handed ones, and to remove objects from sealed bottles. These feats are impossible in three dimensions, but are trivial if one can move objects through the fourth dimension.

52

Mathematicians an d Mystics 5

3

Slade amaze d hi s audience s wit h thes e "miraculous " feat s durin g seances i n the presence of distinguished scientists . (However, Slade als o flunked som e o f the test s that were conducted unde r controlle d condi tions.) Zollner's spirite d defens e o f Slade's feat s wa s sensationalize d throughout Londo n society . (I n fact , thi s wa s actually on e o f severa l highly publicize d incident s involvin g spiritualist s an d medium s i n th e late nineteent h century . Victoria n Englan d wa s apparendy fascinate d with th e occult. ) Scientists , a s well as th e genera l public , quickl y too k sides i n th e matter . Supportin g Zollner' s claim s was his circl e o f repu table scientists , includin g Webe r an d Crookes . Thes e were no t averag e scientists, bu t master s o f th e ar t o f scienc e an d seasone d observer s o f experiment. The y had spent a lifetime working with natural phenomena , and no w before thei r eyes, Slade was performing feats that were possible only if spirits lived in th e fourt h dimension . But detractors o f Zollner pointed out that scientists, because the y are trained t o trus t thei r senses , are th e wors t possible peopl e t o evaluate a magician. A magician is trained specificall y to distract, deceive, and con fuse thos e ver y senses. A scientist ma y carefully observe th e magician' s right hand, but i t is the lef t han d tha t secretly performs th e trick . Critics also pointed ou t tha t onl y another magicia n i s clever enough t o detec t the sleight-of-han d trick s of a fellow magician . Onl y a thief ca n catc h a thief. One particularl y savag e piec e o f criticism, published i n th e scienc e quarterly magazine Bedrock, was made agains t two other prominent phys icists, Sir W. F. Barrett and Si r Oliver Lodge, an d thei r work on telepathy. The articl e was merciless: It i s not necessar y either t o regar d th e phenomen a o f so-called telepathy as inexplicable or t o regard th e menta l condition of Sir W. F. Barrett an d Sir Olive r Lodge a s indistinguishable from idiocy . There i s a thir d possi bility. Th e will to believe has mad e the m read y to accep t evidenc e obtaine d under condition s which they would recogniz e t o b e unsoun d i f they ha d been traine d i n experimental psychology.

Over a centur y later , precisel y th e sam e arguments , pr o an d con , would b e use d i n th e debat e ove r th e feat s o f th e Israel i psychi c Ur i Geller, who convinced tw o reputable scientist s at the Stanfor d Researc h Institute i n Californi a tha t h e coul d ben d key s by mental powe r alon e and perfor m othe r miracles. (Commentin g on this, some scientist s have repeated a saying tha t dates back t o the Romans : "Populus vult decipi,

54 E N T E R I N G T H E F I F T H D I M E N S I O

N

ergo decipiatur " [Peopl e wan t t o b e deceived , therefor e le t the m b e deceived].) The passion s ragin g within the British scientific communit y touched off a lively debate tha t quickly spread acros s the English Channel. Unfortunately, in th e decade s followin g Riemann's death, scientist s lost sight of his original goal, to simplify the law s of nature throug h highe r dimensions. As a consequence, the theor y of higher dimensions wandered int o many interesting bu t questionabl e directions . Thi s i s an importan t les son. Withou t a clear physica l motivation o r a guiding physica l picture, pure mathematica l concept s sometime s drift int o speculation . These decade s wer e not a complet e loss , however, because mathe maticians and mystic s like Charles Hinton would invent ingenious way s in which to "see" th e fourt h dimension. Eventually, the pervasiv e influence o f the fourt h dimension would come full circl e and cross-pollinat e the worl d of physics once again .

3

The Ma n Who "Saw" the Fourt h Dimensio n [T]he fourt h dimensio n ha d becom e almos t a househol d word b y 1910. . . . Ranging from a n idea l Platonic or Kantia n reality—or eve n Heaven—th e answe r t o al l o f th e problem s puzzling contemporary science, th e fourt h dimensio n could be al l things to all people. Linda Dalrympl e Henderso n

W

ITH th e passion s arouse d b y th e tria l o f th e "notoriou s Mr .

Slade," it was perhaps inevitable that the controversy would even-

tually spawn a best-selling novel . In 1884 , afte r a decad e o f acrimoniou s debate , clergyma n Edwi n

Abbot, headmaste r o f the Cit y of London School , wrote th e surprisingly successful an d endurin g nove l Flatland: A Romance o f Many Dimensions by a Square.* Becaus e o f th e intens e publi c fascination wit h highe r dimen *It wasn't surprising that a clergyman wrote the novel , since theologians o f the Churc h of England wer e among th e first to jump into the fra y create d by the sensationalize d trial . For uncounte d centuries , clergyme n ha d skillfull y dodge d suc h perennia l question s a s Where ar e heave n an d hell ? an d Wher e d o angel s live ? Now , the y foun d a convenien t resting place for these heavenl y bodies: th e fourt h dimension . Th e Christia n spiritualist A. T. Schofield , i n hi s 188 8 boo k Another World, argue d a t lengt h tha t Go d an d th e spirit s resided in the fourth dimension. 1 Not to be outdone, i n 1893 the theologian Arthur Willink wrote Th e World o f the Unseen, in which he claime d tha t i t was unworthy of God t o reside in the lowl y fourth dimension . Willink claimed tha t the only domain magnificen t enough fo r God wa s infinite-dimensional space. 2

55

56 E N T E R 1 N G T H E F I F T H D I M E N S I O

N

sions, th e boo k wa s an instan t success in England , with nine successive reprintings by the year 1915, and edition s too numerous t o count today. What was surprising about the novel Flatland was that Abbott, for th e first time, use d th e controvers y surrounding th e fourt h dimensio n a s a vehicle for bitin g socia l criticism and satire . Abbot too k a playfu l swipe at th e rigid , piou s individual s who refuse d t o admi t th e possibilit y of other worlds. The "bookworms " of Gauss became th e Flatlanders . The Boeotians who m Gaus s so feared becam e th e Hig h Priests , who would persecute—with th e vigo r and impartialit y of the Spanis h Inquisition — anyone who dared mention th e unseen thir d dimension . Abbot's Flatland i s a thinl y disguised criticis m of th e subtl e bigotr y and suffocatin g prejudice prevalen t i n Victorian England. Th e her o of the nove l is Mr. Square, a conservative gentleman wh o lives in a socially stratified, two-dimensiona l land wher e everyon e i s a geometri c object . Women, occupyin g th e lowes t rank i n th e socia l hierarchy , ar e mer e lines, th e nobilit y are polygons , while the Hig h Priest s ar e circles . Th e more side s people have, the highe r thei r social rank. Discussion of the thir d dimensio n i s strictly forbidden. Anyone mentioning it is sentenced t o severe punishment. Mr. Square is a smug, selfrighteous perso n wh o would never thin k o f challengin g th e Establish ment for it s injustices. One day , however, his life i s permanently turne d upside dow n whe n h e i s visited b y a mysteriou s Lord Sphere , a three dimensional sphere. Lord Spher e appear s t o Mr. Square as a circle that can magicall y change siz e (Figur e 3.1) Lord Spher e patientl y trie s t o explai n tha t h e come s fro m anothe r world calle d Spaceland , where al l objects have three dimensions. However, Mr . Squar e remain s unconvinced ; h e stubbornl y resist s the ide a that a thir d dimensio n ca n exist . Frustrated , Lor d Spher e decide s t o resort t o deeds, no t mer e words. H e the n peels Mr. Square of f the twodimensional Flatlan d an d hurl s hi m int o Spaceland . I t i s a fantastic , almost mystical experience tha t change s Mr. Square's life . As th e fla t Mr . Squar e float s in th e thir d dimensio n lik e a sheet o f paper drifting in th e wind , he ca n visualize only two-dimensional slices of Spaceland. Mr . Square, seein g only the cros s sections of three-dimensional objects , views a fantastic world where object s chang e shap e an d even appear and disappea r into thi n air . However, when h e trie s to tell his fellow Flatlanders o f the marvels he saw in his visit to the third dimension, the Hig h Priest s consider hi m a blabbering, seditiou s maniac. Mr. Square become s a threa t t o th e Hig h Priest s because h e dare s t o challenge thei r authorit y an d thei r sacre d belie f tha t onl y tw o dimensions can possibl y exist.

Figure 3.1. In Flatland, Mr. Square encounters Lord Sphere. As Lord Sphere passes through Flatland, he appears to be a circle that becomes successivley larger and then smaller. Thus Flatlanders cannot visualize three-dimensional beings, but can understand their cross sections.

57

58 E N T E R I N

G TH E FIFT H DIMENSIO N

The boo k end s on a pessimistic note. Although h e is convinced tha t he did , indeed , visi t th e third-dimensiona l worl d o f Spaceland , Mr . Square i s sent t o jail an d condemne d t o spen d th e res t o f hi s day s in solitary confinement. A Dinne r Part y i n the Fourth Dimension

Abbot's nove l is important becaus e i t was the firs t widel y read popular ization o f a visit t o a higher-dimensiona l world . His description o f Mr. Square's psychedeli c tri p int o Spacelan d i s mathematically correct. I n popular account s an d th e movies , interdimensiona l trave l throug h hyperspace i s ofte n picture d wit h blinkin g light s an d dark , swirlin g clouds. However, the mathematic s of higher-dimensional trave l is much more interestin g tha n th e imaginatio n o f fictio n writers . T o visualize what a n interdimensiona l tri p woul d loo k like , imagin e peelin g Mr . Square of f Flatland an d throwin g him into the air . As he float s throug h our three-dimensiona l world , let' s sa y that h e come s acros s a huma n being. What do w e look like to Mr. Square? Because his two-dimensional eyes can see only flat slices of our world, a huma n woul d look like a singularly ugly and frightenin g object . First, he migh t se e tw o leather circle s hovering i n fron t o f him (ou r shoes) . As he drift s upward , thes e tw o circles change colo r an d tur n int o clot h (our pants). Then thes e tw o circles coalesce int o on e circl e (ou r waist) and spli t int o thre e circle s of cloth an d chang e colo r agai n (ou r shir t and ou r arms) . As he continue s t o floa t upward , thes e thre e circle s of cloth merge into one smaller circle of flesh (ou r neck and head). Finally, this circle of flesh turn s into a mass of hair, and the n abruptly disappears as Mr. Square float s abov e ou r heads . To Mr . Square, thes e mysteriou s "humans" are a nightmarish, maddeningly confusing collection o f constantly changing circle s made o f leather, cloth , flesh , an d hair . Similarly, if we were peeled of f our three-dimensiona l univers e an d hurled int o th e fourt h dimension , w e would fin d tha t commo n sens e becomes useless. As we drift through th e fourth dimension, blobs appear from nowher e in front of our eyes. They constantly change in color, size, and composition , defyin g al l the rule s of logic of our three-dimensiona l world. And they disappear into thin air, to be replaced by other hoverin g blobs. If we were invite d t o a dinne r part y i n th e fourt h dimension , ho w would we tell the creature s apart ? We would have to recognize the m by

The Man Wh o "Saw" the Fourth Dimension 5

9

the differences i n how these blobs change. Eac h person i n higher dimen sions would hav e hi s o r he r ow n characteristi c sequence s o f changin g blobs. Ove r a perio d o f time , w e woul d lear n t o tel l thes e creature s apart b y recognizin g thei r distinctiv e pattern s o f changin g blob s an d colors. Attending dinner partie s i n hyperspace might be a trying exper-

Class Struggl e i n the Fourt h Dimensio n

The concep t o f th e fourt h dimensio n ha d s o pervasivel y infected th e intellectual climate b y the late nineteenth centur y that eve n playwrights poked fun at it. In 1891, Oscar Wilde wrote a spoof on these ghost stories, "The Cantervill e Ghost," whic h lampoons the exploit s of a certain gullible "Psychica l Society" ( a thinly veiled reference to Crookes' s Society for Psychica l Research) . Wild e wrot e o f a long-sufferin g ghost wh o encounters th e newl y arrive d America n tenant s o f Canterville . Wilde wrote, "Ther e was evidently no tim e to be lost , so hastily adopting th e Fourth Dimensio n of Spac e a s a mean s of escape , h e [th e ghost] vanished throug h th e wainscoting and th e house becam e quiet. " A more serious contribution t o the literature of the fourth dimension was the wor k of H. G. Wells. Although he i s principally remembered fo r his works in science fiction, he was a dominant figure in the intellectual life o f London society , noted fo r his literary criticism, reviews, and piercing wit. In hi s 1894 novel, Th e Time Machine, h e combine d severa l mathematical, philosophical, and political diemes. He popularized a new idea in science—that the fourth dimension might also be viewed as time, not necessarily space:* Clearly . . . any rea l bod y mus t hav e extensio n i n four directions : i t mus t have Length , Breadth , Thickness , and—Duration. But through a natura l infirmity o f the fles h . . . we incline to overlook thi s fact. Ther e are really four dimensions , thre e whic h w e cal l th e thre e lane s o f Space , an d a Fourth, Time. There is, however, a tendency to draw an unreal distinctio n between th e forme r thre e dimension s and th e latter , becaus e i t happen s that ou r consciousnes s move s intermittentl y in on e directio n alon g th e latter fro m th e beginnin g t o the en d o f our lives. 3 *Wells was not th e firs t t o speculat e tha t tim e coul d b e viewed as a new type of fourth dimension, different from a spatial one. Jean d'Alemberthad considered tim e as the fourt h dimension i n hi s 1754 article "Dimension. "

60 E N T E R I N G T H E F I F T H D I M E N S I O

N

Like Flalland befor e it , wha t makes Th e Time Machine s o enduring , even a century after it s conception, i s its sharp politica l and socia l critique. England i n th e year 802,701, Wells's protagonist finds, is not th e gleaming citade l o f moder n scientifi c marvels that th e positivist s fore told. Instead, th e futur e Englan d is a land where the clas s struggle went awry. Th e workin g class was cruelly forced t o liv e underground , unti l the workers mutated into a new, brutish species of human, the Morlocks, while th e rulin g class, with its unbridled debauchery , deteriorated an d evolved into the useles s race o f elflike creatures , th e Eloi . Wells, a prominent Fabia n socialist , was using the fourt h dimensio n to revea l th e ultimat e iron y o f th e clas s struggle . Th e socia l contrac t between th e poo r an d th e ric h ha d gon e completel y mad. Th e useless Eloi are fed and clothe d b y the hard-workin g Morlocks, but th e workers get the final revenge: The Morlock s eat the Eloi. The fourth dimension, in othe r words, became a foil fo r a Marxist critique o f modern society, but wit h a novel twist: The workin g class will not brea k the chain s of th e rich, a s Marx predicted. The y will ea t the rich . In a shor t story , "Th e Plattne r Story, " Well s even toye d wit h th e paradox of handedness. Gottfried Plattner, a science teacher, is performing an elaborate chemica l experiment, bu t his experiment blow s up and sends him into another universe. When he returns from the netherworld to the real world, he discovers that his body has been altered in a curious fashion: Hi s heart i s now on hi s righ t side , and h e i s now left handed . When the y examine him , his doctors are stunned to find that Plattner's entire bod y ha s bee n reversed , a biologica l impossibility in ou r three dimensional world: " [T] he curious inversion of Plattner's right and lef t sides is proof tha t he ha s moved out o f our spac e into what is called th e Fourth Dimension , and tha t he has returned agai n to our world." However, Plattner resists the ide a of a postmortem dissectio n afte r hi s death, thereby postponing "perhaps forever, the positiv e proof tha t his entire body had ha d it s left an d righ t sides transposed." Wells wa s well awar e tha t ther e ar e tw o ways t o visualiz e ho w left handed object s ca n b e transforme d int o right-hande d objects . A Flat lander, fo r example , ca n b e lifte d ou t o f hi s world, flipped over , an d then place d bac k in Flatland, thereby reversing his organs. Or th e Flat lander ma y live on a Mobiu s strip, created b y twisting a stri p of pape r 180 degrees an d the n gluin g th e end s together . I f a Flatlande r walk s completely around th e Mobius strip and returns, he finds that his organs have bee n reverse d (Figur e 3.2). Mobius strips have othe r remarkable properties that have fascinated scientists over the past century. For example, i f you wal k completel y around th e surface , yo u will fin d tha t i t ha s

The Man Who "Saw" the Fourth Dimension

61

Figure 3.2. A Mobius strip is a strip with only one side. Its outside and inside are identical. If a Flatlander wanders around a Mobius strip, his internal organs will be reversed.

only one side . Also, if you cut it in half along the cente r strip, it remain s in on e piece . This has given ris e t o the mathematicians ' limerick : A mathematician confided That a Mobius band i s one-sided And you'll get quite a laugh If you cu t i t in half , For it stays in one piec e when divided.

In hi s classi c Th e Invisible Man, Well s speculate d tha t a ma n migh t even becom e invisible by some tric k involving "a formula , a geometrica l expression involving four dimensions." Wells knew that a Flatlander dis appears if he i s peeled off his two-dimensional universe ; similarly , a man could become invisible if he could somehow lea p into the fourth dimen sion. In th e shor t story "Th e Remarkabl e Cas e o f Davidson's Eyes, " Well s explored th e idea that a "kin k i n space" might enable an individua l t o

62 E N T E R I N

G T HE F I F T H D I M E N S I O N

see across vast distances. Davidson , the her o of the story , one day finds he ha s th e disturbin g power o f being abl e t o se e events transpirin g on a distant South Se a island. This "kink in space" is a space warp whereby light from th e Sout h Sea s goes through hyperspac e and enter s his eyes in England . Thu s Well s used Riemann' s wormhole s as a literary device in hi s fiction . In Th e Wonderful Visit, Wells explored th e possibility that heaven exists in a parallel worl d o r dimension . The plo t revolve s around th e predic ament o f a n ange l wh o accidentall y falls fro m heave n an d land s i n a n English country village. The popularit y o f Wells's work opened u p a ne w genr e o f fiction . George McDonald , a friend o f mathematician Lewi s Carroll, also spec ulated about the possibility of heaven being located in the fourth dimension. I n McDonald' s fantasy Lilith, written i n 1895 , th e her o create s a dimensional windo w between ou r univers e and othe r worlds by manipulating mirror reflections . And i n th e 190 1 story Th e Inheritors by Joseph Conrad an d For d Mado x Ford , a rac e o f superme n fro m th e fourt h dimension enter s int o ou r world . Cruel and unfeeling , these superme n begin t o take over the world.

The Fourt h Dimension as Art The year s 189 0 t o 191 0 ma y b e considere d th e Golde n Year s o f th e Fourth Dimension . I t was a tim e durin g whic h the idea s originate d b y Gauss and Rieman n permeated literar y circles, the avant garde, an d th e thoughts o f th e genera l public , affectin g trend s i n art , literature , an d philosophy. Th e ne w branc h o f philosophy , calle d Theosophy , wa s deeply influenced b y higher dimensions . On th e on e hand , seriou s scientist s regrette d thi s developmen t because th e rigorou s result s o f Rieman n wer e no w bein g dragge d through tabloi d headlines. O n the other hand, the popularization of the fourth dimensio n ha d a positive side. Not only did it make the advances in mathematic s availabl e t o th e genera l public , bu t i t als o serve d a s a metaphor tha t coul d enric h an d cross-fertiliz e cultural currents . Art historia n Lind a Dalrympl e Henderson , writin g i n Th e Fourth Dimension and Non-Euclidean Geometry in Modern Art, elaborate s on thi s and argue s that th e fourt h dimension cruciall y influenced the development of Cubism and Expressionis m i n th e ar t world. She writes that "i t was among the Cubist s that the first and most coherent ar t theory based on the new geometries was developed."4 To the avan t garde, the fourt h

The Man Who "Saw" the Fourth Dimension

63

Figure 3.3. One scene in the Bayeux Tapestry depicts frightened English troops pointing to an apparition in the sky (Halley's comet). The figures are flat, as in most of the art done in the Middle Ages. This signified that God was omnipotent. Pictures were thus drawn two dimensionally. (Giraudon/Art Resource) dimension symbolize d the revol t against the excesse s of capitalism. The y saw its oppressiv e positivis m and vulga r materialis m a s stifling creative expression. Th e Cubists , for example , rebelle d agains t th e insufferabl e arrogance o f th e zealot s o f science who m the y perceived a s dehuman izing the creativ e process . The avan t garde seize d on th e fourth dimension a s their vehicle. O n the on e hand , th e fourth dimensio n pushe d th e boundaries o f moder n science t o thei r limit . It was more scientifi c than th e scientists . O n th e other hand , i t wa s mysterious . An d flauntin g th e fourt h dimensio n tweaked th e nose s o f th e stiff , know-it-al l positivists. In particular , thi s took th e for m o f an artisti c revolt against th e law s of perspective . In the Middl e Ages, religious ar t was distinctive for its deliberate lack of perspective. Serfs , peasants, an d king s were depicte d a s though the y were flat, much in the way children dra w people. Thes e paintings largel y reflected th e church' s vie w tha t Go d was omnipotent an d coul d there fore se e al l part s o f ou r worl d equally . Art ha d t o reflec t hi s poin t o f view, s o th e worl d wa s painted tw o dimensionally . Fo r example , th e famous Bayeu x Tapestr y (Figur e 3.3 ) depict s th e superstitiou s soldier s of King Harold II of England pointin g i n frightened wonde r a t an omi -

64

ENTERING TH E FIFT H DIMENSIO N

Figure 3.4. During the Renaissance, painters discovered the third dimension. Pictures were painted with perspective and were viewed from the vantage point of a single eye, not God's eye. Note that all the lines in Leonardo da Vinci's fresco The Las t Supper converge t o a point at the horizon. (Bettmann Archive)

nous comet soaring overhead i n April 1066, convinced that it is an ome n of impendin g defeat . (Si x centurie s later , th e sam e come t woul d b e christened Halley' s comet. ) Harol d subsequentl y los t th e crucia l Battle of Hasting s t o Willia m th e Conqueror , wh o wa s crowned th e kin g o f England, an d a ne w chapte r i n Englis h histor y began. However , th e Bayeux Tapestry, like other medieva l works of art, depicts Harold' s sol diers' arm s and face s as flat, as though a plane of glass had bee n place d over thei r bodies , compressin g the m agains t th e tapestry. Renaissance ar t wa s a revol t against thi s fla t God-centere d perspec tive, and man-centere d ar t bega n t o flourish, with sweeping landscapes and realistic , three-dimensiona l peopl e painte d fro m th e poin t o f view of a person's eye . In Leonard o d a Vinci's powerful studie s o n perspec tive, we see the line s in his sketches vanishing into a single point o n th e horizon. Renaissanc e ar t reflecte d th e wa y the ey e viewed th e world , from th e singula r point o f view of the observer . I n Michelangelo' s fres coes or in da Vinci's sketch book, we see bold, imposing figures jumping out of the second dimension . In other words, Renaissance art discovered the thir d dimensio n (Figur e 3.4). With th e beginnin g o f the machin e ag e an d capitalism , the artistic world revolte d agains t th e col d materialis m tha t seeme d t o dominat e

The Man Wh o "Saw" the Fourth Dimension 6

5

industrial society . To th e Cubists , positivism was a straitjacke t tha t con fined u s t o what could b e measure d i n th e laboratory , suppressin g th e fruits o f our imagination . The y asked : Why must ar t b e clinicall y "realistic"? This Cubist "revolt against perspective" seize d the fourth dimension becaus e i t touche d th e thir d dimensio n fro m all possible perspec tives. Simply put, Cubis t art embrace d th e fourt h dimension . Picasso's paintings are a splendid example , showing a clear rejection of th e perspective , with women's faces viewe d simultaneously from several angles . Instea d o f a singl e poin t o f view, Picasso' s paintings sho w multiple perspectives, as though the y were painted b y someone from the fourth dimension , abl e t o se e al l perspective s simultaneousl y (Figur e 3.5). Picasso wa s once accoste d o n a trai n b y a strange r wh o recognize d him. The stranger complained : Why couldn't h e draw pictures of people the wa y they actually were? Why did h e hav e t o distor t th e wa y people looked? Picasso then aske d th e ma n t o sho w him picture s of his family . After gazin g a t th e snapshot , Picass o replied , "Oh , i s your wif e reall y that small and flat? " T o Picasso, any picture, n o matte r ho w "realistic," depended o n th e perspectiv e of the observer . Abstract painters trie d no t onl y to visualize people's face s a s thoug h painted b y a four-dimensional person, bu t also to treat time as the fourth dimension. I n Marce l Duchamp' s paintin g Nude Descending a Staircase, we see a blurred representatio n o f a woman, with a n infinite number o f her image s superimposed ove r tim e as she walks down the stairs . This is how a four-dimensiona l perso n woul d se e people , viewin g al l tim e sequences a t once , i f time were the fourt h dimension . In 1937 , art critic Meyer Schapiro summarize d the influence of these new geometries o n th e ar t world when h e wrote, ' 'Just as the discovery of non-Euclidea n geometr y gav e a powerfu l impetu s t o th e vie w tha t mathematics wa s independent o f existence , s o abstract paintin g cu t a t the root s o f th e classi c ideas o f artisti c imitation." Or , a s art historia n Linda Henderso n ha s said, "th e fourt h dimensio n an d non-Euclidea n geometry emerg e a s among th e mos t important theme s unifyin g muc h of modern ar t and theory." 5

Bolsheviks an d the Fourt h Dimension

The fourt h dimensio n als o crossed ove r into Czaris t Russia via the writ ings of the mysti c P. D. Ouspensky, who introduced Russia n intellectuals to its mysteries. His influence was so pronounced tha t even Fyodor Dos-

Figure 3.5. Cubism was heavily influenced by the fourth dimension. For example, it tried to view reality through the eyes of a fourth-dimensional person. Such a being, looking at a human face, would see all angles simultaneously. Hence, both eyes would be seen at once by a fourth-dimensional being, as in Picasso's painting Portrait o f Dor a Maar . (Giraudon/Art Resource. ® 1993. Ars, Ne w York/ Spadem, Paris)

66

The Man Wh o "Saw" the Fourth Dimension 6

7

toyevsky, in Th e Brothers Karamazov, ha d hi s protagonist Iva n Karamazov speculate o n th e existenc e o f highe r dimension s an d non-Euclidea n geometries durin g a discussion o n th e existenc e o f God. Because of the historic events unfolding in Russia, the fourth dimension wa s to play a curious rol e i n th e Bolshevi k Revolution. Today, thi s strange interlude i n the history of science is important because Vladimir Lenin would join th e debat e ove r th e fourt h dimension , which woul d eventually exert a powerful influence on the science of the former Soviet Union for th e nex t 7 0 years.6 (Russia n physicists, of course, hav e played key roles in developing th e present-da y ten-dimensional theory.) After th e Cza r brutally crushed th e 190 5 revolution , a faction called the Otzovists , or "God-builders, " developed withi n the Bolshevi k party. They argue d tha t th e peasant s weren' t read y fo r socialism ; to prepar e them, Bolshevik s should appea l t o the m throug h religio n an d spiritu alism. To bolster their heretical views, the God-builders quoted from th e work o f th e Germa n physicis t and philosophe r Erns t Mach , wh o ha d written eloquently about th e fourth dimension an d th e recent discovery of a new , unearthl y propert y o f matte r calle d radioactivity . The God builders pointe d ou t tha t th e discover y of radioactivit y by the Frenc h scientist Henri Becquere l in 189 6 an d th e discover y of radium b y Marie Curie in 189 6 had ignite d a furious philosophica l debat e i n French an d German literar y circles. It appeared that matter could slowly disintegrate and tha t energ y (i n the for m of radiation) coul d reappear . Without question, the new experiments on radiation showe d that the foundation o f Newtonian physics was crumbling. Matter, thought b y the Greeks to be eterna l an d immutable , was now disintegrating befor e ou r very eyes . Uraniu m an d radium , confoundin g accepte d belief , wer e mutating in the laboratory . T o some, Mach was the prophe t wh o would lead the m ou t o f th e wilderness . However , h e pointe d i n th e wron g direction, rejectin g materialism and declarin g tha t space and tim e were products o f our sensations . I n vain, he wrote, "I hop e tha t nobod y wil l defend ghost-stories with the help of what I have said and written on this subject."7 A spli t develope d withi n th e Bolsheviks . Thei r leader , Vladimi r Lenin, was horrified. Are ghosts and demon s compatible with socialism? In exil e i n Genev a i n 1908 , h e wrot e a mammoth philosophica l tome , Materialism an d Empirio-Criticism, defending dialectical materialism fro m the onslaugh t o f mysticism and metaphysics . To Lenin , th e mysteriou s disappearance o f matter an d energ y did not prov e th e existenc e o f spirits. He argue d tha t this meant instead tha t a ne w dialectic was emerging, which would embrace bot h matter an d energy . No longer coul d they be

68 E N T E R I N G T H E F I F T H D I M E N S I O

N

viewed as separat e entities , as Newto n had done . The y mus t now be viewed a s two poles o f a dialectical unity. A new conservation principl e was needed . (Unknow n t o Lenin , Einstei n ha d propose d th e correc t principle 3 year s earlier , i n 1905. ) Furthermore , Leni n questione d Mach's easy embrace of the fourth dimension. First, Lenin praised Mach, who "ha s raise d th e very important an d usefu l questio n o f a space o f n dimensions a s a conceivabl e space. " The n h e too k Mac h t o tas k fo r failing t o emphasiz e tha t onl y the thre e dimension s o f space coul d b e verified experimentally . Mathematics may explore th e fourth dimensio n and th e world of what is possible, and thi s is good, wrot e Lenin, but th e Czar can be overthrow n onl y in th e thir d dimension! 8 Fighting on th e battlegroun d o f the fourt h dimensio n an d th e new theory o f radiation, Leni n neede d years to root ou t Otzovis m from th e Bolshevik party. Nevertheless, he wo n the battl e shortl y before th e out break o f the 191 7 October Revolution.

Bigamists an d the Fourt h Dimension

Eventually, th e idea s of the fourt h dimensio n crosse d th e Atlanti c an d came to America. Their messenge r was a colorful English mathematician named Charle s Howar d Hinton. Whil e Albert Einstein was toiling at his desk job i n th e Swis s patent office i n 1905 , discoverin g th e law s of relativity, Hinto n wa s working at th e Unite d State s Paten t Offic e i n Washington, D.C . Although the y probably never met , their paths would cross in severa l interesting ways. Hinton spen t his entire adul t lif e obsesse d wit h the notio n o f popularizing and visualizing the fourth dimension. H e would go down in th e history o f science a s the ma n who "saw" th e fourt h dimension . Hinton wa s the son of James Hinton , a renowned Britis h ear surgeo n of libera l persuasion . Ove r th e years , th e charismati c elde r Hinto n evolved into a religious philosopher, a n outspoken advocat e of free lov e and ope n polygamy , an d finall y th e leade r o f a n influentia l cul t i n England. H e wa s surrounded b y a fiercel y loya l and devote d circle o f free-thinking followers . One o f his best-known remarks was "Christ was the Savio r of men, bu t I am th e savio r of women, and I don't envy Him a bit!"9 His so n Charles , however , seeme d doome d t o lea d a respectable , boring lif e a s a mathematician. H e wa s fascinated no t b y polygamy, but by polygons ! Havin g graduate d fro m Oxfor d i n 1877 , h e becam e a respectable maste r a t the Uppingha m Schoo l whil e working on his mas-

The Man Wh o "Saw" the Fourth Dimension 6

9

ter's degre e in mathematics. At Oxford, Hinto n becam e intrigue d with trying t o visualize the fourt h dimension . A s a mathematician , h e kne w that on e canno t visualiz e a four-dimensional object in its entirety. However, i t i s possible , h e reasoned , t o visualiz e the cros s sectio n o r th e unraveling of a four-dimensional object. Hinton publishe d hi s notion s i n th e popula r press . H e wrot e th e influential articl e "Wha t is the Fourt h Dimension? " for th e Dublin University Magazine an d th e Cheltenham Ladies' College Magazine, reprinte d i n 1884 with the catch y subtitle "Ghost s Explained. " Hinton's lif e a s a comfortable academic , however, took a sharp turn for th e worse in 188 5 when h e was arrested fo r bigamy and pu t o n trial . Earlier, Hinton had married Mar y Everest Boole, the daughter of a member o f his father's circle, and wido w of the grea t mathematicia n Georg e Boole (founde r of Boolean algebra) . However, he was also the fathe r of twins born t o a certain Maude Weldon. The headmaste r a t Uppingham, noticing Hinto n i n the presence o f his wife , Mary , an d hi s mistress , Maude, had assume d tha t Maud e was Hinton's sister. All was going well for Hinton, until he made th e mistake of marrying Maude a s well. When th e headmaste r learne d tha t Hinto n was a bigamist, it set of f a scandal. H e wa s promptly fire d from hi s jo b at Uppingham an d place d o n tria l for bigamy. He was imprisoned for 3 days, but Mar y Hinton decline d t o pres s charges an d togethe r the y lef t England fo r th e Unite d States. Hinton wa s hired a s an instructo r in the mathematic s department a t Princeton University , where his obsession with the fourth dimension was temporarily sidetracke d whe n h e invente d th e basebal l machine . Th e Princeton basebal l team benefited fro m Hinton' s machine, which could fire baseballs at 70 miles per hour . The descendants of Hinton's creatio n can no w be found o n ever y major baseball field in the world . Hinton wa s eventually fired from Princeton , bu t manage d t o ge t a job a t th e Unite d States Naval Observatory throug h th e influenc e of its director, a devout advocate o f the fourt h dimension. Then, in 1902 , he took a job a t the Paten t Offic e i n Washington .

Hinton's Cube s Hinton spen t years developing ingenious methods b y which the average person an d a growing legion o f followers, not onl y professional mathe maticians, coul d "see " four-dimensiona l objects . Eventually , he per fected specia l cube s that , i f one trie d har d enough , could allo w one t o

70 E N T E R I N G T H E F I F T H D I M E N S I O

N

visualize hypercubes , o r cube s i n fou r dimensions . Thes e woul d eventually b e calle d Hinton' s cubes . Hinto n eve n coine d th e officia l nam e for a n unravele d hypercube , a tesseract , whic h found it s way into th e English language . Hinton's cube s wer e widel y advertised i n women' s magazine s an d were even used i n seances , where they soon becam e object s of mystical importance. By mediating on Hinton' s cubes, it was claimed by members of high society , you coul d catc h glimpse s of the fourt h dimensio n an d hence th e nethe r world of ghosts and th e dearl y departed. Hi s disciples spent hour s contemplatin g an d meditatin g o n thes e cubes , until they attained th e abilit y to mentall y rearrang e an d reassembl e thes e cube s via the fourt h dimensio n into a hypercube . Those wh o could perfor m this mental feat , i t was said, would attain th e highes t stat e of nirvana. As an analogy , take a three-dimensional cube. Although a Flatlander cannot visualize a cube i n its entirety, it is possible for u s to unravel th e cube in thre e dimensions , so that we have a series of six squares making a cross. Of course, a Flatlander canno t reassembl e th e square s to make a cube . I n th e secon d dimension , th e joints betwee n eac h squar e ar e rigid an d canno t b e moved . However , thes e joints ar e eas y to ben d i n the thir d dimension . A Flatlander witnessin g this event would see th e squares disappear , leavin g only one squar e i n his universe (Figur e 3.6). Likewise, a hypercube i n four dimensions canno t b e visualized. But one ca n unravel a hypercube into its lower components, whic h are ordinary three-dimensional cubes . These cubes , in turn, can be arranged i n a three-dimensional cross— a tesseract. It is impossible for us to visualize how t o wrap u p thes e cube s t o for m a hypercube . However , a higher dimensional person ca n "lift" eac h cube off our universe and then wrap up th e cub e t o for m a hypercube . (Ou r three-dimensiona l eyes , witnessing this spectacular event , would only see the other cubes disappear, leaving only one cub e i n ou r universe. ) So pervasive was Hinton's influ ence tha t Salvadore Dal i used Hinton' s tesserac t in his famous painting Christus Hypercubus, o n displa y at th e Metropolita n Museu m o f Ar t i n New York, whic h depicts Chris t bein g crucifie d on a four-dimensional cross (Figur e 3.7). Hinton als o kne w of a secon d wa y to visualiz e higher-dimensional objects: b y looking a t th e shadow s they cas t i n lowe r dimensions. Fo r example, a Flatlander ca n visualiz e a cube b y looking at it s two-dimensional shadow. A cube looks like two squares joined together . Similarly , a hypercube' s shado w cas t o n th e thir d dimensio n become s a cub e within a cube (Figur e 3.8) . In addition t o visualizing unravelings o f hypercubes an d examinin g their shadows , Hinto n wa s aware o f a thir d wa y to conceptualiz e th e

Figure 3.6. Flatlanders cannot visualize a cube, but they can conceptualize a three-dimensional cube by unraveling it. To a Flatlander, a cube, when unfolded, resembles a cross, consisting of six squares. Similarly, we cannot visualize a fourdimensional hypercube, but if we unfold it we have a series of cubes arranged in a crosslike tesseract. Although the cubes of a tesseract appear immobile, a fourdimensional person can "wrap up" the cubes into a hypercube.

71

Figure 3. 7. I n Christu s Hypercubus , Salvadore Dali depicted Christ a s being crucified on a tesseract, an unraveled hypercube. (The Metropolitan Museum of Art. Gift of Chester Dale, Collection, 1955. © 1993. Ars, New York/Demart Pro Arte, Geneva)

72

Figure 3.8. A Flatlander can visualize a cube by examining its shadow, which appears as a square within a square. If the cube is rotated, the squares execute motions that appear impossible to a Flatlander. Similarly, the shadow of a hypercube is a cube within a cube. If the hypercube is rotated in four dimensions, the cubes execute motions that appear impossible to our three-dimensional brains.

73

74 ENTERIN

G TH E FIFT H DIMENSIO N

fourth dimension : b y cross sections . Fo r example , whe n Mr . Square i s sent int o th e thir d dimension , hi s eye s ca n se e onl y two-dimensional cross section s o f th e thir d dimension . Thu s h e ca n se e onl y circle s appear, ge t larger , chang e color , an d the n suddenl y disappear. I f Mr. Square move d pas t a n apple , h e woul d see a red circl e materializ e ou t of nowhere , graduall y expand , the n contract , the n tur n int o a smal l brown circl e (th e stem) , an d finall y disappear . Likewise, Hinto n kne w that i f we were hurled int o th e fourt h dimension , we would see strange objects suddenl y appea r ou t o f nowhere , ge t larger , chang e color , change shape , ge t smaller, and finall y disappear . In summary , Hinton' s contributio n ma y b e hi s popularizatio n o f higher-dimensional figure s usin g thre e methods : b y examinin g thei r shadows, thei r cros s sections, an d thei r unravellings . Even today , thes e three methods ar e th e chie f ways in which professional mathematicians and physicist s conceptualize higher-dimensiona l object s i n thei r work . The scientist s whose diagrams appea r i n today' s physics journals ow e a small debt of gratitude t o Hinton's work. The Contes t o n th e Fourt h Dimensio n In his articles, Hinton ha d answer s for all possible questions. When people aske d hi m t o nam e th e fourt h dimension , h e woul d repl y that th e words an a and kata described movin g in the fourth dimension and were the counterpart s o f the term s u p and down, or left an d right. When asked where the fourt h dimension was , he als o had a ready answer. For the moment , conside r th e motion o f cigarette smoke in a closed room. Becaus e the atom s of the smoke , by the law s of thermodynamics, spread and diffus e int o all possible location s in th e room , we can determine i f there ar e an y regions o f ordinary three-dimensiona l spac e tha t the smok e molecule s miss . However , experimental observation s sho w that ther e ar e n o suc h hidde n regions . Therefore , th e fourt h spatia l dimension i s possible onl y if it is smaller than th e smok e particles. Thus if the fourt h dimensio n actuall y exists, it must be incredibl y small, even smaller than a n atom. Thi s is the philosoph y that Hinton adopted , tha t all objects in ou r three-dimensiona l univers e exist in th e fourt h dimension, but tha t th e fourt h dimension i s so small that it evades any exper imental observation. (W e will fin d tha t physicists today adopt essentially the same philosophy as Hinton and conclude tha t the higher dimensions are to o smal l t o be experimentall y seen. When asked , "What is light?" he als o had a read y answer . Following Riemann, Hinto n believe d tha t

The Man Wh o "Saw" the Fourth Dimension 7

5

light is a vibration o f the unsee n fourt h dimension , whic h is essentially the viewpoin t taken toda y by many theoretical physicists. ) In th e Unite d States , Hinton single-handedl y sparked a n enormou s public interest in the fourt h dimension. Popular magazine s like Harper's Weekly, McClure's, Current Literature, Popular Science Monthly, an d Science all devoted page s t o th e blossomin g interes t i n th e fourt h dimension . Bu t what probably ensured Hinton's fame in America was the famous contest sponsored b y Scientific American in 1909 . Thi s unusua l contes t offere d a $500 prize ( a considerable amoun t o f money in 1909 ) t o "the bes t popular explanatio n o f th e Fourt h Dimension. " Th e magazine' s editor s were pleasantly surprised by the delug e o f letters that poured into thei r offices, includin g entrie s fro m a s far awa y a s Turkey, Austria, Holland , India, Australia, France, an d Germany . The objec t o f th e contes t wa s to "se t fort h i n a n essa y not longe r than twenty-fiv e hundre d word s th e meanin g o f th e ter m s o tha t th e ordinary la y reader coul d understan d it. " I t dre w a larg e numbe r o f serious essays. Some lamented the fact that people like Zollner and Slade had besmirche d th e reputatio n o f the fourt h dimension by confusing it with spiritualism. However, many of the essay s recognized Hinton's pio neering work on the fourth dimension. (Surprisingly , not one essay mentioned th e wor k of Einstein. In 1909 , i t was still far fro m clea r tha t Einstein ha d uncovere d th e secre t o f spac e an d time . I n fact , th e ide a o f time as the fourt h dimension di d no t appea r i n a single essay.) Without experimenta l verification , th e Scientific American contes t could not , o f course , resolv e th e questio n o f th e existenc e o f highe r dimensions. However , th e contes t di d addres s th e questio n o f wha t higher-dimensional object s might look like.

Monsters fro m th e Fourt h Dimensio n What would it be lik e to meet a creature fro m a higher dimension? Perhaps th e bes t wa y to explai n th e wonde r an d excitemen t o f a hypothetical visi t t o othe r dimension s is through scienc e fiction, where writers have tried t o grapple wit h this question. In "Th e Monste r from Nowhere, " write r Nelson Bond tried to imagine wha t would happe n i f an explore r i n th e jungles o f Latin America encountered a beast from a higher dimension . Our her o is Burch Patterson, adventurer, bon vivant , and soldie r of fortune, who hits on th e ide a o f capturing wild animals in th e towering mountains o f Peru . Th e expeditio n wil l b e pai d fo r b y various zoos,

76 E N T E R 1 N G T H E F I F T H D I M E N S I O

N

which pu t u p th e mone y fo r th e tri p i n retur n fo r whateve r animals Patterson ca n find. With much hoopl a an d fanfare , th e pres s covers the progress o f the expeditio n a s it journeys int o unexplore d territory . But after a few weeks, th e expeditio n lose s contac t with th e outsid e worl d and mysteriousl y disappear s withou t a trace . Afte r a lon g an d futil e search, th e authoritie s reluctantly give the explorer s u p fo r dead . Two year s later , Burc h Patterso n abruptl y reappears . H e meet s secretly with reporters an d tell s them an astonishing story of tragedy and heroism. Just before th e expeditio n disappeared , i t encountered a fantastic animal in the Maratan Plateau of upper Peru, an unearthly bloblike creature tha t was constantly changing shape in the mos t bizarre fashion. These blac k blobs hovered i n midair, disappearing and reappearing an d changing shap e an d size . Th e blob s the n unexpectedl y attacke d th e expedition, killin g mos t o f th e men . Th e blob s hoiste d som e o f th e remaining me n of f the ground ; the y screame d an d the n disappeare d into thi n air . Only Burch escaped th e rout. Da/.ed and frightened , he nonetheles s studied thes e blobs from a distance and gradually formed a theory abou t what the y were an d ho w t o captur e them . H e ha d rea d Flatland years before, an d imagine d tha t anyon e stickin g his fingers into an d ou t o f Flatland would startle th e two-dimensiona l inhabitants. The Flatlander s would see pulsating rings of flesh hoverin g in midair (ou r fingers poking through Flatland) , constantly changing size . Likewise , reasone d Patter son, any higher-dimensional creatur e stickin g his foot or arm s throug h our univers e woul d appea r a s three-dimensional , pulsatin g blob s o f flesh, appearing ou t of nowhere and constantly changing shape and size. That woul d als o explai n wh y his tea m member s ha d disappeare d int o thin air : They had bee n dragge d int o a higher-dimensional universe . But on e questio n stil l plague d him : Ho w do yo u captur e a higher dimensional being ? I f a Flatlander , seein g ou r finge r pok e it s way through hi s two-dimensiona l universe , trie d t o captur e ou r finger , h e would be at a loss. If he trie d to lasso our finger, we could simply remove our finger and disappear . Similarly , Patterson reasoned , h e coul d put a net around on e of these blobs, but then the higher-dimensional creature could simply pull his "finger" o r "leg " ou t of our universe, and th e ne t would collapse. Suddenly, th e answe r cam e t o him : I f a Flatlande r were t o tr y t o capture ou r finge r a s it poked it s way into Flatland, the Flatlander could stick a needle through ou r finger, painfully impalin g i t t o th e two-dimensional universe . Thus Patterson' s strateg y was to driv e a spike throug h one of the blobs and impale the creature in our universe !

The Man Who "Saw" the Fourth Dimension 77

After month s o f observin g th e creature , Patterso n identifie d what looked lik e the creature' s "foot " an d drov e a spike right throug h it . It took hi m 2 years to capture th e creatur e an d shi p the writhing , struggling blob back to New Jersey. Finally, Patterson announce s a major press conference where he will unveil a fantastic creature caugh t in Peru. Journalists and scientists alike gasp i n horro r when th e creatur e i s unveiled, writhing and strugglin g against a large stee l rod . Lik e a scene from King Kong, on e newspaper man, agains t th e rules , take s flas h picture s o f th e creature . Th e flas h enrages th e creature , whic h then struggle s so hard against the ro d tha t its flesh begins to tear. Suddenly, the monster is free, and pandemoniu m breaks out . Peopl e ar e tor n t o shreds , an d Patterso n an d other s ar e grabbed by the creature and the n disappear int o the fourth dimension . In th e aftermat h of the tragedy , one o f the survivor s of the massacr e decides to burn al l evidence of the creature. Better to leave this mystery forever unsolved . Building a Four-Dimensional House

In the previou s section, the question o f what happens when we encounter a higher-dimensional bein g was explored. Bu t what happens i n th e reverse situation , whe n w e visit a higher-dimensiona l universe ? As we have seen, a Flatlander cannot possibly visualize a three-dimensional universe in its entirety. However, there are , as Hinton showed, several ways in which the Flatlander can comprehend revealin g fragments of higherdimensional universes . In his classic short stor y "... And He Buil t a Crooked Hous e . . . ," Robert Heinlein explored th e many possibilities of living in an unraveled hypercube. Quintus Teal i s a brash, flamboyan t architect whose ambition i s to build a hous e i n a trul y revolutionary shape: a tesseract , a hypercube that has been unravele d in the thir d dimension. He cons his friends Mr. and Mrs. Bailey into buyin g the house. Built i n Lo s Angeles, the tesserac t is a serie s of eigh t ultramoder n cubes stacked o n to p o f one anothe r i n th e shap e o f a cross. Unfortu nately, just a s Teal i s about t o show off his ne w creation t o th e Baileys, an earthquake strikes southern California , and th e hous e collapses into itself. The cube s begin t o topple , but strangel y only a single cube i s left standing. Th e othe r cube s hav e mysteriousl y disappeared. Whe n Tea l and th e Bailey s cautiously enter th e house , no w just a single cube, they

78 E N T E R 1 N G T H E F I F T H D 1 M E N S I O

N

are amazed that th e othe r missin g rooms are clearly visible through th e windows of the firs t floor . Bu t tha t i s impossible. The hous e is now only a single cube. Ho w can th e interio r o f a single cube b e connecte d t o a series of other cubes tha t cannot b e seen fro m th e outside? They climb the stairs and fin d th e maste r bedroom above the entryway. Instea d o f findin g the thir d floor , however , they find themselves back on the ground floor. Thinking the house is haunted, the frightened Baileys race to the front door. Instead of leading to the outside, the front door jus t leads t o another room . Mrs . Bailey faints. As they explore th e house , the y find that each roo m i s connected t o an impossibl e serie s of othe r rooms . I n th e origina l house , eac h cub e had window s to vie w th e outside . Now , all windows face othe r rooms . There is no outside ! Scared ou t o f thei r wits , the y slowl y tr y all th e door s o f th e house , only to wind up in other rooms . Finally, in the study they decide t o open the fou r Venetia n blind s an d loo k outside . When the y ope n th e first Venetian blind, they find that they are peering down at the Empire State Building. Apparently, that windo w opened u p t o a "window " i n spac e just above th e spir e of the tower . When the y open th e secon d Venetian blind, the y fin d themselve s staring a t a vast ocean , excep t i t i s upside down. Opening the third Venetian blind , they find themselves looking at Nothing. No t empty space. No t inky blackness. Just Nothing. Finally, opening u p th e las t Venetia n blind , the y fin d themselve s gazin g a t a bleak desert landscape, probabl y a scene from Mars. After a harrowing tou r throug h th e room s o f th e house , with eac h room impossibl y connected t o the othe r rooms, Teal finall y figures it all out. The earthquake , he reasons , mus t have collapsed th e joints of various cubes and folde d th e hous e i n th e fourt h dimension. 10 On th e outside , Teal' s hous e originall y looke d lik e a n ordinar y sequence o f cubes . Th e hous e di d no t collaps e becaus e th e joint s between th e cube s were rigid and stabl e in three dimensions . However, viewed from th e fourt h dimension, Teal's hous e i s an unravele d hypercube tha t ca n b e reassemble d o r folde d bac k int o a hypercube . Thu s when the hous e was shaken by the earthquake , it somehow folded up in four dimensions, leaving only a single cube dangling in our third dimension. Anyone walking into th e single remaining cube would view a series of room s connecte d i n a seemingl y impossibl e fashion . B y racin g through th e various rooms, Teal ha s moved throug h th e fourt h dimension without noticing it. Although ou r protagonist s see m doome d t o spen d thei r live s fruit lessly wanderin g i n circle s insid e a hypercube , anothe r violen t earth -

The Ma n Wh o "Saw" the Fourth Dimension 7

9

quake shake s the tesseract . Holding thei r breath, Tea l an d th e terrified Baileys lea p ou t th e neares t window . When the y land, the y find themselves i n Joshu a Tre e Nationa l Monument , mile s fro m Lo s Angeles. Hours later , hitchin g a ride bac k t o th e city , the y return t o the house , only t o fin d tha t th e las t remaining cub e ha s vanished. Where di d th e tesseract go? It is probably drifting somewhere in the fourt h dimension.

The Useles s Fourt h Dimensio n In retrospect, Riemann's famous lecture was popularized to a wide audience vi a mystics, philosophers , an d artists , but di d littl e to furthe r ou r understanding o f nature . Fro m th e perspectiv e o f moder n physics , we can also see why the years 1860 to 1905 did not produce any fundamental breakthroughs i n ou r understandin g o f hyperspace. First, ther e was no attemp t to use hyperspace to simplify th e law s of nature. Without Riemann's original guiding principle—that the law s of nature becom e simpl e i n highe r dimensions—scientist s durin g thi s period were groping in the dark. Riemann's seminal idea of using geometry—that is, crumpled hyperspace—t o explain th e essenc e of a "force " was forgotten durin g thos e years. Second, ther e wa s no attemp t t o exploi t Faraday' s field concept o r Riemann's metri c tenso r t o fin d th e fiel d equation s obeye d b y hyperspace. Th e mathematica l apparatu s develope d b y Riemann becam e a province o f pur e mathematics , contrar y t o Riemann' s origina l inten tions. Without field theory, you cannot make any predictions with hyperspace. Thus b y the tur n o f th e century , the cynic s claime d (wit h justification) tha t there was no experimenta l confirmation of the fourth dimension. Worse , the y claimed , ther e wa s no physica l motivation fo r introducing th e fourt h dimension , othe r tha n t o titillate the genera l public with ghos t stories . Thi s deplorabl e situatio n would soo n change , however. Within a few decades, the theor y of the fourth dimension (o f time) would forever change th e cours e o f human history . It would give us th e atomic bomb an d th e theor y of Creation itself . And the man who would do i t would be a n obscur e physicis t named Albert Einstein.

4

The Secre t o f Light : Vibrations i n the Fift h Dimensio n If [relativity ] should prove to be correct , as I expect it will, h e will be considere d th e Copernicu s of the twentiet h century. Max Planck on Albei t Einstein

T

HE life of Albert Einstein appeared to be one long series of failures and disappointments . Eve n his mother was distressed at how slowly he learned t o talk. His elementary-school teachers thought him a foolish dreamer. The y complaine d tha t he was constantly disrupting classroom discipline with his silly questions. One teache r eve n tol d th e bo y bluntly that he woul d prefer tha t Einstein drop out o f his class. He ha d fe w friend s i n school . Losin g interes t i n hi s courses , h e dropped out o f high school . Withou t a high-school diploma , h e ha d t o take special exam s t o enter college , but h e di d no t pas s them an d ha d to tak e them a second time . He eve n failed the exa m fo r th e Swis s military because h e ha d fla t feet . After graduation , h e coul d no t ge t a job. H e wa s an unemploye d physicist wh o wa s passed ove r fo r a teachin g positio n a t th e universit y and wa s rejected fo r jobs everywher e h e applied . H e earne d barel y 3 francs a n hour— a pittance—b y tutorin g students . H e tol d hi s friend Maurice Solovin e tha t "a n easie r wa y of earnin g a livin g woul d b e t o play the violi n in public places. "

80

The Secret o f Light: Vibrations i n th e Fifth Dimension 8

1

Einstein wa s a ma n wh o rejecte d th e thing s mos t me n chas e after , such as power and money . However, he once note d pessimistically, "By the mere existence of his stomach, everyone is condemned t o participate in tha t chase. " Finally, through th e influenc e of a friend, h e lande d a lowly job a s a clerk at the Swiss patent offic e i n Bern, earning just enoug h money s o his parents woul d no t hav e t o suppor t him . O n hi s meage r salary, he supporte d hi s young wife an d thei r newbor n baby . Lacking financial resources o r connection s wit h the scientifi c estab lishment, Einstei n bega n t o wor k i n solitud e a t th e paten t office . I n between paten t applications , hi s min d drifte d t o problem s tha t ha d intrigued hi m as a youth. He then undertook a task that would eventually change th e cours e o f huma n history . Hi s too l wa s th e fourt h dimension.

Children's Question s Wherein lies the essenc e of Einstein's genius? In Th e Ascent of Man, Jacob Bronowski wrote: "The geniu s of men lik e Newton and Einstei n lies in that: the y as k transparent, innocen t question s whic h turn ou t t o hav e catastrophic answers . Einstein was a man wh o could as k immensely simple questions." 1 As a child, Einstein asked himself the simpl e question : What would a light beam look like if you could catch up with one? Would you see a stationary wave, frozen in time? This question set him on a 50year journey throug h th e mysterie s of space and time . Imagine trying to overtake a train in a speeding car. If we hit th e gas pedal, ou r ca r race s neck-and-nec k wit h th e train . We can pee r insid e the train , which now appears t o be at rest. We can see the seat s and th e people, wh o ar e actin g as though th e trai n weren' t moving. Similarly, Einstein a s a chil d imagine d travelin g alongsid e a ligh t beam . He though t tha t th e ligh t bea m shoul d resembl e a serie s o f stationary waves , froze n i n time ; tha t is , the ligh t beam shoul d appea r motionless. When Einstein was 16 years old, he spotted the fla w in this argument. He recalled later , After te n years of reflection such a principle resulted from a paradox upo n which I had alread y hit a t th e ag e o f sixteen: I f I pursue a bea m o f light with th e velocit y c (velocity of light in a vacuum) I should observ e such a beam o f ligh t as a spatially oscillatory electromagnetic fiel d a t rest . How-

82 E N T E R I N G T H E F I F T H D 1 M E N S I O

N

ever, there seems to be no such thing, whether on th e basis of experience or accordin g to Maxwell's equations.2

In college , Einstei n confirmed hi s suspicions. He learne d tha t ligh t can be expressed i n terms of Faraday's electric and magneti c fields, and that these fields obey the field equations found by James Cler k Maxwell. As he suspected , h e foun d tha t stationary, frozen waves are no t allowed by Maxwell's field equations. I n fact , Einstei n showed that a light beam travels a t th e same velocit y c , n o matte r ho w har d yo u tr y t o catc h up wit h it. At first, this seemed absurd . This meant tha t we could never overtake the trai n (ligh t beam). Worse, no matte r ho w fast we drove ou r car , th e train would always seem t o be travelin g ahead o f us at the same velocity. In other words, a light beam i s like the "ghos t ship" that old sailors love to spin tal l tales about. I t is a phantom vesse l tha t can neve r be caught . No matter ho w fast we sail, the ghos t ship always eludes us, taunting us. In 1905, with plenty of time on his hands at the patent office, Einstein carefully analyzed the field equations of Maxwell and was led to postulate the principl e o f special relativity: Th e spee d o f ligh t i s th e sam e i n al l constantly movin g frames . Thi s innocent-soundin g principl e i s one o f the greates t achievement s of th e huma n spirit . Som e hav e sai d tha t i t ranks wit h Newton' s law of gravitatio n a s on e o f th e greates t scientifi c creations of the human min d in the 2 million years our specie s has been evolving on thi s planet. From it, we can logically unlock the secret of the vast energies release d b y the star s and galaxies . To see how this simple statement can lead t o such profound conclusions, let us return t o the analogy of the ca r trying to overtake the train . Let u s say that a pedestrian o n th e sidewal k clock s our ca r travelin g at 99 miles per hour , an d th e trai n travelin g at 10 0 miles per hour . Naturally, fro m ou r poin t o f view in th e car , we see th e trai n movin g ahead of u s a t 1 mile pe r hour . Thi s i s because velocitie s can b e adde d an d subtracted, just lik e ordinary numbers. Now let us replace th e trai n by a light beam, but kee p the velocity of light at just 10 0 miles per hour . Th e pedestria n stil l clock s our ca r trav eling at 99 miles per hou r i n ho t pursui t of the ligh t beam travelin g at 100 miles per hour . According to th e pedestrian , w e should b e closin g in o n th e ligh t beam . However , accordin g t o relativity , w e in th e ca r actually see the light beam no t travelin g ahead of us at 1 mile per hour , as expected , bu t speedin g ahea d o f us at 10 0 miles per hour . Remark ably, we see the ligh t beam racing ahead o f us as though we were at rest. Not believing our own eyes, we slam on the gas pedal until the pedestria n

The Secret o f Light: Vibrations i n th e Fifth Dimension 8

3

clocks our car racing ahead a t 99.99999 miles per hour. Surely, we think, we must be abou t t o overtak e th e ligh t beam . However , when we look out th e window , we see the ligh t beam stil l speeding ahea d o f us at 100 miles per hour . Uneasily, we reach severa l bizarre, disturbing conclusions . First, n o matter ho w much we gun th e engine s of our car , the pedestria n tells us that we can approac h bu t neve r excee d 10 0 miles per hour . This seems to be th e to p velocity of the car . Second , n o matte r how close we come to 10 0 miles per hour , we still see the ligh t bea m speeding ahea d o f us at 10 0 miles per hour , a s though w e weren't moving at all. But this is absurd. Ho w can both peopl e i n the speedin g car and th e stationary person measur e the velocity of the light beam to be the same ? Ordinarily, this is impossible. It appears t o be nature' s colossal joke. There is only one wa y out o f this paradox. Inexorably , we are le d t o the astonishing conclusio n that shook Einstein to the core when he firs t conceived o f it. Th e onl y solution t o thi s puzzl e i s that time slows down for u s in th e car . If the pedestria n take s a telescope an d peer s into ou r car, h e see s everyone i n th e ca r movin g exceptionally slowly. However, we in the ca r never notice tha t time is slowing down because our brains , too, hav e slowe d down , an d everythin g seems norma l t o us . Further more, h e see s tha t th e ca r ha s becom e flattene d i n th e directio n o f motion. Th e ca r ha s shrunk like an accordion . However , we never fee l this effect becaus e our bodies , too , hav e shrunk. Space an d tim e pla y trick s o n us . I n actua l experiments , scientists have show n tha t th e spee d o f ligh t i s alway s c , no matte r ho w fas t w e travel. This is because the faste r we travel, the slowe r our clock s tick an d the shorte r ou r ruler s become . I n fact , ou r clock s slow dow n and ou r rulers shrin k just enoug h s o tha t wheneve r we measur e th e spee d o f light, it comes out th e same. But why can't we see or feel thi s effect? Sinc e our brain s are thinking more slowly, and ou r bodies are also getting thinner as we approach th e speed o f light, we are blissfull y unawar e that w e are turnin g int o slow witted pancakes. These relativisti c effects, o f course, are to o smal l to be seen in everyday life becaus e th e spee d o f light is so great. Bein g a New Yorker, however, I a m constantl y reminded o f thes e fantasti c distortion s o f spac e and time whenever I ride the subway. When I am on the subway platform with nothin g t o d o excep t wai t fo r th e nex t subwa y train , I sometimes let m y imagination drif t an d wonde r what it would be lik e i f the spee d of light were only, say, 30 miles per hour , th e spee d o f a subwa y train . Then when th e trai n finall y roar s int o th e station , it appears squashed,

84 E N T E R I N G T H E F I F T H D I M E N S I O

N

like a n accordion . Th e train , I imagine , woul d b e a flattene d sla b o f metal 1 foot thick , barreling dow n th e tracks . And everyon e inside th e subway cars would be as thin as paper. They would also be virtually frozen in time , a s though the y were motionles s statues . However , a s the trai n comes t o a grindin g halt , i t suddenly expands , unti l thi s sla b of metal gradually fills the entir e station . As absur d a s thes e distortion s migh t appear , th e passenger s insid e the trai n woul d be totall y oblivious to thes e changes . Thei r bodie s an d space itsel f would be compresse d alon g the directio n o f motion o f th e train; everything would appea r t o hav e it s normal shape . Furthermore , their brains woul d have slowe d down , so that everyon e insid e th e trai n would act normally. Then when th e subwa y train finall y comes to a halt, they ar e totall y unaware tha t thei r train , t o someon e o n th e platform , appears t o miraculousl y expan d unti l i t fill s u p th e entir e platform . When th e passenger s depar t from the train , the y are totall y oblivious to the profoun d change s demande d b y special relativity. *

The Fourt h Dimension and High-School Reunion s

There have been, of course, hundreds o f popular account s of Einstein's theory, stressin g differen t aspect s o f hi s work . However , fe w account s capture th e essenc e behin d th e theor y of special relativity, which is that time i s the fourt h dimensio n an d tha t th e law s of nature ar e simplified and unified in higher dimensions. Introducin g tim e as the fourth dimension overthre w the concep t o f time datin g al l the wa y back t o Aristotle. Space an d tim e would now be foreve r dialectically linked by special relativity. (Zollne r and Hinto n ha d assumed tha t th e next dimension t o be discovered woul d be th e fourt h spatia l dimension . I n thi s respect, they were wron g an d H . G . Well s was correct. Th e nex t dimensio n t o b e discovered woul d b e time , a fourt h tempora l dimension . Progres s i n understanding th e fourt h spatia l dimension woul d hav e t o wait several more decades. ) To se e how higher dimension s simplif y th e law s of nature, w e recall that any object has length, width, and depth . Sinc e we have the freedom *Similarly, passengers riding in the trai n would think that the trai n was at rest and tha t the subway station was coming toward the train. They would see the platfor m and everyon e standing o n i t compresse d lik e an accordian . Then thi s lead s us t o a contradiction , that people on th e train and i n the station each think that the other ha s been compressed. The resolution o f this paradox i s a bit delicate. 3

The Secret o f Light: Vibrations i n th e Fifth Dimension 8

5

to rotate a n objec t by 90 degrees, we can tur n its length int o width and its width into depth. B y a simple rotation, we can interchange any of th e three spatial dimensions. Now if tim e is the fourt h dimension, then i t is possible to make "rotations" that convert space into time and vice versa. These four-dimensiona l "rotations " ar e precisel y th e distortion s o f space and tim e demanded b y special relativity. In other words, space and time have mixed in an essential way, governed by relativity. The meanin g of time as being the fourt h dimension is that time and spac e can rotate into each other in a mathematically precise way. From now on, they must be treate d a s two aspects of the sam e quantity: space-time. Thus adding a higher dimensio n helpe d t o unif y th e law s of nature. Newton, writing 300 years ago, though t tha t tim e bea t a t th e sam e rate everywher e in th e universe . Whether we sat on th e earth , on Mars , or on a distant star, clocks were expected t o tick at the same rate. There was thought t o b e a n absolute , uniform rhythm to th e passag e of time throughout th e entire universe. Rotations between time and space were inconceivable. Time and spac e were two distinct quantities with no relationship betwee n them . Unifyin g the m int o a singl e quantit y was unthinkable. However , according t o specia l relativity , tim e can bea t a t different rates , depending o n ho w fast on e i s moving. Time bein g th e fourth dimensio n means that time is intrinsically linked with movement in space. How fast a clock ticks depends on ho w fast it is moving in space. Elaborate experiment s don e wit h atomi c clock s sent int o orbi t aroun d the earth hav e confirmed that a clock on the earth and a clock rocketing in oute r space tic k at differen t rates . I wa s graphicall y reminde d o f th e relativit y principl e whe n I was invited t o m y twentiet h high-schoo l reunion . Althoug h I hadn' t see n most of my classmates since graduation, I assumed that all of them would show th e sam e telltal e sign s o f aging . As expected, mos t o f u s a t th e reunion wer e relieve d t o fin d tha t th e agin g proces s wa s universal: It seemed tha t all of us sported graying temples, expanding waistlines, and a fe w wrinkles. Althoug h w e were separate d acros s spac e an d tim e by several thousan d mile s and 2 0 years, each o f us had assume d that time had bea t uniforml y for all . We automaticall y assumed that eac h o f u s would ag e at the sam e rate. Then m y mind wandered , and I imagine d what would happen i f a classmate walked into the reunion hal l looking exactly as he had o n graduation day . At first, he woul d probably draw stares from hi s classmates. Was this the sam e person w e knew 20 years ago? When people realized that he was, a panic would surge through th e hall .

86 E N T E R I N G T H E F I F T H D I M E N S I O

N

We would be jolted by this encounter becaus e we tacitly assume that clocks beat th e sam e everywhere, even i f they are separate d b y vast distances. However , if time is the fourt h dimension , the n spac e an d tim e can rotate int o each other and clocks can beat at different rates , depending o n ho w fas t the y move . Thi s classmate , fo r example , ma y hav e entered a rocke t travelin g at near-ligh t speeds . Fo r us , th e rocke t tri p may hav e laste d fo r 2 0 years. However , fo r him , becaus e tim e slowe d down i n th e speedin g rocket , h e age d onl y a few moments fro m grad uation day . To him , h e just entere d th e rocket , spe d int o oute r spac e for a few minutes, an d the n lande d bac k on eart h i n tim e for hi s twentieth high-schoo l reunio n afte r a short , pleasan t journey , stil l lookin g youthful ami d a field of graying hair . I am als o reminded tha t th e fourt h dimensio n simplifie s th e law s of nature whenever I think back to m y first encounte r wit h Maxwell's field equations. Ever y undergraduate studen t learnin g the theor y of electricity an d magnetis m toil s for severa l years to maste r thes e eigh t abstrac t equations, whic h ar e exceptionall y ugl y an d ver y opaque . Maxwell' s eight equations ar e clumsy and difficul t t o memorize because tim e an d space ar e treate d separately . (T o thi s day, 1 have t o loo k the m u p i n a book t o mak e sur e tha t I ge t al l th e sign s an d symbol s correct.) 1 still remember th e relie f 1 felt whe n I learned tha t these equation s collaps e into on e trivial-lookin g equatio n whe n tim e i s treate d a s th e fourt h dimension. I n on e masterfu l stroke , th e fourt h dimensio n simplifie s these equation s i n a beautiful, transparen t fashion. 1 Written in this way, the equation s possess a higher symmetry; tha t is, space and tim e can turn into each other . Like a beautiful snowflak e tha t remains the same when we rotate i t around it s axis, Maxwell's field equations, writte n in relativ istic form, remain th e sam e when we rotate space int o time . Remarkably, this one simple equation, written in a relativistic fashion, contains the sam e physical content as the eight equations originally written dow n b y Maxwell over fO O years ago . Thi s on e equation , i n turn , governs the properties o f dynamos, radar, radio, television, lasers, household appliances , an d th e cornucopi a o f consume r electronic s tha t appear i n everyone' s living room. Thi s was one o f my first exposures t o the concep t o f beauty i n physics—tha t is , tha t th e symmetr y of four dimensional spac e ca n explai n a vast ocean o f physical knowledge that would fill an engineerin g library. Once again, this demonstrates on e o f the mai n theme s of this book, that th e additio n o f higher dimension s help s t o simplif y an d unif y th e laws of nature .

The Secret, of Light: Vibrations i n th e Fifth Dimension 8

7

Matter a s Condense d Energ y This discussio n o f unifyin g th e law s o f nature , s o far, has bee n rathe r abstract, an d woul d hav e remained s o had Einstei n no t take n th e nex t fateful step . H e realize d tha t i f spac e an d tim e ca n b e unifie d int o a single entity , calle d space—time , the n perhap s matte r an d energ y ca n also b e unite d int o a dialectica l relationship. I f ruler s ca n shrin k an d clocks slo w down , he reasoned , the n everythin g that w e measure wit h rulers an d clock s mus t als o change . However , almos t everythin g in a physicist's laboratory i s measured b y rulers and clocks . This meant tha t physicists had t o recalibrate all the laborator y quantities they once too k for grante d t o be constant . Specifically, energ y i s a quantit y that depend s o n ho w w e measur e distances an d tim e intervals . A speeding tes t ca r slamming into a brick wall obviousl y ha s energy . If the speedin g ca r approache s th e spee d o f light, however, its properties becom e distorted . I t shrinks like an accor dion an d clock s in i t slow down. More important , Einstei n foun d tha t th e mas s o f th e ca r als o increases a s it speed s up . Bu t wher e di d thi s exces s mas s com e from ? Einstein conclude d tha t i t came fro m th e energy . This ha d disturbin g consequences . Tw o of th e grea t discoverie s of nineteenth-century physic s were th e conservatio n o f mass and th e con servation o f energy; tha t is , the tota l mas s and tota l energ y o f a close d system, take n separately , d o no t change . Fo r example , i f the speedin g car hit s th e bric k wall , th e energ y o f th e ca r doe s no t vanish , bu t i s converted int o th e soun d energ y of the crash , the kineti c energy of the flying bric k fragments , hea t energy , and s o on. Th e tota l energ y (an d total mass) befor e an d afte r th e cras h i s the same . However, Einstein now said that th e energ y of the ca r could b e con verted int o mass— a ne w conservation principl e tha t sai d tha t th e su m total o f the mas s added t o energy mus t alway s remain th e same . Matter does not suddenl y disappear, no r doe s energ y spring ou t of nothing. In this regard , th e God-builder s wer e wrong an d Leni n wa s right. Matte r disappears onl y to unleash enormou s quantitie s of energy, or vice versa. When Einstei n was 26 years old, h e calculate d precisel y how energy must change i f the relativity principle was correct, and h e discovered th e relation E = rru 2. Since th e spee d o f light squared (c 2 ) i s an astronomi cally large number , a small amount of matter ca n release a vast amoun t of energy. Locked within the smalles t particles of matter is a storehous e of energy, more tha n 1 million time s the energ y released i n a chemical

88 ENTERIN

G TH E FIFT H DIMENSIO N

explosion. Matter, in some sense, can be seen a s an almost inexhaustible storehouse o f energy; that is , matter is condensed energy . In thi s respect, we see the profoun d differenc e between the work of the mathematicia n (Charle s Hinton ) an d tha t o f the physicis t (Alber t Einstein). Hinton spen t mos t of his adult years trying to visualize highe r spatial dimensions . H e ha d n o interes t i n findin g a physical interpretation fo r th e fourt h dimension . Einstei n saw , however, tha t th e fourt h dimension can be taken as a temporal one. H e was guided b y a conviction and physica l intuition tha t higher dimension s hav e a purpose: t o unif y the principle s o f nature. B y adding highe r dimensions , h e coul d unit e physical concept s that , i n a three-dimensiona l world , hav e n o connec tion, such as matter an d energy . From the n on , th e concep t o f matter an d energ y would be take n as a singl e unit: matter-energy . Th e direc t impac t o f Einstein's work on the fourt h dimensio n was , of course , th e hydroge n bomb , whic h ha s proved t o be th e mos t powerful creatio n o f twentieth-century science.

"The Happies t Though t o f M y Life " Einstein, however, wasn't satisfied. Hi s special theor y of relativity alone would hav e guarantee d hi m a plac e amon g th e giant s o f physics . But there wa s something missing. Einstein's ke y insight wa s to us e th e fourt h dimensio n t o unit e th e laws of nature by introducing tw o new concepts: space-time and matterenergy. Althoug h h e ha d unlocke d som e o f th e deepes t secret s o f nature, h e realize d ther e were several gaping hole s i n hi s theory. What was the relationshi p betwee n these tw o new concepts? More specifically , what abou t accelerations , whic h ar e ignore d i n specia l relativity ? An d what about gravitation? His friend Max Planck, the founde r o f the quantu m theory , advised the youn g Einstei n tha t th e proble m o f gravitatio n wa s too difficult . Planck tol d hi m tha t h e wa s too ambitious : "A s an olde r frien d I must advise you against it for in th e first place you will not succeed ; and eve n if you succeed , n o on e wil l believ e you."5 Einstein , however , plunge d ahead t o unravel the myster y of gravitation. Once again , th e ke y to his momentous discover y was to ask questions that onl y children ask. When childre n rid e i n a n elevator , the y sometime s nervousl y ask, "What happen s i f th e rop e breaks? " Th e answe r i s tha t yo u becom e weightless an d floa t insid e th e elevator , a s thoug h i n oute r space , because bot h yo u and th e elevato r ar e fallin g a t th e sam e rate . Eve n

The Secret o f Light: Vibrations i n th e Fifth Dimension 8

9

though bot h you and th e elevato r ar e acceleratin g in th e earth' s gravitational field, the acceleration is the same for both, and hence i t appears that you are weightles s in th e elevato r (a t leas t until you reach th e bot tom o f the shaft) . In 1907 , Einstein realized that a person floating in the elevator might think tha t someon e ha d mysteriousl y turne d of f gravity. Einstei n once recalled, " I wa s sitting in a chair in th e paten t office a t Bern when al l of a sudde n a thought occurre d t o me: 'I f a person fall s freel y h e wil l no t feel hi s ow n weight. ' I was startled. Thi s simpl e though t mad e a dee p impression on me . It impelled me toward a theory of gravitation."6 Einstein would call it "th e happies t thought o f my life." Reversing th e situation , h e kne w tha t someon e i n a n acceleratin g rocket will fee l a force pushing hi m int o hi s seat, as though ther e were a gravitational pull on him . (I n fact, th e forc e o f acceleration felt by our astronauts is routinely measured i n g-'s—that is, multiples of the forc e of the earth' s gravitation. ) The conclusio n h e reache d wa s that someon e accelerating in a speeding rocket may think that these forces were caused by gravity. From thi s children' s question , Einstei n graspe d th e fundamenta l nature o f gravitation : Th e laws o f nature i n a n accelerating frame ar e equivalent t o the laws i n a gravitational field. This simpl e statement, calle d th e equivalence principle, may not mea n muc h to the average person, but onc e again, in th e hand s of Einstein, it became th e foundatio n o f a theory of the cosmos. (The equivalenc e principl e als o give s simpl e answer s t o comple x physics questions. For example, if we are holding a helium balloon while riding in a car, and th e ca r suddenly swerve s to the left , ou r bodie s wil l be jolte d t o th e right , but whic h way will th e balloo n move ? Commo n sense tell s us tha t th e balloon , lik e our bodies , wil l mov e t o th e right . However, the correct resolution of this subtle question has stumped even experienced physicists . The answe r is to us e th e equivalenc e principle. Imagine a gravitational field pulling on th e ca r fro m th e right . Gravity will make us lurch us to the right , so the helium balloon, which is lighter than ai r and alway s floats "up, " opposit e th e pul l o f gravity, mus t floa t to th e left , int o th e directio n o f the swerve , defyin g commo n sense.) Einstein exploited th e equivalenc e principle to solve the long-stand ing proble m o f whether a light beam i s affected b y gravity. Ordinarily, this is a highly nontrivial question . Throug h th e equivalenc e principle, however, th e answe r becomes obvious. If we shine a flashlight insid e an accelerating rocket, the light beam wil l bend downward toward the floo r (because th e rocke t has accelerated beneat h th e ligh t beam durin g th e

90 ENTERIN

G TH E FIFT H DIMENSIO N

time i t take s fo r th e ligh t bea m t o mov e acros s th e room) . Therefore , argued Einstein , a gravitational field will als o bend th e pat h o f light. Einstein kne w that a fundamental principl e o f physics is that a light beam wil l take the pat h requirin g th e leas t amount o f time between two points. (Thi s is called Fermat's least-time principle.) Ordinarily , the pat h with the smallest time between two points i s a straight line, so light beams are straight . (Eve n whe n ligh t bend s upo n enterin g glass , i t stil l obey s the least-tim e principle. Thi s i s because ligh t slow s dow n i n glass , an d the pat h wit h th e leas t tim e throug h a combinatio n o f air an d glas s is now a bent line . This i s called refraction , which is the principl e behind microscopes an d telescopes.) * However, i f ligh t take s th e pat h wit h th e leas t tim e betwee n tw o points, an d ligh t beams ben d unde r th e influenc e of gravity, then th e shortest distanc e betwee n tw o point s i s a curve d line . Einstei n wa s shocked b y thi s conclusion : I f ligh t coul d b e observe d travelin g i n a curved line , it would mean tha t space itself i s curved.

Space Warp s

At th e cor e o f Einstein' s belie f wa s th e ide a tha t "force " coul d b e explained usin g pure geometry. Fo r example, thin k of riding on a merrygo-round. Everyon e knows that if we change horse s o n a merry-go-round, we feel a "force " tuggin g at u s as we walk across th e platform . Because the oute r ri m o f the merry-go-roun d move s faster tha n th e center , th e outer ri m o f the merry-go-roun d mus t shrink , accordin g t o special rel ativity. However , i f th e platfor m o f th e merry-go-roun d no w ha s a shrunken ri m or circumference, the platform as a whole must be curved. To someon e o n th e platform , ligh t n o longe r travel s in a straight line , as though a "force" were pulling it toward th e rim. The usua l theorem s of geometr y n o longe r hold . Thu s th e "force " w e fee l whil e walking between horses on a merry-go-round can be explained as the curvin g of space itself . Einstein independentl y discovere d Riemann' s origina l program , t o give a purely geometric explanatio n o f the concept o f "force." We recall

*For example, imagine being a lifeguard o n a beach, a t some distance from th e water; out o f the corne r of your eye, you spy someone drowning in th e ocea n far off at an angle. Assume tha t you ca n ru n ver y slowl y in th e sof t sand , but ca n swi m swiftl y i n th e water . A straight path to the victi m will spend to o much tim e on th e sand . The path with th e leas t time i s a bent line, on e tha t reduces the tim e spent running on th e san d and maximize s the tim e spent swimming in th e water.

The Secret o f Light: Vibrations in th e Fifth Dimension 9

1

that Riemann used the analogy of Flatlanders living on a crumpled sheet of paper . T o us , i t i s obvious tha t Flatlander s movin g ove r a wrinkled surface wil l b e incapabl e o f walkin g in a straigh t line . Whichever way they walk, they will experience a "force" that tugs at them from lef t an d right. To Riemann, th e bending o r warping of space cause s the appear ance o f a force. Thus forces do no t reall y exist; what is actually happening i s that space itsel f i s being ben t ou t o f shape . The proble m wit h Riemann' s approach , however , was that h e ha d no ide a specificall y ho w gravity or electricity and magnetis m caused th e warping o f space . Hi s approac h wa s purely mathematical, withou t any concrete physica l pictur e o f precisel y ho w th e bendin g o f spac e wa s accomplished. Her e Einstei n succeeded wher e Riemann failed. Imagine, for example, a rock placed o n a stretched bedsheet . Obvi ously the roc k will sink into th e sheet , creatin g a smooth depression . A small marbl e sho t ont o th e bedshee t wil l the n follo w a circula r o r a n elliptical path around th e rock. Someone lookin g from a distance a t the marble orbitin g aroun d th e rock may say that there i s an "instantaneou s force" emanatin g from th e rock that alters the path o f the marble. However, o n clos e inspection i t is easy to se e what is really happening: Th e rock has warped th e bedsheet, an d henc e th e pat h o f the marble . By analogy, if the planet s orbit around th e sun , it is because they are moving in space tha t has been curve d by the presenc e o f the sun . Thus the reaso n w e are standing on th e earth , rathe r tha n bein g hurle d int o the vacuu m o f outer space , is that th e eart h i s constantly warping th e space around u s (Figur e 4.1). Einstein noticed tha t the presenc e of the sun warps the path o f light from th e distan t stars. This simple physical picture therefor e gav e a way in whic h the theor y coul d b e teste d experimentally . First, w e measure the positio n o f the star s at night, when the su n is absent. Then , durin g an eclipse of the sun, we measure th e position o f the stars, when the sun is present (bu t doesn't overwhel m the ligh t from th e stars) . According to Einstein , th e apparen t relativ e positio n o f th e star s shoul d chang e when th e su n i s present, becaus e th e sun' s gravitationa l field will hav e bent th e pat h o f the ligh t of those star s on it s way to the earth . B y comparing th e photograph s o f th e star s a t nigh t an d th e star s durin g a n eclipse, one shoul d b e able t o test this theory. This picture ca n b e summarize d by what is called Mach's principle, the guid e Einstei n use d t o creat e hi s genera l theor y o f relativity . We recall that the warping of the bedshee t was determined b y the presenc e of the rock . Einstein summarized this analogy by stating: The presenc e of matter-energy determine s th e curvature of the space-time surround ing it . This is the essenc e of the physica l principle tha t Rieman n failed

92

ENTERING TH E FIFT H DIMENSIO N

Figure 4.1. To Einstein, "gravity" was an illusion caused by the bending of space. He predicted that starlight moving around the sun would be bent, and hence the relative positions of the stars should appear distored in the presence of the sun. This has been verified by repeated experiments.

to discover, that th e bendin g o f space i s directly related t o the amoun t of energy an d matte r containe d withi n that space . This, i n turn , ca n b e summarize d b y Einstein's famou s equation, 7 which essentiall y states: Matter—energy — > curvatur e o f space—tim e where th e arro w mean s "determines. " This deceptivel y short equatio n is one o f the greates t triumph s of the huma n mind . From i t emerge the principles behind th e motion s o f stars and galaxies , black holes, the Big Bang, an d perhap s the fat e of the univers e itself.

The Secret o f Light: Vibrations i n th e Fifth Dimension 9

3

Nevertheless, Einstein was still missing a piece o f the puzzle . He ha d discovered th e correc t physica l principle, but lacke d a rigorous mathe matical formalism powerful enoug h t o express thi s principle. He lacked a versio n o f Faraday' s field s fo r gravity . Ironically , Rieman n ha d th e mathematical apparatus , bu t no t th e guidin g physica l principle . Ein stein, by contrast, discovered the physical principle, but lacked the mathematical apparatus . Field Theor y o f Gravity

Because Einstein formulated thi s physical principle without knowing of Riemann, he did not have the mathematical language or skill with which to express his principle. He spent 3 long, frustrating years, from 191 2 to 1915, i n a desperat e searc h fo r a mathematica l formalis m powerfu l enough t o express th e principle. Einstein wrote a desperate lette r to his close friend , mathematicia n Marce l Grossman , pleading , "Grossman , you must help m e or els e I'll go crazy!" 8 Fortunately, Grossman , when combing throug h th e librar y for clues to Einstein' s problem, accidentall y stumbled o n th e wor k of Riemann. Grossman showe d Einstein the wor k of Riemann an d hi s metric tensor , which ha d bee n ignore d b y physicists for 6 0 years. Einstein would later recall tha t Grossma n "checke d throug h th e literatur e an d soo n discovered tha t th e mathematica l proble m ha d alread y bee n solve d b y Riemann, Ricci , and Levi-Civita . . . . Riemann's achievement was the great est one. " To hi s shock, Einstein foun d Riemann' s celebrate d 185 4 lectur e t o be th e ke y to th e problem . H e foun d tha t h e coul d incorporat e th e entire bod y o f Riemann' s wor k in th e reformulatio n o f hi s principle . Almost line for line , the grea t work of Riemann foun d its true hom e i n Einstein's principle . Thi s wa s Einstein's proudes t piec e o f work, even more tha n hi s celebrated equatio n E = mi?. The physica l reinterpreta tion of Riemann's famous 1854 lecture is now called general relativity, an d Einstein's fiel d equation s ran k amon g th e mos t profoun d idea s i n scientific history. Riemann's great contribution , w e recall, was that he introduce d th e concept o f the metric tensor, a field that is defined at all points in space. The metri c tenso r i s not a singl e number . A t eac h poin t i n space , i t consists of a collection o f ten numbers . Einstein' s strategy was to follo w Maxwell an d writ e dow n th e fiel d theor y o f gravity . The objec t o f hi s search fo r a fiel d t o describ e gravit y was found practicall y o n th e firs t

94 ENTERIN

G TH E FIFT H DIMENSIO N

page of Riemarin's lecture. In fact, Riemann's metric tensor was precisely the Farada y field fo r gravity ! When Einstein' s equations are full y expresse d i n terms of Riemann's metric tensor , the y assum e a n eleganc e neve r befor e see n i n physics. Nobel laureate Subrahmanya n Chandrasekha r once calle d it "the mos t beautiful theor y ther e ever was." (I n fact , Einstein' s theory is so simple yet so powerful tha t physicists are sometime s puzzle d as to why it works so well. MIT physicist Victor Weisskopf onc e said , "It' s lik e the peasan t who ask s th e enginee r ho w th e stea m engin e works . Th e enginee r explains to th e peasan t exactl y where the stea m goe s an d ho w it moves through th e engin e an d s o on. And then the peasan t says : 'Yes , I understand al l that, bu t wher e is the horse? ' That' s how I feel abou t genera l relativity. I know all the details , I understand wher e the stea m goes, bu t I'm stil l not sur e I know where th e hors e is." 9) In retrospect , w e now see ho w clos e Rieman n cam e t o discoverin g the theor y o f gravity 60 years before Einstein . The entir e mathematica l apparatus wa s in plac e in 1854 . Hi s equations were powerful enough t o describe th e mos t complicated twistin g of space-time in any dimension. However, he lacked the physical picture (tha t matter-energy determines the curvatur e of space—time) an d th e kee n physica l insight that Einstein provided.

Living i n Curve d Space I once attended a hockey game i n Boston . All the action , o f course, was concentrated o n th e hocke y player s a s the y glide d o n th e ic e rink . Because the puck was being rapidly battered bac k and forth between th e various players, it reminded m e of how atoms exchange electron s when they for m chemica l element s o r molecules . I notice d tha t th e skatin g rink, o f course , di d no t participat e i n th e game . I t onl y marke d th e various boundaries; i t was a passive arena on whic h th e hocke y players scored points . Next, I imagined wha t it must be lik e if the skatin g rink actively participated i n the game : What would happen if the player s were forced t o play on a n ice rink whose surface was curved, with rolling hills and stee p valleys? The hocke y gam e woul d suddenl y becam e mor e interesting . Th e players would have to skate along a curved surface. The rink' s curvature would distort their motion, acting like a "force" pullin g the players one

The Secret o f Light: Vibrations i n th e Fifth Dimension 9

5

way o r another . Th e puc k woul d mov e i n a curve d lin e lik e a snake, making the gam e muc h mor e difficult . Then I imagined takin g thi s on e ste p further ; I imagined tha t th e players were forced t o pla y on a skatin g rink shaped lik e a cylinder. If the players could generate enoug h speed , they could skate upside down and mov e entirely around th e cylinder . New strategies could be devised, such a s ambushing an opposin g playe r by skating upside down aroun d the cylinde r and catchin g him unawares . Once th e ic e rink was bent i n the shap e of a circle, space would become the decisive factor in explaining th e motio n o f matter o n it s surface. Another, mor e relevan t example for our univers e might be living in a curve d spac e give n b y a hypersphere, a sphere i n fou r dimensions. 10 If we look ahead, light will circle completely around th e small perimeter of th e hyperspher e an d retur n t o ou r eyes . Thus w e will se e someon e standing in fron t o f us, with his back facing us, a person who is wearing the sam e clothe s a s w e are . W e loo k disapprovingl y a t th e unruly , unkempt mas s of hair o n thi s person's head , an d the n remembe r tha t we forgot to comb our hai r tha t day. Is this person a fake image created by mirrors? To find out, we stretch out ou r han d an d pu t i t on hi s shoulder. We fin d tha t th e perso n i n front o f us is a real person, not just a fake. I f we look into the distance, in fact , w e see an infinit e numbe r o f identical people, eac h facin g forward, each wit h hi s hand o n th e shoulde r of the perso n i n front . But what is most shocking is that we feel someone' s han d sneakin g up from behind , which then grabs our shoulder. Alarmed, we look back, and se e another infinit e sequence o f identical peopl e behin d us , with their faces turne d th e othe r way. What's reall y happening? We, of course, are th e onl y person livin g in thi s hypersphere. The perso n i n front of us is really ourself. We ar e staring at the bac k of our own head. By placing our hand in front of us, we are really stretching our hand around th e hypersphere, until we place our han d o n our ow n shoulder. The counterintuitiv e stunts tha t ar e possibl e i n a hyperspher e ar e physically interestin g becaus e man y cosmologist s believe tha t ou r uni verse is actually a large hypersphere. There are also other equally strange topologies, lik e hyperdoughnuts and Mobiu s strips. Although they may ultimately have no practica l application, they help t o illustrate many of the feature s of living in hyperspace . For example, le t us assume tha t we are livin g on a hyperdoughnut . If we look to our lef t an d right , we see, much t o our surprise , a perso n on eithe r side . Light circles completely around th e large r perimeter o f

% ENTERINGTHEFIFTHDIMENSIO

N

the doughnut, and returns to its starting point. Thus if we turn our head s and loo k to the left , we see the righ t side of someone's body. By turning our head s the othe r way, we see someone's lef t side . No matter how fast we tur n ou r heads , th e peopl e ahea d o f us and t o our side s tur n thei r heads jus t as fast, an d w e can neve r see their faces . Now imagine stretching our arm s to either side . Both the person o n the lef t an d th e on e o n th e righ t will als o stretch their arms . In fact , if you are clos e enough, yo u can grab th e lef t an d righ t hands of the per sons to either side. If you look carefully i n either direction, you can see an infinitel y long , straight line of people al l holding hands. I f you look ahead, there is another infinit e sequence of people standing before you, arranged i n a straight line, all holding hands . What's actuall y happening? I n realit y our arm s are lon g enoug h t o reach aroun d th e doughnut, unti l the arms have touched. Thus we have actually grabbed ou r ow n hands (Figur e 4.2)! Now we find ourselves tiring of this charade. Thes e peopl e see m t o be tauntin g us ; the y ar e copy-cats , doing exactly what we do . W e ge t annoyed—so we get a gun an d poin t it at the perso n i n front of us. Just before w e pull the trigger , we ask ourselves: Is this person a fake mirro r image? If so, then th e bulle t will go right throug h him . But if not, the n the bullet will go completely around th e universe and hi t us in the back. Maybe firing a gun i n thi s universe is not suc h a good idea ! For an even more bizarre universe, imagine living on a Mobius strip, which is like a long strip of paper twisted 180 degrees an d the n reglue d back togethe r int o a circula r strip . Whe n a right-hande d Flatlande r moves completely around the Mobius strip, he finds that he has become left-handed. Orientation s ar e reverse d whe n travelin g around th e uni verse. This is like H. G. Wells's "The Plattne r Story," i n which the her o returns t o eart h afte r a n acciden t t o fin d tha t hi s bod y i s completely reversed; for example , his heart i s on hi s right side. If we lived on a hyper-Mobius strip, and w e peered i n front o f us, we would se e th e bac k o f someone' s head . A t first , w e wouldn't thin k i t could b e our head, becaus e th e part of the hair would be on the wrong side. If we reached ou t and place d our righ t hand on his shoulder, the n he would lift u p his left han d an d place it on the shoulder o f the perso n ahead o f him . I n fact , w e would se e a n infinit e chai n o f peopl e wit h hands on each other's shoulders, except the hands would alternate fro m the lef t t o the righ t shoulders . If w e lef t som e o f ou r friend s a t on e spo t an d walke d completely around thi s universe, we would find that we had returned t o our original spot. Bu t ou r friend s woul d b e shocke d t o fin d tha t ou r bod y was

The Secret of Light: Vibrations in the Fifth Dimension

Figure 4.2. If we lived in a hyperdoughnut, we would see of ourselves repeated in front of us, to the back of us, and because there are two ways that light can travel around the hands with the people to our sides, we are actually holding is, our arms are actually encircling the doughnut.

97

an infinite succession to our sides. This is doughnut. If we hold our own hands; that

reversed. The par t in our hair and th e rings on our finger s would be on the wrong side, and our internal organs would have been reversed . Our friends woul d be amaze d a t th e reversa l of our body , an d woul d ask if we felt well. In fact , we would feel completel y normal; t o us, it would be our friend s wh o ha d bee n completel y turne d around ! A n argumen t would now ensue ove r who was really reversed.

98 E N T E R 1 N G T H E F I F T H D I M E N S I O

N

These an d othe r interestin g possibilitie s open u p whe n we live in a universe wher e spac e an d tim e ar e curved . N o longe r a passiv e arena , space becomes a n active player in th e drama unfoldin g i n our universe . In summary , we see tha t Einstei n fulfille d th e progra m initiate d by Riemann 6 0 years earlier, t o use higher dimension s t o simplif y th e law s of nature. Einstein, however, went beyond Rieman n in several ways. Like Riemann befor e him , Einstein independently realized tha t "force " i s a consequence o f geometry, but unlik e Riemann, Einstein was able to find the physical principle behind thi s geometry, that the curvature of spacetime i s du e t o th e presenc e o f matter-energy . Einstein , als o lik e Riemann, kne w that gravitation can be described b y a field, the metric ten sor, bu t Einstei n was able to find the precis e field equations that these fields obey .

A Univers e Mad e of Marbl e By th e mid-1920s , wit h th e developmen t o f bot h specia l an d genera l relativity, Einstein's place in th e histor y of science was assured. I n 1921 , astronomers ha d verified tha t starlight indeed bends as it travels around the sun , precisely as Einstein ha d predicted. By then, Einstein wa s being celebrated a s the successo r to Isaac Newton. However, Einstein stil l was not satisfied . He woul d tr y one las t time to produc e anothe r world-clas s theory. Bu t o n hi s thir d try , he failed . His third an d fina l theor y was to hav e been th e crownin g achievemen t of his lifetime. He was searching fo r the "theor y of everything," a theory that would explain all the familiar forces found in nature, including light and gravity . He coine d thi s theory th e unified fiel d theory. Alas, his search for a unified theory of light and gravit y was fruitless. When h e died , h e left onl y the unfinishe d idea s of various manuscripts o n hi s desk. Ironically, the source o f Einstein's frustration was the structure of his own equation . Fo r 3 0 years, he wa s disturbed b y a fundamental fla w i n this formulation . O n on e sid e o f th e equatio n wa s th e curvatur e o f space-time, which he likene d t o "marble" becaus e o f its beautiful geometric structure . T o Einstein , the curvatur e o f space-time was like th e epitome o f Greek architecture, beautiful and serene. However , he hate d the other sid e of this equation, describin g matter—energy, which he considered t o be ugly and which he compared t o "wood." While the "mar ble" o f space-time was clean and elegant, th e "wood" of matter-energ y was a horrible jumble of confused, seemingly rando m forms , from subatomic particles , atoms , polymers , an d crystal s to rocks , trees , planets ,

The Secret o f Light: Vibrations i n th e Fifth Dimension 9

9

and stars . But in the 1920 s and 1930s , when Einstein was actively working on the unified field theory, the true nature of matter remained an unsolved mystery. Einstein's gran d strateg y was to tur n woo d int o marble—tha t is , to give a completely geometric origi n to matter. But without more physical clues and a deeper physica l understanding o f the wood, this was impossible. B y analogy, thin k o f a magnificent , gnarled tre e growin g i n th e middle o f a park . Architects have surrounde d thi s grizzle d tree wit h a plaza made o f beautiful pieces of the purest marble . The architect s have carefully assemble d th e marbl e piece s to resemble a dazzling floral pattern wit h vine s an d root s emanatin g fro m th e tree . T o paraphras e Mach's principle : Th e presenc e o f the tre e determine s th e patter n o f the marbl e surroundin g it . But Einstein hate d thi s dichotomy between wood, which seemed t o be ugly and complicated , and marble , which was simple and pure . His dream wa s to turn the tree into marble; he would have liked t o have a plaza completely made o f marble, wit h a beautiful, symmetrical marble statu e of a tree at its center. In retrospect , w e can probabl y spo t Einstein' s error. We recall tha t the law s o f natur e simplif y an d unif y i n highe r dimensions . Einstei n correctly applie d thi s principl e twice , i n specia l an d genera l relativity . However, o n hi s thir d try , he abandone d thi s fundamenta l principle . Very little was known about th e structur e o f atomic and nuclea r matte r in his time; consequently, it was not clea r how to use higher-dimensional space a s a unifying principle. Einstein blindly tried a number of purely mathematical approaches . He apparentl y though t tha t "matter " coul d b e viewe d as kinks, vibrations, or distortions of space-time. In thi s picture, matter was a concentrated distortio n o f space. I n other words , everything we see around us , from th e tree s an d cloud s t o th e star s in th e heavens , was probably an illusion, som e for m of crumpling of hyperspace. However, without any more soli d leads or experimenta l data , this idea le d to a blind alley. It would be lef t t o an obscur e mathematicia n t o tak e th e nex t step , which would lead u s to the fifth dimension. The Birt h o f Kaluza-Klei n Theor y In April 1919, Einstei n received a letter tha t left hi m speechless. It was from an unknown mathematician, Theodr Kaluza, at the University of Konigsberg in Germany , i n what is Kaliningrad i n th e forme r Soviet Union. In a short article, only a few pages long, this obscure math-

100 E N T E R I N

G TH E F I F T H D I M E N S I O N

ematician was proposing a solutio n t o on e o f the greates t problem s of the century . In just a few lines, Kaluza was uniting Einstein' s theory of gravity with Maxwell's theory of light by introducing the fifth dimension (that is, four dimensions o f space an d on e dimensio n o f time). In essence , h e was resurrecting the ol d "fourt h dimension " o f Hinton an d Zollne r an d incorporatin g i t into Einstein' s theory in a fresh fashion a s th e fift h dimension . Lik e Rieman n befor e him , Kaluz a assumed that light is a disturbance cause d by the ripplin g of this higher dimension. Th e ke y differenc e separatin g thi s ne w wor k fro m Rie mann's, Hinton's, and Zollner's was that Kaluza was proposing a genuine field theory. In this short note, Kaluza began, innocently enough, by writing down Einstein's fiel d equation s fo r gravit y in five dimensions, no t th e usua l four. (Riemann' s metric tensor, we recall, can be formulated in any number o f dimensions.) The n h e proceede d t o show that thes e five-dimensional equation s containe d withi n the m Einstein' s earlier four-dimen sional theor y (whic h was to be expected ) wit h an additiona l piece. Bu t what shocked Einstei n was that thi s additional piec e was precisely Maxwell's theor y o f light . I n othe r words , thi s unknow n scientis t was proposing t o combine, i n one stroke , the tw o greatest field theories known to science, Maxwell's and Einstein's , by mixing them in the fifth dimension. This was a theory made o f pure marble—tha t is, pure geometry. Kaluza had foun d the first important clu e in turning wood into marble. I n th e analog y o f th e park , w e recall tha t th e marbl e pla/ a i s two dimensional. Kaluza' s observatio n wa s that w e could buil d a "tree" of marble i f we could mov e the piece s of marble u p into th e thir d dimen sion. To th e averag e layman , light and gravit y have nothin g i n common . After all , light i s a familiar force tha t come s i n a spectacula r variet y of colors an d forms , while gravit y is invisibl e and mor e distant . O n th e earth, it is the electromagnetic force , not gravity , that has helped us tarne nature; i t is the electromagneti c forc e tha t power s our machines , electrifies our cities , lights our neon signs, and brighten s ou r televisio n sets. Gravity, by contrast, operates o n a larger scale; it is the forc e that guides the planet s an d keep s the su n fro m exploding. I t is a cosmic force tha t permeates th e univers e and bind s th e sola r system . (Along with Webe r and Riemann , on e o f th e firs t scientist s to searc h activel y for a lin k between ligh t an d gravit y in th e laborator y wa s Faraday himself . Th e actual experimenta l apparatu s use d b y Farada y t o measur e th e lin k between thes e tw o forces can stil l b e foun d i n th e Roya l Institution i n Piccadilly, London. Although h e faile d experimentall y to find any con-

The Secret o f Light: Vibrations i n th e Fifth Dimension 10

1

nection at all between the two forces, Faraday was confident of the power of unification. He wrote, "If th e hope [o f unification] should prove well founded, ho w great an d might y and sublim e in it s hitherto unchange able characte r i s the forc e I am tryin g to dea l with, and ho w large may be th e ne w domain o f knowledg e tha t ma y be opene d t o th e min d o f man." 11 ) Even mathematically , ligh t and gravit y are lik e oi l and water . Maxwell's fiel d theor y o f ligh t require s fou r fields, while Einstein's metric theory o f gravit y require s ten . Ye t Kaluza' s pape r wa s so elegan t an d compelling tha t Einstei n could not rejec t it. At first, it seemed lik e a cheap mathematica l trick simply to expan d the numbe r o r dimensions of space and tim e from fou r to five. This was because, as we recall, there was no experimental evidenc e for the fourth spatial dimension . Wha t astonishe d Einstei n wa s tha t onc e th e fivedimensional fiel d theor y was broken dow n t o a four-dimensiona l field theory, bot h Maxwell' s and Einstein' s equation s remained . I n othe r words, Kaluza succeede d i n joining th e tw o pieces o f the jigsaw puzzle because bot h o f the m wer e par t o f a large r whole , a five-dimensional space. "Light" wa s emerging a s th e warpin g o f th e geometr y o f higher dimensional space. This was the theor y that seemed t o fulfill Riemann' s old dream o f explaining forces as the crumplin g of a sheet of paper. In his article , Kaluz a claimed tha t hi s theory , which synthesize d th e tw o most important theorie s u p t o that time, possessed "virtuall y unsurpassed formal unity." He furthermore insisted that the sheer simplicity and beauty of his theory could no t "amoun t t o the mer e allurin g play of a capricious accident." 12 What shoo k Einstei n was the audacit y and sim plicity of the article . Like all great ideas, Kaluza's essential argument was elegant and compact . The analog y with piecing togethe r th e part s o f a jigsaw puzzle is a meaningful one . Recal l that the basis of Riemann's and Einstein' s work is the metric tensor—that is, a collection of ten numbers defined at each point in space. This was a natural generalizatio n o f Faraday's field concept. I n Figur e 2.2, we saw how these te n number s ca n b e arrange d a s in the pieces of a checker board with dimensions 4 X 4. We can denote these te n number s as gu, g ]2, . . . . Furthermore, th e field of Maxwell is a collection o f four numbers defined at each point in space. These four numbers can b e represente d b y the symbol s A,, A 2, A 3, A 4. To understand Kaluza's trick, let us now begin with Riemann's theory in five dimensions. The n th e metri c tensor ca n be arrange d i n a 5 X 5 checkerboard. Now , by definition, we will renam e th e component s o f

Figure 4.3. Kaluza's brilliant idea was to write down the Riemann metric in five dimensions. The. fifth column and row are identified as the. electromagnetic field of Maxwell, while the remaining 4X4 block is the old four-dimensional metric of Einstein. In one stroke, Kaluza unified the theory oj gravity with light simply by adding another dimension.

102

The Secret o f Light: Vibrations i n th e Fifth Dimension 10

3

Kaluza's field, so that some of them become Einstein' s original fiel d an d some o f the m becom e Maxwell' s field (Figure 4.3) . Thi s i s the essenc e of Kaluza' s trick, which caugh t Einstei n totall y b y surprise . B y simply adding Maxwell's field to Einstein's, Kaluz a was able to reassemble both of the m int o a five-dimensional field. Notice tha t ther e i s "enoug h room " withi n th e 15 component s o f Riemann's five-dimensiona l gravit y to fi t bot h th e te n component s o f Einstein's field and th e fou r components o f Maxwell's field! Thus Kaluza's brillian t idea ca n b e crudel y summarized as

15 = 1 0 + 4 + 1 (the leftove r componen t i s a scala r particle , whic h i s unimportant fo r our discussion) . When carefull y analyzin g the ful l five-dimensional theory, w e fin d tha t Maxwell' s field is nicely included within th e Rieman n metric tensor , just a s Kaluza claimed. Thi s innocent-lookin g equatio n thus summarized on e o f the semina l ideas o f the century . In summary , the five-dimensional metric tenso r include d bot h Maxwell's field and Einstein's metric tensor. I t seemed incredibl e to Einstein that such a simple idea coul d explai n th e tw o most fundamental forces of nature: gravit y and light . Was it just a parlor trick ? O r numerology ? Or blac k magic? Einstein was deeply shaken b y Kaluza's letter and , i n fact , refuse d t o respond t o the article . He mulle d over the lette r fo r 2 years, an unusually long tim e for someon e t o hol d u p publicatio n o f a n importan t article . Finally, convinced tha t this article was potentially important, h e submitte d i t for publication i n th e Silzungsberichte Preussische Akademie der Wissenschaften. It bore the imposin g titl e "On th e Unit y Problem o f Physics." In th e histor y of physics , no on e ha d foun d an y use fo r th e fourt h spatial dimension . Eve r sinc e Riemann , i t wa s known tha t th e mathe matics of higher dimension s was one o f breathtaking beauty , but without physical application. For the first time, someone ha d found a use for th e fourth spatia l dimension : t o unit e th e law s o f physics ! In som e sense , Kalu/.a wa s proposing tha t th e fou r dimension s o f Einstei n wer e "to o small" t o accommodat e bot h th e electromagneti c an d gravitationa l forces. We can als o se e historicall y that Kaluza' s work was not totall y unexpected. Mos t historians o f science, when the y mention Kaluza' s work at all, sa y that th e idea of a fifth dimension was a bolt out of the blue, totally unexpected an d original . Given the continuit y of physics research, thes e historians ar e startle d t o find a new avenue o f science opening up with-

104 E N T E R I N

G TH E FIFT H D I M E N S I O N

out an y historical precedent . Bu t thei r amazemen t i s probably du e t o their unfamiliarit y wit h th e nonscientifi c wor k o f th e mystics , literati , and avant e garde . A close r loo k a t th e cultura l an d historica l settin g shows that Kaluza' s work was not such a n unexpecte d development . As we have seen, because of Hinton, Zollner , and others , th e possibl e existence o f higher dimension s wa s perhaps th e singl e most popular quasi scientific idea circulating within the arts. From this larger cultural point of view, it was only a matter o f time before som e physicist took seriously Hinton's widel y known idea tha t light is a vibration of the fourth spatial dimension. I n thi s sense, th e wor k of Riemann pollinate d th e worl d of arts an d letter s via Hinton an d Zollner , and the n probabl y cross-pollinated bac k int o th e worl d o f science throug h th e wor k of Kaluza . (I n support o f this thesis, it was recently revealed by Freund tha t Kaluza was actually not th e firs t on e t o propose a five-dimensional theory of gravity. Gunnar Nordstrom , a rival of Einstein, actually published th e first fivedimensional fiel d theory , bu t i t was too primitiv e to includ e both Ein stein's and Maxwell's theories. The fact that both Kaluza and Nordstrom independently trie d t o exploi t th e fift h dimensio n indicate s tha t th e concepts widel y circulating within popular cultur e affecte d thei r thinking.13)

The Fifth Dimension

Every physicist receives quite ajolt when confronting the fifth dimension for th e first time. Peter Freun d remember s clearl y the precis e momen t when h e firs t encountere d th e fift h an d highe r dimensions . It was an event that lef t a deep impression o n hi s thinking. It was 1953 in Romania, th e countr y of Freund's birth. Joseph Stalin had just died , a n importan t even t tha t le d to a considerable relaxatio n of tensions. Freund wa s a precocious college freshman that year, and h e attended a tal k b y Georg e Vranceanu . H e vividl y remember s hearin g Vranceanu discus s the important question : Why should light and gravit y be so disparate? Then th e lecture r mentione d a n ol d theory that could contain both the theor y of light and Einstein' s equations o f gravity. The secret wa s t o us e Kaluza-Klei n theory , whic h wa s formulate d i n fiv e dimensions. Freund wa s shocked. Her e wa s a brillian t idea tha t too k hi m com pletely b y surprise. Althoug h onl y a freshman , h e ha d th e audacit y to pose th e obviou s question: Ho w does thi s Kaluza—Klein theor y explain the othe r forces ? He asked , "Eve n i f you achiev e a unificatio n o f light

The Secret o f Light: Vibrations i n th e Fifth Dimension 10

5

and gravity , yo u wil l no t achiev e anythin g becaus e ther e i s stil l th e nuclear force. " H e realize d that th e nuclea r force wa s outside Kaluza Klein theory. (In fact, th e hydrogen bomb, which hung like a sword over everyone o n th e plane t a t th e heigh t o f th e Col d War , was based o n unleashing the nuclea r force, no t electromagnetis m or gravity. ) The lecture r ha d n o answer . I n hi s youthfu l enthusiasm , Freund blurted out, "What about addin g more dimensions? " "But ho w many more dimensions?" asked the lecturer . Freund wa s caught off guard. H e di d no t wan t to give a low number of dimensions , only to b e scoope d b y someone else . So he propose d a number tha t n o on e coul d possibly top: a n infinit e numbe r o f dimensions!14 (Unfortunatel y for thi s precocious physicist, an infinit e numbe r of dimensions does not seem to be physicall y possible.)

Life on a Cylinder

After the initial shock of confronting the fifth dimension, most physicists invariably begi n t o as k questions. I n fact , Kaluza' s theor y raise d mor e questions tha n i t answered . Th e obviou s questio n t o as k Kaluz a was : Where is the fifth dimension? Since all earthly experiments showed conclusively tha t w e liv e i n a universe with thre e dimension s o f space an d one o f time, the embarrassin g question stil l remained . Kaluza ha d a cleve r response. Hi s solution wa s essentially the sam e as tha t propose d b y Hinton year s before , tha t th e highe r dimension , which wa s not observabl e b y experiment, wa s different fro m th e othe r dimensions. I t had, in fact, collapse d down to a circle so small that even atoms coul d no t fit inside it . Thus th e fifth dimension wa s not a mathematical tric k introduced t o manipulate electromagnetism an d gravity , but a physical dimension tha t provided th e glu e t o unite thes e tw o fundamental force s into on e force , but was just to o small to measure. Anyone walking in th e directio n o f the fift h dimensio n woul d eventually find himself back where he started. This is because the fifth dimension is topologically identical to a circle, and th e universe is topologically identical to a cylinder. Freund explain s it this way: Think o f som e imaginar y people livin g i n Lineland , whic h consist s o f a single line . Throughout thei r history , the y believed tha t thei r worl d was just a single line. Then, a scientist in Lineland propose d tha t thei r world was not just a one-dimensiona l line , but a two-dimensiona l world. Whe n

106 E N T E R I N

G THE FIFT H D I M E N S I O N

asked wher e thi s mysterious and unobservabl e second dimensio n was, he would repl y tha t th e secon d dimensio n wa s curled u p int o a smal l ball. Thus, th e lin e people actuall y live on th e surfac e o f a long, but ver y thin, cylinder. Th e radiu s o f th e cylinde r i s too smal l to b e measured ; i t i s so small, in fact , tha t it appears tha t th e worl d is just a line. 15

If the radiu s of the cylinde r were larger, th e lin e people coul d move off thei r univers e an d mov e perpendicula r t o their line world. In othe r words, they could perfor m interdimensiona l travel . As they moved per pendicular t o Lineland , the y would encounte r a n infinit e numbe r o f parallel lin e world s tha t coexiste d wit h thei r universe . A s the y move d farther into the second dimension, they would eventually return t o their own line world. Now thin k of Flatlanders livin g on a plane. Likewise , a scientist o n Flatland ma y make the outrageous clai m that traveling through th e third dimension i s possible. In principle, a Flatlander coul d rise off the surface of Flatland. As this Flatlander slowl y floated upwar d in th e thir d dimen sion, hi s "eyes " would se e an incredibl e sequenc e o f different parallel universes, each coexistin g with his universe . Becaus e hi s eyes would be able to see only parallel t o the surface of Flatland, he would see differen t Flatland universe s appearin g befor e him . I f the Flatlande r drifte d to o far abov e th e plane , eventuall y he woul d retur n t o his original Flatlan d universe. Now, imagin e tha t ou r presen t three-dimensional worl d actually has another dimension tha t has curled u p into a circle. For the sake of argument, assum e tha t th e fift h dimensio n i s 1 0 feet long. B y leaping int o the fifth dimension, we simply disappear instantl y from our presen t universe. Once we move i n th e fift h dimension , w e find that, after movin g 10 feet, we are bac k where we started from . But why did th e fifth dimension cur l up int o a circle in th e first place? In 1926 , the mathematicia n Oskar Klein made severa l improvements on th e theory , stating that perhaps th e quantu m theor y coul d explai n wh y the fifth dimension rolle d up. O n thi s basis , h e calculate d tha t th e siz e o f th e fift h dimensio n should b e 10~ 33 centimeter s (th e Planc k length) , whic h i s muc h to o small for any earthly experiment t o detect its presence. (Thi s is the same argument use d toda y t o justify th e ten-dimensiona l theory. ) On th e on e hand , thi s meant that th e theor y was in agreement wit h experiment becaus e th e fifth dimension was too smal l t o be measured . On th e othe r hand , it also meant tha t th e fifth dimension wa s so fantastically smal l tha t on e coul d neve r buil d machine s powerfu l enough t o prove th e theor y wa s really correct. (Th e quantu m physicis t Wolfgang

The Secret o f Light: Vibrations in th e Fifth Dimension 10

7

Pauli, in hi s usual caustic way, woul d dismis s theorie s he didn' t lik e by saying, "I t isn' t eve n wrong. " I n othe r words , the y were s o half-bake d that one coul d not eve n determin e i f they were correct . Give n th e fac t that Kaluza's theory could not be tested, one coul d also say that it wasn't even wrong.)

The Deat h o f Kaluza-Klei n Theor y As promising a s Kaluza-Klein theor y was for givin g a purely geometri c foundation t o th e force s o f nature , b y the 1930 s th e theor y was dead. On th e on e hand, physicist s weren't convinced tha t th e fifth dimension really existed. Klein' s conjecture that the fifth dimension wa s curled u p into a tiny circle the size of the Planck length was untestable. The energy necessary t o prob e thi s tiny distanc e ca n b e computed , an d i t is called the Planck energy, o r 10 19 billion electro n volts. Thi s fabulou s energ y is almost beyond comprehension . I t is 100 billion billion times the energ y locked i n a proton, a n energ y beyon d anythin g we will b e abl e t o pro duce withi n the nex t severa l centuries. On th e othe r hand , physicist s lef t thi s are a o f researc h i n drove s because o f th e discover y of a ne w theor y tha t wa s revolutionizing th e world o f science . Th e tidal wav e unleashe d b y this theor y o f th e sub atomic world completely swamped research i n Kaluza-Klein theory. The new theor y wa s called quantu m mechanics , an d i t sounde d th e dead i knell fo r Kaluza-Klei n theor y fo r th e nex t 6 0 years. Worse , quantu m mechanics challenge d th e smooth , geometri c interpretatio n o f forces, replacing i t with discret e packets of energy. Was th e progra m initiate d b y Rieman n an d Einstei n completel y wrong?

This page intentionally left blank

PART II Unification in Ten Dimensions

This page intentionally left blank

5 Quantum Heres y Anyone who is not shocke d b y the quantu m theor y does no t understand it . Niels Bohr

A Univers e Mad e of Wood

I

N 1925, a new theory burst into existence. With dizzying, almost meteoric speed , thi s theory overthrew long-cherished notion s abou t mat ter that ha d been held sinc e the tim e of the Greeks . Almost effortlessly , it vanquishe d score s o f long-standin g fundamenta l problem s tha t ha d stumped physicists for centuries . What is matter made of ? What holds it together? Why does it come i n an infinite variety of forms, such as gases, metals, rocks , liquids , crystals , ceramics, glasses , lightnin g bolts , stars, and s o on? The ne w theor y was christened quantum mechanics, and gav e us th e first comprehensive formulatio n wit h whic h to pr y open th e secret s of the atom . Th e subatomi c world, once a forbidden real m fo r physicists , now began t o spill its secrets into th e open . To understan d th e spee d wit h which this revolution demolishe d it s rivals, w e note tha t i n th e earl y 1920 s some scientist s still hel d seriou s reservations abou t th e existenc e o f "atoms." What couldn't b e seen o r measured directl y in th e laboratory , the y scoffed , didn' t exist . But by 1925 and 1926 , Erwin Schrodinger, Werner Heisenberg, and others ha d

111

112 U N I F I C A T I O

N I N TE N D I M E N S I O N S

developed a n almos t complete mathematica l description o f the hydro gen atom. With devastating precision, the y could now explain nearl y all the propertie s o f the hydroge n ato m fro m pure mathematics . By 1930, quantum physicist s such a s Pau l A. M . Dirac were declarin g tha t al l of chemistry coul d b e derive d fro m firs t principles . The y eve n mad e th e brash clai m that , give n enoug h tim e o n a calculatin g machine, the y could predic t al l th e chemica l propertie s o f matte r foun d i n th e uni verse. T o them , chemistr y would n o longe r b e a fundamental science. From no w on, it would b e "applie d physics." Not only did it s dazzling rise include a definitive explanatio n o f th e bizarre propertie s o f th e atomi c world ; bu t quantu m mechanic s als o eclipsed Einstein's work for many decades: One o f the first casualties of the quantum revolution was Einstein's geometric theor y of the universe. In th e hall s of the Institut e for Advanced Study , young physicists began to whisper that Einstei n was over th e hill , that the quantu m revolutio n had bypasse d hi m completely . Th e younge r generatio n rushe d t o read the lates t paper s writte n abou t quantu m theory , no t thos e abou t th e theory of relativity. Even th e directo r o f the institute , J. Rober t Oppen heimer, confide d privatel y to hi s clos e friend s that Einstein' s work was hopelessly behind th e times . Even Einstein began t o think of himself as an "ol d relic. " Einstein's dream, w e recall, was to creat e a universe mad e o f "mar ble"—that is, pure geometry . Einstein was repelled b y the relativ e ugliness of matter, with its confusing, anarchistic jumble of forms, which he called "wood. " Einstein's goal was to banis h thi s blemish fro m hi s theories forever, to turn wood into marble. Hi s ultimate hope was to creat e a theory of the universe based entirely on marble. To his horror, Einstein realized tha t th e quantu m theor y wa s a theor y mad e entirely o f wood! Ironically, i t now appeared tha t h e ha d mad e a monumenta l blunder, that th e univers e apparently preferre d woo d t o marble. In th e analog y betwee n woo d an d marble , w e recall tha t Einstei n wanted t o convert th e tre e i n th e marbl e plaz a to a marble statue, creating a park completel y made o f marble. Th e quantu m physicists , however, approache d th e proble m fro m th e opposit e perspective . Thei r dream wa s to tak e a sledge hamme r an d pulveriz e all the marble . Afte r removing the shattere d marbl e pieces , they would cover th e par k com pletely with wood. Quantum theory , in fact, turned Einstein on his head. In almost every sense o f the word , quantum theor y is the opposit e o f Einstein's theory. Einstein's genera l relativit y is a theor y o f th e cosmos , a theor y of stars

Quantum Heresy 11

3

and galaxie s hel d togethe r vi a th e smoot h fabri c o f spac e an d time . Quantum theory , by contrast, i s a theory of the microcosm , where subatomic particle s are hel d togethe r b y particlelike forces dancing o n th e sterile stage of space—time, which is viewed as an empty arena, devoid of any content . Thu s th e tw o theories ar e hostil e opposites . I n fact , th e tidal wav e generate d b y the quantu m revolutio n swampe d al l attempts at a geometric understandin g o f forces for over a half-century . Throughout thi s book, w e have developed th e them e tha t th e law s of physic s appear simpl e an d unifie d i n highe r dimensions . However , with th e appearanc e o f the quantu m heres y after 1925 , we see the firs t serious challeng e t o this theme. I n fact , fo r th e nex t 60 years, until th e mid-1980s, th e ideolog y o f th e quantu m heretic s would dominat e th e world o f physics , almost buryin g th e geometri c idea s o f Rieman n an d Einstein unde r a n avalanch e o f undeniabl e successe s an d stunnin g experimental victories. Fairly rapidly , quantu m theor y bega n t o giv e u s a comprehensiv e framework i n which t o describ e th e visibl e universe : The materia l universe consists of atoms and its constituents. There are about 100 different types o f atoms , o r elements , ou t o f which we ca n buil d al l th e know n forms of matter found on earth an d even in outer space. Atoms, in turn , consist of electrons orbitin g around nuclei , which in turn are compose d of neutrons an d protons . I n essence , th e ke y differences betwee n Einstein's beautiful geometric theory and quantum theor y can now be summarized a s follows . 1. Forces are create d b y the exchang e o f discrete packet s of energy, called quanta. In contrast to Einstein's geometric pictur e of a "force," i n quantum theory light was to be chopped u p into tiny pieces. These packets of light were name d photons, an d the y behav e ver y muc h lik e poin t particles . When tw o electrons bum p int o eac h other , the y repe l eac h othe r no t because o f the curvatur e of space, but becaus e the y exchange a packet of energy, the photon . The energ y of these photons is measured in units of something called Planck's constant (h ~ 10~ 27 er g sec) . Th e almos t infinitesima l siz e o f Planck's constan t mean s tha t quantu m theor y give s tiny corrections t o Newton's laws. These are called quantum corrections, and ca n be neglecte d when describin g ou r familiar , macroscopic world . That i s why we can, for th e mos t part, forget about quantu m theor y when describin g everyday phenomena . However , whe n dealin g wit h th e microscopi c sub -

114 U N I F I C A T I O

N I N TE N D I M E N S I O N S

atomic world, these quantum corrections begin to dominate any physical process, accountin g for th e bizarre , counterintuitive properties o f subatomic particles. 2. Different force s are cause d b y the exchang e o f different quanta . The wea k force, for example, is caused by the exchange o f a differen t type o f quantum , called a W particle (VFstand s fo r "weak") . Similarly , the strong force holding th e proton s and neutron s togethe r withi n the nucleus o f th e ato m i s caused b y the exchang e o f subatomi c particles called I T mesons . Bot h W bosons an d T T meson s hav e been see n experi mentally i n th e debri s o f ato m smashers , thereb y verifyin g th e funda mental correctnes s o f thi s approach. And finally, the subnuclea r force holding the protons and neutrons an d eve n the n mesons together are called gluons. In this way, we have a new "unifying principle" for the laws of physics. We ca n unit e th e law s o f electromagnetism , th e wea k force, an d th e strong forc e b y postulating a variet y of differen t quant a tha t mediat e them. Three of the fou r force s (excludin g gravity) ar e therefor e unite d by quantu m theory , givin g u s unificatio n withou t geometry , whic h appears t o contradic t th e them e o f thi s boo k an d everythin g we have considered s o far. 3. We can neve r kno w simultaneously the velocit y and positio n o f a subatomic particle . This is the Heisenberg Uncertainty Principle, which is by far the most controversial aspec t o f the theory , but on e tha t ha s resisted ever y challenge i n th e laborator y fo r hal f a century . There i s no know n experimental deviatio n t o this rule. The Uncertaint y Principle mean s tha t w e can neve r b e sur e wher e an electro n i s or wha t its velocity is. The bes t we can d o i s to calculate the probabilit y tha t th e electro n wil l appea r a t a certai n plac e wit h a certain velocity . The situatio n i s not a s hopeless a s one migh t suspect, because we can calculate with mathematical rigor the probabilit y of finding tha t electron. Although th e electro n i s a point particle , it is accompanied b y a wav e tha t obey s a well-defined equation , th e Schrodinge r wave equation . Roughl y speaking, th e large r th e wave , th e greate r th e probability of finding the electro n a t that point . Thus quantum theory merges concepts of both particle and wave into a nice dialectic: The fundamental physical objects of nature are particles, but th e probabilit y of finding a particle a t any given place in space an d

Quan turn Heresy 11

5

time i s give n b y a probabilit y wave . This wave , i n turn , obey s a well defined mathematica l equation give n by Schrodinger. What is so crazy abou t th e quantu m theor y is that it reduce s everything to these baffling probabilities . We can predict with great precision how many electrons in a beam will scatter when moving through a screen with hole s i n it . However , we can neve r kno w precisely which electro n will scatter in which direction. This is not a matter of having crude instruments; according t o Heisenberg, i t is a law of nature . This formulation , o f course , ha d unsettlin g philosophical implications. The Newtonia n vision held tha t the univers e was a gigantic clock, wound at the beginning of time and ticking ever since because it obeyed Newton's thre e law s o f motion ; thi s pictur e o f th e univers e was now replaced by uncertainty and chance. Quantum theor y demolished, once arid fo r all , th e Newtonia n drea m o f mathematicall y predictin g th e motion o f all the particle s in th e universe . If quantu m theor y violate s our commo n sense , i t i s onl y becaus e nature doe s not see m to care muc h about our commo n sense . As alien and disturbin g as these idea s may seem, the y can b e readil y verified i n the laboratory . Thi s i s illustrated b y the celebrate d double-sli t experiment. Let u s say we fire a beam o f electrons at a screen wit h two small slits. Behind the screen, there is sensitive photographic paper. According to nineteenth-centur y classica l physics, there shoul d b e tw o tiny spots burned int o th e photographi c pape r b y the bea m o f electrons behin d each hole . However , when th e experimen t i s actually performed i n th e laboratory, w e find an interference pattern ( a series of bright and dar k lines) o n th e photographi c paper , which is commonly associated with wavelike, no t particlelike , behavior (Figur e 5.1). (Th e simples t way of creating an interferenc e patter n i s to take a quiet bat h an d the n rhyth mically splash wave s o n th e water' s surface. The spiderweblik e patter n of waves criss-crossing the surfac e o f the water is an interference pattern caused b y the collisio n of many wave fronts. ) Th e patter n o n th e pho tographic sheet correspond s t o a wav e tha t ha s penetrated bot h hole s simultaneously and the n interfere d wit h itsel f behind th e screen . Since the interferenc e patter n i s create d b y th e collectiv e motion o f man y individual electrons , an d sinc e th e wav e ha s gon e throug h bot h hole s simultaneously, naively we come t o the absurd conclusion that electrons can someho w enter both holes simultaneously. But ho w ca n a n electro n be i n tw o places a t th e sam e time ? According to quantu m theory, th e electron i s indeed a point particl e tha t went through on e o r th e othe r hole, but th e wave function of the electro n sprea d ou t ove r space, went

116

UNIFICATION I N TE N DIMENSION S

Figure 5.1. A beam of electrons is shot through two small holes and exposes some film. We expect to see two dots on the film. Instead, we find an undulating interference pattern. How can this be? According to quantum theory, Ihe electron is indeed a pointlike particle and cannot go through both holes, but the Schrodinger wave associated with each electron can pass through both holes and interfere with itself.

through bot h holes , and the n interacte d wit h itself. As unsettling as this idea is , it ha s bee n verifie d repeatedl y b y experiment. A s physicist Sir James Jean s onc e said , "I t i s probably a s meaningles s t o discus s how much roo m a n electro n take s u p a s it i s to discus s ho w much roo m a fear, a n anxiety, or an uncertaint y takes up."1 ( A bumper sticke r I once saw in German y summe d thi s up succinctly . It read , "Heisenber g may have slept here.") 4. There i s a finite probability that particle s may "tunnel" throug h or make a quantum lea p throug h impenetrabl e barriers . This is one o f more stunnin g predictions of quantum theory . On th e atomic level , this predictio n ha s had nothin g less than spectacula r suc cess. "Tunneling, " o r quantu m leap s throug h barriers , ha s survived every experimental challenge . In fact , a world without tunneling is now unimaginable. One simpl e experimen t tha t demonstrates th e correctnes s o f quantum tunnelin g starts by placing an electron i n a box. Normally, the electron doe s no t hav e enoug h energ y t o penetrate th e wall s of the box . If

Quan turn Heresy 11

7

classical physics is correct, the n th e electro n woul d never leave the box . However, according t o quantum theory , the electron' s probabilit y wave will spread throug h th e bo x and see p int o th e outsid e world . The seep age through th e wal l can be calculate d precisel y wit h the Schrodinge r wave equation ; tha t is , ther e i s a smal l probabilit y that th e electron' s position i s somewhere outside the box . Another way of saying this is that there is a finite but smal l probability that the electro n wil l tunnel its way through th e barrie r (th e wall o f the box ) an d emerg e fro m th e box . I n the laboratory , whe n on e measure s th e rat e a t which electrons tunne l through thes e barriers , th e number s agre e precisel y with the quantu m theory. This quantum tunneling is the secret behind the tunnel diode, which is a purely quantum-mechanical device . Normally, electricity might no t have enoug h energ y to penetrat e pas t th e tunne l diode . However , the wave function o f these electron s ca n penetrat e throug h barrier s i n th e diode, s o there is a non-negligible probability that electricity will emerg e on th e other side of the barrier by tunneling through it . When you listen to the beautiful sounds of stereo music, remember tha t you are listening to th e rhythm s o f trillion s of electron s obeyin g thi s an d othe r bizarr e laws of quantum mechanics. But i f quantu m mechanic s wer e incorrect , the n al l o f electronics , including televisio n sets , computers , radios , stereo , an d s o on , woul d cease to function. (I n fact, if quantum theor y were incorrect, the atom s in ou r bodie s woul d collapse , an d w e woul d instantl y disintegrate . According t o Maxwell' s equations , th e electron s spinnin g i n a n ato m should los e thei r energ y withi n a microsecon d an d plung e int o th e nucleus. This sudden collaps e is prevented b y quantum theory. Thus th e fact tha t w e exis t i s livin g proo f o f th e correctnes s o f quantu m mechanics.) This als o mean s tha t ther e i s a finite , calculabl e probabilit y tha t "impossible" event s will occur. For example, I can calculat e the proba bility tha t I will unexpectedl y disappea r an d tunne l throug h th e eart h and reappea r i n Hawaii . (The tim e w e would have to wai t fo r suc h a n event t o occur , it should b e pointe d out , is longer tha n th e lifetim e of the universe. So we cannot use quantum mechanics to tunnel to vacation spots around th e world.) The Yang-Mills Field, Successo r t o Maxwel l Quantum physics , after a n initia l flush o f success in th e 1930 s and 1940 s unprecedented in th e histor y o f science, bega n t o run ou t o f steam by

118 U N I F I C A T I O

N I N TE N D I M E N S I O N S

the 1960s . Powerfu l ato m smasher s built to break up th e nucleus of the atom foun d hundred s o f mysterious particles among th e debris . Physicists, in fact , wer e deluged b y mountains o f experimental dat a spewing from thes e particl e accelerators . While Einstei n guesse d th e entir e framewor k o f genera l relativit y with onl y physical intuition, particle physicists were drowning in a mass of experimental dat a i n th e 1960s . As Enrico Fermi, one o f the builders of the atomi c bomb, confessed , "I f I could remembe r th e name s of all these particles , I would have become a botanist." 2 As hundreds o f "ele mentary" particle s were discovered i n the debris of smashed atoms, particle physicists would propose innumerable schemes to explain them , all without luck. So great were the numbe r o f incorrect schemes that it was sometimes said that the half-lif e o f a theory of subatomic physics is only 2 years. Looking back at all the blin d alleys and fals e start s in particle physics during that period, on e i s reminded o f the stor y of the scientist and th e flea. A scientist once trained a flea to jump whenever he rang a bell. Using a microscope, h e the n anesthetize d one o f the flea' s legs and ran g th e bell again. The fle a stil l jumped. The scientis t then anesthetize d anothe r leg and the n ran g th e bell. The fle a stil l jumped. Eventually, the scientist anesthetized more and more legs , each tim e ringing the bell , and eac h tim e recording tha t the fle a jumped. Finally, the flea had only one leg left. When the scientist anesthetized the las t leg and ran g th e bell , he foun d t o his surprise tha t th e fle a n o longer jumped. Then th e scientis t solemnly declared hi s conclusion, based o n irref utable scientifi c data: Fleas hear throug h thei r legs! Although high-energ y physicists have ofte n fel t lik e th e scientis t in that story, over th e decade s a consistent quantu m theor y of matter ha s slowly emerged. I n 1971 , the ke y development tha t propelled a unifie d description o f thre e o f th e quantu m force s (excludin g gravity ) an d changed th e landscap e of theoretical physics was made b y a Dutch graduate student , Gerard ' t Hooft , who was still in hi s twenties. Based o n th e analog y with photons , th e quant a o f light , physicist s believed tha t th e wea k and stron g force s were caused b y the exchang e of a quantum of energy, called the Yang-Mills field. Discovered by C. N. Yang and hi s studen t R . L. Mill s i n 1954 , th e Yang-Mill s fiel d i s a gen eralization o f the Maxwel l field introduced a century earlier t o describ e light, except tha t th e Yang-Mills field has many more component s an d

Quantum Heresy 11

9

can have an electrical charge (th e photon carrie s no electrica l charge). For the weak interactions, the quantum corresponding to the Yang-Mills field i s th e W particle, which can hav e charge +1 , 0 , an d — 1. For th e strong interactions, the quantum corresponding t o the Yang-Mills field, the "glue" that holds the protons and neutrons together, was christened the gluon. Although thi s genera l pictur e wa s compelling, th e proble m tha t bedeviled physicists in the 1950 s and 1960 s was that the Yang-Mills field is not "renormali/able" ; that is, it does not yield finite, meaningful quantities when applied t o simple interactions. This rendered quantu m theory useles s i n describin g th e wea k and stron g interactions . Quantum physics had hi t a brick wall. This problem arose because physicists, when they calculate what happens whe n tw o particles bump int o eac h other , us e somethin g calle d perturbation theory, which is a fancy way of saying they use cleve r approximations. For example , in Figure 5.2(a), we see what happens when an electron bumps into another weakly interacting particle, the elusive neutrino. A s a firs t guess , thi s interaction ca n b e describe d b y a diagra m (called a Feynman diagram) showin g that a quantu m o f th e wea k interactions, the Wparticle , is exchanged betwee n the electron an d th e neutrino. T o a first approximation, thi s gives us a crude bu t reasonabl e fi t to the experimental data . But according t o quantum theory , we must also add smal l quantum corrections to our first guess. To make our calculation rigorous, we must also add in the Feynman diagrams for all possible graphs, including ones that hav e "loops" in them , as in Figure 5.2(b) . Ideally, these quantu m corrections should b e tiny . After all , as we mentioned earlier , quantu m theory was meant to give tiny quantum corrections to Newtonian physics. But muc h t o th e horro r o f physicists , these quantu m corrections , o r "loop graphs, " instea d o f bein g small , were infinite . N o matte r ho w physicists tinkered with their equations or tried to disguise these infinite quantities, thes e divergences were persistently found in an y calculation of quantum corrections . Furthermore, th e Yang-Mill s fiel d ha d a formidabl e reputatio n o f being devilishly hard to calculate with, compared wit h th e simpler Maxwell field. There wa s a mythology surrounding the Yang-Mills field that held tha t i t was simply too complicate d fo r practica l calculations. Perhaps it was fortunate that't Hooft was only a graduate studen t and wasn't influenced b y the prejudices of more "seasoned " physicists. Using techniques pioneere d b y hi s thesi s adviser , Martinu s Veltman, ' t Hoof t showed tha t wheneve r w e hav e "symmetr y breaking" (whic h we wil l

Figure 5.2. (a) In quantum theory, when subatomic particles bump into one another, they exchange packets of energy, or quanta. Electrons and neutrinos interact by exchanging a quantum of the weak force, called the Vjparticle, (b) To calculate the complete interaction oj electrons and neutrinos, we must add up an infinite series of graphs, called Feynman diagrams, where the quanta are exchanged in increasingly complicated geometric patterns. This process of adding up an infinite series of Feynman graphs is called perturbation theory.

120

Quantum Heresy 12

1

explain later) , the Yang-Mills field acquires a mass but remain s a finite theory. He demonstrate d tha t th e infinitie s due t o the loo p graphs can all be cancele d o r shuffle d aroun d unti l the y become harmless . Almost 20 years after it s being propose d b y Yang and Mills , ' t Hoof t finally showed tha t th e Yang—Mill s fiel d i s a well-defined theory o f particle interactions. New s of 't Hooft's work spread lik e a flash fire . Nobe l laureate Sheldon Glashow remembers that when h e heard the news, he exclaimed, "Eithe r thi s guy's a total idiot, or he' s th e biggest genius t o hit physic s in years!" 3 Development s cam e thic k an d fast . A n earlie r theory of th e wea k interactions, propose d i n 196 7 b y Steven Weinber g and Abdus Salam, was rapidly shown to be the correct theory of the weak interactions. B y the mid-1970s , th e Yang-Mill s field was applied t o th e strong interactions . In th e 1970 s came th e stunning realization tha t th e secret of all nuclear matte r coul d b e unlocke d b y the Yang-Mills field. This wa s the missin g piec e i n th e puzzle . Th e secre t o f wood tha t bound matte r togethe r wa s the Yang-Mill s field , no t th e geometr y o f Einstein. It appeared a s though this , and no t geometry , was the centra l lesson of physics.

The Standar d Model Today, th e Yang-Mills field has made possibl e a comprehensive theor y of all matter. I n fact , w e are s o confident o f thi s theory tha t we blandly call it th e Standard Model. The Standar d Mode l ca n explai n ever y piece o f experimental dat a concerning subatomi c particles , u p t o about 1 trillion electro n volt s in energy (th e energ y create d b y acceleratin g a n electro n b y 1 trillio n volts). Thi s i s about th e limi t of th e ato m smasher s currentl y o n line . Consequently, i t is no exaggeratio n t o stat e that th e Standar d Mode l is the mos t successfu l theor y in th e histor y of science. According t o th e Standar d Model , eac h o f th e force s bindin g th e various particles i s created by exchanging differen t kind s of quanta. Le t us no w discuss each forc e separately, an d the n assembl e the m int o th e Standard Model . The Strong Force The Standar d Mode l states tha t the protons , neutrons , an d othe r heavy particles ar e no t fundamenta l particle s a t all , but consis t of some even tinier particles , calle d quarks. Thes e quarks , i n turn , com e i n a wid e

122 U N I F I C A T I O N I N T E N D I M E N S I O N

S

variety: three "colors " and si x "flavors." (Thes e name s have nothing to do with actual color s an d flavors. ) Ther e ar e als o th e antimatte r coun terparts o f the quarks , called antiquarks. (Antimatte r is identical to matter in all respects, except that , the charges ar e reversed an d i t annihilates on contac t wit h ordinar y matter. ) Thi s give s us a tota l o f 3 X 6 X 2 = 36 quarks. The quarks, in turn, are held together b y the exchange o f small packets of energy, called gluons. Mathematically, these gluon s ar e describe d by th e Yang-Mills field, which "condenses " int o a sticky , taffylik e sub stance tha t "glues " th e quark s permanentl y together . Th e gluo n field is so powerful an d bind s th e quark s so tightly together tha t th e quark s can never be torn away from one another . This is called quark confinement, and ma y explain why free quark s have never been seen experimentally . For example, the proton and neutron ca n be compared t o three steel balls (quarks ) hel d togethe r b y a Y-shaped string (gluon ) in th e shap e of a bola. Other strongly interactin g particles , suc h as the t r meson, can be compared t o a quark and an antiquark held together b y a single string (Figure 5.3). Obviously, by kicking this arrangement o f steel balls, we can se t this contraption vibrating . In the quantu m world, only a discrete set of vibrations is allowed. Each vibration o f this set of steel ball s or quark s corre sponds t o a different typ e of subatomic particle . Thu s thi s simple (bu t powerful) pictur e explain s the fac t tha t ther e ar e an infinite number o f strongly interacting particles . Thi s par t o f the Standar d Mode l describ ing th e stron g forc e i s called quantu m chromodynamic s (QCD)—tha t is, the quantu m theor y of the colo r force . The Wea k Forc e

In th e Standar d Model , th e wea k force govern s th e propertie s o f "leptons," suc h a s the electron , th e muon , an d th e ta u meson , an d thei r neutrino partners . Lik e th e othe r forces , th e lepton s interac t b y exchanging quanta , calle d W an d Z bosons . Thes e quant a ar e als o described mathematicall y b y th e Yang-Mill s field . Unlik e th e gluo n force, the force generated b y exchanging the Wan d Zbosons is too weak to bind the leptons into a resonance, so we do not see an infinite number of leptons emergin g fro m ou r ato m smashers . The Electromagneti c Force

The Standard Model includes the theory of Maxwell interacting with the other particles . This part o f the Standar d Mode l governing th e interac -

Quantum Heresy

123

Figure 5.3. Strongly interacting particles are actually composites of even smaller particles, called quarks, which are bound together by a taffylike "glue, " which is described by the Yang-Mills field. The proton and neutron are each made up of three quarks, while mesons are made up of a quark and an antiquark. don o f electron s an d ligh t i s called quantu m electrodynamic s (QED) , which ha s been experimentall y verifie d to be correct to within one par t in 1 0 million, technicall y making i t the mos t accurat e theor y in history . In sum, the fruitio n of 50 years of research, and severa l hundred millio n dollars i n governmen t funds , ha s given u s th e followin g picture o f subatomic matter : Al l matter consists o f quarks an d leptons, which interact by exchanging different types of quanta, described by the Maxwell and Yang—Mills

124 U N I F I C A T I O

N I N TE N D I M E N S I O N S

fields. I n one sentence , we have captured th e essenc e o f the pas t century of frustrating investigation into th e subatomi c realm. From thi s simple picture on e ca n derive , from pur e mathematic s alone , al l th e myria d and bafflin g propertie s o f matter . (Althoug h it al l seems s o eas y now, Nobel laureat e Steve n Weinberg , one o f th e creator s o f th e Standar d Model, once reflecte d o n ho w tortuous th e 50-yea r journey t o discover the mode l ha d been. H e wrote, "There's a long tradition o f theoretical physics, which by no mean s affected everyon e but certainl y affected me , that said th e strong interactions [were ] too complicated fo r the huma n mind.'"1)

Symmetry i n Physic s The detail s of the Standard Mode l are actually rather borin g and unimportant. The most interesting feature o f the Standard Model is that it is based o n symmetry . What ha s propelle d thi s investigatio n into matte r (wood) is that we can see the unmistakable sign of symmetry within each of these interactions . Quarks an d lepton s are not random , bu t occu r in definite pattern s i n the Standar d Model . Symmetry, of course, is not strictl y the provinc e o f physicists. Artists, writers, poets, and mathematician s have long admired th e beauty that is to be found in symmetry. To the poet William Blake, symmetry possessed mystical, even fearful qualities , as expressed in the poem "Tyger ! Tyger! burning bright": Tyger! Tyger ! burning brigh t In th e forest s of the nigh t What immorta l han d or eye Could fram e th y fearful symmetry? 5 To mathematicia n Lewi s Carroll , symmetr y represente d a familiar , almost playful concept . In the "The Hunting of the Snark," he captured the essenc e of symmetry when h e wrote : You boil i t in sawdust: You sal t it in glue : You condense it with locusts i n tape: Still keepin g on e principa l objec t in view —

To preserv e it s symmetrical shape .

Quantum Heresy 12

5

In othe r words , symmetry is the preservatio n o f th e shap e o f an objec t even afte r w e defor m o r rotat e it . Severa l kind s o f symmetrie s occu r repeatedly i n nature . Th e firs t i s the symmetr y of rotations an d reflec tions. Fo r example , a snowflak e remain s th e sam e i f we rotate i t b y 60 degrees. Th e symmetr y of a kaleidoscope, a flower, or a starfish i s of this type. We call these space-time symmetries, which are create d by rotating the objec t through a dimension of space or time . The symmetry of special relativit y is of thi s type , since i t describe s rotation s betwee n spac e and time . Another typ e of symmetry is created b y reshuffling a series of objects. Think of a shell game, where a huckster shuffles thre e shell s with a pe a hidden beneat h one of them. What makes the game difficult i s that ther e are man y ways in which the shell s can be arranged. I n fact, ther e ar e six different way s i n whic h thre e shell s can b e shuffled . Sinc e th e pe a i s hidden, thes e si x configurations are identica l t o th e observer . Mathe maticians like to give names to these various symmetries. The nam e fo r the symmetrie s of a shell game i s called S, 5, which describes th e numbe r of ways that thre e identica l object s may be interchanged. If w e replace th e shell s with quarks , the n th e equation s o f particle physics must remain th e same if we shuffle th e quarks among themselves. If we shuffle thre e colore d quark s and th e equation s remai n th e same , then we say that the equations possess something called SU(3) symmetry. The 3 represents th e fac t tha t we have three type s of colors, and th e SU stands fo r a specifi c mathematica l propert y o f th e symmetry. * W e say that ther e are thre e quarks in a multipkt. Th e quark s in a multiplet can be shuffle d amon g on e anothe r withou t changin g th e physic s o f th e theory. Similarly, the wea k force governs the propertie s o f two particles, the electron an d th e neutrino . Th e symmetr y that interchange s thes e par ticles, yet leaves the equatio n th e same , is called SU(2) . This means tha t a multiplet of the weak force contains an electron and a neutrino, which can b e rotate d int o eac h other . Finally , th e electromagneti c forc e ha s U ( l ) symmetry , which rotates th e component s o f the Maxwel l field into itself. Each o f these symmetrie s is simple an d elegant . However , the mos t controversial aspec t of the Standar d Mode l is that it "unifies" th e thre e fundamental force s by simply splicing al l three theorie s int o on e larg e symmetry, SU(3 ) X SU(2 ) X U(l) , whic h i s just th e produc t o f th e *SU stands for "special unitary" matrices—that is, matrices that have unit determinant and ar e unitary.

126 U N I F I C A T I O

N I N TE N D I M E N S I O N S

symmetries o f th e individua l forces . (Thi s can b e compare d t o assem bling a jigsaw puzzle. If we have three jigsaw pieces tha t don' t quite fit, we ca n alway s take Scotch tap e an d splic e the m togethe r b y hand. Thi s is how the Standar d Mode l is formed, by taping three distinc t multiplets together. Thi s ma y not b e aestheticall y pleasing, bu t a t leas t the thre e jigsaw puzzles now hang togethe r b y tape.) Ideally, on e migh t have expected tha t "th e ultimat e theory" would have al l the particle s insid e jus t a singl e multiplet . Unfortunately , the Standard Mode l has three distinc t multiplets, which cannot b e rotate d among on e another . Beyond th e Standard Model

Promoters o f the Standar d Mode l can say truthfully tha t it fits all known experimental data. They can correctly point out that there are no exper imental result s tha t contradic t th e Standar d Model . Nonetheless , nobody, not eve n its most fervent advocates, believes it is the final theory of matter . Ther e ar e severa l dee p reason s wh y it canno t b e th e fina l theory. First, the Standard Mode l does not describe gravity, so it is necessarily incomplete. When attempts are made to splice Einstein's theory with the Standard Model , th e resultin g theor y give s nonsensical answers. When we calculate , say , the probabilit y o f a n electro n bein g deflecte d b y a gravitational field , th e hybri d theor y give s u s a n infinit e probability , which make s n o sense . Physicist s say that quantu m gravit y is nonrenormalizabk, meanin g tha t i t canno t yiel d sensible , finit e number s t o describe simpl e physical processes. Second, an d perhaps most important, it is very ugly because it crudely splices three very different interaction s together. Personally, I think that the Standar d Mode l can be compare d t o crossing three entirel y dissimilar type s of animals, such a s a mule, an elephant , an d a whale. In fact , it i s so ugly an d contrive d tha t eve n it s creators ar e a bit embarrassed . They ar e th e firs t t o apologiz e fo r it s shortcomings an d admi t tha t i t cannot b e the final theory. This ugliness is obvious when we write down the details of the quarks and leptons . T o describ e ho w ugly the theor y is , let u s lis t th e various particles and force s within the Standar d Model : 1. Thirty-si x quarks, coming i n si x "flavors " an d thre e "colors, " and thei r antimatte r counterpart s t o describe th e stron g interactions

Quantum Heresy 12

7

2. Eigh t Yang-Mills fields to describ e th e gluons , which bind th e quarks 3. Fou r Yang-Mills fields to describe the weak and electromagneti c forces 4. Si x types of leptons to describe the weak interactions (including the electron , muon , ta u lepton , an d thei r respectiv e neutrin o counterparts) 5. A larg e numbe r of mysteriou s "Higgs " particle s necessar y to fudge th e masse s arid th e constant s describing th e particles 6. A t least 1 9 arbitrary constants tha t describ e th e masse s of th e particles and th e strength s of the various interactions. These 19 constants must be pu t i n b y hand; the y are no t determine d b y the theor y in an y way Worse, this long list of particles can be broken dow n into three "families" of quarks and leptons, which are practically indistinguishable from one another . In fact , thes e thre e families o f particles appear t o be exac t copies o f one another , giving a threefold redundanc y in the numbe r of supposedly "elementary " particles (Figure 5.4). (I t is disturbing t o realize tha t we now have vastly mor e "elementary " particle s tha n th e tota l number o f subatomi c particle s tha t wer e discovere d b y th e 1940s . I t makes on e wonde r ho w elementar y thes e elementar y particle s reall y are.) The uglines s of th e Standar d Mode l ca n b e contraste d t o th e sim plicity o f Einstein' s equations , i n whic h everythin g was deduced fro m first principles. To understand th e aestheti c contrast between th e Stan dard Mode l an d Einstein' s theory o f general relativity , w e must realize that when physicists speak of "beauty" in their theories, they really mean that thei r theory possesses at least two essential features: 1. A unifying symmetr y 2. Th e abilit y t o explai n vas t amounts o f experimenta l dat a wit h the mos t economical mathematical expressions The Standar d Mode l fails o n bot h counts . It s symmetry, as we have seen, i s actually formed b y splicin g three smalle r symmetries , one fo r each o f the thre e forces. Second, th e theor y is unwieldy and awkwar d in form. I t i s certainl y no t economica l b y any means . Fo r example , Ein stein's equations , writte n ou t i n thei r entirety , are onl y about a n inc h long and wouldn't even fill up on e lin e of this book. From thi s one lin e of equations , w e can g o beyon d Newton' s laws an d deriv e th e warpin g

128

UNIFICATION I N TEN DIMENSION S

Figure 5.4. In the Standard Model, the first generation of particles consists of the "up" and "down" quark (in three colors, with their associated antipartides) and the electron and neutrino. The embarrassing feature of the Standard Model is that there are three generation of such particles, each generation being nearly an exact copy of the previous generation. It's hard to believe that nature would be so redundant as to create, at a fundamental level, three identical copies of particles.

of space, the Big Bang, and other astronomically important phenomena . However, just t o writ e dow n th e Standar d Mode l i n it s entirety woul d require two-third s of this page and would look like a blizzard of complex symbols. Nature, scientists like to believe, prefers economy in its creations an d always seem s t o avoi d unnecessar y redundancie s i n creatin g physical ,

Quantum Heresy 12

9

biological, an d chemica l structures . When natur e create s pand a bears , protein molecules, or black holes, it is sparing in its design. Or, as Nobel laureate C . N. Yang once said , "Natur e seem s t o tak e advantag e o f th e simple mathematica l representation s o f th e symmetr y laws. When on e pauses t o conside r th e eleganc e an d th e beautifu l perfectio n o f th e mathematical reasonin g involve d and contras t i t with th e comple x an d far-reaching physica l consequences , a dee p sens e o f respec t fo r th e power o f th e symmetr y laws neve r fail s t o develop." 6 However , a t th e most fundamenta l level , we now fin d a gross violation o f this rule. Th e existence o f three identica l families, eac h on e wit h a n od d assortmen t of particles, is one o f the most disturbing features of the Standard Model, and raise s a persisten t proble m fo r physicists : Shoul d th e Standar d Model, the mos t spectacularly successful theor y in the histor y of science, be throw n out just because i t is ugly?

Is Beaut y Necessary ? I onc e attende d a concert i n Boston , where people wer e visibly moved by th e powe r an d intensit y of Beethoven's Nint h Symphony . After th e concert, wit h th e ric h melodie s still fres h i n m y mind, I happened t o walk past the empt y orchestra pit , where I noticed som e people starin g in wonder a t the shee t music left b y the musicians. To th e untraine d eye , I thought, th e musica l score o f even the mos t moving musica l piece mus t appea r t o b e a ra w mass o f unintelligibl e squiggles, bearin g mor e resemblanc e t o a chaoti c jumble o f scratches than a beautiful work of art. However, to the ea r o f a trained musician , this mas s of bars , clefs , keys , sharps, flats , an d note s come s aliv e an d resonates i n the mind. A musician can "hear" beautiful harmonies an d rich resonances by simply looking at a musical score . A sheet of music, therefore, i s more tha n jus t the su m o f its lines. Similarly, it would b e a disservice to defin e a poem a s "a shor t col lection o f word s organize d accordin g t o som e principle. " No t onl y is the definitio n sterile, bu t i t i s ultimately inaccurate becaus e i t fail s t o take into account the subtle interaction betwee n the poem and the emotions tha t i t evoke s in th e reader . Poems , becaus e the y crystallize an d convey th e essenc e o f th e feeling s and image s o f th e author , hav e a reality much greate r tha n th e word s printed o n a sheet of paper. A few short words of a haiku poem, for example, may transport th e reader into a new realm o f sensations and feelings.

130 U N I F I C A T I O I M I N T E N D I M E N S I O N

S

Like musi c o r art , mathematica l equations ca n hav e a natura l pro gression an d logi c tha t ca n evok e rare passion s in a scientist. Although the lay public considers mathematica l equations t o be rather opaque, t o a scientis t an equatio n i s very much lik e a movemen t i n a large r symphony. Simplicity. Elegance. These ar e the qualities that have inspired some of the greatest artists to create their masterpieces, and they are precisel y the same qualities that motivate scientists to search for the laws of nature. Like a wor k o f art o r a hauntin g poem , equation s hav e a beaut y an d rhythm all their own. Physicist Richard Feynma n expressed thi s when he said, You can recognize truth by its beauty and simplicity . When you get it right, it i s obvious that it is right—at least if you hav e any experience—becaus e usually what happens is that mor e come s ou t tha n goe s in . ... Th e inex perienced, th e crackpots , and peopl e like that, make guesses that are simple, bu t yo u ca n immediatel y see tha t the y ar c wrong , s o tha t doe s no t count. Others , th e inexperience d students , mak e guesse s tha t ar e ver y complicated, and i t sort of looks as if it is all right , bu t I know it is not tru e because th e trut h alway s turns out t o be simpler than you thought. 7

The Frenc h mathematicia n Henr i Poincar e expresse d i t even mor e frankly whe n h e wrote , "The scientis t does no t stud y Nature because i t is useful ; h e studie s i t becaus e h e delight s i n it , an d h e delight s i n i t because i t is beautiful. If Nature were not beautiful , it would not be worth knowing, and i f Nature were not worth knowing, life would not be worth living." I n som e sense , th e equation s o f physics are lik e the poem s o f nature. The y are short an d ar e organized accordin g t o some principle , and th e most beautiful of them convey the hidden symmetries of nature. For example , Maxwell' s equations, w e recall, originall y consisted of eight equations . These equations are not "beautiful. " The y do not pos sess much symmetry . In thei r origina l form, they are ugly , but the y are the bread and butte r o f every physicist or engineer wh o has ever earned a living working with radar, radio, microwaves , lasers, or plasmas. These eight equation s ar e wha t a tor t i s to a lawye r o r a stethoscop e i s to a doctor. However , when rewritte n usin g tim e a s th e fourt h dimension , this rather awkward set of eigh t equations collapse s into a single tenso r equation. Thi s i s what a physicis t calls "beauty," becaus e bot h criteri a are no w satisfied . B y increasing th e numbe r o f dimensions , w e reveal the true , tour-dimensional symmetr y of the theory and ca n now explain vast amounts o f experimental dat a with a single equation .

Quan turn Heresy 13

1

As we have repeatedly seen, the addition of higher dimensions causes the law s of nature t o simplify . One o f the greates t mysteries confronting science toda y is the expla nation o f th e origi n o f thes e symmetries , especially in th e subatomi c world. When ou r powerfu l machine s blow apart th e nucle i of atoms by slamming the m wit h energie s beyon d 1 trillion electro n volts , w e find that th e fragment s ca n b e arrange d accordin g t o thes e symmetries . Something rar e an d preciou s i s unquestionabl y happening whe n we probe down to subatomic distances. The purpos e o f science, however, is not t o marvel at the eleganc e of natural laws, but t o explain them. The fundamental problem facing subatomic physicist s is that, historically , w e had n o ide a o f wh y these symmetries were emerging in our laboratorie s and ou r blackboards . And her e i s precisely why the Standar d Mode l fails . N o matte r ho w successful th e theor y is , physicist s universall y believe tha t i t mus t b e replaced b y a highe r theory . It fail s bot h "tests " fo r beauty. I t neithe r has a single symmetry group no r describes the subatomic world economically. But more important , the Standard Mode l does not explai n where these symmetries originally came from. They are just spliced together by fiat, without any deeper understanding o f their origin .

GUIs

Physicist Ernes t Rutherford , who discovere d th e nucleu s of th e atom , once said, "All science is either physic s or stam p collecting." 8 By this, he mean t that science consists of two parts. The firs t i s physics, which is based o n th e foundatio n o f physical laws or principles. The second i s taxonom y ("bu g collecting " o r stam p collecting) , whic h is giving erudite Greek names for objects you know almost nothing abou t based o n superficial similarities. In this sense, the Standard Mode l is not real physics ; it i s more lik e stam p collecting , arrangin g th e subatomi c particles according to some superficial symmetries, but without the vaguest hint of where the symmetrie s come from . Similarly, when Charles Darwin named hi s boo k O n the Origin o f Species, he was going far beyond taxonomy by giving the logica l explanation for th e diversit y o f animal s i n nature . Wha t i s neede d i n physic s is a counterpart o f thi s book, t o b e calle d O n th e Origin o f Symmetry, whic h explains the reason s why certain symmetries are found in nature. Because th e Standar d Mode l is so contrived, over the year s attempts have bee n mad e t o g o beyond it , with mixe d success . One prominen t

132 U N I F I C A T I O N I N T E N D I M E N S I O N

S

attempt wa s called th e Gran d Unifie d Theor y (GUT) , popular i n th e late 1970s , which tried t o unite the symmetrie s of the strong , weak, and electromagnetic quanta by arranging them into a much larger symmetry group [fo r example, SU(5) , O(10), or E(6)] , Instead of naively splicing the symmetry groups of the three forces, GUTs tried to start with a larger symmetry that required fewe r arbitrary constants and fewer assumptions. GUTs vastl y increase d th e numbe r o f particle s beyon d th e Standar d Model, but th e advantag e was that th e ugl y SU(3) X SU(2) X U ( l) was now replaced b y a single symmetry group. Th e simples t of these GUTs, called SU(5), used 2 4 Yang-Mills fields, but a t least all these Yang-Mills fields belonged t o a single symmetry, not thre e separate ones . The aestheti c advantage o f the GUT s was that the y put th e strongl y interacting quarks and th e weakl y interacting leptons on th e same foot ing. I n SU(5) , fo r example , a multiple ! of particle s consiste d o f thre e colored quarks , an electron , and a neutrino. Unde r an SU(5 ) rotation , these five particles could rotat e int o one anothe r withou t changing th e physics. At first, GUTs were met wit h intens e skepticism, because th e energ y at whic h th e thre e fundamenta l force s wer e unifie d wa s around 10 15 billion electron volts, just a bit smaller than th e Planc k energy. This was far beyon d th e energ y of any atom smashe r o n th e earth , an d tha t was discouraging. However , physicists gradually warmed u p t o th e ide a o f GUTs when i t was realized tha t the y made a clear, testabl e prediction : the deca y of the proton . We recall that in the Standar d Model , a symmetry like SU(3) rotate s three quark s int o on e another ; tha t is , a multiple ! consist s of thre e quarks. Thi s mean s tha t eac h o f th e quark s can tur n int o on e o f th e other quarks under certain conditions (suc h as the exchange of a YangMills particle). However, quarks cannot tur n int o electrons . The multi plets d o no t mix . But in SU(5 ) GUT , ther e ar e five particles withi n a multiple! that ca n rotat e int o on e another : three quarks , the electron , and the neutrino. This means that one can, under certain circumstances, turn a proton (mad e of quarks) into an electron or a neutrino. In othe r words, GUT s sa y that th e proton , whic h was long hel d t o b e a stabl e particle with an infinit e lifetime , i s actually unstable. In principle, it also means tha t al l atom s i n th e univers e wil l eventuall y disintegrate int o radiation. I f correct , i t mean s tha t th e chemica l elements , whic h ar e taught in elementary chemistry classes to be stable, are actually all unstable. This doesn't mea n tha t we should expec t th e atom s in ou r bod y to disintegrate int o a burs t o f radiatio n anytim e soon . Th e tim e fo r th e

Quantum Heresy 13

3

proton t o deca y into lepton s wa s calculated t o be o n th e orde r o f 10 31 years, fa r beyon d th e lifetim e o f th e univers e (1 5 t o 2 0 billion years). Although thi s tim e scal e wa s astronomically long , thi s didn' t faz e th e experimentalists. Since an ordinary tan k of water contains an astronom ical amount o f protons, ther e i s a measurable probabilit y that some proton withi n the tan k will decay , even i f the proton s o n the average decay on a cosmological tim e scale. The Searc h for Proto n Deca y

Within a fe w years, thi s abstrac t theoretica l calculatio n wa s put t o th e test: Several expensive, multimillion-dollar experiments were conducted by severa l group s o f physicist s around th e world . The constructio n o f detectors sensitiv e enough t o detect proton deca y involved highly expensive and sophisticate d techniques . First, experimentalists needed to construct enormous vat s in which to detect proto n decay . Then they had t o fill the vat s with a hydrogen-rich flui d (suc h as water or cleanin g fluid ) that ha d bee n filtere d with specia l technique s i n orde r t o eliminate all impurities an d contaminants . Mos t important , the y the n ha d t o bur y these giganti c tank s dee p i n th e eart h t o eliminat e an y contaminatio n from highl y penetrating cosmi c rays. And finally , the y had t o construc t thousands o f highly sensitive detectors t o record th e fain t track s of subatomic particle s emitted fro m proto n decay. Remarkably, by the late 1980s six gigantic detectors were in operatio n around th e world , such a s the Kamiok a detector i n Japan an d th e 1M B (Irvine, Michigan , Brookhaven ) detecto r nea r Cleveland , Ohio . The y contained vas t amounts o f pure flui d (suc h as water) rangin g i n weight from 6 0 t o 3,30 0 tons . (Th e 1M B detector , fo r example , i s the world' s largest and i s contained i n a huge 20-mete r cub e hollowe d ou t o f a salt mine underneat h Lak e Erie. Any proton tha t spontaneously decaye d i n the purifie d water would produce a microscopic burs t of light, which in turn woul d be picked up b y some o f the 2,04 8 photoelectric tubes. ) To understand ho w these monstrou s detectors ca n measure th e proton lifetime, by analogy think of the American population. W e know that the averag e American ca n expec t t o live on th e orde r of 70 years. However, we don't have to wait 70 years to fin d fatalities . Because ther e ar e so man y Americans, i n fac t mor e tha n 25 0 million , w e expect t o fin d some America n dyin g every few minutes. Likewise , the simples t SU(5) GUT predicted tha t the half-life of the proton should be about 10 29 years; that is , afte r 10 29 years , half o f th e proton s i n th e univers e wil l hav e

134 U N I F I C A T I O N I N T E N D I M E N S I O N

S

decayed.* (B y contrast, thi s is about 1 0 billion billion times longer tha n the lif e o f th e univers e itself. ) Althoug h thi s seem s lik e a n enormou s lifetime, these detectors shoul d have been abl e to see these rare, fleeting events simpl y because ther e wer e so many protons i n th e detector . I n fact, eac h to n o f water contains ove r 10 29 protons. Wit h that many protons, a handful o f protons wer e expected t o decay every year. However, n o matte r ho w long th e experimentalist s waited, they saw no clear-cu t evidenc e o f an y proto n decays . At present , i t seem s tha t protons mus t have a lifetime larger tha n 10 32 years, which rules out th e simpler GUTs , but stil l leaves open th e possibilit y of more complicate d GUTs. Initially, a certain amoun t o f excitement ove r the GUT s spilled ove r into the media . The ques t for a unified theory of matter an d th e searc h for th e deca y o f the proto n caugh t th e attentio n o f science producer s and writers . Public television' s "Nova" devoted severa l shows to it, an d popular book s and numerou s article s in science magazines were written about it . Nevertheless, the fanfar e die d ou t by the late 1980s . No matte r how lon g physicist s waited fo r th e proto n t o decay , th e proto n simpl y didn't cooperate . After ten s of millions of dollars were spent by various nations lookin g for this event, it has not ye t been found . Public interest in th e GUT s began t o fizzle . The proto n may still decay, and GUT s may still prove t o be correct , but physicist s are no w much mor e cautiou s abou t toutin g th e GUT s as the "fina l theory, " fo r severa l reasons . A s with th e Standar d Model , GUTs mak e n o mentio n o f gravity . If we naivel y combine GUT s with gravity, the theor y produce s number s tha t are infinit e an d henc e mak e no sense. Like the Standard Model , GUTs are nonrenormalizable. Moreover, th e theor y i s defined a t tremendou s energies , wher e we certainly expected gravitationa l effect s t o appear . Thu s th e fac t tha t gravit y is missing in th e GU T theory is a serious drawback. Furthermore, i t is also plagued b y the mysteriou s presence o f three identical carbo n copie s o r families o f particles. An d finally, the theor y coul d no t predic t th e fundamental constants , such a s th e quar k masses . GUT s lacke d a large r physical principl e tha t woul d fix the quar k masse s an d th e othe r con stants from first principles. Ultimately, it appeared tha t GUTs were also stamp collecting . The fundamenta l proble m wa s that the Yang-Mills field was not suf* Half-life i s the amount of time it takes for hal f of a substance to disintegrate. After two half-lives, onl y one-quarter of the substanc e remains.

Quantum Heresy 13

5

ficient to provide the "glue" to unite all four interactions. Th e world of wood, as described b y the Yang-Mills field, was not powerfu l enough t o explain th e world of marble. After hal f a century of dormancy, the tim e had com e for "Einstein' s revenge."

6

Einstein's Revenge Supersymrnetry is the ultimat e proposal for a complete unifi cation o f all particles. Abdus Sala m

The Resurrectio n of Kaluza-Klei n

I

T'S bee n calle d "th e greates t scientifi c proble m o f al l time." Th e press ha s dubbe d i t th e "Hol y Grail " o f physics , th e ques t t o unite the quantu m theor y with gravity , thereb y creatin g a Theor y o f Every thing. Thi s i s the proble m tha t ha s frustrate d th e fines t mind s o f th e twentieth century. Without question, the person who solves this problem will win the Nobe l Prize. By th e 1980s , physic s was reaching a n impasse . Gravit y alon e stub bornly stoo d apar t an d aloo f fro m th e othe r thre e forces . Ironically , although th e classica l theory o f gravit y was the firs t t o b e understoo d through th e wor k of Newton, the quantu m theor y of gravity was the last interaction t o be understoo d b y physicists. All th e giant s of physics have had thei r crack at this problem, and al l have failed . Einstein devoted th e las t 30 years of hi s lif e t o hi s unifie d field theory. Even the great Werner Heisenberg , on e o f the founders of quantum theory , spent th e las t years of his life chasin g after hi s version of a unified theor y of fields, even publishing a book o n th e subject . I n 1958, Heisenber g eve n broadcas t o n radi o tha t h e an d hi s colleagu e

136

Einstein's Revenge 13

7

Wolfgang Pauli had finall y succeeded i n finding the unified field theory, and tha t onl y th e technica l detail s wer e missing . (Whe n th e pres s go t wind of this stunning declaration, Pauli was furious tha t Heisenberg ha d prematurely made tha t announcement. Paul i send a letter t o his collaborator, consistin g of a blank sheet of paper wit h th e caption , "This is to show th e worl d tha t I ca n pain t lik e Titian . Onl y technica l detail s ar e missing."1) Later tha t year , when Wolfgan g Pauli finally gave a lectur e o n th e Heisenberg-Pauli unified field theory, many eager physicists were in th e audience, anxiou s t o hea r th e missin g details . When h e wa s finished, however, the tal k received a mixed response. Niel s Bohr finally stood u p and said , "W e ar e al l agree d tha t you r theor y i s crazy . Th e questio n which divides us is whether it is crazy enough."2 In fact, so many attempts have bee n mad e a t th e "fina l synthesis " tha t i t has create d a backlash of skepticism . Nobel laureat e Julian Schwinge r has said , "It' s nothin g more tha n anothe r sympto m of th e urg e tha t afflict s ever y generatio n of physicist—the itch to have all the fundamental questions answered in their ow n lifetimes." 3 However, by the 1980s , th e "quantu m theor y of wood," after a half century o f almos t uninterrupte d success , was beginning t o ru n ou t o f steam. I ca n vividl y remembe r th e sens e o f frustratio n amon g jaded young physicists during thi s period. Everyon e sensed tha t th e Standar d Model was being killed by its own success. It was so successful tha t every international physic s conference seemed lik e just another rubbe r stam p of approval. All the talk s concerned findin g ye t another borin g experi mental succes s fo r th e Standar d Model . A t on e physic s conference , I glanced bac k at th e audienc e an d foun d tha t hal f o f the m wer e slowly do/ing of f to sleep ; th e speake r wa s droning o n wit h char t afte r char t showing ho w th e lates t dat a coul d b e fi t accordin g t o th e Standar d Model. I felt like the physicist s at the tur n o f the century . They, too, seeme d to be facin g a dead end . The y spent decade s tediousl y filling up table s of figures for th e spectra l line s of various gases, or calculatin g the solu tions to Maxwell's equations for increasingly complicated metal surfaces. Since th e Standar d Mode l ha d 1 9 free parameter s tha t coul d b e arbi trarily "tuned " t o an y value, like th e dial s o n a radio , I imagined tha t physicists woul d spen d decade s findin g th e precis e value s o f al l 1 9 parameters. The tim e had com e fo r a revolution. What beckoned th e nex t gen eration o f physicists was the worl d of marble . Of course, several profound problem s stoo d i n th e wa y of a genuin e

138 U N I F I C A T I O N 1 N T E N D I M E N S I O N

S

quantum theor y o f gravity . On e proble m wit h constructing a theory of gravity i s that th e forc e i s so maddeningly weak . For example , i t take s the entir e mas s of the eart h t o keep piece s o f paper o n m y desk. However, by brushing a comb throug h m y hair, I can pick up thes e piece s of paper, overwhelmin g the force of the planet earth . Th e electron s in my comb are more powerful than th e gravitational pull of the entire planet . Similarly, if I were to try to construct an "atom " with electron s attracted to th e nucleu s b y the gravitationa l force , an d no t th e electrica l force , the ato m woul d be th e siz e of the universe. Classically, we see that the gravitationa l force is negligible compare d with th e electromagneti c force , and henc e i s extraordinarily difficult t o measure. Bu t i f we attempt t o write down a quantum theor y of gravity, then th e table s are turned. Th e quantu m correction s du e t o gravity are on th e orde r o f th e Planc k energy , o r 10 19 billio n electro n volts , fa r beyond anythin g achievabl e o n th e plane t eart h i n thi s century . Thi s perplexing situation deepens when we try to construct a complete theory of quantum gravity . We recall that when quantum physicist s try to quantize a force, they break i t up int o tin y packets of energy, called quanta . If you blindly try to quantiz e th e theor y of gravity, you postulate tha t i t functions by the exchang e o f tiny packets of gravity, called gravitom. The rapid exchang e o f gravitons between matter is what binds them togethe r gravitationally. In thi s picture, what holds us to th e floor , an d keep s us from flyin g into outer spac e a t a thousand mile s per hour, is the invisible exchange o f trillion s o f tin y graviton particles . Bu t whenever physicists tried t o perfor m simpl e calculation s t o calculat e quantum correction s to Newton's and Einstein' s laws o f gravity, they found tha t th e resul t is infinite, whic h is useless. For example, le t us examine what happens when two electrically neutral particles bump into each other . To calculate the Feynman diagrams for thi s theory , we have t o mak e a n approximation , s o we assume tha t the curvatur e o f space-tim e i s small, an d henc e th e Rieman n metri c tensor i s close to 1 . For a first guess, we assume that space-time is close to bein g flat , no t curved , s o we divide th e component s o f th e metri c tensor a s g-,, = 1 + h n, where 1 represents fla t spac e i n our equation s and /} , | is the graviton field. (Einstein, of course, was horrified that quantum physicist s would mutilat e his equation s i n thi s way by breaking u p the metri c tensor . Thi s i s like takin g a beautifu l piec e o f marbl e an d hitting it with a sledge hammer in order t o break it.) After this mutilation is performed , w e arrive a t a conventional-lookin g quantu m theory . I n Figure 6.1 (a), we see that the tw o neutral particles exchange a quantum of gravity, labeled b y the fiel d h .

Figure 6.1. (a) In quantum theory, a quantum of the gravitational force, labeled h, i s called th e graviton, which i s formed b y breaking up Riemann's metric. In this theory, objects interact by exchanging this packet of gravity. In this way, we completely lose the beautiful geometric picture of Einstein, (b) Unfortunately, all the diagrams with loops in them are infinite, which has prevented a unification of gravity with the quantum theory for the past half-century. A quantum theory of gravity that unites it with the other forces is the Holy Grail of physics.

139

140 U N I F I C A T I O N I N T E N D I M E N S I O N

S

The proble m arise s whe n we sum ove r al l loop diagrams : W e fin d that they diverge, as in Figure 6. 1 (b). For the Yang-Mills field, we could use clever sleight-of-hand tricks to shuffl e aroun d thes e infinit e quanti ties until they either cance l or are absorbed int o quantities that can't be measured. However , it can be shown that the usual renormalization prescriptions fai l completel y when we apply them t o a quantu m theor y of gravity. I n fact , th e effort s o f physicists over hal f a century to eliminate or absor b thes e infinitie s ha s bee n i n vain . I n othe r words , th e brute force attemp t t o smash marble int o piece s faile d miserably. Then, in th e earl y 1980s, a curious phenomenon occurred . Kaluza — Klein theory , w e recall , ha d bee n a dorman t theor y fo r 6 0 years. But physicists were s o frustrate d i n thei r attempt s t o unif y gravit y with th e other quantum forces that they began t o overcome their prejudice about unseen dimension s and hyperspace . They were ready for an alternative, and tha t was Kaluza-Klein theory . The late physicist Heinz Pagels summarized this excitement over the re-emergence of Kaluza—Klein theoiy : After th e 1930s , the Kaluza-Klei n idea fell out o f favor, and fo r many years it lay dormant. But recently , as physicists searched ou t ever y possible avenue fo r th e unificatio n o f gravity with other forces, i t has again sprung to prominence. Today, in contras t with th e 1920s , physicists are challenge d to d o mor e tha n unif y gravit y wit h just electromagnetism—the y want t o unify gravit y with th e wea k an d stron g interactions as well. This requires even mor e dimensions, beyond the fifth.'1 Even Nobe l laureat e Steve n Weinberg wa s swept up b y the enthusi asm generated b y Kaluza-Klein theory. However , there were still physicists skeptica l o f th e Kaluza-Klei n renaissance . Harvard' s Howar d Georgi, remindin g Weinber g ho w difficul t i t is to measur e experimen tally these compactifie d dimension s tha t have curled up , compose d th e following poem : Steve Weinberg, returning from Texa s brings dimensions galore to perplex us But the extr a ones all are rolled up in a ball so tiny it never affects us. 5 Although Kaluza-Klei n theor y wa s stil l nonrenormalizable , wha t sparked th e intens e interes t i n the theor y was that it gave the hope of a

Einstein's Revenge 14

1

theory made of marble. Turning th e ugly, confused jumble of wood into the pure , elegan t marbl e of geometry was, of course, Einstein' s dream . But in the 1930 s and 1940s , almost nothing was known about the natur e of wood . However , b y th e 1970s , th e Standar d Mode l ha d finall y unlocked th e secret of wood: that matter consists of quarks and lepton s held togethe r b y the Yang-Mill s field , obeyin g th e symmetr y SU(3) X SU(2) X U(l). The proble m was how to derive these particle s and mysterious symmetries from marble . At first, that seeme d impossible . After all , these symmetrie s are th e result of interchanging point particles among one another . I f N quarks within a multiple! are shuffled amon g on e another , the n th e symmetry is SU(JV) . These symmetrie s seemed t o be exclusivel y the symmetrie s of wood, not marble . What did SU(AJ ) hav e to do with geometry? Turning Wood int o Marbl e

The first small clue came in th e 1960s , when physicists found, much t o their delight, tha t ther e is an alternativ e way in which to introduce symmetries into physics. When physicists extended th e old five-dimensional theory of Kaluza-Klein t o N dimensions, the y realized that ther e i s th e freedom t o impose a symmetry on hyperspace. When the fifth dimension was curled up , the y saw that the Maxwell field popped ou t o f Riemann's metric. Bu t whe n N dimension s wer e curle d up , physicist s found th e celebrated Yang-Mills field, the key to the Standard Model , popping ou t of their equations ! To se e ho w symmetrie s emerg e fro m space , conside r a n ordinar y beach ball . I t ha s a symmetry: We can rotat e i t around it s center, an d the beac h bal l retain s it s shape . Th e symmetr y of a beac h ball , o r a sphere, i s calle d O(3) , o r rotation s i n thre e dimension . Similarly , in higher dimensions , a hypersphere can also be rotated aroun d it s center and maintai n it s shape. Th e hyperspher e has a symmetry called O(N). Now consider vibrating die beach ball . Ripples form on th e surface of the ball . If we carefully vibrate the beac h ball in a certain way, we can induce regula r vibration s o n i t tha t ar e calle d resonances. Thes e reso nances, unlik e ordinary ripples, can vibrate at only certain frequencies. In fact , i f w e vibrate the beac h bal l fast enough , w e can creat e musical tones of a definite frequency. These vibrations, in turn, can be cataloge d by the symmetr y O(3). The fac t tha t a membrane, lik e a beach ball , can induce resonanc e frequencies i s a common phenomenon. Th e vocal chords in our throat ,

142 U N I F I C A T I O N I N T E N D I M E N S I O N

S

for example , ar e stretche d membrane s tha t vibrate at definite frequencies, o r resonances , an d ca n thereb y produc e musica l tones . Anothe r example i s our hearing . Soun d wave s o f al l type s impinge o n ou r ear drums, which then resonat e a t definite frequencies. These vibrations are then turne d int o electrica l signal s tha t ar e sen t int o ou r brain , which interprets the m a s sounds . Thi s i s als o th e principl e behin d th e tele phone. The metalli c diaphragm containe d i n an y telephone i s set into motion b y electrical signals in the telephon e wire . This creates mechan ical vibrations or resonances i n the diaphragm, which in turn create th e sound wave s we hear o n th e phone . Thi s i s also th e principl e behin d stereo speaker s a s well a s orchestral drums . For a hypersphere , th e effec t i s the same . Lik e a membrane , i t ca n resonate a t various frequencies, which in tur n ca n b e determine d b y its symmetry O(N). Alternatively , mathematicians have dreame d u p mor e sophisticated surface s in highe r dimension s tha t ar e describe d b y complex numbers . (Comple x number s us e th e squar e roo t o f — 1, ; —1.) Then i t i s straightforward t o sho w that th e symmetr y corresponding t o a comple x "hypersphere " i s SU(A/). The ke y point i s now this : If the wav e functio n o f a particle vibrates along thi s surface, i t will inheri t thi s SU(A/ ) symmetry . Thus th e mysterious SU(jV ) symmetrie s arisin g i n subatomi c physic s ca n no w be see n as by-products o f vibrating hyperspace! I n other words, we now have an expla nation fo r th e origi n o f th e mysteriou s symmetrie s of wood: The y ar e really th e hidde n symmetrie s coming fro m marble . If we no w tak e a Kaluza—Klei n theor y defined i n 4 + A ^ dimensions and the n cur l up N dimensions, we will find that the equations split into two pieces. The first piece is Einstein's usual equations, which we retrieve as expected. Bu t the secon d piec e wil l not b e th e theor y of Maxwell. We find tha t th e remainde r i s precisely the Yang-Mills theory, which forms the basi s o f al l subatomi c physics ! Thi s i s th e ke y to turnin g th e sym metries o f wood int o th e symmetrie s of marble. At first, i t seems almos t mystica l that th e symmetrie s of wood, which were discovere d painfull y b y tria l an d error—tha t is , b y painstakingly examining th e debri s fro m ato m smashers—emerg e almos t automati cally from higher dimensions . It is miraculous that the symmetries found by shufflin g quark s an d lepton s amon g themselve s shoul d aris e fro m hyperspace. A n analog y ma y hel p u s understan d this . Matte r ma y b e likened t o clay, which is formless and lumpy . Clay lacks any of the beau tiful symmetrie s that ar e inheren t i n geometri c figures . However , clay may be presse d int o a mold , whic h can hav e symmetries. For example , the mol d ma y preserve it s shape i f it is rotated by a certain angle . The n

Einstein's Revenge

143

the cla y wil l als o inheri t th e symmetr y of th e mold . Clay , lik e matter, inherits its symmetry because the mold, like space-time, ha s a symmetry. If correct, then this means that the strange symmetries we see among the quark s and leptons , which were discovered largely by accident over several decades , can no w be see n a s by-products of vibrations in hyperspace. For example, if the unseen dimensions have the symmetry SU(5), then w e can writ e SU(5 ) GU T as a Kaluza-Klein theory. This can also be seen fro m Riemann's metric tensor . W e recall that it resembles Faraday's field excep t tha t it has many more components . It can be arranged lik e the squares of a checkerboard. By separating out the fifth column and ro w of the checkerboard , we can split off Maxwell's field fro m Einstein' s field. Now perform th e sam e tric k with Kaluza Klein theory in ( 4 + N) -dimensional space. If you split off the JVcolumn s and row s from th e first four column s and rows , then you obtain a metric tensor tha t describes bot h Einstein' s theor y an d Yang-Mills theory . I n Figure 6.2, we have carved up the metric tensor of a (4 + N ) -dimensional

figure 6.2. If we go to the Nth dimension, then the metric tensor is a series of N 2 numbers that ca n b e arranged i n a n N X N black. B y slicing of f th e fift h and higher columns and rows, we can extract the Maxwell electromagnetic field and the Yang-Mills field. Thus, in one stroke, the hyperspace theory allows us to unify the Einstein field (describing gravity), the Maxwell field (describing the electromagnetic force), and the Yang-Mitts field (describing the weak and strong force). The fundamental forces fit together exactly like a jigsaw puzzle.

144 U N I F I C A T I O

N IN T E N D I M E N S I O N S

Kaluza-Klein theory , splittin g of f Einstein' s fiel d fro m th e Yang-Mill s field. Apparently, on e o f the first physicists to perfor m thi s reduction was University o f Texa s physicis t Bryce DeWitt, who ha s spen t man y years studying quantum gravity . Once this trick of splitting up th e metric ten sor was discovered, th e calculatio n for extractin g th e Yang—Mill s field is straightforward. DeWit t felt that extracting the Yang-Mills field from Ndimensional gravit y theory was such a simple mathematical exercise that he assigne d i t as a homework problem at the Le s Houches Physic s Summer School i n France i n 1963 . [Recently , it was revealed b y Peter Freun d that Oskar Klei n had independentl y discovere d th e Yang-Mills field in 1938, precedin g th e wor k of Yang, Mills, and other s b y several decades. In a conferenc e hel d i n Warsaw titled "Ne w Physica l Theories," Klein announced tha t he was able to generalize the work of Maxwell to include a higher symmetry, O(3). Unfortunately, because of the chaos unleashed by World Wa r I I an d becaus e Kaluza-Klei n theor y wa s buried b y th e excitement generate d b y quantum theory , thi s important wor k was forgotten. I t is ironic that Kaluza-Klein theor y was killed by the emergenc e of quantum theory , whic h i s now based o n th e Yang-Mill s field, which was first discovered b y analyzing Kaluza-Klein theory. In the excitemen t to develo p quantu m theory , physicist s had ignore d a centra l discovei y coming fro m Kaluza-Klein theory. ] Extracting th e Yang-Mills field out o f Kaluza-Klein theor y wa s only the firs t step . Althoug h th e symmetrie s o f wood coul d no w be see n a s arising from the hidden symmetrie s of unseen dimensions , the next step was t o creat e woo d itsel f (mad e o f quarks an d leptons ) entirel y out o f marble. Thi s next step would be calle d supergravity.

Supergravity Turning woo d int o marbl e stil l face d formidabl e problem s because , according t o th e Standar d Model , all particles ar e "spinning. " Wood , for example , we now know is made o f quarks and leptons . They, in turn, have 'A unit of quantum spi n (measure d i n unit s of Planck's constant fi. Particles with half-integra l spin ('A, ? 'A, %, and s o on ) ar e calle d fermions (named afte r Enric o Fermi, who first investigated thei r strang e proper ties). However , force s ar e describe d b y quanta wit h integra l spin . Fo r example, the photon, the quantum of light, has one unit of spin. So does the Yang-Mill s field . Th e graviton , th e hypothetica l packe t o f gravity , has tw o units of spin. They ar e calle d bosons (afte r th e India n physicist Satyendra Bose) .

Einstein's Revenge 14

5

Traditionally, quantu m theor y kep t fermion s an d boson s strictl y apart. Indeed , an y attempt t o tur n woo d int o marbl e woul d inevitably come t o grip s with th e fac t tha t fermion s an d boson s ar e world s apar t in thei r properties . Fo r example, SU(JV ) ma y shuffle quark s among on e another, bu t fermion s and boson s wer e never supposed t o mix. It came as a shock , therefore , whe n a ne w symmetry , called supersymmetry, wa s discovered, tha t di d exactl y that . Equation s tha t ar e supersymmetri c allow the interchang e o f a fermion with a boson an d stil l keep the equa tions intact . In othe r words, one multiple t of supersymmetry consists of equal number s o f boson s an d fermions . B y shuffling th e boson s an d fermions withi n th e sam e multiplet , th e supersymmetri c equation s remain th e same . This give s us the tantalizin g possibility of putting al l the particle s in the univers e int o on e multiplet ! A s Nobel laureat e Abdu s Sala m ha s emphasized, "Supersymmetr y i s th e ultimat e proposal fo r a complet e unification o f all particles. " Supersymmetry is based o n a new kind of number syste m that would drive any schoolteacher insane . Mos t of the operation s o f multiplication and divisio n that we take for grante d fai l fo r supersymmetry . For example, i f a and b are tw o "super numbers," the n a X b= —b X a . This, o f course, is strictly impossible for ordinary numbers. Normally, any schoolteacher would throw these supe r number s ou t th e window, because you can sho w that a X a = — a X a , or, i n othe r words , a X a = 0 . I f thes e were ordinar y numbers , the n thi s mean s tha t a = 0 , and th e numbe r system collapses. However, with super numbers, the syste m does not collapse; w e hav e th e rathe r astonishin g statemen t tha t a X a = 0 eve n when a + 0 . Although thes e supe r number s violat e almost everything we have learned abou t number s sinc e childhood, they can be show n to yield a self-consisten t and highl y nontrivia l system . Remarkably , a n entirely new system o f super calculu s can b e base d o n them . Soon, thre e physicist s (Daniel Freedman, Sergi o Ferrara, and Pete r van Nieuwenhuizen, at the Stat e University of New York at Stony Brook) wrote down the theor y of supergravity in 1976 . Supergravity was the firs t realistic attempt to construct a world made entirely of marble. In a supersymmetric theory, all particles have super partners , calle d sparticles. Th e supergravity theor y o f th e Ston y Broo k grou p contain s just tw o fields: the spin-tw o graviton fiel d (whic h i s a boson) an d it s spin-3/2 partner , called th e gravitino (whic h mean s "littl e gravity") . Sinc e thi s i s no t enough particle s to include the Standard Model , attempts were made t o couple th e theor y t o more complicate d particles. The simples t way to include matter i s to write down the supergravity theory i n 11-dimensiona l space . I n orde r t o writ e dow n th e supe r

146

UNIFICATION I N TEN DIMENSION S

Figure 6.3. Supergravity almost fulfills Einstein's dream of giving a purely geometric derivation of all the forces and particles in the universe. To see this, notice, that if we add supersymmetry to the Riemann metric tensor, the metric doubles in size, giving us the super Riemann metric. The new components of the super Riemann tensor correspond to quarks and leptons. By slicing the super Riemann tensor into its components, we find that it includes almost all the fundamental particles and forces in nature: Einstein's theory of gravity, the Yang-Mills and Maxwell fields, and the, quarks and leptons. But the fact that certain particles are missing in this picture forces us to go a more powerful formalism: superstring theory.

Kaluza-Klein theor y in 1 1 dimensions, one ha s t o increase th e compo nents within the Rieman n tenso r vastly , which now becomes th e supe r Riemann tensor . T o visualize how supergravity converts wood into marble, let us write down the metric tensor an d sho w how supergravity manages to fit the Einstei n field, the Yang-Mills field, and th e matte r fields into on e supergravit y fiel d (Figur e 6.3) . Th e essentia l featur e o f thi s diagram i s tha t matter , alon g wit h th e Yang-Mill s and Einstei n equations, i s no w include d i n th e sam e 11-dimensiona l supergravit y field. Supersymmetry i s th e symmetr y that reshuffle s th e woo d int o marbl e and vic e vers a withi n the supergravit y field. Thus the y are al l manifestations o f th e sam e force , th e superforce . Woo d n o longe r exist s as a single, isolated entity . It is now merged wit h marble , to form supermar ble (Figur e 6.4)!

Einstein's Revenge

147

Physicist Pete r va n Nieuwenhuizen , on e o f supergravity' s creators , was deepl y impresse d b y the implicatio n o f thi s super-unification . H e wrote that supergravity "may unif y gran d unifie d theories . . . with gravity, leading to a model with almost no fre e parameters. I t is the uniqu e theory wit h a local gauge symmetr y between fermions an d bosons . I t is the most beautiful gauge theor y known, so beautiful, in fact, that Nature should b e aware of it!"6 I fondl y remember attendin g an d givin g lectures a t man y o f thes e supergravity conferences. There was an intense, exhilarating feelin g that we were on th e verg e o f something important . At one meetin g i n Moscow, I remember well, a series of lively toasts were made to the continued success of the supergravit y theory. It seemed tha t we were finally o n th e verge of carrying out Einstein' s dream of a universe of marble afte r 60 years of neglect. Som e o f us jokingly called it "Einstein's revenge. " On April 29, 1980, when cosmologist Stephen Hawkin g assumed th e Lucasian Professorshi p (previousl y held b y som e o f th e immortal s o f physics, includin g Isaa c Newton and P . A. M. Dirac), he gav e a lectur e with th e auspiciou s titl e "I s th e En d i n Sigh t for Theoretical Physics?"

Figure 6.4, In supergravity, we almost get a unification of all the known forces (marble) with matter (wood). Like a jigsaw puzzle, they fit inside Riemann 's metric tensor. This almost fulfills Einstein's dream.

148 U N 1 F I C A T I O N I N T E N D I M E N S I O N

S

A studen t rea d fo r him : "[W] e hav e mad e a lot o f progress i n recen t years and, as I shall describe, ther e ar e some ground s fo r cautiou s optimism tha t we may see a complete theor y within the lifetim e of some o f those presen t here." Supergravity's fam e graduall y sprea d int o th e genera l publi c an d even bega n t o hav e a following among religiou s groups . Fo r example , the concept o f "unification" i s a central belief within the transcendental meditation movement . It s followers therefore publishe d a larg e poste r containing th e complet e equation s describin g 11-dimensiona l super gravity. Each term in the equation, the y claimed, represented somethin g special, suc h a s "harmony," "love, " "brotherhood, " an d s o on. (Thi s poster hangs on the wall of the theoretical institut e at Stony Brook. This is the first time that I am awar e of that an abstrac t equation fro m theo retical physics has inspired a following among a religious group!) Super Metri c Tensors

Peter van Nieuwenhuizen cuts a rather dashin g figure in physics circles. Tall, tanned , athleti c looking , and wel l dressed , h e look s more lik e an actor promotin g sunta n lotio n o n televisio n than on e o f th e origina l creators of supergravity. He i s a Dutch physicist who i s now a professor at Ston y Brook; h e wa s a student o f Veltman, as was 't Hooft , an d was therefore lon g interested i n the questio n of unification. He is one of the few physicist s I hav e eve r me t wit h a trul y inexhaustible capacit y for mathematical punishment . Workin g wit h supergravit y require s a n extraordinary amount o f patience. We recall that th e simple metric tensor introduced b y Riemann in the nineteenth centur y had only ten components. Riemann' s metri c tenso r ha s now been replace d b y the supe r metric tenso r o f supergravity , which ha s literall y hundreds o f compo nents. This is not surprising, since any theory that has higher dimensions and make s the clai m of unifying al l matter ha s to have enough compo nents to describe it, but this vastly increases the mathematical complexity of th e equations . (Sometime s I wonde r wha t Rieman n woul d think, knowing tha t afte r a centur y hi s metri c tenso r woul d blosso m int o a super metri c man y time s large r tha n anythin g a nineteenth-century mathematician coul d conceive.) The comin g of supergravity and super metric tensors has meant that the amoun t o f mathematic s a graduat e studen t mus t maste r ha s exploded withi n the pas t decade. A s Steven Weinberg observes , "Look what's happened wit h supergravity. The people who've been working on

Einstein's Revenge 14

9

it fo r th e pas t te n year s ar e enormousl y bright . Som e o f the m ar e brighter tha n anyon e I knew in m y early years."7 Peter i s not onl y a superb calculator , but als o a trendsetter. Becaus e calculations for a single supergravity equation ca n easily exceed a sheet of paper, h e eventually started using large, oversize artist's sketch boards. I went to hi s house on e day , and sa w how he operated . H e would start at the upper-left-han d corne r o f the pad, and star t writing his equations in hi s microscopic handwriting . He would then procee d t o work across and dow n th e sketc h pad unti l i t was completely filled , an d the n tur n the pag e an d star t again. This process would then g o on for hours, until the calculatio n wa s completed. Th e onl y time h e woul d eve r b e inter rupted wa s when h e inserte d hi s penci l int o a nearb y electri c penci l sharpener, an d the n withi n second s h e woul d resum e hi s calculation without missing a symbol. Eventually, he would store thes e artist' s notepads on his shelf, as though they were volumes of some scientific journal. Peter's sketc h pad s graduall y became notoriou s aroun d campus . Soon , a fa d started ; al l th e graduat e student s i n physic s began t o bu y thes e bulky artist' s sketch pad s an d coul d b e see n o n campu s haulin g the m awkwardly but proudl y under thei r arms . One time , Peter, his friend Paul Townsend (no w at Cambridge University), an d I wer e collaboratin g o n a n exceptionall y difficul t super gravity problem. Th e calculatio n was so difficult tha t it consumed several hundred pages . Sinc e non e o f u s totall y truste d ou r calculations , we decided t o meet in my dining room an d collectivel y check our work. We faced a dauntin g challenge : Severa l thousan d term s had t o sum up t o exactly zero. (Usually , we theoretical physicist s can "visualize " block s of equations i n ou r head s an d manipulat e the m withou t havin g t o us e paper. However , because o f the shee r lengt h an d delicac y of this prob lem, we had t o check ever y single minus sign in th e calculation. ) We the n divide d th e proble m int o severa l larg e chunks . Sittin g around th e dining-room table, each of us would busily calculate the same chunk. After a n hou r o r so , we would then cross-chec k our results . Usually two out o f three woul d get it right, and th e thir d woul d be asked t o find hi s mistake. Then we would go to th e nex t chunk , and repea t th e same process until all three of us agreed o n the same answer. This repet itive cross-checking went on lat e into th e night . We knew that even on e mistake i n severa l hundre d page s woul d giv e u s a totall y worthless calculation. Finally, well past midnigh t w e checked th e las t and fina l term . It was zero, as we had hoped . We then toaste d our result . (Th e arduou s calculation mus t hav e exhauste d eve n a n indefatigabl e workhors e lik e Peter. After leaving m y apartment, he promptl y forgo t wher e hi s wife's

ISO U N I F I C A T I O N I N T E N D I M E N S I O N

S

new apartmen t wa s in Manhattan . H e knocke d o n severa l door s o f an apartment house , but got only angry responses; he had chosen the wrong building. After a futile search , Pete r an d Pau l reluctantl y headed bac k to Ston y Brook . Bu t becaus e Pete r ha d forgotte n t o replac e a clutch cable, the cable snapped , and the y had t o push hi s car. They eventually straggled int o Stony Brook in thei r broke n ca r at 5:00 in the morning!) The Declin e of Supergravit y

The critics , however, gradually began to see problems with supergravity. After a n intensiv e search, sparticle s were no t see n i n an y experiment . For example, the spin-1/2 electron doe s not hav e any spin-0 partner. In fact, ther e is, at the present, no t one shred of experimental evidence for sparticles in our low-energ y world. However, the firm belief of physicists working in this area is that, at the enormous energie s found at the instant of Creation, all particles were accompanied b y their super partners. Only at this incredible energ y do we see a perfectly supersymmetri c world. But afte r a few years of fervent interest an d score s o f international conferences, i t becam e clea r tha t thi s theor y coul d no t b e quantize d correctly, thu s temporaril y derailin g th e drea m o f creatin g a theor y purely out o f marble. Lik e every other attempt t o construct a theory of matter entirely from marble, supergravity failed for a very simple reason: Whenever we tried t o calculate numbers fro m these theories , we would arrive at meaningless infinities. The theory , although i t had fewe r infin ities than th e origina l Kaluza-Klein theory, was still nonrenormalizable. There were other problems. Th e highest symmetry that supergravity could include was called O(8), which was too small to accommodate th e symmetry o f th e Standar d Model . Supergravity , i t appeared , wa s just another step in the long journey toward a unified theory of the universe. It cured on e proble m (turnin g wood into marble), only to fall victi m t o several other diseases. However, just as interest in supergravity began to wane, a new theory came along tha t was perhaps th e strangest but mos t powerful physical theory ever proposed: the ten-dimensional superstring theory.

7 Superstrings String theor y i s twenty-first centur y physic s that fel l acciden tally into th e twentiet h century. Edward Witten

E

DWARD Witten, of the Institut e for Advanced Stud y in Princeton , New Jersey , dominate s th e worl d o f theoretica l physics . Witten i s currently the "leade r of the pack, " th e mos t brillian t high-energy physicist, who sets trends i n the physics community the way Picasso would set trends in the art world. Hundreds o f physicists follow his work religiously to ge t a glimme r o f hi s path-breaking ideas . A colleague a t Princeton , Samuel Treiman , says , "He' s hea d an d shoulder s abov e th e rest . He' s started whol e group s o f peopl e o n ne w paths . H e produce s elegant , breathtaking proof s whic h peopl e gas p at , whic h leav e the m i n awe. " Treiman the n concludes , "W e shouldn't toss comparisons wit h Einstein around to o freely , bu t whe n i t comes t o Witten . . ."' Witten comes fro m a family o f physicists. His father is Leonard Wit ten, professo r o f physic s at th e Universit y of Cincinnat i an d a leadin g authority o n Einstein' s theor y o f general relativity . (His father, in fact , sometimes boasts that his greatest contributio n t o physics was producing his son. ) Hi s wif e i s Chiar a Nappi , als o a theoretica l physicis t a t th e institute. Witten is not lik e other physicists . Most of them begin thei r romanc e with physic s at an earl y age (suc h as in junior hig h schoo l or eve n ele mentary school). Witten ha s defied mos t conventions , startin g ou t a s a

151

152 U N I F I C A T I O

N I N TE N D I M E N S I O N S

history major at Brandeis Universit y with a n intens e interes t i n linguistics. After graduatin g i n 1971 , h e worked on Georg e McGovern' s presidential campaign . McGover n even wrot e hi m a lette r o f recommenda tion for graduate school . Witten has published article s in Th e Nation arid the Ne w Republic. (Scientific American, in a n intervie w with Witten, commented, "yes , a man wh o is arguably th e smartes t perso n i n th e world is a liberal Democrat." 2) But once Witte n decided tha t physic s was his chosen profession , h e learned physic s wit h a vengeance . H e becam e a graduat e studen t a t Princeton, taught a t Harvard, an d the n rockete d a full professorshi p a t Princeton a t th e ag e of 28. He als o receive d th e prestigiou s MacArthur Fellowship (sometime s dubbe d th e "genius " awar d b y the press) . Spinoffs fro m hi s work have also deeply affected th e worl d of mathematics. In 1990 , he wa s awarded th e Field s Medal, which is as prestigious as the Nobel Priz e in th e worl d of mathematics. Most o f th e time , however , Witten sit s an d stare s ou t th e window, manipulating an d rearrangin g vas t arrays of equations i n hi s head. Hi s wife notes , "H e neve r doe s calculation s excep t i n hi s mind . I wil l fil l pages with calculations before I understand wha t I'm doing . Bu t Edward will sit down only to calculate a minus sign, or a factor of two." 3 Witten says, "Mos t peopl e wh o haven't bee n traine d i n physic s probably think of what physicists do as a question of incredibly complicated calculations, but that' s no t reall y the essenc e o f it. The essenc e o f it is that physics is about concepts , wantin g to understan d th e concepts , th e principle s by which th e worl d works." 4 Witten's next project is the mos t ambitious and darin g of his career . A ne w theor y calle d superstrin g theor y ha s create d a sensatio n i n th e world o f physics , claiming t o b e th e theor y tha t ca n unit e Einstein' s theory o f gravity with the quantu m theory . Witte n is riot content , however, with the wa y superstring theory is currently formulated. H e ha s set for himself the problem of finding the origin of superstring theory, which may prov e t o b e a decisiv e developmen t towar d explainin g th e ver y instant of Creation. The ke y aspect of this theory, the facto r tha t gives it its powe r a s wel l a s uniqueness , i s it s unusua l geometry : String s ca n vibrate self-consistentl y onl y in te n an d 2 6 dimensions.

What I s a Particle?

The essenc e o f strin g theor y i s that i t ca n explai n th e natur e o f bot h matter an d space-time—tha t is, the natur e o f wood an d marble . Strin g

Superstrings 15

3

theory answers a series of puzzling questions about particles, such as why there ar e s o many of the m i n nature . Th e deepe r w e probe int o th e nature o f subatomic particles , th e mor e particle s we find. The curren t "7,00" o f subatomic particles numbers several hundred, an d thei r properties fill entir e volumes. Even with the Standar d Model, we are lef t with a bewildering number o f "elementary particles. " Strin g theory answers this question becaus e th e string , abou t 10 0 billion billion time s smaller than a proton, is vibrating; each mod e o f vibration represent s a distinct resonance or particle . The strin g is so incredibly tin y that, fro m a distance, a resonance o f a string and a particle are indistinguishable. Only when we somehow magnify th e particl e ca n we see that it is not a point at all, but a mode o f a vibrating string. In this picture, each subatomic particle corresponds t o a distinct resonance that vibrates only at a distinct frequency. The idea of a resonance is a familiar on e fro m dail y life. Thin k of the exampl e o f singing in th e shower. Although our natural voice maybe frail, tinny, or shaky, we know that we suddenly blossom into opera stars in the privacy of our showers. This is because our sound waves bounce rapidl y back and forth between the wall s of the shower . Vibrations that ca n fit easil y within the showe r walls ar e magnifie d man y times, producin g tha t resonan t sound . Th e specific vibration s are calle d resonances , while other vibration s (whose waves are o f an incorrec t size ) are cancele d out. Or think of a violin string, which can vibrate at different frequencies, creating musical notes lik e A, B, and C . The onl y modes that can survive on th e strin g are thos e tha t vanish at th e endpoin t o f the violi n string (because it is bolted down at the ends) and undulate an integral number of time s between the endpoints . I n principle , th e strin g ca n vibrate a t any o f a n infinit e numbe r o f differen t frequencies . We know that th e notes themselve s ar e no t fundamental . The not e A is no mor e funda mental than th e note B. However, what is fundamental is the string itself. There is no need to study each note in isolation of the others. By understanding ho w a violi n string vibrates, we immediately understand th e properties o f an infinit e number o f musical notes. Likewise, th e particle s o f th e univers e are not , b y themselves, fundamental. An electro n i s no mor e fundamenta l tha n a neutrino. The y appear t o b e fundamenta l only because ou r microscope s ar e no t powerful enoug h t o reveal their structure . According to string theory, if we could someho w magnif y a point particle , we would actually see a small vibrating string. I n fact , accordin g t o thi s theory, matter i s nothing bu t the harmonies created by this vibrating string. Since there are an infinite number o f harmonies tha t can be composed fo r the violin, there ar e an

154 U N I F I C A T I O

N I N TE N D I M E N S I O N S

infinite number o f forms of matter that can be constructed out of vibrating strings. This explains the richness of the particles in nature. Likewise, the law s of physics can be compare d t o the law s of harmony allowed on the string . The univers e itself, compose d o f countless vibrating strings, would the n b e comparable t o a symphony. String theor y ca n explai n no t onl y the natur e o f particles, bu t tha t of space-tim e a s well. As a strin g move s i n space-time , i t execute s a complicated se t of motions. Th e strin g can, in turn , break int o smaller strings or collide with other strings to form longer strings. The key point is tha t al l thes e quantu m correction s o r loo p diagram s ar e finit e an d calculable. Thi s i s the firs t quantu m theor y o f gravity in th e histor y of physics to have finite quantum corrections. (Allknown previou s theories, we recall—includin g Einstein' s origina l theory , Kaluza-Klei n theory , and supeigravity—faile d thi s key criterion.) In order to execute thes e complicated motions , a string must obey a large set of self-consistency conditions. These self-consistenc y conditions are s o stringent that the y place extraordinaril y restrictive conditions on space-time. I n othe r words , the strin g cannot self-consistentl y trave l in any arbitrary space-time, like a point particle . When th e constraint s that th e string places on space-time were first calculated, physicist s were shocke d t o fin d Einstein' s equations emerg ing fro m the string . This was remarkable; without assuming any of Einstein's equations , physicist s found tha t the y emerged ou t o f the strin g theory, as if by magic. Einstein's equations were no longe r foun d t o b e fundamental; the y could b e derive d from string theory. If correct, then strin g theory solves the long-standin g myster y about the nature o f wood and marble. Einstein conjectured tha t marble alone would one da y explain all the propertie s o f wood. To Einstein, wood was just a kink or vibration of space-time, nothing mor e o r less . Quantu m physicists, however , though t th e opposite . The y though t tha t marbl e could b e turne d int o wood—that is, that Einstein's metric tensor coul d be turne d int o a graviton, the discret e packet of energy that carries th e gravitational force. These ar e tw o diametrically opposite point s o f view, and i t was long thought tha t a compromise betwee n them wa s impossible. Th e string , however, is precisely the "missin g link " between wood and marble . String theory can derive th e particle s of matter a s resonances vibrating on th e string . And string theory can also derive Einstein's equations by demandin g tha t th e strin g mov e self-consistentl y in space-time . I n this way , we have a comprehensiv e theor y o f both matter—energ y an d space—time.

Superstrings 15

5

These self-consistency constraints are surprisingly rigid. For example, they forbid th e strin g to move in thre e o r fou r dimensions . We will see that thes e self-consistenc y conditions forc e th e strin g to mov e in a specific number of dimensions. I n fact, th e only "magic numbers" allowed by string theory are te n an d 2 6 dimensions. Fortunately, a string theory denned in these dimension s has enough "room " to unify all fundamental forces . String theory, therefore , is rich enough t o explain all the fundamental law s o f nature . Startin g from a simpl e theor y o f a vibratin g string, one ca n extrac t th e theor y of Einstein, Kaluza-Klein theory , supergravity, th e Standar d Model , an d eve n GU T theory . I t seem s nothin g les s than a miracl e that , startin g fro m som e purel y geometri c argument s from a string, on e i s able t o rederiv e th e entir e progres s o f physics for the pas t 2 millennia. All the theorie s s o far discusse d in thi s book ar e automatically included i n string theory . The curren t interes t i n strin g theor y stem s fro m th e wor k of John Schwarz o f th e Californi a Institute o f Technology an d hi s collaborato r Michael Gree n of Queen Mary' s College in London . Previously , it was thought tha t th e strin g might possess defects that would prevent a full y self-consistent theory . Then in 1984 , these tw o physicists proved tha t all self-consistency condition s o n th e strin g ca n b e met . This , i n turn , ignited the current stampede among young physicists to solve the theory and win potential recognition . B y the late 1980s , a veritable "gold rush" began amon g physicists . (Th e competitio n amon g hundred s o f th e world's brightes t theoretica l physicist s to solv e th e theor y ha s become quite fierce. In fact, the cove r of Discover recently featured string theorist D. V. Nariopoulous of Texas, who openly boasted tha t he was hot o n th e trail o f winning the Nobe l Priz e in physics . Rarely has suc h an abstrac t theory aroused suc h passions.)

Why Strings ? I onc e ha d lunc h wit h a Nobe l Priz e winne r i n physic s at a Chines e restaurant in New York. While we were passing the swee t and sou r pork , the subjec t o f superstrin g theor y cam e up . Withou t warning , h e launched int o a long personal discussio n of why superstring theor y was not th e correc t path fo r young theoretical physicists . It was a wild-goose chase, he claimed . There ha d neve r been anythin g like it in the history of physics, so he foun d i t too bizarr e for hi s tastes. It was too alien , to o

156 U N I F I C A T I O N I N T E N D I M E N S 1 O N

S

orthogonal to all the previous trends in science. After a long discussion, it boiled down to one question : Why strings? Why not vibrating solids or blobs? The physica l world , h e reminde d me , uses the sam e concept s over and ove r again. Nature is like a work by Bach or Beethoven, often starting with a central them e an d makin g countless variations on i t that are scattered throughou t th e symphony . By this criterion , i t appear s tha t strings are not fundamenta l concepts i n nature . The concep t o f orbits, fo r example , occur s repeatedl y i n natur e i n different variations ; since th e wor k of Copernicus, orbits have provided an essentia l theme tha t is constantly repeated throughout nature in different variations , from th e larges t galax y to th e atom , t o th e smalles t subatomic particle. Similarly, Faraday's field s hav e proved t o be on e o f nature's favorite themes. Fields can describe the galaxy's magnetism and gravitation, or they can describe th e electromagneti c theory of Maxwell, the metri c theor y of Riemann an d Einstein , and th e Yang-Mills fields found i n th e Standar d Model . Field theory, in fact, ha s emerged a s the universal languag e o f subatomi c physics , and perhap s th e univers e as well. It is the singl e most powerful weapon i n th e arsena l of theoretical physics. All known forms of matter an d energ y have been expresse d i n terms o f fiel d theory . Patterns , then , lik e theme s an d variation s in a symphony, are constantly repeated. But strings? Strings do not seem t o be a pattern favore d by nature in designing the heavens . We do not se e strings in outer space. In fact, my colleague explaine d t o me, we do no t se e strings anywhere. A moment' s thought , however , will revea l tha t natur e ha s reserve d the strin g for a specia l role , a s a basi c building bloc k for othe r forms . For example , th e essentia l feature of life o n eart h i s the stringlik e DNA molecule, whic h contains th e comple x informatio n an d codin g o f lif e itself. When building the stuf f of life, as well as subatomic matter, strings seem t o b e th e perfec t answer. In bot h cases , we want to pac k a larg e amount o f information int o a relatively simple, reproducible structure . The distinguishing feature of a string is that it is one of the most compact ways of storing vast amounts o f data in a way in which information ca n be replicated . For livin g things, nature use s the doubl e strand s o f the DN A molecule, whic h unwin d an d for m duplicat e copie s o f eac h other . Also , our bodie s contai n billion s upo n billion s o f protei n strings , formed o f amino aci d buildin g blocks . Our bodies , i n som e sense , ca n be viewe d a s a vas t collectio n o f strings—protei n molecule s drape d around our bones.

Superstrings

15

The Strin g Quartet Currently, the most successful version of string theory is the one create d by Princeton physicist s David Gross, Emil Martinec, Jeffrey Harvey , and Ryan Rohm, who are sometimes called the Princeton string quartet. The most senior of them i s David Gross. At most seminars in Princeton, Witten ma y ask questions in his soft voice, but Gross' s voice is unmistakable: loud, booming, and demanding . Anyon e who gives a seminar at Princeton live s in fea r o f th e sharp , rapid-fir e questions tha t Gros s will shoo t at them. What is remarkable is that his questions are usually on the mark. Gross an d hi s collaborator s propose d wha t is called th e heterotic string. Today, it is precisely the heteroti c string, of all the various Kaluza-Kleintype theorie s tha t have been propose d i n the past , that has the greates t potential of unifying al l the law s of nature int o on e theory . Gross believes that string theory solves the problem o f turning wood into marble: "T o buil d matte r itsel f fro m geometry—tha t i n a sense is what strin g theory does . I t ca n b e though t o f tha t way , especially in a theory like the heteroti c strin g which is inherently a theory of gravity in which the particles of matter as well as the othe r forces of nature emerg e in th e sam e way that gravity emerges from geometry." 5 The mos t remarkabl e featur e o f strin g theory , a s we have empha sized, is that Einstein's theory o f gravity is automatically contained i n it. In fact , th e gravito n (th e quantu m o f gravity ) emerge s a s th e smalles t vibration of the closed string. While GUTs strenuously avoided any mention of Einstein's theory of gravity, the superstring theories demand tha t Einstein's theor y be included. Fo r example, if we simply drop Einstein's theory of gravity as one vibration of the string , then th e theory becomes inconsistent an d useless . This , i n fact , i s th e reaso n wh y Witten was attracted t o strin g theor y i n th e firs t place . I n 1982 , h e rea d a review article by John Schwar z and wa s stunned t o realize that gravit y emerge s from superstrin g theor y fro m self-consistenc y requirements alone . H e recalls that it was "the greates t intellectual thrill of my life." Witten says, "String theory is extremely attractive because gravit y is forced upo n us . All known consisten t strin g theorie s includ e gravity , so while gravit y is impossible i n quantu m fiel d theor y a s we have known it, it's obligatory in strin g theory." 6 Gross take s satisfactio n in believin g tha t Einstein , i f h e wer e alive, would lov e superstring theory . H e woul d love th e fac t tha t th e beaut y and simplicit y of superstring theory ultimatel y come fro m a geometri c principle, whose precise nature is still unknown. Gross claims, "Einstein would have been pleased wit h this, at least with th e goal , if not th e real-

158 U N I F I C A T I O

N I N TE N D I M E N S I O NS

ization. . . . H e woul d hav e like d th e fac t tha t ther e i s a n underlyin g geometrical principle—which , unfortunately , w e don' t reall y under stand."7 Witten eve n goe s s o far a s t o sa y that "al l th e reall y great idea s i n physics" ar e "spinoffs " o f superstring theory . By this, he mean s that all the great advances in theoretical physics are included within superstring theory. H e eve n claim s tha t Einstein' s genera l relativit y theor y bein g discovered before superstring theory was "a mer e acciden t o f the development o n plane t Earth. " H e claim s that, somewher e i n oute r space , "other civilization s in th e universe " migh t have discovere d superstrin g theory first, an d derive d genera l relativit y as a by-product. 8

Compactification an d Beauty

String theory is such a promising candidat e for physic s because i t gives a simpl e origi n o f th e symmetrie s found i n particl e physic s a s well a s general relativity . We saw in Chapte r 6 that supergravit y was both nonrenormalizabl e and to o smal l t o accommodat e th e symmetr y of th e Standar d Model . Hence, i t wa s no t self-consisten t an d di d no t begi n t o realisticall y describe th e know n particles. However , strin g theor y does both . As we shall soon see , it banishes th e infinitie s found in quantum gravity, yielding a finite theor y of quantum gravity . That alon e woul d guarantee tha t string theory should be take n a s a serious candidat e fo r a theory o f th e universe. However, there i s an adde d bonus. When we compactify som e of the dimension s of the string , we find that there is "enough room" to accommodate th e symmetrie s o f th e Standar d Mode l an d eve n th e GUTs. The heteroti c string consists of a closed strin g that has two types of vibrations, clockwis e an d counterclockwise , which ar e treate d differ ently. The clockwis e vibrations live in a ten-dimensional space. The counterclockwise live in a 26-dimensional space, of which 16 dimensions have been compactified . (W e recall tha t i n Kaluza's original five-dimensional theory, the fifth dimension wa s compactified by being wrapped up int o a circle.) The heterotic strin g owes its name t o the fact that the clockwise and th e counterclockwis e vibrations live in two different dimension s but are combine d t o produc e a single superstrin g theory . Tha t i s why it is named afte r th e Gree k word for heterosis, which means "hybri d vigor. " The 16-dimensiona l compactified space is by far the most interesting. In Kaluza-Klei n theory , we recall tha t th e compactifie d jV-dimensiona l

Superstrings 15

9

space ca n hav e a symmetr y associated wit h it , muc h lik e a beac h ball . Then al l the vibration s (o r fields ) denne d o n th e A Ldimensional spac e automatically inheri t thes e symmetries . If th e symmetr y is SU(jV), the n all the vibration s o n th e spac e must obey SU(N) symmetr y (i n the sam e way that cla y inherits the symmetrie s of the mold) . In thi s way, KaluzaKlein theory could accommodate th e symmetries of the Standard Model . However, i n thi s way it coul d als o b e determine d tha t th e supergravity was "to o small " t o contai n al l the particle s of the symmetrie s found i n the Standar d Model . Thi s wa s sufficient t o kil l th e supergravit y theory as a realistic theory of matter an d space-time . But whe n th e Princeto n strin g quarte t analyze d th e symmetrie s of the 16-dimensiona l space, the y found tha t it is a monstrously large symmetry, called E(8) X E(8), which is much larger than an y GUT symmetry that has ever been tried. 9 Thi s was an unexpecte d bonus . I t meant tha t that al l th e vibration s o f th e strin g would inheri t th e symmetr y of th e 16-dimensional space , whic h was more tha n enoug h t o accommodat e the symmetr y of the Standar d Model . This, then , i s the mathematica l expressio n o f th e centra l them e o f the book , tha t th e law s of physics simplify i n highe r dimensions. I n thi s case, the 26-dimensiona l space o f the counterclockwis e vibrations of the heterotic strin g ha s roo m enoug h t o explai n al l the symmetrie s found in bot h Einstein' s theor y an d quantu m theory . So , fo r th e firs t time , pure geometr y ha s give n a simpl e explanatio n o f wh y the subatomi c world mus t necessarily exhibit certain symmetries that emerg e fro m th e curling u p o f higher-dimensiona l space : Th e symmetries o f th e subatomic realm are but remnants of the symmetry of higher-dimensional space. This mean s tha t th e beaut y and symmetr y found i n natur e ca n ultimately be trace d bac k t o higher-dimensional space . For example, snowflakes creat e beautiful , hexagonal patterns , non e of which are precisely the same . Thes e snowflake s an d crystals , in turn , hav e inherite d thei r structure from th e way in which their molecules have been geometrically arranged. Thi s arrangement i s mainly determined b y the electro n shell s of the molecule , which in turn tak e us back to the rotationa l symmetries of th e quantu m theory , give n b y O(3). Al l the symmetrie s o f th e low energy univers e tha t w e observe i n chemica l element s ar e du e t o th e symmetries cataloge d b y th e Standar d Model , whic h i n tur n ca n b e derived b y compactifying th e heteroti c string . In conclusion , th e symmetrie s that we see around us , from rainbow s to blossomin g flower s t o crystals , may ultimately be viewe d as manifestations o f fragments o f th e origina l ten-dimensiona l theory. 10 Riemann and Einstein had hoped to find a geometric understanding of why forces

160 U N I F I C A T I O N I N T E N D i M E N S I O N

S

can determine th e motion an d the nature o f matter. But they were missing a key ingredient in showing the relationship between wood and marble. Thi s missin g link i s most likel y superstrin g theory . Wit h th e ten dimensional strin g theory , we see tha t th e geometr y o f th e strin g may ultimately be responsible for both the forces and the structure of matter. A Piec e o f Twenty-First-Century Physic s Given th e enormou s powe r o f it s symmetries , it i s not surprisin g tha t superstring theor y is radically different from an y other kin d o f physics. It was, in fact , discovere d quit e b y accident. Man y physicists have commented tha t i f thi s fortuitou s acciden t ha d neve r occurred , the n th e theory woul d no t hav e bee n discovere d unti l th e twenty-firs t century . This i s because i t is such a shar p departur e fro m al l th e idea s trie d i n this century . It i s not a continuou s extensio n o f trend s an d theorie s popular in thi s century; it stands apart . By contrast, the theor y of general relativit y had a "normal" and logical evolution. First, Einstein postulated the equivalence principle. Then he reformulate d thi s physica l principle i n th e mathematic s of a field theory o f gravitatio n base d o n Faraday' s field s an d Riemann' s metri c tensor. Later cam e the "classica l solutions, " suc h as the blac k hole an d the Bi g Bang. Finally, the las t stage is the curren t attemp t t o formulate a quantum theor y of gravity. Thus general relativit y went through a logical progression, fro m a physical principle t o a quantum theory : Geometry — > fiel d theor y — > classical theory — » quantum theor y By contrast, superstrin g theor y ha s been evolvin g backward sinc e its accidental discover y i n 1968 . That' s wh y superstrin g theor y look s s o strange an d unfamilia r to mos t physicists . We are stil l searching fo r it s underlying physical principle, the counterpart t o Einstein's equivalence principle. The theor y was born quite by accident in 196 8 when two young theoretical physicists , Gabriel Venezian o an d Mahik o Suzuki , were inde pendently leafin g through mat h books , lookin g for mathematical func tions tha t woul d describ e th e interaction s o f strongl y interactin g particles. While studying at CERN, th e Europea n cente r fo r theoretica l physics i n Geneva , Switzerland , they independentl y stumble d o n th e Euler beta function, a mathematical function written down in the nine teenth century by the mathematician Leonhard Euler. They were aston-

Superstrings 16

1

ished t o fin d tha t th e Eule r bet a functio n fit almost al l the propertie s required t o describe th e stron g interactions of elementary particles. Over lunc h at the Lawrenc e Berkeley Laboratory in California , wit h a spectacula r vie w o f the su n bla/in g down ove r Sa n Francisco harbor , Suzuki once explained t o me the thril l of discovering, quite by accident, a potentially important result. Physics was not suppose d t o happen tha t way. After findin g th e Eule r beta functio n in a mat h book , h e excitedly showed hi s resul t t o a senio r physicis t at CERN . Th e senio r physicist, after listenin g to Suzuki , was not impressed . In fact, h e tol d Suzuki that another youn g physicist (Veneziano ) had discovere d the identica l function a fe w weeks earlier. H e discourage d Su/.uk i fro m publishin g his result. Today , thi s bet a functio n goe s b y th e nam e o f th e Venezian o model, which has inspired several thousand research papers, spawned a major school of physics, and no w makes the claim of unifying all physical laws. (I n retrospect, Suzuki , of course, should have published his result. There is a lesson to all this, I suspect: Never take too seriously the advice of your superiors.) In 1970 , th e myster y surrounding th e Veneziano-Suzuki mode l was partly explained when Yoichiro Nambu at the University of Chicago and Tetsuo Got o at Niho n Universit y discovere d tha t a vibrating string lies behind it s wondrous properties . Because string theory was discovered backward and b y accident, physicists still do not know the physical principle that underlies string theory. The las t ste p i n th e evolutio n o f th e theor y (an d th e firs t ste p i n th e evolution o f general relativity ) is still missing. Witten adds tha t human being s on plane t Earth neve r ha d th e conceptua l framework that would lead the m t o invent string theor y on purpose . . . . No one invente d it o n purpose , i t wa s invented i n a luck y accident . B y rights, twentiethcentury physicists shouldn't hav e had th e privileg e of studying this theory. By rights, string theory shouldn't hav e been invented until our knowledge of some o f the idea s that are prerequisit e fo r string theory had develope d to the poin t tha t it was possible for us to have the righ t concept of what it was all about." Loops

The formul a discovered b y Veneziano an d Suzuki , which they hoped would describe the properties o f interacting subatomic particles, was still

162 U N 1 F I C A T I O N I N T E N D I M E N S I O N

S

incomplete. It violated one o f the propertie s o f physics: unitarity, o r th e conservation o f probability . B y itself , th e Veneziano-Suzuk i formul a would giv e incorrect answers for particl e interactions . So the nex t step in th e theory' s evolutio n wa s to ad d smal l quantu m correctio n term s that would restore thi s property. I n 1969 , eve n before th e strin g interpretation o f Namb u an d Goto , thre e physicist s (Keij i Kikkawa , Bunj i Sakita, an d Migue l A. Virasoro, then al l at th e Universit y of Wisconsin) proposed th e correc t solution : adding increasingl y smaller term s to the Vene/iano-Suzuki formula in order to restore unitarity. Although thes e physicists had t o guess at how to construct the serie s from scratch , today it is most easily understood i n th e framewor k of th e string picture of Nambu. For example, when a bumblebee flie s in space, its path ca n be described a s a wiggly line. When a piece of string drifting in th e ai r move s in space , it s path ca n b e likene d t o a n imaginar y two dimensional sheet . When a closed strin g floats i n space, its path resem bles a tube . Strings interact b y breaking into smalle r strings and b y joining wit h other strings . When thes e interactin g string s move, they trace ou t th e configurations show n in Figure 7.1 . Notice that two tubes com e in fro m the left , wit h one tub e fissioning in half, exchange th e middle tube , an d then vee r of f t o th e right . Thi s i s how tube s interac t wit h eac h other . This diagram, of course, is shorthand fo r a very complicated mathematical expression . Whe n w e calculat e th e numerica l expressio n corre sponding t o these diagrams , we get back the Euler beta function . In the string picture, the essential trick proposed b y Kikkawa-SakitaVirasoro (KSV ) amounte d t o adding al l possible diagrams wher e string s can collide and brea k apart. Ther e are, o f course, an infinite number of these diagrams . Th e proces s o f adding a n infinit e numbe r o f "loop " diagrams, with each diagra m comin g close r t o th e fina l answer , is perturbation theor y and i s one o f most importan t weapon s i n th e arsena l of an y quantu m physicist . (Thes e strin g diagram s posses s a beautiful symmetry that has never been see n i n physics before, which is known as conformal symmetry i n tw o dimensions. Thi s conforma l symmetr y allows us to trea t thes e tube s and sheet s a s though the y were made o f rubber : We ca n pull , stretch, bend , an d shrin k these diagrams . Then , becaus e of conformal symmetry, we can prove that all these mathematical expressions remain th e same.) KSV claimed that the su m total of all these loop diagrams would yield the precis e mathematica l formul a explainin g ho w subatomi c particle s interact. However , th e KS V program consiste d o f a serie s o f unproven conjectures. Someon e ha d t o construc t thes e loop s explicitly , or els e these conjecture s were useless.

+ .. . Figure 7.1. In string theory, the gravitational force is represented by the exchange of closed strings, which sweep out tubes in space-time. Even if we add up an infinite series of diagrams with a large number of holes, infinities never appear in the theory, giving us a finite theory of quantum gravity.

163

164 U N I F I C A T I O N I N T E N D I M E N S I O N

S

Intrigued by the progra m bein g initiated by KSV, I decided t o try my luck at solving the problem . This was a bit difficult , becaus e I was dodging machine-gun bullets at the time .

Boot Cam p

I remembe r clearl y when th e KS V paper cam e ou t i n 1969 . KS V wa s proposing a program fo r future work , rather tha n givin g precise details. I decided then t o calculate all possible loops explicitly and complet e th e KSV program. It's hard t o forget those times . There was a war raging overseas , and the universit y campuses from Ken t State t o the Universit y of Paris, were in a stat e o f turmoil . I ha d graduate d fro m Harvar d th e yea r before , when Presiden t Lyndo n Johnson revoke d deferment s for graduate students, sending panic throughout graduate school s in the country. Chaos gripped th e campuses . Suddenly, my friends were dropping ou t o f college, teachin g hig h school , packin g their bags and headin g t o Canada, or tryin g to ruin thei r health i n order to flunk th e arm y physical. Promising careers wer e being shattered. On e o f my good friend s i n physics fro m MI T vowed tha t h e woul d go t o jai l rathe r tha n figh t i n Vietnam. He tol d us to send copie s of the Physical Review to his jail cell so he could kee p up with developments in the Vene/iano model. Othe r friends, wh o quit college t o teac h i n hig h school s rather tha n figh t i n the war , terminate d promisin g scientifi c careers . (Man y o f the m stil l teach i n thes e high schools.) Three day s after graduation , I lef t Cambridg e an d foun d mysel f in the Unite d States Army stationed a t Fort Benning, Georgi a (th e largest infantry trainin g cente r i n th e world), and late r a t Fort Lewis, Washington. Tens of thousands of raw recruits with no previous military training were bein g hammere d int o a fighting force an d the n shippe d t o Vietnam, replacing the 50 0 GIs who were dying every week. One day , while throwing live grenade s unde r th e gruelin g Georgia sun and seein g the deadly shrapnel scatter in all directions, my thoughts began t o wander. How many scientists throughout histor y had t o fac e the punishin g ravage s o f war ? Ho w man y promisin g scientist s were snuffed ou t b y a bullet in th e prim e o f their youth? I remembered tha t Karl Schwarzschild had die d in the kaiser' s army on th e Russia n fron t durin g Worl d Wa r I just a fe w months afte r h e found th e basic solution to Einstein's equations used in every black hole calculation. (Th e Schwarzschil d radius o f a black hole i s named i n hi s

Superstrings 16

5

honor. Einstein addressed th e Prussia n Academ y in 191 6 t o commem orate Schwar/schild' s work after hi s untimely death a t the fron t lines.) And how many promising people were cut down even before they could begin their careers ? Infantry training , I discovered, is rigorous; it is designed t o toughe n the spirit and dul l the intellect . Independence o f thought is ground ou t of you. After all , the militar y does not necessaril y want some wit who will question th e sergeant's order s i n the middle of a firefight. Understanding this , I decided t o bring alon g som e physic s papers. I needed some thing t o kee p m y min d activ e whil e peelin g potatoe s i n K P or firin g machine guns, so I brought along a copy of the KS V paper. During night infantr y training , I had t o go past a n obstacl e course , which mean t dodgin g liv e machine-gu n bullets , frogleggin g unde r barbed wire , and crawlin g through thic k brown mud. Because the automatic fire had tracer s on them, 1 could see the beautiful crimson streaks made b y thousand s o f machine-gu n bullet s sailing a fe w feet ove r m y head. However , my thoughts kep t driftin g bac k t o th e KS V paper an d how their progra m coul d b e carried out . Fortunately, the essentia l feature of the calculatio n was strictly topological. It was clear to me tha t these loop s were introducing an entirely new language t o physics, the languag e o f topology. Never before in th e history of physics had Mobiu s strips or Klei n bottles been use d i n a fundamental way. Because I rarel y ha d an y pape r o r pencil s whil e practicin g wit h machine guns, I forced mysel f to visualize in my head how strings could be twiste d into loop s an d turne d insid e out . Machine-gu n training was actually a blessing in disguis e because i t forced me t o manipulate larg e blocks o f equation s i n m y head. B y the tim e I finishe d th e advance d machine-gun-training program, I was convinced tha t I coul d complet e the progra m o f calculating all loops. Finally, I managed t o squeeze time from th e arm y to go to the Uni versity of California at Berkeley, where I furiously worked out the detail s that were racin g i n m y head. I san k severa l hundre d hour s o f intens e thought int o th e question . This , in fact, becam e m y Ph.D. dissertation . By 1970, the final calculation too k up severa l hundred densel y filled notebook pages . Unde r th e carefu l supervisio n o f m y adviser, Stanle y Mandelstam, my colleague Loh-pin g Yu and I successfully calculate d a n explicit expressio n fo r al l possible loo p diagram s know n a t tha t time . However, I wasn't satisfied wit h this work. The KS V program consiste d of a hodge-podge o f rules of thumb an d intuition , not a rigorous se t of basic principles from which these loop s could b e derived. String theory ,

166 U N I F I C A T I O N I N T E N D I M E N S I O N

S

we saw , was evolving backward, sinc e it s accidenta l discover y by Veneziano and Suzuki . The nex t step in the backwar d evolution of the strin g was to follow in the footsteps of Faraday, Riemann, Maxwell, and Einstein and construc t -afield theory o f strings.

Field Theory of String s Ever sinc e th e pioneerin g wor k of Faraday , ever y physica l theory ha d been writte n in term s o f fields . Maxwell' s theory o f ligh t was based o n field theory . S o was Einstein's. In fact , al l o f particl e physic s was based on fiel d theory . Th e onl y theor y no t based o n fiel d theor y wa s string theory. The KS V program wa s more a set of convenient rules than a field theory. My next goa l wa s to rectif y tha t situation . Th e proble m wit h a field theory o f strings , however , was that man y o f th e pioneerin g figure s i n physics argued agains t it . Their argument s were simple. These giant s of physics, such a s Hideki Yukaw a an d Werne r Heisenberg , ha d labore d for year s t o create a fiel d theor y tha t wa s not base d o n poin t particles . Elementary particles, the y thought, migh t be pulsatin g blobs o f matter, rather tha n points . However , no matte r ho w hard the y tried, fiel d the ories base d o n blob s alway s violated causality. If w e were t o shak e th e blo b a t on e point , th e interaction s woul d spread faste r tha n th e spee d o f light throughout th e blob , violating special relativit y an d creatin g al l sorts o f tim e paradoxes . Thu s "nonloca l field theories" based o n blob s were known to be a monstrously difficul t problem. Man y physicists, in fact , insiste d tha t onl y local fiel d theorie s based o n poin t particle s coul d b e consistent . Nonloca l fiel d theorie s must violate relativity. The secon d argumen t wa s even mor e convincing . Th e Venezian o model ha d man y magical propertie s (includin g something calle d duality) tha t ha d neve r bee n seen befor e i n fiel d theory . Years earlier, Richard Feynman had give n "rules" tha t any field theory should obey. However, these Feynman rules were in direct violation of duality. Thus many string theorist s were convinced that a field theory o f strings was impossible becaus e strin g theor y necessaril y violate d th e propertie s o f th e Veneziano model . Strin g theory , the y said, was unique i n al l of physics because i t could no t b e recas t as a field theory. I collaborated wit h Keiji Kikkawa on this difficult but important problem. Ste p b y step we built ou r fiel d theory , i n muc h th e sam e way that our predecessors ha d constructe d field theories for other forces . Follow-

Superstrings 16

7

ing Faraday, we introduced a field at every point in space-time. However, for a field theory of strings, we had t o generalize the concep t of Faraday and postulat e a fiel d that was defined for al l possible configuration s of a string vibrating in space-time. The secon d ste p was to postulate th e field equations tha t th e strin g obeyed. Th e fiel d equatio n fo r a single strin g movin g alone i n space time was easy. As expected, our fiel d equation s reproduced an infinit e series of string resonances, each correspondin g t o a subatomic particle. Next, w e foun d tha t th e objection s o f Yukaw a an d Heisenber g wer e solved by string field theory. If we jiggled th e string , the vibration s traveled dow n th e strin g at less than th e spee d o f light. Soon, however, we hit a brick wall. When we tried t o introduce inter acting strings , w e could no t reproduc e th e Venezian o amplitud e cor rectly. Dualit y an d th e countin g o f graph s give n b y Feynman fo r an y field theory were in direct conflict. Just as the critics expected, the Feyn man graph s wer e incorrect . Thi s wa s disheartening. I t appeare d tha t field theory , which had forme d th e foundatio n o f physic s for th e pas t century, was fundamentally incompatible wit h strin g theory . Discouraged, I remembe r mullin g ove r th e proble m lat e int o th e night. For hours , I began systematicall y t o chec k all the possibl e alter natives to this problem. But the conclusion that duality had to be broke n seemed inescapable . The n I remembere d wha t Sherloc k Holmes , i n Ardiur Conan Doyle' s "The Sig n of Four," said to Watson: "How ofte n have I said to you that when you have eliminate d th e impossible , whatever remains, however improbable, must be th e truth. " Encouraged by this idea, I eliminate d al l the impossibl e alternatives. The onl y improbable alternative remainin g wa s to violat e th e propertie s o f th e Veneziano Suzuki formula. At about 3:0 0 A.M., the resolutio n finally hit me . I realized tha t physicist s had overlooke d th e obviou s fact tha t on e ca n split the Veneziano-Suzuk i formul a into tw o pieces. Eac h par t the n corre sponds t o on e o f Feynman' s diagrams , an d eac h par t violate s duality, but th e su m obeys all the correc t properties o f a field theory. I quickly took out some paper and went over the calculation. I spent the nex t 5 hours checking and recheckin g the calculatio n from al l possible directions . Th e conclusio n was inescapable: Fiel d theory does violate duality, as everyone expected, but this is acceptable because the final sum reproduces th e Veneziano-Suzuki formula. I had now solved most of the problem. However , one more Feynman diagram, representing th e collisio n of four strings, was still lacking. That year, I wa s teaching introductor y electricit y and magnetis m t o under graduates a t the Cit y University of New York, and w e were studying Far-

168 U N I F 1 C A T I O N I N T E N D I M E N S I O N

S

aday's lines offeree. I would ask the student s t o draw the line s offere e emanating from different configurations of charges, repeating th e sam e steps pioneere d b y Farada y i n th e nineteent h century . Suddenly , i t dawned o n m e tha t th e squiggl y lines that I was asking my students t o draw ha d exactl y th e sam e topologica l structur e a s th e collisio n o f strings. Thu s b y rearranging charge s i n a freshma n laboratory , I ha d found th e correct configuration describin g the collision of four strings. Was it that simple? I rushe d hom e t o chec k m y hunch, ari d I was right . B y employing pictorial techniques that even a freshman can use, I could show that the four-string interactio n mus t b e hidde n withi n th e Venezian o formula. By th e winte r of 1974 , usin g methods datin g bac k t o Faraday , Kikkaw a and I completed th e fiel d theor y o f strings, the firs t successfu l attemp t to combine strin g theory with the formalis m of field theory. Our fiel d theory , althoug h i t correctl y embodie d th e entir e infor mation containe d withi n strin g theory , stil l neede d improvement . Because w e were constructin g th e fiel d theor y backward , many of th e symmetries wer e stil l obscure. Fo r example , th e symmetrie s of special relativity were present bu t no t i n an obviou s way. Much more wor k was needed t o streamline th e fiel d equation s w e had found . Bu t just a s we were beginning t o explore th e properties o f our field theory, the mode l unexpectedly suffere d a severe setback. That year, physicist Claude Lovelace of Rutgers University discovered that th e bosoni c strin g (describin g integral spins ) is self-consistent only in 2 6 dimensions. Othe r physicist s verified this result ari d showe d that the superstrin g (describin g both integra l an d half-integra l spin) i s selfconsistent onl y in te n dimensions . I t was soon realize d that , i n dimen sions othe r tha n te n o r 2 6 dimensions, th e theor y completel y loses all its beautiful mathematical properties . Bu t no one believe d that a theory defined i n te n o r 2 6 dimension s ha d anythin g t o d o wit h reality . Research i n string theor y abruptly ground t o a halt. Lik e Kaluza-Klein theory befor e it , strin g theor y lapse d int o a dee p hibernation . Fo r 1 0 long years , the mode l was banished t o obscurity. (Although most string physicists, myself included, abandone d th e mode l lik e a sinking ship, a few die-hards, like physicists John Schwarz and th e late Joel Scherk, tried to keep the mode l aliv e by steadily making improvements. For example, string theor y wa s originally though t t o b e just a theor y o f th e stron g interactions, with each mod e o f vibration corresponding t o a resonanc e of the quark model. Schwarz and Scherk correctly showed that the string model wa s really a unified theory of all forces, not jus t the stron g interactions.)

Superstrings 16

9

Research i n quantu m gravit y went into othe r direction . From 197 4 to 1984 , when string theory was in eclipse, a large number of alternative theories o f quantu m gravit y wer e successivel y studied . Durin g thi s period, th e origina l Kaluza-Klei n theor y and the n th e supergravity theory enjoyed great popularity, but eac h tim e th e failures of these model s also became apparent . For example, both Kaluza-Klein and supergravity theories were shown t o be nonrenormalizable. Then somethin g strange happened durin g that decade. O n the one hand, physicist s became frustrate d b y the growin g lis t of model s tha t were tried an d then discarde d during this period. Everything failed. The realization came slowly that Kaluza-Klein theory and supergravity theory were probably on th e right track, but they weren't sophisticated enoug h to solve the proble m o f nonrenormalizability. But the onl y theory complex enoug h t o contain bot h Kaluza-Klei n theor y and th e supergravity theory wa s superstring theory . O n th e othe r hand , physicist s slowl y became accustome d t o working in hyperspace . Becaus e o f the Kaluza Klein renaissance, the idea of hyperspace didn't seem that farfetched or forbidding anymore. Over time, even a theory defined in 26 dimensions didn't see m tha t outlandish . Th e origina l resistance t o 2 6 dimensions began t o slowl y melt away with time . Finally, in 1984 , Gree n an d Schwar z proved tha t superstring theor y was the only self-consistent theory of quantum gravity , and th e stamped e began. I n 1985 , Edwar d Witten mad e a significant advanc e in th e fiel d theory of strings, which many people think is one o f the mos t beautiful achievements o f the theory . He showe d tha t our ol d field theory could be derived usin g powerful mathematica l and geometric theorem s (coming from something calle d cohomology theory) wit h a fully relativisti c form. With Witten' s ne w field theory, th e tru e mathematica l eleganc e o f string field theory, which was concealed in our formalism , was revealed. Soon, almos t a hundre d scientifi c paper s wer e written t o explor e th e fascinating mathematica l propertie s o f Witten's field theory. 12 No On e I s Smart Enoug h Assuming tha t strin g fiel d theor y i s correct, i n principl e w e should b e able t o calculate the mas s of the proto n from first principles and mak e contact wit h know n data, suc h a s the masse s of the variou s particles. If the numerica l answer s are wrong, then w e will have to throw the theor y out th e window. However, if the theory is correct, it will rank among th e most significant advances in physic s in 2,00 0 years.

170 U N I F I C A T I O

N I N TE N D I M E N S I O N S

After th e intense , euphori c fanfar e o f th e lat e 1980 s (whe n i t appeared tha t the theor y would be completely solved within a few years and th e Nobel Prizes handed ou t by the dozen), a certain degre e o f cold realism ha s se t in. Although th e theor y i s well denned mathematically, no on e ha s been abl e t o solve the theory . N o one . The proble m i s tha t n o on e i s smart enough t o solve th e fiel d theory o f strings or an y other nonperturbative approac h t o string theory. This is a well-defined problem , bu t th e iron y is that solvin g field theory require s techniques tha t ar e currentl y beyon d th e skil l o f an y physicist. This is frustrating. Sittin g before u s is a perfectly well-defined theor y of strings. Within i t i s th e possibilit y o f settlin g al l th e controvers y surroundin g higher-dimensional space . Th e drea m o f calculatin g everythin g fro m first principles i s staring u s i n th e face . The proble m i s how t o solv e it. One i s reminded o f Julius Caesar's famous remark in Shakespeare's play: "The fault , dear Brutus, is not in our stars, but in ourselves." Fo r a string theorist, the fault is not in the theory , but in our primitive mathematics . The reaso n fo r thi s pessimis m i s that ou r mai n calculationa l tool , perturbation theory , fails . Perturbatio n theor y begins with a Venezianolike formula and the n calculate s quantum correction s t o it (whic h hav e the shap e o f loops). I t was the hop e o f string theorist s tha t the y coul d write dow n a mor e advance d Veneziano-lik e formul a define d i n fou r dimensions tha t woul d uniquely describe th e know n spectrum o f particles. In retrospect, the y were too successful. Th e proble m i s that millions upon million s o f Veneziano-lik e formula s hav e no w bee n discovered . Embarrassingly, strin g theorist s ar e literall y drowning i n thes e pertur bative solutions. The fundamenta l proble m tha t ha s stalle d progres s i n superstrin g theory in the past few years is that no one know s how to select the correc t solution ou t o f th e million s tha t hav e bee n discovered . Som e o f thes e solutions come remarkably close to describing th e real world. With a few modest assumptions , i t i s eas y t o extrac t th e Standar d Mode l a s on e vibration o f the string. Several groups hav e announced, i n fact, that they can find solutions tha t agree wit h th e know n data abou t subatomi c particles. The problem , w e see, is that there ar e als o millions upon million s of other solution s describin g universe s tha t d o no t appea r anythin g lik e our universe . In som e o f these solutions , the univers e ha s no quark s or too man y quarks. I n mos t o f them, lif e a s we know it canno t exist . Ou r universe may be lost somewhere among the millions of possible universes that hav e been foun d i n strin g theory. T o fin d th e correc t solution , we

Superstrings 17

1

must us e nonperturbativ e techniques , whic h ar e notoriousl y difficult . Since 99% o f what we know about high-energ y physics is based o n per turbation theory , thi s mean s tha t w e are a t a total los s t o fin d th e on e true solution to the theory . There i s some roo m fo r optimism , however . Nonperturbativ e solu tions tha t hav e bee n foun d fo r muc h simple r theorie s sho w that man y of the solutions are actually unstable. After a time, these incorrect, unstable solutions wil l make a quantum lea p t o the correct , stable solution . If this is true fo r string theory , then perhaps th e million s of solutions tha t have bee n foun d ar e actuall y unstabl e an d wil l deca y over tim e t o th e correct solution . To understan d th e frustratio n tha t w e physicists feel, think , fo r a moment, o f how nineteenth-century physicist s might react if a portabl e computer wer e given t o them . The y could easil y learn t o tur n th e dial s and pres s th e buttons. They could learn t o master video games o r watch educational program s o n th e monitor. Bein g a century behind i n tech nology, the y woul d marve l a t th e fantasti c calculationa l abilit y o f th e computer. Withi n its memory coul d easil y be stored al l known scientific knowledge o f that century . In a short perio d o f time, the y could lear n to perform mathematica l feat s that would amaze any of their colleagues . However, once the y decide t o open up th e monitor t o see what is inside, they would be horrified . Th e transistor s an d microprocessor s woul d be totally alien t o anythin g they could understand . Ther e would b e reall y nothing i n thei r experienc e t o compare wit h th e electroni c computer . It would be beyond thei r ken . They could onl y stare blankly at the com plicated circuitry , not knowin g i n th e slightes t ho w it works o r wha t it all means . The sourc e o f thei r frustratio n woul d b e tha t th e compute r exist s and i s sitting there i n fron t o f their noses , bu t the y would hav e n o reference fram e fro m whic h t o explai n it . Analogously , strin g theor y appears to be twenty-first-century physics that was discovered accidentally in ou r century . Strin g fiel d theory , too , seem s t o includ e al l physical knowledge. With little effort, w e are abl e t o turn a few dials and pres s a few button s wit h th e theory , an d ou t pop s th e supergravit y theory , Kaluza-Klein theory , and th e Standar d Model . But we are at a total loss to explain why it works. String field theor y exists, but it taunts us because we are no t smar t enough t o solve it. The proble m i s that while twenty-first-century physics fell accidentally into th e twentieth century, twenty-first-century mathematic s hasn't bee n invented yet . It seem s tha t w e may have t o wai t for twenty-first-centur y

172 U N I F I C A T I O N I N T E N D I M E N S 1 O N

S

mathematics before w e can mak e an y progress, o r th e curren t genera tion o f physicists must invent twenty-first-century mathematic s on thei r own.

Why Te n Dimensions ? One o f the deepes t secret s of string theory, which is still not wel l understood, i s why it i s denned i n onl y ten an d 2 6 dimensions. I f the theor y were three dimensional , it would not b e able t o unify th e know n laws of physics in any sensible manner. Thus it is the geometry of higher dimensions that i s the centra l featur e o f the theory . If we calculate how strings break and re-for m in jV~dimensional space, we constantly find meaningless terms cropping up tha t destroy the marvelous propertie s o f th e theory . Fortunately , thes e unwante d term s appear multiplie d b y ( N — 10). Therefore , t o mak e thes e anomalie s vanish, we have no choic e but t o fix N to b e ten . Strin g theory, in fact , is th e onl y know n quantu m theor y tha t specificall y demand s tha t th e dimension o f space-time be fixed at a unique number . Unfortunately, string theorists are, at present, at a loss to explain why ten dimension s are singled out . The answe r lies deep within mathematics, i n an area called modular functions. Whenever we manipulate the KS V loop diagrams create d b y interacting strings, we encounter thes e strange modular functions , wher e th e numbe r te n appear s i n th e stranges t places. These modular functions are as mysterious as the ma n who investigated them , th e mysti c from the East . Perhaps i f we better understoo d the wor k of this Indian genius, w e would understand wh y we live in ou r present universe . The Myster y o f Modula r Function s Srinivasa Ramanujan was the stranges t man i n al l of mathematics, probably in the entire history of science. He has been compare d t o a bursting supernova, illuminatin g the darkest , most profound corner s o f mathematics, before being tragically struck down by tuberculosis at the ag e of 33, like Riemann befor e him . Working in tota l isolation fro m th e mai n currents of his field, he was able to rederive 10 0 years' worth of Western mathematics on hi s own. The traged y of his life i s that much of his work was wasted rediscovering known mathematics. Scattered throughout th e

Superstrings 17

3

obscure equation s i n his notebooks ar e thes e modula r functions , which are amon g th e stranges t eve r foun d i n mathematics . The y reappea r i n the most distant and unrelated branche s o f mathematics. One function, which appear s again an d agai n i n th e theor y o f modular functions , i s today called th e Ramanujan function in his honor. Thi s bizarre function contains a term raise d t o the twenty-fourt h power. In th e wor k of Ramanujan, the numbe r 2 4 appears repeatedly . Thi s is an exampl e o f what mathematicians call magic numbers , which continually appear , wher e w e least expec t them , fo r reason s tha t n o on e understands. Miraculously, Ramanujan's functio n also appears i n strin g theory. Th e numbe r 2 4 appearing i n Ramanujan' s function i s also th e origin o f th e miraculou s cancellation s occurrin g i n strin g theory . I n string theory , eac h o f the 2 4 modes i n th e Ramanuja n function corre sponds t o a physical vibration of the string. Whenever the string executes its complex motions in space-time by splitting and recombining, a large number o f highl y sophisticate d mathematica l identitie s mus t b e satis fied. Thes e ar e precisel y th e mathematica l identitie s discovere d b y Ramanujan. (Sinc e physicists add two more dimensions when they count the tota l numbe r o f vibration s appearin g i n a relativisti c theory , this means tha t space-time mus t have 24 + 2 = 2 6 space-time dimensions.13) When th e Ramanuja n functio n i s generalized , th e numbe r 2 4 i s replaced b y the numbe r 8 . Thus th e critica l number for the superstrin g is 8 + 2 , or 10 . This i s the origi n o f th e tent h dimension . Th e strin g vibrates in te n dimension s becaus e i t requires thes e generalize d Rama nujan functions in order to remain self-consistent. In other words, physicists have not the slightest understanding of why ten and 26 dimensions are singled out as the dimension o f the string. It's as though ther e i s some kind of deep numerology bein g manifeste d i n thes e function s tha t n o on e under stands. It is precisely these magic numbers appearing i n the elliptic modular functio n tha t determine s th e dimensio n o f space-time t o b e ten. In th e fina l analysis , the origi n o f th e ten-dimensiona l theor y i s as mysterious as Ramanujan himself . When asked by audiences wh y nature might exist in ten dimensions, physicists are forced to answer, "We don't know." W e know, in vagu e terms , wh y some dimensio n o f space-time must b e selecte d (o r els e th e strin g canno t vibrat e i n a self-consisten t quantum fashion), but we don't know why these particular numbers ar e selected. Perhap s th e answe r lie s waiting to b e discovere d i n Ramanu jan's lost notebooks .

174 U N I F I C A T I O

N I N TE N D I M E N S I O N S

Reinventing 10 0 Years o f Mathematic s Ramanujan wa s born i n 188 7 i n Erode , India , nea r Madras . Althoug h his family was Brahmin, the highes t o f th e Hind u castes , they were destitute, living off the meage r wages of Ramanujan's father's job as a clerk in a clothing merchant's office . By the ag e o f 10 , it was clear tha t Ramanuja n was not lik e the othe r children. Lik e Riemann before him, he became well known in his village for hi s awesom e calculationa l powers . As a child , he ha d alread y rederived Euler's identity between trigonometric functions and exponentials. In ever y young scientist' s life , ther e i s a turnin g point , a singular event tha t help s t o chang e th e cours e o f his or he r life . Fo r Einstein, it was the fascinatio n o f observing a compass needle . Fo r Riemann, i t was reading Legendre' s boo k o n numbe r theory . Fo r Ramanujan , i t was when h e stumble d o n a n obscure , forgotte n boo k o n mathematic s by George Carr . Thi s book ha s since bee n immortalize d by the fac t tha t it marked Ramanujan' s only known exposure t o modern Western mathe matics. Accordin g t o hi s sister , "I t wa s this boo k whic h awakene d hi s genius. He set himself to establish the formula e given therein. As he was without the ai d of other books, each solutio n was a piece o f research so far a s he was concerned. . . . Ramanujan used t o say that the goddes s of Namakkal inspired hi m with the formulae in dreams. "H Because o f hi s brilliance , h e wa s able t o wi n a scholarshi p t o hig h school. Bu t becaus e h e wa s bored wit h th e tediu m o f classwor k an d intensely preoccupied wit h the equation s tha t were constantl y dancing in hi s head, h e faile d t o ente r hi s senior class , and hi s scholarshi p was canceled. Frustrated, h e ran awa y from home. H e did finally return , but only t o fal l il l and fai l hi s examinations again . With the hel p of friends, Ramanujan managed t o become a low-level clerk i n th e Por t Trus t o f Madras. I t wa s a menia l job, payin g a paltry £20 a year, but i t freed Ramanujan, like Einstein before hi m at the Swis s patent office , t o follo w hi s dream s i n hi s spare time . Ramanujan the n mailed som e o f the result s of his "dreams" t o thre e well-know n Britis h mathematicians, hopin g fo r contac t wit h othe r mathematica l minds . Two of the mathematicians , receiving this letter written by an unknown Indian cler k with no formal education, promptly threw it away. The thir d one wa s th e brillian t Cambridg e mathematicia n Godfre y H . Hardy . Because o f hi s statur e i n England , Hardy wa s accustomed t o receiving crank mai l an d though t diml y of th e letter . Amid th e dens e scribblin g he noticed man y theorems o f mathematics that were already well known.

Superstrings 17

5

Thinking i t th e obviou s work of a plagiarist, h e als o threw it away . Bu t something wasn' t quite right . Something nagge d a t Hardy; he couldn' t help wonderin g abou t thi s strange letter . At dinner tha t night, January 16 , 1913, Hard y and his colleague John Littlewood discusse d thi s od d lette r an d decide d t o tak e a second loo k at it s contents. I t began , innocentl y enough, wit h " 1 beg t o introduc e myself t o you as a clerk i n th e Account s Departmen t of the Port Trus t Office o f Madras on a salary of only 20 pounds pe r annum." 13 But th e letter fro m th e poo r Madra s clerk containe d theorem s tha t were totally unknown to Western mathematicians . In all, it contained 12 0 theorems. Hardy wa s stunned. H e recalle d tha t provin g som e o f thes e theorem s "defeated m e completely." H e recalled , " I ha d neve r seen anythin g in the leas t like them before. A single look at them i s enough t o show that they coul d onl y b e writte n dow n b y a mathematicia n o f th e highes t class."16 Litdewood an d Hard y reache d th e identica l astounding conclusion : This was obviously the wor k of a genius engaged i n rederiving 100 years of European mathematics . "H e ha d bee n carryin g an impossible hand icap, a poor an d solitar y Hindu pittin g hi s brains agains t th e accumu lated wisdo m of Europe," recalle d Hardy. 17 Hardy sen t fo r Ramanuja n and , afte r muc h difficulty , arrange d fo r his stay in Cambridge i n 1914 . For the first time, Ramanujan could communicate regularl y with hi s peers, th e communit y of European mathe maticians. Then bega n a burst o f activity : 3 short, intens e year s of collaboration wit h Hardy at Trinity College in Cambridge . Hardy later trie d t o estimate th e mathematica l skil l tha t Ramanujan possessed. H e rate d Davi d Hilbert, universall y recognized a s one o f th e greatest Wester n mathematician s o f the nineteent h century , an 80 . To Ramanujan, h e assigne d a 100 . (Hard y rated himsel f a 25.) Unfortunately, neithe r Hard y nor Ramanuja n seemed intereste d i n the psycholog y o r thinkin g proces s b y whic h Ramanuja n discovere d these incredibl e theorems , especiall y when thi s floo d o f material cam e pouring ou t o f hi s "dreams " wit h suc h frequency . Hard y noted , "I t seemed ridiculou s t o worr y him abou t ho w he ha d foun d thi s or tha t known theorem, when he was showing me half a dozen new ones almost every day." 18 Hardy vividly recalled , I remember going to see him onc e when h e wa s lying ill in Putney . I ha d ridden in taxi-cab No. 1729, and remarke d tha t the numbe r seemed to be

176 U N I F I C A T i O N I N T E N D I M E N S I O N

S

rather a dull one, and tha t I hoped tha t it was not a n unfavorabl e omen. "No," h e replied, "it is a very interesting number; it is the smallest number expressible as a sum o f two cubes in tw o different ways." 19

(It i s the su m o f 1 X 1 X 1 and 1 2 X 1 2 X 12 , and als o the su m o f 9 X 9 X 9 and 10X 10 X 10. ) On the spot, he could recite complex theorems in arithmetic that would require a modern compute r t o prove . Always in poor health, the austerit y of the war-tor n British economy prevented Ramanuja n fro m maintainin g his strict vegetarian diet , an d he wa s constantly in an d ou t o f sanitariums . After collaboratin g wit h Hardy for 3 years, Ramanujan fell il l and neve r recovered . Worl d War 1 interrupted trave l between Englan d an d India , an d i n 191 9 h e finally managed t o return home , where he die d a year later.

Modular Functions

Ramanujan's legac y is his work, which consists of 4,000 formulas on 40 0 pages fillin g thre e volume s of notes, al l densely packed wit h theorem s of incredible power but without any commentary or, which is more frus trating, an y proof . I n 1976 , however , a ne w discover y was made. On e hundred an d thirt y pages o f scrap paper , containin g th e outpu t o f the last year of his life, was discovered b y accident in a box at Trinity College. This is now called Ramanujan's "Lost Notebook." Commentin g on th e Lost Notebook , mathematicia n Richar d Aske y says , "Th e wor k o f tha t one year , while he was dying, was the equivalen t of a lifetime o f work for a very great mathematician. What he accomplishe d wa s unbelievable. If it were a novel, nobody would believe it. " T o underscor e th e difficult y of their arduous tas k of deciphering th e "notebooks, " mathematicians Jonathan Borwein and Pete r Borwei n have commented, "T o ou r knowl edge no mathematical redactio n of this scope or difficulty has ever been attempted."20 Looking at the progressio n o f Ramanujan's equations, it's as though we have been traine d for years to listen to the Western musi c of Beethoven, an d the n suddenl y we are expose d t o anothe r typ e o f music , an eerily beautifu l Easter n musi c blending harmonie s an d rhythm s never heard befor e i n Wester n music . Jonathan Borwei n says , "H e seem s t o have functioned in a way unlike anybody else we know of. He ha d suc h a feel fo r things that they just flowed ou t o f his brain. Perhaps he didn' t see them i n any way that's translatable. It's like watching somebody at a feast yo u haven't been invite d to. "

Superstrings 17

7

As physicist s know, "accidents " d o no t appea r withou t a reason . When performin g a lon g an d difficul t calculation , an d the n suddenl y having thousands o f unwanted term s miraculously add up t o zero, physicists know that thi s does no t happe n without a deeper, underlyin g reason. Today, physicists know that these "accidents" are an indication that a symmetr y is a t work . Fo r strings , th e symmetr y is calle d conforma l symmetry, the symmetr y of stretching and deformin g the string' s world sheet. This is precisely where Ramanujan's work comes in. In order to protect the origina l conformal symmetry from being destroyed by quantum theory, a numbe r o f mathematical identitie s mus t b e miraculousl y satisfied. Thes e identitie s are precisel y the identitie s of Ramanujan's modular function. In summary , we have said tha t ou r fundamenta l premis e i s that th e laws of nature simplif y whe n expressed i n higher dimensions . However, in ligh t o f quantum theory , we must ho w amend thi s basic theme. Th e correct statemen t shoul d no w read : Th e law s o f natur e simplif y whe n self-consistently expresse d in higher dimensions. The addition o f the word self-consistently i s crucial . This constrain t force s u s t o us e Ramanujan' s modular functions , which fixes the dimensio n o f space-time t o be ten . This, in turn , ma y give us the decisiv e clue t o explai n the origi n o f th e universe. Einstein often asked himself whether God had any choice in creating the universe . According to superstring theorists, once we demand a unification of quantum theor y and genera l relativity , Go d ha d n o choice . Self-consistency alone , the y claim, must have forced Go d t o creat e th e universe a s he did . Although the mathematical sophistication introduced b y superstring theory has reached dizzyin g heights and has startled the mathematicians, the critic s o f th e theor y still poun d i t a t it s weakest point. An y theory, they claim , mus t b e testable . Sinc e an y theor y define d a t th e Planc k energy of 10 19 billion electron volt s is not testable , superstring theory is not reall y a theory at all! The mai n problem, a s we have pointed out, is theoretical rather tha n experimental. If we were smart enough, w e could solve the theory exactly and fin d th e tru e nonperturbative solutio n of the theory . However, this does no t excus e u s fro m findin g som e mean s b y which t o verif y th e theory experimentally. To tes t the theory , we must wait for signals from the tent h dimension .

8 Signals from the Tenth Dimension How strange it would be if the fina l theor y were to be discovered in our lifetimes ! The discovery of the final law s of natur e will mar k a discontinuit y i n huma n intellectua l history , th e sharpest that has occurred since the beginning o f modern science i n th e seventeent h century . Can w e now imagin e wha t that would be like ? Steven Wcinberg

Is Beaut y a Physical Principle ?

A

LTHOUGH superstring theory give s us a compelling formulation of the theor y of the universe , the fundamental problem i s that an experimental tes t of the theor y seems beyond ou r present-da y technology. In fact , th e theor y predicts tha t th e unificatio n of all forces occur s at th e Planc k energy , o r 10 19 billio n electro n volts , which i s abou t 1 quadrillion time s larger tha n energie s currentl y available in ou r accel erators. Physicist Davi d Gross , commentin g o n th e cos t o f generatin g thi s fantastic energy , says , "Ther e i s not enoug h mone y in th e treasurie s of all the countrie s i n th e world put together . It' s trul y astronomical." 1 This i s disappointing, becaus e i t mean s tha t experimenta l verifica tion, the engine that drives progress in physics, is no longer possible with our curren t generation o f machines or with any generation o f machines

178

Signals fro m th e Tenth Dimension 17

9

in the conceivabl e future. This , in turn, means tha t the ten-dimensiona l theory is not a theor y i n th e usua l sense, becaus e i t is untestable given the presen t technologica l stat e o f our planet . We are the n lef t wit h th e question: Is beauty, by itself, a physical principle tha t can be substituted for th e lac k of experimental verification? To some , th e answe r is a resoundin g no . The y derisivel y cal l thes e theories "theatrical physics" o r "recreationa l mathematics." The mos t caustic of the critic s is Nobel Prize winner Sheldon Glasho w of Harvard University. He has assumed th e rol e o f gadfly i n this debate, leading th e charge agains t the claims of other physicists that higher dimension s may exist. Glasho w rails against these physicists , comparing th e curren t epidemic t o th e AID S virus; tha t is , it's incurable . H e als o compare s th e current bandwagon effect with former President Reagan's Star Wars program: Here's a riddle : Nam e tw o grand design s tha t ar e incredibl y complex , require decades o f researc h t o develop , an d ma y never wor k i n th e rea l world? Star s Wars and strin g theory . . . . Neither ambitio n ca n b e accom plished wit h existing technology, and neithe r ma y achieve its stated objectives. Both adventure s ar e costl y in terms of scarce huma n resources . And, in bot h cases , the Russian s are tryin g desperately to catch up. 2

To sti r up mor e controversy , Glashow even penned a poem, which ends: The Theor y of Everything, if you dare to be bold , Might be somethin g more tha n a string orbifold. While some o f your leaders hav e got old and sclerotic , Not to be trusted alon e wit h thing s heterotic, Please hee d ou r advic e that you are not smitten — The Boo k is not finished , th e las t word i s not Witten. 3

Glashow has vowed (unsuccessfully ) t o kee p these theorie s out o f Harvard, where he teaches. But he does admit that he is often outnumbere d on thi s question. He regrets , "I find mysel f a dinosau r in a worl d of upstart mammals.'" 1 (Glashow' s views are certainl y not share d by other Nobel laureates, such as Murray Gell-Mann and Steve n Weinberg. Physicist Weinberg , i n fact , says , "Strin g theor y provides our onl y present source of candidates for a final theory—how could anyon e expect that many of the brightes t young theorists would not work on it?" 5) To understan d th e implication s of this debate concernin g the uni-

180 U N I F I C A T I O N I N T E N D I M E N S I O N

S

fication of all forces, and als o the problem s wit h its experimental verification, i t is instructive to consider th e followin g analogy, the "parabl e of the gemstone. " In th e beginning , le t u s say , was a gemstone o f great beauty , which was perfectly symmetrical in thre e dimensions . However , this gemston e was unstable. One day , it burst apart and sent fragments in all directions; they eventuall y rained dow n on th e two-dimensiona l world of Flatland. Curious, th e resident s o f Flatland embarke d o n a ques t t o reassembl e the pieces. They called th e original explosio n th e Big Bang, but did no t understand why these fragments were scattered throughou t thei r world. Eventually, two kinds of fragments were identified. Some fragments were polished an d smoot h o n on e side , an d Flatlander s compare d the m t o "marble." Other fragments were entirely jagged an d ugly , with no reg ularity whatsoever, and Flatlander s compare d thes e piece s to "wood." Over th e years , th e Flatlander s divide d int o tw o camps. Th e firs t camp bega n t o piece togethe r th e polishe d fragments . Slowly, some of the polishe d piece s begi n t o fi t together . Marvelin g at ho w these pol ished fragments were being assembled, these Flatlanders were convinced that somehow a powerful ne w geometry must be operating . Thes e Flatlanders calle d thei r partiall y assembled piece "relativity. " The secon d grou p devote d thei r effort s t o assemblin g th e jagged , irregular fragments . They, too, ha d limite d succes s in findin g patterns among thes e fragments . However , the jagge d piece s produce d onl y a larger bu t eve n mor e irregula r clump , which the y calle d th e Standar d Model. No one was inspired by the ugly mass called the Standard Model . After year s o f painstakin g wor k tryin g t o fi t thes e variou s piece s together, however , it appeare d a s though ther e wa s no wa y to pu t th e polished piece s together wit h the jagged pieces . Then on e da y an ingeniou s Flatlande r hi t upo n a marvelou s idea . He declared tha t the tw o sets of pieces could b e reassembled into on e piece if they were moved "up"—tha t is, in something he called the third dimension. Mos t Flatlander s wer e bewildere d b y thi s ne w approach , because n o on e coul d understan d wha t "up" meant . However , he was able to show by computer tha t the "marble " fragments could be viewed as outer fragment s of some object , and wer e hence polished , while the "wood" fragment s were th e inne r fragments . When bot h set s of frag ments were assembled i n the thir d dimension, the Flatlanders gasped at what wa s revealed i n th e computer : a dazzlin g gemstone wit h perfec t three-dimensional symmetry . I n on e stroke , th e artificia l distinctio n between th e tw o sets of fragments was resolved b y pure geometry . This solution, however, left several questions unanswered. Some Flat-

Signals from th e Tenth Dimension 18

1

landers still wanted experimental proof, notjust theoretical calculations, that the pieces could really be assembled into this gemstone. This theory gave a concrete numbe r fo r the energ y it would take to build powerfu l machines that could hoist these fragments "up" of f Flatland and assemble th e piece s in three-dimensiona l space . But the energ y required was about a quadrillion times the largest energy source availabl e to the Flatlanders. For some , th e theoretica l calculatio n wa s sufficient . Eve n lacking experimental verification , they felt tha t "beauty " wa s more tha n suffi cient to settle the question of unification. History had alway s shown, they pointed out , tha t th e solutio n to the mos t difficult problem s in natur e had been the ones with the most beauty. They also correctly pointed ou t that th e three-dimensiona l theor y had n o rival. Other Flatlanders , however , raised a howl. A theory tha t cannot b e tested i s not a theory , they fumed. Testing thi s theory would drain th e best mind s an d wast e valuable resource s o n a wild-goos e chase, the y claimed. The debat e i n Flatland , as well a s in th e rea l world, will persis t fo r some time, which is a good thing. As the eighteenth-century philosopher Joseph Joubert once said, "I t i s better t o debate a question without settling it than t o settle a question without debating it. " The Superconducting Supercollider: Windo w o n Creation

The eighteenth-centur y Englis h philosophe r Davi d Hume , wh o wa s famous for advancing the thesi s that every theory must be grounded on the foundatio n o f experiment , wa s at a los s t o explai n ho w on e ca n experimentally verify a theory of Creation. The essenc e of experiment , he claimed , is reproducibility. Unless an experimen t can b e duplicate d over and over , in different location s and a t different time s with the same results, the theory is unreliable. But how can one perform an experiment with Creation itself ? Sinc e Creation, by definition, i s not a reproducible event, Hum e ha d t o conclud e tha t i t i s impossible to verif y an y theory of Creation. Science , h e claimed , can answe r almost all questions concerning the universe except for one, Creation, the only experiment that cannot be reproduced . In some sense, we are encountering a modern version of the problem identified b y Hume i n th e eighteent h century . Th e proble m remain s the same : The energ y necessary to re-create Creatio n exceed s anythin g available o n th e plane t earth . However , althoug h direc t experimenta l

182 U N I F I C A T I O N I N T E N D I M E N S I O N

S

verification o f the ten-dimensiona l theor y in our laboratorie s i s not pos sible, ther e ar e severa l way s t o approac h thi s questio n indirectly . The most logica l approac h wa s to hop e tha t th e superconductin g supercol lider (SSC ) woul d fin d subatomi c particle s tha t show the distinctiv e signature o f th e superstring , suc h a s supersymmetry . Althoug h th e SS C could not hav e probed the Planc k energy, it might have given us strong, indirect evidence o f the correctnes s o f superstring theory . The SS C (killed off by formidable politica l opposition ) woul d hav e been a trul y monstrous machine , th e las t o f its type. When complete d outside Dallas, Texas, aroun d th e yea r 2000 , it would have consisted o f a gigantic tube 5 0 miles in circumference surrounde d b y huge magnets . (If i t wer e centere d i n Manhattan , i t woul d hav e extende d wel l int o Connecticut and Ne w Jersey.) Ove r 3,00 0 full-time an d visitin g scientists and staf f would have conducted experiment s an d analyze d the data fro m the machine . The purpos e o f the SS C was to whi p two beams o f protons aroun d inside thi s tub e unti l the y reached a velocity very close t o th e spee d o f light. Becaus e thes e beam s woul d b e travelin g clockwis e and counter clockwise, i t woul d hav e bee n a simpl e matte r t o mak e the m collid e within the tub e when they reached thei r maximu m energy. The proton s would have smashed into one another a t an energy of 40 trillion electro n volts (TeV) , thereb y generatin g a n intens e burs t o f subatomi c debri s analy/ed by detectors. Thi s kin d o f collision ha s not occurre d sinc e the Big Ban g itsel f (henc e th e nicknam e fo r th e SSC : "windo w o n crea tion"). Amon g th e debris , physicist s hope d t o fin d exoti c subatomi c particles that would have shed ligh t on th e ultimat e form of matter. Not surprisingly, the SSC was an extraordinary engineering and physics project, stretching the limit s of known technology. Because the mag netic fields necessar y t o ben d th e proton s an d antiproton s withi n th e tube are so exceptionally large (o n the order of 100,000 times the earth's magnetic field), extraordinary procedures woul d have been necessar y to generate an d maintai n them . Fo r example , t o reduce th e heatin g an d electrical resistanc e withi n th e wires , th e magnet s woul d hav e bee n cooled dow n nearl y to absolute zero . The n the y would have bee n spe cially reinforce d becaus e th e magneti c field s ar e s o intense tha t other wise the y would have warped th e meta l o f the magne t itself . Projected t o cost $1 1 billion, th e SS C became a prized plu m an d a matter o f intens e politica l jockeying. I n th e past , th e site s fo r ato m smashers were decided b y unabashed politica l horse trading . For example, th e stat e o f Illinoi s was able t o lan d th e Fermila b accelerato r i n Batavia, just outside Chicago, because (accordin g to Physics Today) Pres -

Signals from th e Tenth Dimension 18

3

ident Lyndon Johnson neede d Illinoi s senator Everet t Dirkson's crucial vote on th e Vietnam War. The SS C was probably no different . Althoug h many states vigorously competed fo r the project , it probably came as no surprise tha t in 198 8 th e grea t state o f Texas lande d th e SSC , especially when bot h th e president-elec t o f the Unite d State s an d th e Democrati c vice-presidential candidat e cam e fro m Texas . Although billions of dollars have been spent on th e SSC , it will never be completed . T o th e horro r o f th e physic s community, th e Hous e o f Representatives voted i n 199 3 t o cancel th e projec t completely . Intense lobbying faile d t o restor e fundin g fo r th e project . T o Congress , a n expensive ato m smashe r ca n be seen i n two ways. It can be a juicy plum, generating thousand s o f jobs and billion s of dollars i n federal subsidies for th e stat e that has it. Or it can be viewed as an incredible boondoggle , a wast e o f money tha t generate s n o direc t consume r benefits . I n lea n times, they argue, a n expensiv e to y for high-energy physicists is a luxury the countr y cannot afford . (I n al l fairness, though , fundin g for the SSC project mus t be pu t int o prope r perspective. Sta r Wars funding for just I yea r cost s $4 billion. It cost s abou t $ 1 billion t o refurbis h a n aircraf t carrier. A single space-shuttle missio n cost s $1 billion. And a single B-2 stealth bombe r cost s almost $1 billion.) Although th e SS C is dead, wha t might we have discovered wit h it ? At the ver y least, scientists hoped t o fin d exoti c particles , such a s the mysterious Higg s particl e predicte d b y the Standar d Model . I t is the Higg s particle tha t generates symmetr y breaking and i s therefore th e origi n of the mas s of the quarks . Thu s w e hoped tha t th e SS C would have found the "origi n o f mass." Al l objects surrounding u s tha t hav e weight owe their mas s to the Higg s particle . The bettin g amon g physicists , however, wa s that ther e wa s an eve n chance tha t th e SS C would fin d exoti c particle s beyon d th e Standar d Model. (Possibilitie s include d "Technicolor " particles , whic h li e just beyond th e Standar d Model , o r "axions, " whic h ma y help t o explai n the dar k matte r problem.) Bu t perhaps th e mos t exciting possibility was the sparticles , which ar e th e supersymmetri c partner s o f ordinar y par ticles. The gravitino , for example , i s the supersymmetri c partne r o f th e graviton. The supersymmetri c partners o f the quark and lepton , respec tively, are the squar k an d th e slepton . If supersymmetric particles are eventuall y discovered, then ther e is a fighting chanc e tha t w e will b e seein g th e remnant s o f th e superstrin g itself. (Supersymmetry , a s a symmetr y of a fiel d theory , was first discovered i n superstrin g theor y in 1971 , eve n befor e th e discover y of super gravity. I n fact , th e superstrin g i s probably th e onl y theor y i n whic h

184 U N I F I C A T I O

N IN TE N D I M E N S I O N S

supersymmetry and gravit y can b e combine d i n a totally self-consistent way.) An d even though th e potential discovery of sparticles will not prov e the correctnes s o f superstrin g theory , i t will hel p t o quie t th e skeptics who hav e sai d tha t ther e i s no t on e shre d o f physica l evidenc e fo r superstring theory .

Signals fro m Oute r Space Since th e SS C will neve r be built , and henc e wil l neve r detec t particle s that ar e low-energ y resonances o f th e superstring , the n anothe r possi bility is to measure the energ y of cosmic rays, which are highl y energetic subatomic particle s whose origin i s still unknown, but mus t lie deep in outer spac e beyon d ou r galaxy . For example , althoug h n o on e knows where they come from, cosmic rays have energies much larger than anything foun d i n our laboratories . Cosmic rays , unlike th e controlle d ray s produced i n atom smashers , have unpredictabl e energie s an d canno t produc e precis e energie s o n demand. In som e sense , it' s like trying to put ou t a fire by either usin g hose wate r o r waitin g for a rainstorm . Th e hos e wate r i s much mor e convenient: W e ca n tur n i t o n an y tim e w e please, w e can adjus t th e intensity of the water at will, and al l the water travels at the same uniform velocity. Water from a fire hydrant therefore correspond s t o producin g controlled beam s i n ato m smashers. However , water fro m a rainstor m may be much more intense and effectiv e tha n water from a fire hydrant. The problem , o f course, i s that rainstorms , like cosmic rays, are unpre dictable. You cannot regulate th e rainwater, nor can you predict its velocity, whic h may fluctuate wildly. Cosmic ray s were firs t discovere d 8 0 years ag o i n experiment s per formed b y the Jesuit priest Theodor Wulf atop th e Eiffe l Tower in Paris. From th e 1900 s to the 1930s , courageous physicist s sailed in balloons o r scaled mountain s t o obtain th e bes t measurements o f cosmic rays . But cosmic-ray research bega n t o fad e durin g th e 1930s , whe n Ernes t Lawrence invente d th e cyclotro n an d produce d controlle d beam s i n th e laboratory mor e energetic tha n mos t cosmic rays. For example, cosmi c rays, which are a s energetic as 100 million electron volts, are as common as rain drops ; the y hit th e atmospher e o f the eart h a t th e rat e o f a few per squar e inc h pe r second . However , Lawrence' s inventio n spawne d giant machines tha t coul d excee d tha t energy by a factor of 10 to 100 . Cosmic-ray experiments , fortunately , hav e change d dramaticall y since Father Wulf first placed electrifiedjars o n the Eiffe l Tower. Rockets

Signals from th e Tenth Dimension 18

5

and eve n satellite s ca n no w sen d radiatio n counter s hig h abov e th e earth's surface , s o tha t atmospheri c effect s ar e minimized . When a highly energetic cosmic ray strikes the atmosphere, it shatters the atoms in it s wake. These fragments , in turn , create a shower of broken atoms, or ions , whic h ca n the n b e detecte d o n th e groun d b y thi s serie s o f detectors. A collaboratio n betwee n th e Universit y o f Chicag o and th e University of Michigan ha s inaugurate d th e mos t ambitiou s cosmic-ray project yet, a vast array of 1,08 9 detector s scattered ove r about a square mile of desert, waiting for the cosmic-ra y showers to trigger them. These detectors ar e locate d i n a n ideal , isolate d area : th e Dugwa y Provin g Grounds, 8 0 miles southwest of Salt Lake City, Utah. The Uta h detector is sensitive enough t o identify th e poin t o f origin of some of the mos t energetic cosmic rays . So far, Cygnus X-3 and Her cules X-l hav e been identified as powerful cosmic-ra y emitters. They are probably large , spinnin g neutro n stars , o r eve n blac k holes , tha t ar e slowly eating up a companion star , creating a large vortex of energy and spewing giganti c quantitie s o f radiatio n (fo r example , protons ) int o outer space . So far, the mos t energetic cosmic ray ever detected ha d a n energy of 20 10 electro n volts . Thi s figur e i s a n incredibl e 1 0 millio n time s th e energy that would have been produce d i n the SSC . We do not expec t to generate energie s approachin g thi s cosmi c energ y wit h ou r machine s within th e century . Although thi s fantasti c energ y i s stil l 10 0 million times smaller tha n th e energ y necessary to probe th e tent h dimension, we hope tha t energie s produce d dee p within blac k holes i n ou r galaxy will approac h th e Planc k energy . Wit h large , orbitin g spacecraft , w e should be able to probe deeper into the structure of these energy sources and detec t energie s even larger tha n this . According t o on e favore d theory , th e larges t energ y sourc e within our Milk y Way galaxy—far beyon d anything produced by Cygnus X-3 or Hercules X-l—lies at th e center , which may consist of millions of black holes. So , because th e SS C was canceled b y Congress, we may find tha t the ultimat e probe fo r explorin g the tent h dimensio n ma y lie in oute r space.

Testing the Untestabl e Historically speaking, there have been many times when physicists have solemnly declared certai n phenomen a t o be "untestable " o r "unprov able." Bu t there is another attitud e tha t scientists can take concernin g

186 U N I F I C A T I O N I N T E N D I M E N S I O N

S

the inaccessibilit y of the Planc k energy—unforeseen breakthrough s will make indirect experiments possibl e near th e Planck energy. In th e nineteent h century , some scientist s declared tha t th e com position o f the star s would forever b e beyond the reac h of experiment. In 1825, the French philosopher and social critic Auguste Comte, writing in Cows de philosophie, declared tha t we would never know the stars other than as unreachable points of light in the sky because of their enormou s distance from us. The machine s o f the nineteent h century , or an y century, he argued, wer e not powerful enoug h t o escape from th e earth an d reach th e stars. Although determinin g wha t the star s were made o f seemed beyon d the capabilitie s of an y science, ironicall y at almos t th e sam e time , th e German physicis t Joseph vo n Fraunhofe r wa s doing just that . Usin g a prism and spectroscope , h e could separate th e white light emitted from the distant stars and determin e th e chemical composition o f those stars. Since each chemical within the stars emits a characteristic "fingerprint, " or spectrum o f light, it was easy for Fraunhofer t o perform th e "impos sible" and t o determine tha t hydrogen i s the most abundant element in the stars. This, i n turn, inspired poe t Ian D . Bush to write: Twinkle, twinkle little star I don't wonder wha t you are , For b y spectroscopic ken , I know that you are hydrogen.' '

Thus although the energ y necessary to reach th e stars via rockets was far beyond anythin g available to Comte (or , for that matter, anything available t o moder n science) , the crucia l ste p di d no t involv e energy . The key observation wa s that signals from th e stars , rather tha n direc t measurement, were sufficient t o solve the problem . Similarly , we can hop e that signal s from th e Planc k energy (perhap s fro m cosmi c rays or per haps an as yet unknown source), rather than a direct measurement from large ato m smashers , ma y b e sufficien t t o prob e th e tent h dimension. Another example of an "untestable" idea was the existence of atoms. In th e nineteent h century , the atomi c hypothesis proved t o be the decisive ste p i n understandin g th e law s o f chemistr y and thermodynamics . However, man y physicist s refused t o believ e tha t atom s actuall y exist. Perhaps the y were just a mathematical device that, by accident, gave the correct descriptio n o f the world . Fo r example , th e philosophe r Erns t

Signals from th e Tenth Dimension 18

7

Mach di d no t believ e in th e existenc e o f atoms , othe r tha n a s a calcu lational tool . (Eve n today , we are stil l unabl e t o tak e direc t picture s of the ato m becaus e o f th e Heisenber g Uncertaint y Principle, althoug h indirect method s no w exist.) I n 1905 , however , Einstein gav e the mos t convincing, although indirect , evidence of the existenc e o f atoms when he showe d tha t Brownia n motion (tha t is , the rando m motio n o f dust particles suspended i n a liquid) ca n b e explaine d a s random collision s between th e particles an d atom s i n the liquid. By analogy , w e migh t hop e fo r experimenta l confirmatio n o f th e physics of the tent h dimensio n usin g indirect methods tha t have not yet been discovered. Instead o f photographing the object we desire, perhaps we should b e satisfie d wit h a photograph o f its "shadow." The indirec t approach woul d be t o examine carefull y low-energ y data fro m a n ato m smasher, and tr y to see if ten-dimensional physics affects th e data in some way. The thir d "untestable " ide a i n physics was the existenc e of the elu sive neutrino . In 1930 , physicis t Wolfgang Pauli hypothesize d a new, unseen particle calle d th e neutrino in orde r t o accoun t fo r th e missin g componen t of energy in certain experiment s on radioactivit y that seemed t o violate the conservatio n o f matter an d energy . Pauli realized, though, tha t neutrinos woul d b e almos t impossibl e to observ e experimentally , becaus e they woul d interac t s o weakly , an d henc e s o rarely , wit h matter . Fo r example, if we could construct a solid block of lead that stretched several light-years from our sola r syste m to Alpha Centauri an d place d i t in th e path o f a beam o f neutrinos, som e would still com e ou t th e othe r end . They ca n penetrat e th e eart h a s thoug h i t doesn' t eve n exist , and , i n fact, trillion s of neutrinos emitte d fro m th e su n ar e alway s penetratin g your body, even at night. Pauli admitted, "I have committed th e ultimate sin, I hav e predicte d th e existenc e o f a particl e tha t ca n neve r b e observed."7 So elusive and undetectabl e wa s the neutrin o tha t it even inspired a poem b y John Updike, calle d "Cosmi c Gall" : Neutrinos, they are very small. They have n o charg e and hav e n o mas s And do not interac t at all. The eart h is just a silly ball To them , through which the y simply pass, Like dustmaids down a drafty hal l Or photon s though a sheet of glass. They snub the mos t exquisite gas,

188 U

N I F I C A T I ON I N T E N D

1 M E N S I O NS

Ignore th e mos t substantial wall, Cold-shoulder stee l and soundin g brass, Insult the stallio n in hi s stall, And scornin g barriers o f class, Infiltrate yo u and me ! like tall And painles s guillotines, they fall Down throug h ou r head s int o th e grass . At night, the y enter a t Nepal And pierc e th e love r and hi s lass From underneat h th e bed—yo u call It wonderful; I call it crass." Although th e neutrino , becaus e i t barel y interacts with other mate rials, was once considere d th e ultimat e "untestable" idea, today we regularly produc e beam s o f neutrino s i n ato m smashers , perform experi ments wit h th e neutrino s emitte d fro m a nuclea r reactor , an d detec t their presence within mines far below the earth's surface. (I n fact, whe n a spectacula r supernov a li t up th e sk y in th e souther n hemispher e i n 1987, physicist s noticed a burs t o f neutrino s streamin g throug h thei r detectors deep in these mines. This was the first time that neutrino detectors were used t o make crucial astronomical measurements.) Neutrinos, in 3 short decades , hav e bee n transforme d fro m a n "untestable " ide a into on e o f the workhorse s of modern physics.

The Proble m I s Theoretical , No t Experimenta l Taking th e lon g vie w o n th e histor y of science, perhap s ther e i s some cause fo r optimism . Witten i s convinced tha t scienc e wil l som e da y b e able to probe dow n to Planck energies. H e says, It's no t alway s so easy to tel l which ar e th e eas y questions an d whic h ar e the har d ones . I n th e 19t h century, the questio n of why water boils at 10 0 degrees wa s hopelessly inaccessible. If you told a 19th-century physicist that by th e 20t h centur y yo u woul d b e abl e t o calculat e this , i t woul d hav e seemed lik e a fairy tale. . . . Quantum field theory is so difficult tha t nobod y fully believe d i t for 2 5 years. In hi s view, "goo d ideas alway s ge t tested." 9 The astronome r Arthu r Eddingto n eve n questione d whethe r scientists wer e no t overstatin g th e cas e whe n the y insiste d tha t everythin g should b e tested. He wrote: " A scientist commonl y professe s t o base his

Signals from th e Tenth Dimension 18

9

beliefs o n observations , no t theories . . . . I have neve r com e acros s any one wh o carrie s thi s profession int o practice. . . . Observation i s not sufficient . . . theory has an important share in determining belief." 10 Nobel laureate Pau l Dira c said i t even mor e bluntly , "It i s more importan t t o have beauty in one's equations tha n t o have them fit experiment."" Or, in th e word s o f CER N physicis t John Ellis , "i n th e word s o f a cand y wrapper I opened a few years ago: 'I t is only the optimist s who achiev e anything in this world.' " Nonetheless, despit e argument s tha t uphold a certain degre e o f optimism , th e experimenta l situatio n look s bleak . I share, alon g wit h the skeptics , the ide a that th e bes t we can hope for is indirect tests often-dimensional theor y into the twenty-firs t century. This is because, i n th e fina l analysis , this theory is a theor y of Creation, an d hence testin g i t necessarily involves re-creating a piece o f the Bi g Bang in ou r laboratories . Personally, I don' t thin k tha t w e hav e t o wai t a centur y unti l ou r accelerators, spac e probes , an d cosmic-ra y counter s wil l b e powerfu l enough t o probe the tent h dimensio n indirectly. Within a span of years, and certainl y within th e lifetim e of today' s physicists , someone wil l b e clever enough t o either verify o r disprove th e ten-dimensiona l theory by solving th e fiel d theor y o f string s o r som e othe r nonperturbativ e for mulation. The proble m i s thus theoretical , no t experimental . Assuming that som e bright physicist solves the field theory of strings and derive s the know n properties o f our universe , there i s still th e prac tical proble m o f whe n w e might b e abl e t o harnes s th e powe r o f th e hyperspace theory . There are tw o possibilities: 1. Wai t until ou r civilizatio n attains th e abilit y to maste r energie s trillions of times larger tha n anythin g we can produc e today 2. Encounte r extraterrestria l civilization s that hav e mastere d th e art o f manipulating hyperspac e We recal l tha t i t too k abou t 7 0 years, between th e wor k of Farada y and Maxwel l to th e wor k o f Edison an d hi s co-workers , to exploi t th e electromagnetic forc e fo r practica l purposes . Ye t modern civili/atio n depends crucially on th e harnessin g o f this force. The nuclea r forc e was discovered nea r th e tur n o f th e century , an d 8 0 years late r w e still d o not hav e th e mean s t o harnes s i t successfull y wit h fusion reactors . Th e next leap , t o harnes s th e powe r o f th e unifie d fiel d theory , require s a much greate r jump i n ou r technology , bu t on e tha t wil l probabl y hav e vastly more important implications. The fundamenta l proble m i s that w e are forcin g superstrin g theor y

190 U N I F I C A T I O N I N T E N D I M E N S I O N

S

to answer questions abou t everyda y energies, whe n it s "natural home " lies at th e Planc k energy. This fabulous energy was released onl y at th e instant of Creation itself . In other words, superstring theor y is naturally a theor y o f Creation. Lik e the cage d cheetah , w e are demandin g tha t this superb animal dance and sing for our entertainment. The real hom e of the cheeta h i s the vas t plains of Africa. Th e rea l "home" of superstring theor y i s th e instan t of Creation . Nevertheless , give n th e sophisti cation o f our artificia l satellites , ther e i s perhaps on e las t "laboratory " in which we may experimentally probe the natura l hom e o f superstring theory, and thi s is the ech o o f Creation !

9 Before Creatio n In the beginning, was the great cosmic egg. Inside the egg was chaos, and floatin g i n chao s was P'an Ku , th e divin e Embryo. P'ari K u myth (China , third century)

If God create d th e world , where was He befor e Creation ? . . . Know tha t th e worl d i s uncreated , a s tim e itsel f is , without beginning and end . Mahapurana (India , ninth century)

* * T | ID God nave a mother?" -L/ Children , when told tha t God made th e heavens and the earth, innocentl y ask whether Go d had a mother. This deceptively simple question has stumped the elders of the church an d embarrassed th e finest theologians , precipitatin g som e o f th e thornies t theologica l debates ove r the centuries. All the great religions have elaborate mythologies surroundin g th e divin e ac t o f Creation , bu t non e o f the m ade quately confronts th e logica l paradoxes inheren t i n th e question s tha t even childre n ask. God may have created th e heavens and th e eart h i n 7 days, but what happened before th e first day? If one concede s tha t God had a mother, then on e naturall y asks whethe r she , too , ha d a mother , an d s o on , forever. However , if God did no t hav e a mother, the n thi s answer raises even mor e questions : Where di d Go d com e from ? Wa s God alway s i n existence sinc e eternity, or i s God beyon d tim e itself ?

191

192 U N I F I C A T I O N I N T E N D I M E N S I O N

S

Over the centuries, even great painters commissione d by the churc h grappled wit h thes e ticklis h theologica l debate s i n thei r work s of art : When depicting God or Adam and Eve , do you give them belly buttons? Since th e nave l marks th e poin t o f attachment o f th e umbilica l cord , then neither God nor Adam and Eve could be painted with belly buttons. For example, Michelangelo faced this dilemma in his celebrated depic tion o f Creation an d th e expulsio n o f Adam and Ev e from th e Garde n of Eden whe n he painte d th e ceilin g of the Sistin e Chapel. The answer to this theological question is to be found hanging in any large museum: God and Adam and Ev e simply have no belly buttons, because they were the first .

Proofs o f the Existenc e of Go d Troubled by the inconsistencies in church ideology, St. Thomas Aquinas, writing in the thirteenth century, decided t o raise the level of theological debate fro m th e vaguenes s of mythology to th e rigo r o f logic. He pro posed t o solve these ancien t questions in hi s celebrated "proof s o f th e existence of God. " Aquinas summarized his proofs in th e followin g poem : Things are i n motion, hence there is a first mover Things are caused , hence there is a first cause Things exist, hence there is a creator Perfect goodnes s exists, hence it has a source Things are designed, hence they serve a purpose.1 (The first three lines are variations of what is called the cosmological proof; the fourth argues on moral grounds; and th e fifth is called the teleological proof. Th e mora l proo f i s by far th e weakest , because moralit y can b e viewed in term s of evolving social customs.) Aquinas's "cosmological" and "teleological" proofs of the existence of God hav e been use d b y the churc h fo r th e pas t 700 years to answer this stick y theological question . Although thes e proof s have since been shown to be flawe d in ligh t of the scientifi c discoverie s made ove r the past 7 centuries, they were quite ingeniou s fo r their tim e and sho w the influence o f the Greeks , who were the first to introduce rigo r into thei r speculations about nature . Aquinas began th e cosmologica l proo f b y postulating tha t God was the Firs t Move r an d Firs t Maker . H e artfull y dodge d th e questio n o f

Before Creation 19

3

"who mad e God " b y simply asserting that the question mad e n o sense . God had n o make r becaus e h e wa s the First . Period. Th e cosmologica l proof state s tha t everythin g that moves mus t have had somethin g pus h it, which in turn mus t have had somethin g pus h it , and s o on. But what started th e first push? Imagine, for the moment, idl y sitting in the park and seeing a wagon moving i n fron t o f you . Obviously , you think , ther e i s a youn g child pushing th e wagon . You wait a moment , onl y t o fin d anothe r wago n pushing the first wagon. Curious, you wait a bit longer fo r the child, but there i s a third wagon pushing the first two wagons. As time goes by, you witness hundreds o f wagons, each one pushin g the others, with no child in sight . Puzzled, you loo k ou t int o th e distance . You are surprise d t o see a n infinit e sequenc e o f wagon s stretchin g int o th e horizon , eac h wagon pushing th e others, wit h no chil d at all. If it takes a child to push a wagon , the n ca n a n infinit e sequenc e o f wagons be pushe d without the Firs t Pusher? Ca n a n infinit e sequenc e o f wagons pus h itself ? No . Therefore, Go d must exist. The ideologica l proo f i s even mor e persuasive . I t state s tha t ther e has to b e a First Designer. For example , imagin e walking on th e sand s of Mars, where the winds and dust storms have worn even the mountain s and gian t craters . Ove r ten s o f million s of years, nothing ha s escape d the corrosive , grinding effect o f the sand storms. Then, to your surprise, you find a beautiful camera lying in the sand dunes. The lens is smoothly polished an d th e shutte r mechanis m delicatel y crafted . Surely , you think, th e sand s of Mars could no t hav e create d suc h a beautiful piece of craftsmanship. You conclude tha t someone intelligent obviously made this camera. Then, afte r wanderin g on th e surfac e of Mars some more , you com e acros s a rabbit . Obviously , the ey e of th e rabbi t i s infinitel y more intricate tha n th e ey e of the camera . Th e muscle s of the rabbit' s eye ar e infinitel y mor e elaborat e tha n th e shutte r o f th e camera . Therefore, th e make r o f thi s rabbi t mus t b e infinitel y mor e advance d than th e make r o f th e camera . Thi s make r mus t therefor e b e God. Now imagine th e machine s o n th e earth . Ther e is no questio n tha t these machines were made by something even greater, suc h as humans. There i s no questio n tha t a human i s infinitely mor e complicated tha n a machine. Therefore, th e person who created us must be infinitely more complicated tha n we are. So therefore God must exist. In 1078 , St . Anselm, the archbisho p o f Canterbury, cooke d u p per haps the most sophisticated proo f o f the existenc e of God, die ontological proof, whic h does no t depen d o n Firs t Movers or Firs t Designers a t all.

194 U N I F I C A T I O

N I N TE N D I M E N S I O N S

St. Anselm claimed tha t he could prov e the existenc e of God from pure logic alone . H e define d Go d a s the mos t perfect, most powerfu l bein g imaginable. It is, however, possible to conceive of two types of God. Th e first God , w e imagine , doe s no t exist . Th e secon d God , w e imagine , actually doe s exis t an d ca n perfor m miracles , suc h a s partin g th e rivers an d raisin g the dead . Obviously , the secon d Go d (wh o exists) is more powerfu l an d mor e perfec t tha n th e firs t Go d (wh o does no t exist). However, we defined God to be the most perfect and powerfu l bein g imaginable. By the definition of God, the second Go d (wh o exists) is the more powerfu l an d more perfect one. Therefore, the secon d God is the one who fits the definition. The first God (wh o does not exist ) is weaker and les s perfect tha n th e secon d God , an d therefor e doe s no t fi t th e definition o f God. Hence Go d mus t exist . In othe r words , if we defin e God as "that being nothing greater tha n which can be conceived," the n God must exist because if he didn't , it' s possible to conceive of a much greater Go d who does exist . This rather ingeniou s proof , unlik e those of St. Thomas Aquinas, is totally independent o f the ac t of Creation an d rests solely on th e definitio n of the perfec t being . Remarkably, these "proofs " o f the existenc e o f God laste d for ove r 700 years, defyin g th e repeate d challenge s o f scientist s and logicians . The reason fo r this is that not enough was known about the fundamental laws o f physic s and biology . In fact , only within th e pas t centur y have new laws o f nature bee n discovere d tha t ca n isolat e th e potentia l flaw s in thes e proofs. The fla w i n th e cosmologica l proof, fo r example, is that th e conser vation of mass and energ y is sufficient t o explain motio n without appealing t o a Firs t Mover . For example , ga s molecules ma y bounce agains t the wall s o f a containe r withou t requiring anyon e o r anythin g t o ge t them moving. In principle, these molecules can move forever, requiring no beginnin g o r end . Thu s ther e i s no necessit y for a Firs t or a Las t Mover as long as mass and energ y are conserved . For th e ideologica l proof , th e theor y o f evolutio n show s that i t is possible to create higher an d mor e complex life forms from more primitive ones through natura l selection and chance. Ultimately, we can trace the origi n o f lif e itsel f bac k t o th e spontaneou s formatio n o f protei n molecules in the early earth's oceans without appealing t o a higher intelligence. Studies performed b y Stanley L. Miller in 195 5 have shown that sparks sen t throug h a flas k containin g methane , ammonia , an d othe r gases foun d i n th e earl y earth's atmospher e ca n spontaneousl y creat e complex hydrocarbon molecules and eventually amino acids (precursor s

Before Creation 19

5

to protei n molecules ) an d othe r comple x organi c molecules . Thu s a First Designer is not necessar y to create th e essential s for life , which can apparently emerge naturall y out o f inorganic chemicals if they are given enough time . And, finally , Immanue l Kant was the first to isolat e th e erro r i n th e ontological proo f afte r centurie s o f confusion . Kan t pointe d ou t tha t stating that an object exists does not make it more perfect . For example, this proof ca n be used to prove the existence of the unicorn. If we define the unicor n t o b e th e mos t perfec t hors e imaginable , an d i f unicorns don't exist , then it' s possibl e t o imagine a unicorn tha t doe s exist . But saying that it exists does not mea n tha t it is more perfec t than a unicor n that does not exist . Therefore, unicorn s do not necessarily have to exist. And neithe r doe s God . Have w e made an y progres s sinc e th e tim e o f St . Thomas Aquinas and St . Anselm? Yes and no . We can say that present-day theories of Creation ar e built on tw o pillars: quantum theor y and Einstein' s theory of gravity. We can say that, for th e first time in a thousand years , religious "proofs " o f the existence o f Go d ar e bein g replace d b y our understandin g o f thermo dynamics and particl e physics . However, by replacing God' s ac t o f Creation with the Big Bang, we have supplanted one problem with another. Aquinas though t h e solve d th e proble m o f what came befor e Go d by defining hi m a s the Firs t Mover. Today, we are stil l strugglin g with th e question o f what happened before th e Bi g Bang. Unfortunately, Einstein' s equation s brea k dow n a t th e enormousl y small distance s an d larg e energie s foun d a t th e origi n o f the universe . At distances on th e order of 10~ 33 centimeter, quantu m effects tak e over from Einstein' s theory . Thu s t o resolv e th e philosophica l question s involving th e beginnin g o f time , w e mus t necessaril y invoke th e ten dimensional theory . Throughout this book, we have emphasized th e fac t tha t th e law s of physics unif y whe n we add highe r dimensions . Whe n studyin g the Bi g Bang, we see th e precis e revers e o f this statement. The Bi g Bang, as we shall see , perhap s originate d i n th e breakdow n o f th e origina l ten dimensional univers e into a four- and a six-dimensional universe . Thu s we can vie w the histor y of the Bi g Bang as the histor y of the breaku p o f ten-dimensional spac e and hence the breakup o f previously unified symmetries. This, in turn , is the them e o f this book i n reverse . It is no wonder, therefore, tha t piecing togethe r th e dynamics of the Big Bang ha s been so difficult . I n effect , b y going backwar d in time , we are reassemblin g th e piece s of the ten-dimensiona l universe .

196 U N I F I C A T I O N I N T E N D I M E N S I O N

S

Experimental Evidenc e for th e Bi g Bang Every year , w e fin d mor e experimenta l evidenc e tha t th e Bi g Ban g occurred roughly 1 5 to 20 billion year s ago. Le t us review some of these experimental results . First, the fact that the stars are receding from us at fantastic velocities has been repeatedl y verifie d b y measuring th e distortio n o f thei r star light (calle d the re d shift) . (Th e starlight of a receding sta r is shifted t o longer wavelengths—that is, toward the red end of the spectrum—in the same way that the whistle of a receding trai n sounds higher than normal when approaching and lowe r when receding. Thi s is called the Dopple r effect. Also , Hubble' s La w states tha t th e farthe r fro m u s th e sta r o r galaxy, the faste r i t is receding from us . This fact, first announced b y the astronomer Edwi n Hubbl e i n 1929 , ha s bee n experimentall y verified over th e pas t 50 years.) We do no t se e any blue shif t o f the distan t galaxies, which would mean a collapsing universe . Second, w e know that th e distributio n o f the chemica l element s in our galax y are i n almos t exact agreement wit h th e predictio n o f heavyelement production i n the Big Bang and i n the stars. In th e original Big Bang, because of the enormous heat , elemental hydrogen nuclei banged into one anothe r a t large enough velocitie s to fuse them , forming a new element: helium. The Bi g Bang theory predicts tha t the rati o of helium to hydroge n i n th e univers e should b e approximatel y 25 % helium t o 75% hydrogen . Thi s agrees wit h th e observationa l resul t fo r th e abun dance o f helium in th e universe. Third, th e earlies t objects in th e univers e date back 1 0 to 1 5 billion years, i n agreemen t wit h th e roug h estimat e fo r th e Bi g Bang. We d o not se e any evidence fo r object s older tha n th e Bi g Bang. Sinc e radio active materials decay (fo r example, via the weak interactions) a t a precisely known rate, it is possible to tell the ag e of an object by calculating the relativ e abundanc e o f certai n radioactiv e materials . Fo r example , half of a radioactive substance called carbon-14 decays every 5,730 years, which allows us to determine th e ag e of archeological artifact s tha t contain carbon. Othe r radioactive element s (lik e uranium-238, with a half life o f over 4 billion years) allo w us to determine the ag e of moon rocks (from th e Apollo mission). The oldes t rocks and meteor s found on eart h date t o abou t 4 to 5 billion years, which is the approximat e ag e of th e solar system . By calculating th e mas s of certain star s whose evolution is known, we can show that th e oldes t star s in ou r galax y date bac k abou t 10 billion years.

Before Creation 19

7

Fourth, an d mos t important , th e Bi g Ban g produce d a cosmi c "echo" reverberatin g throughou t th e univers e that shoul d b e measur able b y our instruments . In fact , Arno Penzias and Rober t Wilson of the Bell Telephone Laboratorie s wo n the Nobe l Prize in 197 8 for detecting this ech o o f th e Bi g Bang, a microwav e radiation tha t permeate s th e known universe . The fac t tha t th e ech o o f the Bi g Bang shoul d be cir culating aroun d th e univers e billions o f years after th e even t was firs t predicted b y George Gamo w and hi s students Ralph Alpher and Rober t Herman, bu t n o on e too k the m seriously . The ver y ide a o f measurin g the ech o o f Creation seeme d outlandis h whe n the y first proposed thi s idea soon afte r Worl d War II. Their logic, however, was very compelling. Any object, when heated, gradually emits radiation. This is the reaso n wh y iron get s red ho t when placed i n a furnace . Th e hotte r th e iron , th e highe r th e frequenc y of radiation i t emits . A precis e mathematica l formula , th e Stefan-Boltz mann law , relates th e frequenc y of ligh t (o r th e color , i n thi s case) t o the temperature . (I n fact , thi s i s how scientist s determine th e surfac e temperature o f a distant star, by examining its color.) Thi s radiation is called blackbody radiation. When th e iro n cools , th e frequenc y o f th e emitte d radiatio n als o decreases, until th e iron no longer emits i n the visible range. The iron returns t o it s normal color , bu t i t continue s t o emi t invisibl e infrared radiation. Thi s i s how th e army' s night glasse s operate i n th e dark . At night, relativel y warm objects such a s enemy soldiers and tan k engine s may b e conceale d i n th e darkness , bu t the y continue t o emi t invisibl e blackbody radiatio n i n th e for m o f infrare d radiation , whic h ca n b e picked u p b y special infrared goggles . Thi s i s also why your sealed ca r gets ho t durin g th e summer . Sunligh t penetrate s th e glas s of your ca r and heat s the interior . As it gets hot, i t begins to emit blackbod y radiation in the form of infrared radiation . However, infrare d radiatio n does not penetrat e glas s very well, and henc e i s trapped insid e your car, dramatically raising its temperature. (Similarly , blackbody radiation drive s the greenhous e effect . Lik e glass, rising levels of carbon dioxid e i n th e atmosphere, cause d b y the burnin g of fossil fuels , ca n tra p the infrare d blackbody radiatio n o f th e eart h an d thereb y graduall y hea t th e planet.) Gamow reasoned tha t the Big Bang was initially quite hot, and henc e would b e a n idea l blackbod y emitter o f radiation . Althoug h th e tech nology o f th e 1940 s was too primitiv e to pic k u p thi s fain t signa l fro m Creation, h e coul d calculat e th e temperatur e o f this radiation an d con -

198 U N I F I C A T I O

N I N TE N D I M E N S I O N S

fidently predict that one day our instruments would be sensitive enough to detec t thi s "fossil " radiation . Th e logi c behin d hi s thinkin g was as follows: About 300,000 years after th e Bi g Bang, the univers e cooled t o the point where atoms could begin t o condense; electrons could begin to circle protons and form stable atoms that would no longer be broken up b y the intens e radiatio n permeating th e universe . Before this time, the univers e wa s so ho t tha t atom s wer e continuall y ripped apar t b y radiation as soon as they were formed. This meant that the univers e was opaque, lik e a thick , absorbing , an d impenetrabl e fog . Afte r 300,00 0 years, however, the radiatio n was no longe r sufficientl y stron g to brea k up the atoms, and hence light could travel long distances without being scattered. In other words, the universe suddenly became black and transparent after 300,00 0 years. (We are so used to hearing about the "black ness of outer space" that we forget that the early universe was not transparent a t all, but filled with turbulent , opaqu e radiation.) After 300,00 0 years, electromagnetic radiatio n n o longe r interacte d so strongly with matter, an d henc e becam e blackbod y radiation. Gradually, as the univers e cooled, th e frequenc y of this radiation decreased . Gamow and his students calculated that the radiation would be far below the infrared range, into the microwave region. Gamow reasoned tha t by scanning the heaven s for a uniform, isotropic source of microwave radiation, one should be able to detect this microwave radiation and discover the ech o o f the Bi g Bang. Gamow's prediction was forgotten for many decades, until the microwave backgroun d radiatio n wa s discovered quit e b y accident i n 1965 . Penzias and Wilso n found a mysterious background radiatio n permeat ing al l space when they turned o n thei r new horn reflecto r antenna i n Holmdel, New Jersey. At first, they thought this unwanted radiation was due t o electrical static caused b y contaminants, such as bird dropping s on thei r antenna . Bu t when the y disassembled an d cleane d larg e por tions of the antenna, they found that the "static" persisted. At the same time, physicists Robert Dicke and James Peebles at Princeton University were rethinkin g Gamow' s old calculation . Whe n Penzia s and Wilso n were finally informed of the Princeto n physicists ' work, it was clear that there was a direct relationship between their results. When they realized that thi s background radiatio n migh t b e th e ech o o f th e origina l Big Bang, they are sai d to have exclaimed, "Either we'v e seen a pile of bird s1 , o r th e creatio n o f the universe! " They discovered tha t this uniform background radiation was almost exactly what had bee n predicte d years earlier by George Gamow and hi s collaborators i f the Big Bang had left a residual blanket of radiation tha t had coole d down to 3°K .

Before Creation 19

9

COBE and th e Bi g Bang Perhaps th e mos t spectacula r scientifi c confirmatio n o f th e Bi g Bang theory cam e i n 199 2 wit h th e result s o f th e COBE (Cosmic Background Explorer) satellite . On Apri l 23, newspaper headline s acros s th e countr y heralded th e finding s o f a team o f scientists at the Universit y of Califor nia a t Berkeley , led b y George Smoot , wh o announce d th e mos t dra matic, convincin g argumen t fo r th e Bi g Bang theory . Journalists an d columnists, wit h n o backgroun d i n physic s or theology , were suddenly waxing eloquent abou t the "fac e o f God" i n their dispatches . The COBE satellit e wa s able t o improv e vastl y th e earlie r wor k o f Penzias, Wilson, Peebles, an d Dick e by many orders o f magnitude, sufficient t o rul e ou t al l doubt tha t th e fossi l radiatio n emitte d b y the Big Bang ha d bee n conclusivel y found. Princeton cosmologis t Jeremiah P . Ostriker declared , "Whe n fossil s wer e found i n th e rocks , i t made th e origin o f specie s absolutel y clear-cut . Well , COBE foun d it s fossils." 2 Launched i n lat e 1989 , th e COBE satellite was specifically designe d t o analyze th e microscopi c detail s in th e structur e o f the microwav e background radiatio n first postulated b y George Gamo w and hi s colleagues. The missio n of COBE also had a new task : to resolv e an earlie r puzzl e arising from th e backgroun d radiation . The origina l work of Penzias and Wilson was crude; the y could show only that th e backgroun d radiatio n wa s smooth t o 10% . When scientists analyzed the background radiatio n i n more detail, they found that it was exceptionally smooth , wit h no apparen t ripples , kinks , or blotches . I n fact, i t wa s to o smooth. Th e backgroun d radiatio n wa s like a smooth , invisible fog filling up th e universe , so uniform tha t scientist s had diffi culty reconciling i t with known astronomical data . In th e 1970s , astronomers turne d thei r grea t telescope s t o systematically map enormou s collection s o f galaxies acros s large portion s o f th e sky. To their surprise , they found that, 1 billion years after the Big Bang, the universe had already exhibited a pattern o f condensing int o galaxie s and eve n larg e cluster s of galaxies and huge , empty spaces called voids. The cluster s were enormous , containin g billion s o f galaxies at a time , and th e void s stretched acros s million s of light-years. But her e la y a cosmi c mystery : If th e Bi g Ban g wa s exceptionally smooth an d uniform , the n 1 billio n year s wa s not enoug h tim e t o develop th e dumpines s tha t w e see amon g th e galacti c clusters . Th e gross mismatc h betwee n th e origina l smoot h Bi g Bang and th e lumpi ness o f th e univers e 1 billion year s late r wa s a naggin g proble m tha t gnawed at every cosmologist. Th e Bi g Bang theory itself was never in any

200 U N I F I C A T I O

N I N TEN DIMENSION S

doubt; wha t was in troubl e was our understandin g o f the post-Big Bang evolution 1 billion year s afte r Creation . Bu t without sensitiv e satellite s that coul d measur e th e cosmi c background radiation , th e proble m fes tered ove r th e years . In fact , b y 1990, journalist s without a rigorous science backgroun d bega n t o write sensational article s saying erroneousl y that scientist s had foun d a fatal fla w i n th e Bi g Bang theor y itself. Many journalists wrote tha t th e Bi g Bang theor y was about to be overthrown . Long-discredited alternative s t o the Bi g Bang theory began t o resurface in th e press . Eve n th e Ne w York Times published a majo r articl e saying that th e Bi g Bang theory was in seriou s trouble (whic h was scientifically incorrect). This pseudocontrovers y surroundin g th e Bi g Bang theor y mad e th e announcement o f the COBE data al l the mor e interesting . With unprecedented accuracy , capable o f detecting variations as small as one par t i n 100,000, th e COBE satellite was able t o scan th e heaven s and radi o bac k the mos t accurat e ma p o f th e cosmi c backgroun d radiatio n eve r con structed. The COBE results reconfirmed th e Bi g Bang theory, and more . COBE& data , however , wer e no t eas y t o analyze . The tea m le d b y Smoot ha d t o face enormou s problems . Fo r example , the y had t o subtract carefull y th e effec t o f the earth' s motio n i n th e backgroun d radi ation. The sola r syste m drifts at a velocity of 370 kilometers per secon d relative t o th e backgroun d radiation . Ther e i s also th e relativ e motio n of the sola r syste m with respec t t o the galaxy, and th e galaxy' s comple x motions wit h respect t o galactic clusters. Nevertheless, after painstaking computer enhancement , severa l stunning results came ou t o f the anal ysis. First, the microwave background fit the earlie r prediction of George Gamow (adjuste d with mor e accurate experimenta l numbers ) t o within 0.1% (Figur e 9.1). The solid line represents th e prediction ; the x' s mark the dat a point s measure d b y th e COBE satellite . Whe n thi s graph wa s flashed o n the screen for the first time t o a meeting o f about a thousand astronomers, everyon e in th e roo m erupte d i n a standing ovation . This was perhaps th e firs t tim e in th e histor y of science that a simple grap h received suc h a thunderou s applaus e fro m s o man y distinguishe d sci entists. Second, Smoot' s tea m was able to show that tiny, almost microscopic blotches did , i n fact , appea r i n th e microwav e background. Thes e tin y blotches wer e precisely what was needed to explain th e dumpines s an d voids found 1 billion years after th e Big Bang. (I f these blotches had no t been found by COBE, then a major revision in the post—Bi g Bang analysis would hav e had t o be made. ) Third, th e result s wer e consisten t with , bu t di d no t prove , th e so-

Before Creation

201

Figure 9.1. The solid line represents the prediction made by the Big Bang theory, which predicts that the background cosmic radiation should resemble blackbody radiation in the microwave region. The x's represent the actual data collected by the COBE satellite, giving us on e of th e most convincing proofs o f th e Big Bang theory.

called inflation theory. (Thi s theory, proposed by Alan Guth of MIT, states that ther e was a much mor e explosiv e expansion o f the univers e at th e initial instant of Creation tha n th e usual Big Bang scenario; it holds tha t the visibl e universe we see with our telescope s i s only the tinies t part o f a muc h bigge r univers e whose boundarie s li e beyond ou r visibl e horizon.)

Before Creation: Orbifolds ?

The COBE results hav e give n physicist s confidence tha t w e understan d the origi n o f the univers e t o within a fraction of a second afte r th e Big Bang. However, we are still left with the embarrassin g question s of what preceded th e Bi g Bang and wh y it occurred. Genera l relativity , if taken to its limits, ultimately yields nonsensical answers . Einstein, realizing that general relativel y simpl y break s dow n a t thos e enormousl y smal l dis -

202 U N I F I C A T I O

N I N TE N DIMENSION S

tances, tried t o extend general relativit y into a more comprehensiv e the ory that coul d explai n thes e phenomena . At the instan t o f the Bi g Bang, we expect quantu m effect s t o be th e dominant force , overwhelming gravity. The ke y to the origi n o f the Big Bang, therefore , i s a quantum theor y o f gravity. So far, the onl y theory that ca n clai m t o solv e th e myster y of wha t happene d befor e th e Bi g Bang i s the ten-dimensiona l superstrin g theory . Scientist s are jus t no w conjecturing ho w the ten-dimensiona l univers e spli t int o a four- an d a six-dimensional universe . What doe s ou r twi n univers e loo k like ? One physicis t who i s struggling with these cosmi c question s i s Gumrum Vafa , a Harvard professo r wh o has spent severa l years studying how our ten-dimensiona l univers e ma y have been tor n int o tw o smaller universes. He is , ironically, also a physicist torn betwee n tw o worlds. Living in Cambridge , Massachusetts , Vaf a i s originall y fro m Iran , whic h ha s been racke d b y political convulsion s for th e pas t decade . O n th e on e hand, h e wishe s eventually to retur n t o hi s native Iran , perhap s whe n the socia l tumul t ha s calme d down . O n th e othe r hand , hi s researc h takes him fa r from tha t troubled regio n o f the world , all the way to th e far reache s o f six-dimensional space , long before th e tumul t in the early universe ha d a chance t o stabilize. "Imagine a simple video game, " he says . A rocket shi p can trave l in the vide o screen , h e point s out , unti l i t veers to o fa r t o th e right . Any video-game playe r know s tha t th e rocke t shi p the n suddenl y appear s from th e lef t sid e o f the screen , a t exactly the sam e height . Similarly , if the rocke t ship wanders to o far and fall s of f the botto m o f the screen , it rematerializes at th e to p o f the screen . Thus , Vafa explains , ther e i s an entirely self-contained universe in that video screen. You can never leave the univers e defined b y that screen. Eve n so, most teenagers hav e never asked themselve s what that univers e is actually shaped like . Vafa point s out, surprisingly enough, tha t the topolog y of the video screen i s that of an inner tube ! Think of the video screen a s a sheet of paper. Since points at the to p of the scree n ar e identica l t o th e point s a t th e bottom , w e can sea l th e top an d botto m side s togethe r wit h glue. We now have rolled th e shee t of pape r int o a tube . Bu t th e point s o n th e lef t sid e o f th e tub e ar e identical t o th e point s o n th e righ t sid e o f th e tube . On e wa y to glu e these tw o ends i s to ben d th e tub e carefull y int o a circle , and sea l th e two open end s togethe r wit h glu e (Figur e 9.2). What we have don e i s to tur n a sheet o f paper int o a doughnut. A rocket shi p wandering on th e video screen ca n be describe d a s moving on th e surfac e of an inne r tube . Ever y tim e th e rocke t vanishes off th e

Figure 9.2. If a rocket disappears off the right side of a video-game screen, it reemerges on the left. If it disappears at the top, it re-emerges at the bottom. Let us now wrap the screen so that identical points match. We first match the top and bottom points by wrapping up the screen. Then we match the points on the leftand right-hand sides by rolling up the screen like a tube. In this way, we can show that a video-game screen has the topology of a doughnut.

203

204 U N I F I C A T I O

N I N TEN DIMENSION S

video scree n an d reappear s o n th e othe r sid e of the screen , thi s corre sponds t o th e rocke t shi p movin g acros s th e glue d join t of th e inne r tube. Vafa conjecture s that our siste r universe has th e shap e o f some sor t of twisted six-dimensional torus. Vafa and hi s colleagues have pioneered the concep t tha t ou r siste r universe ca n b e describe d b y what mathe maticians call an arbifold. I n fact, his proposal tha t our sister universe has the topolog y o f an orbifol d seems to fit the observe d dat a rathe r well. ' To visualiz e an orbifold , thin k o f movin g 36 0 degree s i n a circle. Everyone know s that we come bac k t o th e sam e point . I n othe r words , if I dance 36 0 degrees around a May pole, I know that I will come back to the sam e spot . In an orbifold , however , if we mov e les s tha n 360 degrees aroun d the Ma y pole, we will still come bac k to the sam e point. Although thi s may sound preposterous , i t is easy to construct orbifolds . Think o f Flatlanders living on a cone. If they move less than 360 degrees around th e ape x o f th e cone , the y arriv e a t th e sam e spot . Thu s a n orbifold i s a higher-dimensional generalization of a cone (Figur e 9.3). To get a feel for orbifolds, imagine that some Flatlanders live on what is called a Z-orbifold, which is equivalent to the surfac e of a square bea n bag (lik e thos e foun d a t carnival s and countr y fairs). A t first , nothin g seems different fro m living in Flatland itself. As they explore th e surface, however, they begin t o fin d strang e happenings . Fo r example , if a Flatlander walk s i n an y direction lon g enough , h e return s t o hi s original position as though he walked in a circle. However, Flatlanders also notice that ther e i s something strang e abou t certai n point s i n thei r univers e (the fou r point s o f th e bea n bag) . When walkin g around an y of thes e four point s b y 180 degrees (no t 36 0 degrees), the y return t o th e sam e place fro m whic h they started. The remarkabl e thin g abou t Vafa' s orbifold s is that, wit h just a few assumptions, w e can deriv e man y of th e feature s o f quark s and othe r subatomic particles. (Thi s is because, as we saw earlier, the geometr y of space in Kaluza-Klein theor y forces the quark s to assume th e symmetry of that space. ) Thi s give s us confidence tha t we are o n th e righ t track. If these orbifold s gave us totally meaningless results , then ou r intuition would tell us that there is something fundamentally wrong with this construction. If none of the solutions of string theory contains the Standard Model, then w e must thro w away superstring theor y as another promisin g bu t ultimately incorrec t theory . However , physicists are excite d b y the fac t that i t i s possible t o obtai n solution s that ar e tantalizingl y close t o th e Standard Model .

Figure 9.3. If we join points A and B, then we form a cone, which is the simplest example of an orbifold. In string theory, our four-dimensional universe may have a six-dimensional twin, which has the topology of an orbifold. However, the sixdimensional universe is so small that it is unobservable.

205

206 U N I F I C A T I O

N I N TEN D I M E N S I O N S

Mathematicians for the past 80 years have been working out the properties of these weird surfaces in higher dimensions, ever since the French mathematician Henr i Poincar e pioneered th e subject o f topology in th e early twentieth century. Thus the ten-dimensional theory is able to incorporate a large body of modern mathematic s that previously seemed quite useless. Why Are Ther e Thre e Generations ? In particular , th e ric h storehous e o f mathematica l theorems compile d by mathematicians over th e pas t century are no w being use d t o explain why there are three families o f particles. As we saw earlier, one disastrous feature o f the GUT s is that ther e ar e thre e identica l familie s o f quarks and leptons . However , orbifolds may explain thi s disconcerting feature of the GUTs / Vafa an d hi s co-worker s have discovere d man y promising solutions to the string equations that appear to resemble th e physical world. With a remarkabl y small se t o f assumptions , i n fact , the y ca n rederiv e th e Standard Model , which i s an importan t ste p fo r th e theory . Thi s is , in fact, bot h th e strengt h and th e weakness of superstring theory. Vafa an d his co-workers have been, in a way, too successful : The y have found millions of other possibl e solutions to the strin g equations. The fundamenta l proble m facin g superstrin g theor y i s this : O f the millions of possible universes that can be mathematically generated by superslring theory, which is the correct one? As Davi d Gros s has said , [T]here ar c million s an d million s o f solution s tha t hav e thre e spatia l dimensions. Ther e is an enormou s abundanc e o f possibl e classica l solutions. . . . This abundanc e o f riche s wa s originally very pleasin g because i t provided evidenc e tha t a theor y lik e th e heteroti c strin g could loo k very much like the real world. These solutions, in addition t o having four spacetime dimensions , ha d man y other propertie s tha t resembl e ou r world — the righ t kind s of particles such as quarks and leptons , and th e righ t kinds of interactions. . . . That was a source o f excitement tw o years ago. 5 Gross caution s that althoug h som e o f these solution s are ver y clos e to th e Standar d Model , othe r solution s produc e undesirabl e physical properties: "I t is , however, slightly embarrassing tha t we have so many solutions but n o good way of choosing among them . It seems even more embarrassing tha t thes e solution s have , i n additio n t o man y desire d properties, a few potentially disastrous properties." 6 A layperson, hear-

Before Creation 20

7

ing thi s for th e firs t time , may be puzzle d and ask : Why don't you just calculate which solution th e strin g prefers? Since string theory is a welldefined theory , i t seem s puzzlin g that physicist s cannot calculat e th e answer. The proble m i s that th e perturbatio n theory , one o f the mai n tool s in physics, is of no use. Perturbation theor y (whic h adds up increasingly small quantu m corrections ) fail s t o brea k th e ten-dimensiona l theor y down t o four and si x dimensions. Thus we are forced to use nonperturbative methods, which are notoriously difficult t o use. This, then, is the reason why we cannot solve string theory. As we said earlier, string field theory, developed b y Kikkawa and m e an d furthe r improved by Witten, cannot at present be solved nonperturbatively. No one i s smart enough . I onc e ha d a roommat e wh o wa s a graduat e studen t i n history . 1 remember on e day he warned me about the computer revolution, which eventually might put physicist s out o f a job. "Afte r all," h e said , "com puters can calculate everything, can't they?" To him, it was only a matter of time before mathematicians put al l physics questions in the computer and physicist s got on th e unemploymen t line . I was taken aback by the comment, because to a physicist a computer is nothin g mor e tha n a sophisticate d addin g machine , an impeccabl e idiot. I t make s u p i n spee d wha t it lack s in intelligence . You have t o input th e theor y int o th e compute r befor e i t ca n mak e a calculation. The compute r canno t generat e ne w theories b y itself. Furthermore, eve n i f a theory i s known, the compute r ma y take a n infinite amoun t o f tim e t o solv e a problem . I n fact , computin g all th e really interesting questions in physic s would take an infinit e amoun t o f computer time . Thi s i s the proble m wit h string theory . Although Vaf a and his colleagues have produced millions of possible solutions, it would take an infinit e amoun t o f time t o decide whic h of the million s of possibilities was the correc t one , o r to calculate solutions to quantum problems involving the bizarre process of tunneling, one o f the most difficult of quantum phenomen a to solve.

Tunneling Through Space and Time

In th e fina l analysis , we are askin g the sam e questio n pose d b y Kaluza in 1919—Wher e did the fifth dimension go?—except on a much higher level. As Klein pointed ou t i n 1926 , th e answe r to thi s question ha s t o do with quantum theory. Perhaps the most startling (and complex) phenomenon in quantum theor y is tunneling.

208 U N I F I C A T I O

N I N TEN DIMENSION S

For example , I am no w sitting in a chair . Th e though t o f my body suddenly zappin g throug h th e molecule s o f th e wal l nex t t o m e an d reassembling, uninvited , in someone else' s living room is an unpleasan t one. Also an unlikely one. However , quantum mechanic s postulates tha t there is a finite (althoug h small) probability that even the most unlikely, bizarre events—such as waking up one mornin g and finding our be d in the middl e o f the Amazo n jungle—will actuall y happen. All events, n o matter ho w strange, ar e reduced b y quantum theor y t o probabilities. This tunnelin g proces s sound s mor e lik e scienc e fictio n tha n rea l science. However , tunneling can be measure d i n th e laborator y and, in fact, solve s the riddl e o f radioactive decay. Normally, the nucleu s of an atom i s stable. The proton s an d neutron s within the nucleus are boun d together b y the nuclea r force. However, there i s a small probability that the nucleus might fall apart, that the protons and neutrons might escape by tunneling past the large energ y barrier, th e nuclea r force, tha t bind s the nucleus together. Ordinarily , we would say that all nuclei must therefore b e stable . Bu t i t i s an undeniabl e fac t tha t uraniu m nucle i do , i n fact, deca y when they shouldn't; i n fact , th e conservatio n o f energy law is briefly violated as the neutrons in the nucleus tunnel their way through the barrier . The catch , however , is that thes e probabilitie s ar e vanishingl y small for larg e objects , suc h a s humans . Th e probabilit y o f ou r tunnelin g through a wall within the lifetime of the know n universe is infinitesimally small. Thu s I ca n safel y assum e tha t I wil l no t b e ungraciousl y trans ported throug h th e wall , at least within m y own lifetime. Similarly , ou r universe, which originally might have begun a s a ten-dimensiona l universe, wa s not stable ; i t tunnele d an d explode d int o a four - an d a sixdimensional universe. To understand thi s form of tunneling, think of an imaginary Charlie Chaplin film , i n whic h Chaplin i s trying to stretch a bed shee t aroun d an oversize bed. Th e shee t is the kind with elastic bands o n the corners . But i t is too small, so he ha s t o strain t o wrap the elasti c bands aroun d each corne r o f th e mattress , on e a t a time . H e grin s wit h satisfaction once h e ha s stretched th e be d shee t smoothl y around al l four corner s of th e bed . Bu t th e strai n i s too great ; on e elasti c band pop s of f on e corner, an d th e be d shee t curl s up . Frustrated , h e pull s thi s elastic around the corner , onl y to have another elastic pop off another corner. Every time he yanks an elastic band around on e corner , anothe r elasti c pops of f another corner . This process i s called symmetry breaking. The smoothl y stretched be d sheet possesse s a high degre e o f symmetry. You can rotat e th e be d 18 0

Before Creation 20

9

degrees alon g any axis, and th e bed sheet remain s the same. This highly symmetrical stat e i s called th e false vacuum. Although th e fals e vacuu m appears quit e symmetrical , it is not stable . Th e shee t doe s no t wan t to be in thi s stretched condition. There is too much tension . The energ y is too high. Thu s on e elasti c pops off , and th e be d shee t curl s up. Th e symmetry is broken, an d th e bed shee t ha s gone t o a lower-energy state with les s symmetry . By rotating the curled-u p bed shee t 180 degrees around a n axis , we no longe r retur n t o the same sheet . Now replac e th e be d shee t wit h ten-dimensiona l space-time, th e space-time of ultimate symmetry . At the beginning o f time, the univers e was perfectly symmetrical. If anyone wa s around a t tha t time , he coul d freely pas s through an y of the te n dimension s withou t problem. A t that time, gravit y and th e weak , th e strong , an d th e electromagneti c force s were al l unifie d by the superstring . Al l matter an d force s wer e par t o f the same string multiplet. However, this symmetry couldn't last . The tendimensional universe , although perfectl y symmetrical, was unstable, just like th e be d sheet , an d i n a fals e vacuum . Thu s tunnelin g t o a lowerenergy stat e wa s inevitable. Whe n tunnelin g finally occurred, a phas e transition too k place , an d symmetr y was lost. Because the univers e began t o split up int o a four- and a six-dimensional universe, the univers e was no longer symmetrical . Six dimensions have curle d up , i n th e sam e wa y that th e be d shee t curl s up whe n on e elastic pop s of f a corne r o f a mattress . Bu t notic e tha t ther e ar e fou r ways in which th e be d shee t ca n cur l up , dependin g o n whic h corne r pops of f first . Fo r th e ten-dimensiona l universe , however , ther e ar e apparently millions of ways in which to curl up. To calculate which state the ten-dimensiona l univers e prefers, we need t o solv e the fiel d theor y of strings using th e theor y of phase transitions , th e mos t difficul t prob lem in quantum theory . Symmetry Breakin g

Phase transition s are nothin g new . Think o f our ow n lives. In her boo k Passages, Gail Sheehy stresses that life is not a continuous strea m of experiences, a s it ofte n appears , bu t actuall y passes throug h severa l stages, characterized b y specific conflict s tha t mus t be resolve d an d goal s tha t must be achieved . The psychologis t Erik Erikson even proposed a theory of the psychological stages of development. A fundamental conflict characterizes each phase. When thi s conflic t i s correctly resolved, we move o n t o the nex t

210 U N I F I C A T I O

N I N TE N D I M E N S I O N S

phase. If this conflict is not resolved , it may fester and eve n cause regression t o an earlie r period . Similarly , the psychologist Jean Piaget showed that earl y childhood menta l developmen t i s also no t a smooth proces s of learning, bu t i s actually typified b y abrupt stage s in a child's ability to conceptualize. One month , a child may give up lookin g for a ball once it has rolled ou t o f view, not understandin g tha t a n objec t exists even if you ca n no longe r se e it. The nex t month, thi s is obvious to the child . This i s th e essenc e o f dialectics . According t o thi s philosophy , al l objects (people , gases, the univers e itself ) g o through a series of stages. Each stag e i s characterized b y a conflic t between tw o opposing forces . The natur e o f this conflict, i n fact , determine s th e natur e o f the stage . When th e conflic t is resolved, th e objec t goes t o a higher stage , calle d the synthesis , where a new contradiction begins , an d th e proces s starts over again a t a higher level. Philosophers cal l thi s th e transitio n fro m "quantity " t o "quality. " Small quantitative changes eventually build up until there is a qualitative rupture wit h the past . This theor y applies to societies as well. Tension s in a society can ris e dramatically, as they did i n Franc e i n th e lat e eighteenth century . The peasant s face d starvation , spontaneou s foo d riot s took place, and th e aristocracy retreated behind it s fortresses. When th e tensions reached th e breaking point, a phase transition occurred fro m the quantitativ e t o th e qualitative : The peasant s too k u p arms , seize d Paris, and storme d th e Bastille. Phase transition s ca n als o b e quit e explosiv e affairs . Fo r example , think o f a rive r tha t ha s bee n damme d up . A reservoir quickl y fills u p behind th e da m wit h wate r unde r enormou s pressure . Becaus e i t i s unstable, the reservoi r is in th e fals e vacuum. The wate r would prefer to be i n it s true vacuum , meaning i t woul d prefe r t o burs t th e da m an d wash downstream , t o a stat e o f lowe r energy. Thu s a phas e transitio n would involv e a dam burst, which coul d hav e disastrous consequences. An eve n mor e explosiv e example i s an atomi c bomb. Th e fals e vacuum correspond s t o stabl e uraniu m nuclei . Althoug h th e uraniu m nucleus appears stable , there are enormous, explosive energies trappe d within th e uraniu m nucleu s tha t ar e a millio n time s mor e powerful , pound fo r pound , tha n a chemica l explosive . Onc e i n a while , th e nucleus tunnel s t o a lower state, which means tha t th e nucleu s sponta neously splits apart all by itself. This is called radioactive decay. However, it i s possible, by shooting neutron s a t th e uraniu m nucleus , t o releas e this pent-up energy all at once . This , o f course, i s an atomi c explosion . The ne w featur e discovere d b y scientists about phas e transition s is that the y are usuall y accompanied b y a symmetr y breaking. Nobe l lau-

Before Creation 21

1

reate Abdu s Sala m like s the followin g illustration : Conside r a circula r banquet table , where al l the guest s are seate d wit h a champagne glas s on eithe r side . There is a symmetry here. Looking at the banque t tabl e through a mirror, we see the sam e thing : each guest seated around th e table, wit h champagne glasse s o n eithe r side . Similarly , we can rotat e the circula r banquet table, and th e arrangemen t i s still th e same. Now break th e symmetry . Assume that th e firs t dine r pick s u p th e glass o n hi s o r he r right . B y custom, al l th e othe r guest s pick u p th e champagne glas s to thei r right . Notic e tha t th e imag e o f th e banque t table a s seen in the mirro r produces th e opposite situation . Every diner has picked u p th e glas s to his or he r left . Thu s left-righ t symmetr y has been broken. Another exampl e of symmetry breaking comes from an ancient fair y tale. This fable concern s a princess who is trapped o n to p o f a polished crystal sphere . Althoug h ther e ar e n o iro n bar s confinin g her t o th e sphere, sh e i s a prisone r becaus e i f she make s th e slightes t move, she will slip off the spher e an d kil l herself. Numerous princes have tried t o rescue th e princess , but eac h ha s failed to scale the spher e becaus e it is too smoot h an d slippery . This i s a n exampl e o f symmetr y breaking. While th e princes s is atop th e sphere , sh e i s in a perfectly symmetrical state. There is no preferre d direction fo r the sphere. We can rotate th e sphere at any angle, and th e situation remains the same . Any false move off th e center , however, will cause the princes s to fall, thereb y breaking the symmetry . If she falls to the west , for example, the symmetr y of rotation i s broken. Th e westerl y direction i s now singled out. Thus th e stat e of maximum symmetry is often also an unstabl e state, and henc e correspond s t o a false vacuum . Th e tru e vacuu m state cor responds t o th e princes s fallin g of f the sphere . S o a phas e transitio n (falling off the sphere) correspond s t o symmetry breaking (selectin g the westerly direction). Regarding superstrin g theory , physicist s assume (bu t canno t ye t prove) tha t the original ten-dimensional universe was unstable and tunneled its way to a four- and a six-dimensional universe. Thus the original universe wa s in th e stat e o f th e fals e vacuum , th e stat e o f maximu m symmetry, while toda y we are i n th e broke n stat e of the tru e vacuum. This raises a disturbing question: What would happen if our universe were actually not th e tru e vacuum? What would happen i f the superstring only temporarily chose our universe, but th e true vacuum lay among the million s o f possibl e orbifolds ? This woul d hav e disastrou s conse quences. I n man y other orbifolds , we fin d tha t th e Standar d Mode l is not present . Thu s i f the tru e vacuu m wer e actuall y a stat e wher e th e

212 U N I F I C A T I O

N I N TE N D

1 M E N S I O NS

Standard Mode l was not present , then all the laws of chemistry and physics, as we know them, would come tumblin g down . If this occurred, a tiny bubble might suddenly appear i n our universe. Within thi s bubble , th e Standar d Mode l woul d n o longe r hold , s o a different se t o f chemica l an d physica l laws woul d apply . Matter inside the bubbl e woul d disintegrat e an d perhap s re-for m i n differen t ways . This bubbl e woul d the n expan d a t th e spee d o f light , swallowin g up entire star systems, galaxies, and galactic clusters, until it gobbled u p th e entire universe . We would never see it coming. Traveling at the speed o f light, it could never be observe d beforehand . W e would never kno w what hit us. From Ic e Cubes t o Superstring s Consider an ordinary ice cube sitting in a pressure cooker in our kitchen. We all know what happens i f we turn o n th e stove . But what happens t o an ic e cube i f we heat it up t o trillions upon trillions of degrees? If we heat the ice cube on the stove, first it melts and turn s into water; that is, it undergoes a phase transition . No w let us heat th e water until it boils. It then undergoes anothe r phase transition and turns into steam. Now continue t o heat th e stea m to enormous temperatures . Eventually, the water molecules break up. The energ y of the molecules exceeds th e binding energ y of the molecules, which are ripped apar t int o elemental hydrogen an d oxyge n gas . Now we continue t o heat it past 3,000°K, until the atoms of hydrogen and oxygen are ripped apart . The electrons are pulled from the nucleus, and w e now have a plasma (a n ionized gas), often called the fourth state of matter (afte r gases, liquids, and solids) . Although a plasma is not par t of common experience , w e can se e i t every time we look at th e sun . In fact, plasma is the mos t commo n state of matter in the universe . Now continu e t o heat th e plasm a o n th e stov e t o 1 billion°K, until the nucle i o f hydroge n an d oxyge n ar e rippe d apart , an d w e hav e a "gas" o f individual neutron s an d protons , simila r t o th e interio r o t a neutron star . If we heat th e "gas " o f nucleons even furthe r to 1 0 trillion0K, thes e subatomic particles will turn int o disassociated quarks. We will now have a gas of quarks and lepton s (th e electrons and neutrinos) . If we heat thi s gas to 1 quadrillion°K, the electromagneti c forc e an d the wea k forc e wil l becom e united . Th e symmetr y SU(2) X U ( l ) wil l emerge a t this temperature. A t 1028 °K, the electroweak and strong forces

Before Creation 21

3

become united , an d th e GU T symmetrie s [SU(5) , O(10) , o r E(6) ] appear. Finally, a t a fabulous 10 32 °K, gravity unite s with th e GU T force, an d all th e symmetrie s o f th e ten-dimensiona l superstrin g appear. We now have a gas of superstrings. At that point, so much energy will have gone into the pressur e cooker that the geometr y of space-time may very wel l begin t o distort, and th e dimensionalit y of space-time may change. The space around our kitchen may very well become unstable, a rip may form in th e fabri c o f space , an d a wormhole ma y appear i n th e kitchen . At this point, it may be advisabl e to leav e the kitchen .

Cooling th e Big Bang

Thus by heating a n ordinar y ice cube t o fantastic temperatures , we can retrieve th e superstring . Th e lesso n her e i s that matte r goe s throug h definite stage s of developmen t a s we heat i t up . Eventually , mor e an d more symmetr y becomes restore d a s we increase th e energy . By reversin g thi s process , w e ca n appreciat e ho w th e Bi g Ban g occurred a s a sequenc e o f differen t stages . Instea d o f heatin g a n ic e cube, we now cool the superhot matter in the universe through differen t stages. Beginnin g wit h th e instan t o f Creation , we have th e followin g stages in th e evolutio n of our universe. 10~'t3 seconds Th e ten-dimensiona l univers e breaks down to a four and a six-dimensional universe. The six-dimensiona l universe collapses down to 10~ 32 centimeter i n size. The four-dimensional universe inflates rapidly. The temperatur e i s 10 s2 °K. 10"3r' seconds Th e GU T forc e breaks ; the stron g force i s no longe r united with the electroweak interactions. SU(3) breaks off from th e GUT symmetry. A small spec k i n th e large r univers e become s inflate d by a factor o f 10 r'°, eventually becoming our visibl e universe. l(r9 seconds Th e temperatur e i s now 10 15 °K, and th e electrowea k symmetry breaks into SU(2 ) an d U ( l ) . lO^* seconds Quark s begin t o condense int o neutrons and protons . The temperatur e i s roughly lO 14 °K. 3 minutes Th e protons and neutrons are now condensing into stable nuclei. The energ y of random collision s is no longe r powerfu l enoug h to break up th e nucleu s of the emergin g nuclei. Space is still opaque t o light because ions do no t transmi t light well. 300,000 years Electron s begin t o condens e aroun d nuclei . Atoms

214 U N I F I C A T I O

N I N T E N D I M E N S I O NS

begin to form. Because light is no longer scattered or absorbed as much, the univers e becomes transparent t o light. Outer spac e becomes black. 3 billion years Th e first quasars appear . 5 billion years The first galaxie s appear . 10 t o 1 5 billion years Th e sola r syste m i s born. A fe w billion years after that , the first forms of life appea r o n earth . It seems almost incomprehensible tha t we, as intelligent apes on th e third plane t o f a minor sta r in a minor galaxy , would be able t o reconstruct the histor y of our univers e going back almost to the instan t of its birth, where temperatures an d pressures exceeded anythin g ever found in ou r sola r system . Yet the quantu m theor y o f th e weak , electromagnetic, and stron g interactions reveals this picture to us. As startling as this picture of Creation is , perhaps stranger still is the possibility tha t wormholes ca n ac t a s gateways to anothe r univers e an d perhaps eve n as time machine s into th e pas t and future . Arme d with a quantum theor y of gravity, physicists may be abl e to answer the intriguing questions: Are there paralle l universes? Can the pas t be changed?

PART III

Wormholes: Gateways t o Another Universe ?

This page intentionally left blank

10 Black Hole s and Parallel Universe s Listen, there's a hell of a universe next door: let' s go! e. e. cummings

Black Holes : Tunnels Through Spac e an d Time

B

LACK hole s hav e recentl y seize d th e public' s imagination . Book s and documentarie s hav e bee n devote d t o explorin g thi s strang e prediction o f Einstein's equations , th e fina l stag e in th e deat h o f a collapsed star . Ironically, the public remain s largely unaware of perhaps the most peculia r featur e o f blac k holes , tha t the y ma y b e gateways t o an alternative universe. Furthermore, ther e is also intense speculatio n i n th e scientific communit y that a black hole may open up a tunnel i n time . To understand black holes and ho w difficult the y are to find, we must first understan d wha t make s th e star s shine , ho w the y grow , an d ho w they eventually die. A star is born when a massive cloud of hydrogen ga s many times the siz e of our sola r system is slowly compressed b y the forc e of gravity. The gravitationa l forc e compressin g th e ga s gradually heat s up th e gas , a s gravitational energ y i s converted int o th e kineti c energ y of th e hydroge n atoms . Normally , the repulsiv e charg e o f th e proton s within the hydrogen ga s is sufficient t o keep the m apart . But at a certain point, whe n th e temperatur e rise s t o 1 0 to 10 0 million°K, th e kineti c

217

218 W O R

M H O L E S: G A T E W A Y S T O A N O T H E R U N I V E R S E ?

energy of the protons (whic h are hydrogen nuclei ) overcomes their electrostatic repulsion, and the y slam into on e another . The nuclea r force then take s over from th e electromagneti c force , and th e tw o hydrogen nuclei "fuse " int o helium, releasing vast quantities of energy. In othe r words , a sta r i s a nuclea r furnace , burning hydroge n fue l and creatin g nuclear "ash " i n th e for m of waste helium. A star is also a delicate balancing act between the force of gravity, which tends to crush the sta r int o oblivion , and th e nuclea r force , which tend s t o blo w the star apar t wit h th e forc e o f trillion s of hydroge n bombs . A sta r the n matures and age s as it exhausts its nuclear fuel . To see how energy is extracted fro m the fusion proces s and t o understand th e stage s in th e lif e o f a sta r leadin g t o a blac k hole , w e must analyze Figur e 10.1 , whic h shows one o f th e mos t importan t curve s in modern science , sometime s called th e binding energy curve. On th e hor i/ontal scale is the atomic weight of the various elements, from hydroge n to uranium. O n th e vertica l scale, crudely speaking, i s the approximat e average "weight " of each proto n i n th e nucleus . Notice tha t hydroge n and uraniu m hav e protons tha t weigh, on average , mor e tha n th e pro tons o f other element s in th e cente r o f the diagram . Our su n i s an ordinar y yello w star , consistin g mainl y of hydrogen . Like th e origina l Bi g Bang, it fuse s hydroge n ari d form s helium. However, because th e proton s i n hydrogen weig h more tha n th e protons in helium, ther e i s an exces s o f mass , whic h i s converted int o energ y via Einstein's E = mi? formula. This energy is what binds the nuclei together. This i s also th e energ y release d whe n hydroge n i s fused int o helium . This is why the su n shines . However, as the hydroge n i s slowly used up ove r several billion years, a yellow star eventually builds up to o much waste helium, and its nuclear furnace shut s off. When tha t happens, gravit y eventually takes over an d crushes th e star . A s temperature s soar , th e sta r soo n become s ho t enough t o bur n wast e helium an d conver t i t into th e othe r elements , like lithiu m an d carbon . Notic e tha t energ y ca n stil l b e release d a s we descend dow n the curve to the higher elements. In other words, it is still possible t o bur n wast e helium (i n th e sam e wa y that ordinar y as h ca n still b e burne d unde r certai n conditions) . Althoug h th e sta r ha s decreased enormously in size, its temperature i s quite high, and its atmosphere expand s greatl y in size . I n fact , whe n ou r ow n sun exhaust s its hydrogen suppl y and start s to burn helium , its atmosphere ma y extend out t o th e orbi t of Mars. This is what is called a red giant. This means, of course, tha t th e eart h wil l b e vaporized i n th e process . Thu s th e curv e also predict s th e ultimat e fat e o f th e earth . Sinc e ou r su n i s a middle -

Figure 10.1. The average "weight" of each proton of lighter elements, such as hydrogen and helium, is relatively large. Thus if we fuse hydrogen to form helium inside a star, we have excess mass, which is converted to energy via Einstein's equation E = me 2. This i s the energy that lights u p th e stars. But a s stars fuse heavier and heavier elements, eventually we reach iron, and we cannot extract any more energy. The star then collapses, and the tremendous heat of collapse creates a supernova. This colossal explosion rips the star apart and seeds the interstellar space, in which new stars are formed. The process then starts all over again, like a pinball machine.

219

220 WORMHOLES

: GATEWAY S T O ANOTHE R UNIVERSE ?

aged sta r abou t 5 billio n year s old , i t stil l ha s anothe r 5 billio n years before i t consumes th e earth . (Ironically , the eart h wa s originally born out o f th e sam e swirlin g gas cloud tha t create d ou r sun . Physic s now predicts tha t th e earth , which wa s created wit h th e sun , wil l retur n t o the sun.) Finally, when the heliu m i s used up , th e nuclea r furnac e again shuts down, and gravit y takes over t o crush th e star . The re d gian t shrinks to become a white dwarf, a miniatur e sta r wit h th e mas s o f a n entir e sta r squeezed dow n t o about th e siz e of the plane t earth. 1 White dwarf s ar e not very luminous because, after descending t o the bottom o f the curve, there i s only a little excess energy one ca n squeeze from it through E = mi2. The whit e dwarf burns wha t little ther e i s left a t th e botto m o f th e curve. Our su n wil l eventuall y turn int o a white dwarf and, ove r billions of years, slowly die a s it exhausts its nuclear fuel . I t will eventuall y become a dark , burned-ou t dwar f star . However , i t i s believed tha t i f a sta r i s sufficiently massiv e (several times the mass of our sun), then mos t of the elements i n th e whit e dwarf will continu e t o be fuse d int o increasingly heavier elements , eventuall y reaching iron . Onc e we reach iron , we are near th e ver y bottom o f the curve . We can n o longe r extrac t an y moreenergy from th e exces s mass, so th e nuclea r furnac e shut s off . Gravity once agai n take s over , crushin g th e sta r unti l temperature s ris e explo sively a thousandfold , reachin g trillion s o f degrees . A t thi s point , th e iron cor e collapse s an d th e oute r laye r o f th e whit e dwar f blow s off, releasing th e larges t burst of energy known in th e galaxy , an explodin g star calle d a supernova. Just on e supernov a ca n temporaril y outshin e a n entire galax y of 10 0 billion stars. In th e aftermat h o f th e supernova , we find a totally dead star , a neutron star about th e siz e of Manhattan. The densitie s in a neutron sta r are so grea t that , crudel y speaking , al l th e neutron s ar e "touching " on e another. Althoug h neutro n star s are almos t invisible, we can still detec t them with ou r instruments. Because they emit some radiation while they are rotating , the y ac t lik e a cosmi c lighthous e i n oute r space . W e see them a s a blinkin g star, o r pulsar. (Althoug h thi s scenari o sound s lik e science fiction , well ove r 40 0 pulsar s hav e bee n observe d sinc e thei r initial discover y in 1967. ) Computer calculation s have shown that most of the heavier elements beyond iro n ca n be synthesized in the heat and pressure of a supernova. When th e sta r explodes , i t releases vas t amounts o f stellar debris, con sisting o f th e highe r elements , int o th e vacuu m o f space . Thi s debri s eventually mixe s wit h other gases , unti l enough hydroge n ga s is accu-

Black Holes and Parallel Universes 22

1

mulated t o begin th e gravitationa l contraction proces s once again . Second-generation star s that are born ou t of this stellar gas and dust contain an abundanc e o f heavy elements. Som e of these star s (lik e our sun ) will have planets surrounding them tha t also contain thes e heav y elements. This solve s a long-standin g myster y in cosmology . Ou r bodie s ar e made o f heavy elements beyond iron , but ou r su n is not ho t enoug h t o forge them . I f th e eart h an d th e atom s o f ou r bodie s wer e originally from th e sam e ga s cloud, the n wher e di d th e heav y element s o f ou r bodies com e from ? Th e conclusio n is inescapable: Th e heav y elements in ou r bodie s wer e synthesize d i n a supernov a tha t ble w up before ou r sun was created. In other words, a nameless supernova exploded billions of years ago, seeding th e original gas cloud tha t created ou r solar system. The evolutio n of a star can be roughly pictured as a pinball machine, as in Figur e 10.1 , wit h the shap e o f the bindin g energ y curve. The bal l starts at the top and bounces from hydrogen , to helium, from th e lighter elements to the heavier elements. Each time it bounces along the curve, it becomes a different typ e of star. Finally, the ball bounces to the botto m of the curve , where it lands on iron, and i s ejected explosively in a supernova. Then a s this stellar material is collected again into a new hydrogenrich star, the proces s start s all over agai n o n th e pinball. Notice, however , that ther e ar e tw o ways fo r th e pinbal l t o bounc e down th e curve . I t ca n als o star t at th e othe r sid e o f th e curve , at ura nium, an d g o down th e curv e in a single bounc e b y fissioning th e ura nium nucleu s int o fragments . Sinc e the averag e weigh t of the proton s in fission products, lik e cesium and krypton , is smaller than th e averag e weight of the proton s i n uranium , the exces s mass has been converte d into energ y via E = mi 2. This is the sourc e o f energy behind th e atomi c bomb. Thus th e curv e o f binding energ y no t onl y explains th e birt h an d death o f stars and th e creatio n o f the elements , but als o makes possibl e the existence of hydrogen an d atomic bombs! (Scientist s are often asked whether i t woul d b e possibl e t o develo p nuclea r bomb s othe r tha n atomic and hydroge n bombs. From th e curve of binding energy, we can see that th e answe r is no. Notic e that th e curv e excludes the possibility of bombs mad e o f oxygen o r iron. These element s are near th e botto m of the curve , so there is not enoug h exces s mass to create a bomb. Th e various bombs mentioned i n the press , such as neutron bombs , are only variations on uranium an d hydroge n bombs. ) When on e first hears th e lif e histor y of stars, one ma y be a bit skeptical. Afte r all , n o on e ha s eve r live d 1 0 billion year s t o witnes s their evolution. However, since there are uncountable star s in the heavens, it

222 W O R M H O L E S

: GATEWAYS T O ANOTHE R UNIVERSE ?

is a simple matter t o see stars at practically every stage in their evolution. (For example , th e 198 7 supernova , whic h was visible to th e nake d ey e in th e souther n hemisphere , yielde d a treasur e trov e o f astronomica l data tha t matched th e theoretica l prediction s o f a collapsing dwar f with an iro n core . Also , the spectacula r supernov a observe d b y ancient Chi nese astronomer s o n July 4, 1054, left behin d a remnant, whic h has now been identified as a neutron star. ) In addition , ou r compute r program s hav e becom e s o accurate tha t we can essentiall y predict th e sequenc e o f stellar evolution numerically. I once had a roommate i n graduate schoo l who was an astronomy major. He would invariabl y disappear i n th e earl y morning an d retur n lat e a t night. Just before he would leave, he would say that he was putting a star in th e ove n t o watc h i t grow . A t first , I though t h e sai d thi s i n jest. However, when I pressed hi m o n thi s point, h e sai d with all seriousness that h e wa s putting a sta r int o th e compute r an d watchin g i t evolve during th e day . Sinc e th e thermodynami c equation s an d th e fusio n equations wer e wel l known , it was just a matter o f telling the compute r to start with a certain mas s of hydrogen ga s and the n lettin g it numeri cally solv e for th e evolutio n o f thi s gas . I n thi s way, we can chec k tha t our theor y of stellar evolutio n ca n reproduc e th e know n stage s o f star life tha t we see in th e heaven s with our telescopes .

Black Hole s

If a star was ten t o 50 times the siz e of our sun , then gravity will continu e to squeeze it even afte r it becomes a neutron star . Without the forc e of fusion t o repel th e gravitational pull, there is nothing to oppose the final collapse o f the star . At this point, it becomes th e famou s black hole. In som e sense , blac k hole s mus t exist . A star , w e recall , i s th e by product o f two cosmic forces: gravity, which tries t o crus h th e star , an d fusion, whic h trie s to blo w the sta r apar t lik e in a hydrogen bomb . All the variou s phases i n th e lif e histor y of a star are a consequence o f this delicate balancing ac t between gravity and fusion. Sooner or later, when all th e nuclea r fue l i n a massive star i s finally exhauste d an d th e sta r is a mass of pure neutrons, ther e is nothing know n that can then resis t the powerful force of gravity. Eventually, the gravitationa l force will take over and crus h th e neutro n sta r int o nothingness . Th e sta r ha s com e ful l circle: It was born whe n gravity first began t o compress hydroge n ga s in the heavens into a star, and i t will die when the nuclear fue l i s exhausted and gravit y collapses it.

Black Holes an d Parallel Universes 22

3

The densit y o f a blac k hol e i s s o larg e tha t light , lik e a rocke t launched from the earth, will be forced to orbit around it . Since no light can escap e fro m th e enormou s gravitationa l field , th e collapse d sta r becomes blac k i n color . I n fact , tha t i s the usua l definitio n o f a blac k hole, a collapsed sta r from whic h no ligh t can escape . To understan d this , we not e tha t al l heavenl y bodies hav e wha t is called a n escape velocity. Thi s is th e velocit y necessar y t o escap e perma nently th e gravitationa l pull of tha t body . For example , a spac e prob e must reach a n escape velocity of 25,000 miles per hou r i n order t o leave the gravitationa l pul l o f th e eart h an d g o into dee p space . Ou r spac e probes lik e th e Voyager tha t hav e venture d int o dee p spac e an d hav e completely lef t th e sola r syste m (carryin g good-wil l message s t o an y aliens who might pick them up) hav e reached th e escape velocity of our sun. (Th e fac t tha t w e breathe oxyge n i s because th e oxyge n atoms d o not hav e enoug h velocit y to escap e th e earth' s gravitationa l field . Th e fact tha t Jupiter an d th e othe r ga s giants ar e mad e mainl y of hydrogen is because their escape velocity is large enough t o capture th e primordia l hydrogen of the earl y solar system. Thus escape velocity helps to explain the planetar y evolutio n o f the planet s o f our sola r syste m over th e pas t 5 billion years.) Newton's theor y o f gravity , i n fact , give s th e precis e relationshi p between th e escap e velocit y and th e mas s o f the star . Th e heavie r th e planet o r star an d th e smalle r its radius, th e large r th e escap e velocity necessary t o escap e it s gravitational pull. A s early as 1783 , th e Englis h astronomer John Michel l used thi s calculation t o propose tha t a supe r massive sta r migh t hav e a n escap e velocit y equa l t o th e spee d o f light. The ligh t emitted b y such a massive star could neve r escape , bu t would orbit aroun d it . Thus , t o a n outsid e observer , th e sta r woul d appea r totally black. Using the bes t knowledg e available in th e eighteent h cen tury, h e actuall y calculated th e mas s o f suc h a blac k hole. * Unfortu nately, hi s theor y was considered t o b e craz y an d wa s soon forgotten . Nevertheless, today we tend t o believe that black holes exist because ou r telescopes an d instrument s have seen whit e dwarfs an d neutro n star s in the heavens . There ar e tw o ways t o explai n wh y black holes ar e black . Fro m th e *In th e Philosophical Transactions o f the Royal Society, h e wrote , "I f th e semi-diamete r o f a sphere of the sam e density with th e Su n were to exceed that of the Sun in the proportion of 50 0 t o 1 , a bod y falling fro m a n infinit e heigh t toward s it , woul d hav e acquire d a t it s surface greate r velocity tha n tha t of light, and consequentl y supposing light to be attracted by th e sam e force i n proportio n t o it s vis inertiae, with other bodies, all light emitted fro m such a body would be mad e t o retur n t o i t by its own proper gravity." 2

224 WORMHOLES

: G A T E W A Y S T O A N O T H E R U N I V E R S E?

pedestrian poin t o f view, the "force " betwee n the star and a light bea m is so grea t tha t it s path i s bent int o a circle . O r on e ca n tak e th e Ein steinian point of view, in which case the "shortes t distance betwee n two points i s a curved line. " Bendin g a light beam int o a ful l circl e mean s that spac e itsel f has bee n ben t ful l circle . Thi s ca n happe n onl y if th e black hole ha s completely pinched a piece o f space-time alon g wit h it , so th e ligh t beam i s circulating in a hypersphere. Thi s piec e o f spacetime ha s no w disconnected itsel f from the space—tim e around it . Space itself has now "ripped."

The Einstein-Rose n Bridge

The relativistic descriptio n o f th e blac k hol e come s fro m th e wor k of Karl Schwarzschild . In 1916 , barel y a fe w months afte r Einstei n wrote down his celebrated equations , Schwarzschild was able to solve Einstein's equations exactly and calculat e the gravitationa l field of a massive, stationary star. Schwarzschild's solutio n ha s severa l interestin g features . First , a "point o f no return " surround s th e blac k hole. An y object that come s closer tha n thi s radius will inevitably be sucked into th e black hole, with no possibilit y of escape. Inexorably , any person unfortunat e enoug h t o come withi n the Schwarzschil d radius woul d b e capture d b y the blac k hole an d crushe d t o death. Today , thi s distance from th e blac k hole is called the Schwarzschild radius, or th e horizon (th e farthest visible point). Second, anyon e wh o fel l withi n th e Schwarzschil d radius would b e aware of a "mirror universe" o n the "other side" of space-time (Figure 10.2). Einstein was not worried about the existence of this bizarre mirror universe becaus e communicatio n wit h i t wa s impossible . An y spac e probe sen t int o th e cente r o f a blac k hol e woul d encounte r infinit e curvature; that is, the gravitationa l field would be infinite, and an y material object would be crushed . The electron s woul d be rippe d of f atoms, and eve n th e proton s an d neutron s withi n th e nucle i themselves would be torn apart . Also, to penetrate throug h t o the alternative universe, the probe woul d hav e t o g o faste r tha n th e spee d o f light , whic h i s no t possible. Thus although thi s mirror universe is mathematically necessary to make sense of the Schwarzschil d solution, it could neve r be observed physically. Consequently, th e celebrate d Einstein—Rosen bridge connecting thes e two universe s (name d afte r Einstei n an d hi s collaborator , Natha n Rosen) wa s considered a mathematical quirk. The bridg e was necessary

Black Holes and Parallel Universes

225

Figure 10.2. The Einstein—Rosen bridge connects two different universes. Einstein believed that any rocket that entered the bridge would be crushed, thereby making communication between these two universes impossible. However, more recent calculations show that travel through the bridge might be very difficult, but perhaps possible.

to have a mathematically consistent theory of the blac k hole, bu t i t was impossible t o reac h th e mirro r univers e by traveling through th e Einstein-Rosen bridge . Einstein-Rose n bridge s wer e soon foun d i n othe r solutions of the gravitational equations, such as the Reissner-Nordstro m solution describing an electrically charged blac k hole. However, the Ein-

226 WORMHOLES

: GATEWAY S T O ANOTHE R UNIVERSE ?

stein—Rosen bridg e remaine d a curiou s bu t forgotte n footnot e i n th e lore o f relativity. Things bega n t o chang e wit h the wor k of New Zealand mathemati cian Ro y Kerr, who i n 196 3 found another exact solutio n t o Einstein's equations. Ker r assumed tha t any collapsing star would be rotating. Like a spinnin g skate r who speeds u p whe n bringin g in hi s or he r hands , a rotating star would necessaril y accelerate a s it began t o collapse . Thu s the stationar y Schwarzschild solution fo r a black hole was not th e mos t physically relevant solution o f Einstein's equations. Kerr's solutio n create d a sensation i n th e fiel d o f relativit y when i t was proposed. Astrophysicis t Subrahmanyan Chandrasekha r onc e said, In m y entire scientifi c life , extending ove r forty-fiv e years, th e mos t shat tering experience ha s been th e realizatio n tha t a n exac t solutio n o f Einstein's equations o f general relativity, discovered b y the New Zealand mathematician Ro y Kerr , provide s th e absolutely exact representation o f untol d numbers o f massiv e blac k hole s tha t populat e th e universe . Thi s "shud dering before the beautiful," thi s incredible fact that a discovery motivated by a search after th e beautifu l i n mathematics should find its exact replica in Nature , persuade s m e t o sa y that beaut y i s that t o whic h th e huma n mind respond s at its deepest and mos t profound level. '

Kerr found, however , that a massive rotating sta r doe s no t collaps e into a point. Instead, th e spinning star flattens until it eventually is compressed int o a ring , whic h ha s interestin g properties . I f a prob e wer e shot into th e black hole from the side, it would hit the ring and be totally demolished. The curvatur e of space-time i s still infinite when approach ing th e rin g from th e side . There is still a "ring of death," so to speak, surrounding th e center . However , i f a spac e prob e were sho t int o th e ring from th e to p or bottom, i t would experience a large bu t finite curvature; that is , the gravitationa l force would not b e infinite. This rathe r surprisin g conclusio n fro m Kerr' s solution mean s tha t any spac e prob e sho t throug h a spinnin g blac k hol e alon g it s axi s of rotation might , i n principle , surviv e the enormou s bu t finit e gravita tional fields at the center, and go right on through t o the mirror universe without bein g destroye d b y infinit e curvature . Th e Einstein-Rose n bridge act s lik e a tunne l connectin g tw o region s o f space-time ; it i s a wormhole . Thu s th e Ker r blac k hol e i s a gatewa y t o anothe r universe. Now imagin e tha t you r rocke t ha s entere d th e Einstein-Rose n bridge. A s your rocke t approache s th e spinnin g blac k hole , i t see s a

Black Holes and Parallel Universes 227

ring-shaped spinnin g star. At first, it appears tha t th e rocke t is heade d for a disastrous crash landing as it descends toward the black hole fro m the nort h pole . However , as we get close r t o th e ring , ligh t from th e mirror universe reaches our sensors. Since all electromagnetic radiation, including radar, orbit s th e blac k hole, ou r rada r screens ar e detectin g signals tha t hav e been circulatin g around th e blac k hol e a number o f times. Thi s effec t resemble s a hall of mirrors, in which we are fooled b y the multipl e image s tha t surroun d us . Ligh t goe s ricochetin g acros s numerous mirrors, creating the illusion that there are numerous copies of ourselves in th e hall . The sam e effec t occur s a s w e pas s throug h th e Ker r blac k hole . Because the same light beam orbit s th e black hole numerous times, our rocket's rada r detect s images that have gone spinning around th e black hole, creatin g the illusio n of objects that aren't reall y there. Warp Facto r 5

Does this mean tha t blac k holes ca n b e use d fo r trave l throughout th e galaxy, as in Star Trek an d othe r science-fictio n movies? As we saw earlier, th e curvatur e in a certain spac e i s determined b y the amoun t o f matter-energ y containe d i n tha t spac e (Mach' s principle). Einstein' s famous equation give s us th e precis e degre e o f spacetime bending cause d b y the presenc e of matter—energy. When Captai n Kir k take s u s soarin g throug h hyperspac e a t "war p factor 5," the "dilithium crystals" that power the Enterprise must perform miraculous feat s o f warping space an d time . Thi s mean s tha t th e dili thium crystal s have the magica l powe r o f bending th e space-tim e con tinuum into pretzels; that is, they are tremendous storehouse s of matter and energy . If the Enterprise travels from th e eart h t o the neares t star, it does no t physically mov e t o Alph a Centauri—rather , Alph a Centaur i come s t o the Enterprise. Imagin e sittin g on a rug an d lassoin g a table severa l fee t away. If we are stron g enough an d th e floo r is slick enough, we can pul l the lass o until th e carpe t begin s to fold underneath us . If we pull har d enough, th e tabl e come s t o us , and th e "distance " betwee n th e tabl e and u s disappears into a mas s of crumpled carpeting . The n w e simply hop acros s this "carpet warp." I n othe r words , we have hardly moved; the spac e betwee n u s an d th e tabl e ha s contracted , an d w e just ste p across thi s contracted distance . Similarly , th e Enterprise doe s no t reall y cross the entire space to Alpha Centauri; it simply moves across the crum-

228 WOR

M HOLES: GATEWAY S T O ANOTHE R UNIVERSE ?

pled space-time—through a wormhole. To better understan d wha t happens when one fall s down the Einstein-Rose n bridge , le t us now discuss the topolog y o f wormholes. To visualiz e thes e multipl y connected spaces , imagin e tha t w e ar e strolling down New York's Fifth Avenu e one brigh t afternoon, mindin g our ow n business, when a strange floatin g window opens u p i n front of us, muc h lik e Alice' s lookin g glass . (Neve r mind fo r th e momen t tha t the energ y necessar y to ope n thi s window might be enoug h t o shatte r the earth . Thi s i s a purely hypothetical example.) We ste p u p t o th e hoverin g windo w to tak e a close r look , an d ar e horrified t o fin d ourselve s staring a t th e hea d o f a nasty-looking Tyrannosaurus rex. We are abou t t o run fo r ou r lives , when we notice tha t th e tyrannosaur ha s n o body . H e can' t hur t u s becaus e hi s entir e bod y is clearly on the other sid e of the window. When we look below the window to fin d th e dinosaur' s body , w e can se e al l th e wa y down th e street , as though th e dinosau r an d th e windo w weren't ther e a t all . Puzzled, we slowly circle the windo w and ar e relieve d to find that th e tyrannosau r is nowhere t o be found. However, when we peer int o the window from the back side, we see the hea d o f a brontosaur starin g us in the fac e (Figur e 10.3)! Frightened, we walk around th e windo w once more , staring a t th e window sideways . Much t o ou r surprise , al l trace s o f th e window , the tyrannosaur, and the brontosaur ar e gone. We now take a few more turns around th e floatin g window. From one direction , we see the head o f the tyrannosaur. Fro m th e othe r direction , w e see the hea d o f the bronto saur. And when we look from the side , we find that both th e mirror an d the dinosaur s hav e disappeared . What's happening ? In som e farawa y universe , the tyrannosau r and th e brontosau r hav e squared off in a life-and-death confrontation . A s they face each other, a floating window suddenly appears betwee n them. When the tyrannosaur peers int o th e floatin g mirror , h e i s startled t o see the hea d o f a puny, skinny-looking mammal, with frizz y hai r an d a tiny face: a human. Th e head i s clearly visible, but it has no body. However, when the brontosau r stares into th e sam e window from the othe r direction , he sees Fifth Avenue, wit h it s shop s an d traffic . The n th e tyrannosau r find s tha t thi s human creatur e i n th e windo w has disappeared, onl y to appear o n th e side o f the windo w facing the brontosaur . Now let us say that suddenly the wind blows our hat into the window. We see the hat sailing into the sky of the other universe, but it is nowhere to b e see n alon g Fift h Avenue . W e tak e on e lon g gulp , an d then , i n

Black Holes and Parallel Universes

229

Figure 10.3. In this purely hypothetical example, a "window" or wormhole has opened up in our universe. If we look into the window from one direction, we see one dinosaur. If we look into the other side of the window, we see another dinosaur. As seen from the other universe, a window has opened up between the two dinosaurs. Inside the window, the dinosaurs see a strange small animal (us).

desperation, we stick our han d int o the windo w to retrieve the hat . As seen b y the tyrannosaur , a ha t blow s ou t th e window , appearin g from nowhere. Then h e see s a disembodied hand reachin g out th e window, desperately groping for the hat. The wind now changes direction, and th e hat is carried in the othe r

230

WORMHOLES: GATEWAY S TO ANOTHE R UNIVERSE ?

Figure 10.4. If we insert our hands into the window from two different directions, then it appears as though our hands have disappeared. We have a body, but no hands. In the alternative universe, two hands have emerged from either side of the window but they are not attached to a body.

direction. We stick our other han d int o the window, but fro m th e othe r side. We are no w in a n awkwar d position. Bot h ou r hand s ar e sticking into th e window , but fro m different sides . But we can't se e our fingers . Instead, it appears t o us that both hand s have disappeared . How does thi s appear t o th e dinosaurs ? The y see two wiggling, tiny hands dangling from th e window, from eithe r side . But there is no body (Figure 10.4). This exampl e illustrate s som e o f th e deliciou s distortion s o f spac e and tim e that on e ca n invent with multiply connected spaces .

Black Holes and Parallel Universes 23

1

Closing the Wormhol e It seem s remarkabl e tha t suc h a simpl e idea—tha t highe r dimension s can unif y spac e wit h time, and tha t a "force " ca n b e explaine d b y the warping of that space-time—would lead t o such a rich diversity of physical consequences. However , with the wormhole and multiply connected spaces, w e ar e probin g th e ver y limits of Einstein' s theor y o f genera l relativity. I n fact , th e amoun t o f matter-energ y necessar y to creat e a wormhole o r dimensiona l gatewa y is so large tha t w e expect quantu m effects t o dominate . Quantu m corrections , i n turn , ma y actually close the opening of the wormhole, making travel through th e gateway impossible. Since neithe r quantu m theor y no r relativit y is powerful enough t o settle this question, we will have to wait until the ten-dimensiona l theory is completed t o decide whether these wormholes are physicall y relevant or just anothe r craz y idea . However , before w e discuss the questio n o f quantum correction s an d th e ten-dimensiona l theory , let us now pause and conside r perhap s th e most bizarre consequence o f wormholes. Just as physicist s ca n sho w tha t wormhole s allo w for multipl y connecte d spaces, we can als o show that the y allow for tim e travel as well. Let u s now consider perhap s th e mos t fascinating, and speculative , consequence o f multiply connected universes : building a time machine.

II To Build a Time Machin e People lik e us , who believ e in physics , know that th e distinction betwee n past , present , an d futur e i s onl y a stubbornly persistent illusion. Albert Einstein

Time Travel

C

AN we go backward i n time? Like the protagonis t i n H . G . Wells's Th e Time Machine, ca n we spin th e dia l o f a machine an d lea p hundred s o f thousands o f years to the yea r 802,701 ? Or , lik e Michae l J. Fox , ca n w e hop int o ou r pluto nium-fired car s an d g o back t o the future ? The possibilit y of tim e trave l open s u p a vas t world o f interestin g possibilities. Lik e Kathlee n Turne r i n Peggy Su e Go t Married, everyon e harbors a secret wish somehow to relive the past and correc t some small but vita l mistake i n one' s life . I n Rober t Frost' s poe m "Th e Roa d No t Taken," we wonder wha t might have happened, a t key junctures i n ou r lives, i f we had mad e differen t choices an d take n anothe r path . Wit h time travel, we could go back to our youth and erase embarrassing events from ou r past , choose a different mate, o r enter differen t careers; or we could eve n chang e th e outcom e o f ke y historical event s an d alte r th e fate o f humanity. For example, in the clima x of Superman, our her o is emotionally devastated whe n an earthquak e ravage s most o f California an d crushe s his

232

To Build a Time Machine 23

3

lover under hundreds o f tons of rock and debris. Mourning her horribl e death, he i s so overcome b y anguish tha t he rocket s into spac e an d violates hi s oat h no t t o tampe r wit h th e cours e o f huma n history . H e increases hi s velocity until h e shatter s th e ligh t barrier , disruptin g th e fabric o f spac e an d time . B y traveling a t th e spee d o f light , h e force s time t o slo w down, the n t o stop , an d finall y t o g o backward, to a tim e before Loi s Lane wa s crushed t o death . This trick , however, is clearly not possible . Althoug h tim e doe s slo w down whe n yo u increas e you r velocity , you canno t g o faste r tha n th e speed of light (an d hence make tim e go backward) because special relativity states that your mass would become infinit e in th e process . Thu s the faster-than-light travel method preferred by most science-fiction writers contradicts th e specia l theory of relativity. Einstein himsel f was well aware of thi s impossibility, as was A. H. R . Buller when he publishe d th e followin g limerick in Punch: There was a young lady girl named Bright , Whose speed wa s far faster tha n light , She traveled one day , In a relative way, And returne d o n th e previous night.

Most scientists, who have not seriousl y studied Einstein' s equations , dismiss time travel as poppycock, with as much validity as lurid account s of kidnapping s b y space aliens . However, the situatio n is actually quite complex. To resolv e th e question , w e must leave the simple r theor y of special relativity, which forbids tim e travel , and embrac e th e ful l powe r o f th e general theor y o f relativity , which ma y permi t it . Genera l relativit y has much wide r validit y tha n specia l relativity . Whil e specia l relativit y describes only objects moving at constant velocity far away from any stars, the genera l theor y o f relativit y i s muc h mor e powerful , capabl e o f describing rockets accelerating near supermassive stars and black holes. The general theor y therefore supplant s some of the simpler conclusions of th e specia l theory . Fo r an y physicist who ha s seriousl y analyzed th e mathematics of time trave l within Einstein's general theor y of relativity, the fina l conclusio n is , surprisingly enough, fa r from clear . Proponents o f time travel point out that Einstein's equations for general relativit y do allo w some form s o f tim e travel . The y acknowledge, however, tha t th e energie s necessar y to twis t tim e int o a circl e ar e s o great tha t Einstein's equations brea k down . In th e physically interesting

234 WORMHOLES

: GATEWAY S T O ANOTHE R U N I V E R S E ?

region wher e tim e trave l becomes a serious possibility, quantum theory takes over from general relativity. Einstein's equations , we recall, stat e tha t th e curvatur e o r bendin g of spac e an d tim e i s determined b y the matter-energ y conten t o f th e universe. It : is, in fact , possibl e t o fin d configuration s of matter-energ y powerful enoug h to force the bendin g o f time and allo w for time travel. However, th e concentration s o f matter-energy necessary to bend tim e backward ar e s o vast tha t genera l relativit y break s dow n an d quantu m corrections begi n t o dominat e ove r relativity . Thus th e fina l verdic t o n time travel cannot be answered within th e framework of Einstein's equations, whic h brea k dow n i n extremel y large gravitationa l fields, where we expect quantu m theor y to become dominant . This is where the hyperspac e theory can settle the question . Because both quantu m theor y and Einstein' s theory of gravity are unite d in teri dimensional space , w e expec t tha t th e questio n o f tim e trave l wil l b e settled decisivel y by the hyperspac e theory . As in th e cas e of wormholes and dimensiona l windows , the fina l chapte r wil l b e writte n whe n we incorporate th e ful l powe r o f the hyperspac e theory . Let us now describe the controvers y surrounding tim e travel and th e delicious paradoxes tha t inevitabl y arise .

Collapse o f Causalit y Science-fiction writer s have often wondered what might happen if a single individual wen t back i n time . Many of these stories , on th e surface, appear plausible . Bu t imagin e th e chao s tha t woul d aris e i f tim e machines were as common a s automobiles, with tens of millions of them commercially available . Havo c would soo n brea k loose , tearin g a t th e fabric of our universe . Millions of people would go back in time to meddle with their ow n past and th e pas t of others, rewritin g history in th e process. A few might eve n g o bac k i n tim e arme d wit h gun s t o shoo t down the parent s of their enemies before they were born. I t would thus be impossible to take a simple census to see how many people there were at any given time. If tim e trave l is possible, the n th e law s of causalit y crumble. I n fact , all o f histor y a s we kno w it migh t collaps e a s well . Imagin e th e chao s caused b y thousands o f peopl e goin g bac k i n tim e t o alte r ke y events that change d th e cours e o f history . Al l of a sudden , th e audienc e a t Ford's Theate r woul d be cramme d wit h peopl e fro m th e futur e bickering among themselve s to see who would have th e hono r o f preventing

To Build a Time Machine 23

5

Lincoln's assassination . Th e landin g a t Normandy woul d be botched a s thousands o f thrill seeker s with cameras arrive d t o take pictures . The key battlefields of history would be changed beyon d recognition . Consider Alexande r th e Great' s decisiv e victory over the Persians , led by Darius III, in 33 1 B.C. a t the Battl e of Gaugamela. Thi s battl e le d t o th e collapse o f th e Persia n force s an d ende d thei r rivalry with th e West , which helpe d allo w the flourishin g o f Western civilizatio n and cultur e over the world for the next 1,00 0 years. But consider what would happe n if a small ban d of armed mercenarie s equipped with smal l rocket s an d modern artiller y were to enter the battle. The slightest display of modern firepower would rou t Alexander' s terrifie d soldiers . Thi s meddlin g i n the past would cripple th e expansion o f Western influenc e in the world. Time trave l woul d mea n tha t an y historica l even t coul d neve r b e completely resolved . Histor y book s coul d neve r b e written . Some die hard woul d alway s b e tryin g to assassinat e General Ulysse s S . Grant o r give the secre t o f the atomi c bom b t o the German s i n th e 1930s . What would happen i f history could be rewritten as casually as erasing a blackboard? Ou r past would be like the shiftin g sands at the seashore , constantly blown thi s way or tha t b y the slightes t breeze. Histor y would be constantl y changin g ever y tim e someon e spu n th e dia l o f a tim e machine an d blundere d hi s or her way into the past. History, as we know it, would be impossible . I t would ceas e t o exist. Most scientists obviously do not relish this unpleasant possibility . Not only would it be impossible for historians t o make an y sense out of "his tory," bu t genuin e paradoxe s immediatel y arise whenever we enter th e past o r future . Cosmologist Stephe n Hawking , in fact , has used thi s situation t o provid e "experimental " evidenc e tha t tim e trave l i s no t possible. H e believe s tha t tim e trave l i s no t possibl e b y "th e fact tha t w e hav e no t bee n invade d b y hoarde s o f tourist s fro m th e future."

Time Paradoxe s

To understand the problem s with tim e travel, it is first necessary to classify th e variou s paradoxes. In general , mos t ca n b e broke n dow n int o one o f two principal types: 1. Meetin g your parents before you are born 2. Th e ma n wit h no pas t

236 WORMHOLES

: GATEWAY S TO ANOTHE R UNIVERSE ?

The firs t typ e of tim e trave l doe s th e mos t damag e t o th e fabri c of space-time becaus e i t alter s previousl y recorded events . Fo r example , remember tha t i n Back t o the Future, our youn g hero goe s bac k in tim e and meet s hi s mother a s a young girl, just befor e sh e fall s i n lov e with his father. T o hi s shock an d dismay , he find s tha t h e ha s inadvertently prevented th e fatefu l encounte r betwee n hi s parents. T o mak e matter s worse, his young mother ha s now become amorousl y attracte d t o him ! If he unwittingly prevents his mother an d father from falling in love an d is unable t o diver t his mother's misplace d affections , h e wil l disappea r because hi s birth wil l neve r happen . The secon d parado x involve s event s withou t an y beginning . Fo r example, let' s sa y that a n impoverished , strugglin g inventor is trying to construct th e world's first time machin e i n his cluttered basement . Ou t of nowhere, a wealthy, elderly gentleman appear s an d offer s hi m ampl e funds an d th e complex equations and circuitr y to make a time machine. The invento r subsequently enriches himself with the knowledg e o f time travel, knowing beforehand exactl y when stock-market booms an d busts will occur before the y happen. H e makes a fortune betting on the stoc k market, horse races , and othe r events. Decades later, as a wealthy, aging man, h e goe s bac k i n tim e t o fulfil l hi s destiny. He meet s himsel f as a young man working in his basement, and gives his younger self the secret of time trave l and th e mone y t o exploi t it . The questio n is : Where di d the ide a o f time travel come from ? Perhaps the craziest of these time travel paradoxes of the second type was cooke d u p b y Rober t Heinlei n i n hi s classi c short stor y "Al l You Zombies—." A baby girl is mysteriously dropped off at an orphanag e i n Cleveland in 1945 . "Jane " grow s u p lonel y an d dejected , no t knowin g who he r parents are , unti l one da y in 196 3 sh e is strangely attracted t o a drifter. She falls in love with him. But just when things are finally looking up fo r Jane, a serie s o f disaster s strike . First , sh e become s pregnan t b y th e drifter, wh o then disappears . Second , durin g th e complicate d delivery, doctors fin d tha t Jan e ha s both set s of sex organs, an d t o sav e he r life , they ar e force d t o surgicall y convert "her " t o a "him. " Finally , a mysterious stranger kidnap s her bab y from th e deliver y room. Reeling fro m thes e disasters , rejecte d b y society , scorned b y fate , "he" become s a drunkard an d drifter. Not only has Jane lost her parents and he r lover , but h e ha s lost his only child as well. Years later, in 1970 , he stumble s int o a lonel y bar , calle d Pop' s Place , an d spill s ou t hi s pathetic story to an elderly bartender. Th e sympatheti c bartender offer s the drifter the chanc e to avenge the stranger wh o left he r pregnant an d

To Build a Time Machine 23

7

abandoned, o n th e conditio n tha t h e join th e "tim e traveler s corps." Both o f the m ente r a tim e machine , an d th e bartende r drop s of f th e drifter i n 1963 . Th e drifte r i s strangel y attracte d t o a youn g orpha n woman, who subsequently becomes pregnant . The bartende r the n goe s forwar d 9 months , kidnap s th e bab y girl from th e hospital, and drop s off the baby in an orphanage bac k in 1945 . Then th e bartende r drop s of f the thoroughl y confused drifte r i n 1985 , to enlis t in th e tim e traveler s corps. The drifte r eventuall y gets his lif e together, becomes a respected an d elderly member of the time travelers corps, an d the n disguise s himself as a bartender an d ha s his most diffi cult mission: a date with destiny, meeting a certain drifte r at Pop's Place in 1970 . The questio n is : Who i s Jane's mother , father , grandfather, grand mother, son , daughter , granddaughter , an d grandson ? Th e girl , th e drifter, an d th e bartender , o f course , ar e al l th e sam e person . Thes e paradoxes ca n mad e you r hea d spin , especiall y if you tr y t o untangl e Jane's twiste d parentage. I f we draw Jane's famil y tree , we fin d tha t all the branche s ar e curle d inwar d back on themselves , as in a circle . We come t o th e astonishin g conclusio n tha t sh e i s her ow n mothe r an d father! Sh e is an entir e family tre e unto herself.

World Line s Relativity gives us a simple method t o sort through th e thorniest of these paradoxes. W e will make use of the "worl d line " method, pioneere d by Einstein. For example , sa y our alar m cloc k wakes us up on e da y at 8:0 0 A.M., and w e decide t o spen d th e mornin g i n bed instea d o f going t o work. Although it appears that we are doin g nothing by loafing i n bed, we are actually tracing out a "world line. " Take a sheet o f graph paper , an d o n th e horizonta l scal e put "dis tance" and o n the vertical scale put "time. " If we simply lie in bed fro m 8:00 to 12:00 , our worl d line is a straight vertical line. We went 4 hours into th e future , bu t travele d no distance . Even engaging in our favorit e pastime, doing nothing , create s a world line. (I f someone eve r criticizes us for loafing, we can truthfully claim that, according to Einstein's theory of relativity, we are tracin g out a world line in four-dimensiona l spacetime.) Now let's say that we finally get out o f bed a t noon an d arriv e at work at 1:0 0 P.M . Ou r worl d line becomes slanted becaus e w e are movin g in

238 WORMHOLES

: GATEWAY S TO ANOTHE R UNIVERSE ?

space a s well as time. I n th e lowe r lef t corne r i s our home , an d o n th e upper rightis our offic e (Figur e 11.1).If we take the car to work, though, we arriv e a t th e offic e earlier , a t 12:30 . Thi s mean s tha t th e faste r we travel, th e mor e ou r worl d lin e deviate s from th e vertical . (Notic e that there i s also a "forbidde n region " i n th e diagra m tha t ou r worl d lin e can't ente r because w e would hav e to be goin g faste r than th e spee d o f light.) One conclusio n i s immediate. Ou r worl d line never reall y begins o r ends. Eve n when we die, the worl d line s of the molecule s i n our bodie s keep going . Thes e molecule s may disperse int o th e ai r or soil , but the y will trace ou t thei r ow n never-ending world lines. Similarly, when we are born, the world lines of the molecules coming from our mother coalesc e into a baby. At no point do world lines break off or appear from nothing. To see how this all fits together, tak e th e simpl e example o f our own personal worl d line. In 1950 , say, our mothe r and father met, fell in love, and produced a baby (us). Thus the world lines of our mother and father collided and produced a third world line (ours) . Eventually, when someone dies , th e worl d line s formin g th e perso n dispers e int o billion s of world line s o f our molecules . From thi s poin t o f view, a huma n bein g can b e define d a s a temporar y collectio n o f world line s o f molecules . These world lines were scattered befor e we were born, cam e together t o form ou r bodies , an d wil l rescatte r afte r we die. Th e Bibl e says, "fro m dust to dust." In this relativistic picture, we might say, "from worl d lines to world lines." Our worl d lin e thu s contain s th e entir e bod y o f informatio n con cerning our history. Everything that ever happened to us—from ou r first bicycle, to our firs t date , t o our first job—is recorded in our world line. In fact , th e grea t Russia n cosmologist Georg e Gamow , who was famous for approachin g Einstein' s wor k wit h wi t an d whimsy , aptl y title d hi s autobiography M y World Line. With th e ai d o f th e worl d line, w e ca n no w pictur e wha t happen s when we go back in time . Let's sa y we enter a tim e machin e and mee t our mothe r befor e w e are born . Unfortunately , she fall s i n love with us and jilt s ou r father . D o w e reall y disappear, a s depicte d i n Back t o the Future? O n a worl d line , w e no w se e wh y this i s impossible . When w e disappear, ou r worl d lin e disappears . However , accordin g t o Einstein, world line s canno t b e cut . Thus alterin g the pas t i s not possibl e in rel ativity. The second paradox , involving re-creating the past, poses interesting problems, however . For example, by going back in time, we are fulfillin g the past , no t destroyin g it. Thus th e worl d lin e o f the invento r o f time

Figure 11.1. Our world line summarizes our entire history, from birth to death. For example, if we lie in bed from 8:00 A.M. to 12:00, our world line is a vertical line. If we travel by car to work, then our world becomes a, slanted line. The faster we move, the more slanted our world line becomes. The fastest we can travel, however, is the speed of light. Thus part of this space-time diagram is "forbidden"; that is, we would have to go faster than the speed of light to enter into this forbidden zone.

239

240 WORMHOLES

: GATEWAY S T O ANOTHE R UNIVERSE ?

travel i s a close d loop . Hi s world lin e fulfills, rathe r tha n changes , th e past. Much more complicate d i s the world line of "Jane," the woman who is her ow n mother an d fathe r and so n and daughte r (Figur e 11.2) . Notice, onc e again , tha t w e cannot alte r th e past . Whe n ou r worl d line goes back in time , it simply fulfills wha t is already known. In suc h a universe, therefore, i t is possible t o mee t yourself in th e past . I f we live through on e cycle, then sooner or later we meet a young man or woman who happens t o be ourselves when we were younger. We tell this young person tha t h e o r sh e look s suspiciousl y familiar. Then , thinkin g a bit, we remember tha t when we were young, we met a curious, older perso n who claimed tha t we looked familiar . Thus perhap s w e can fulfil l th e past , bu t neve r alte r it. World lines , as we have stressed , canno t b e cu t an d canno t end . The y ca n perhap s perform loop s i n time , but neve r alter it. These light cone diagrams, however , have been presented onl y in the framework o f specia l relativity , which ca n describ e wha t happens i f we enter the past, but is too primitive to settle th e question o f whether time travel makes any sense. To answer thi s larger question, w e must turn t o the general theor y of relativity, where the situation becomes much mor e delicate. With th e ful l powe r o f genera l relativity , we se e tha t thes e twiste d world line s migh t b e physicall y allowed. Thes e close d loop s g o b y th e scientific nam e closed timelike curves (CTCs) . The debat e i n scientifi c circles i s whethe r CTC s ar e allowe d b y genera l relativit y and quantu m theory. Spoiler o f Arithmetic an d General Relativity

In 1949 , Einstei n was concerned abou t a discover y by one o f hi s clos e colleagues and friends , the Viennese mathematicia n Kurt Godel, als o at the Institut e fo r Advanced Stud y at Princeton , wher e Einstei n worked. Godel foun d a disturbing solutio n t o Einstein's equations tha t allowe d for violation s of the basi c tenets o f common sense : Hi s solution allowed for certai n form s of time travel . For th e firs t tim e i n history , time travel was given a mathematical foundation . In some quarters , Godel wa s known as a spoiler. I n 1931 , he becam e famous (or , actually, infamous) when he proved, contrary to every expectation, tha t yo u canno t prov e th e self-consistenc y of arithmetic. I n th e process, h e ruine d a 2,000-year-ol d dream , datin g bac k t o Eucli d an d

Figure 11.2. If time travel is possible, then our world line becomes a closed loop. In 1945, the girl is born. In 1963, she has a baby. In 1970, he is a drifter, who goes back to 1945 to meet himself. In 1985, he is a time traveler, who picks himself up in a bar in 1970, takes himself back to 1945, kidnaps the baby and takes her back to 1945, to start all over again. The girl is her own mother, father, grandfather, grandmother, son, daughter, and so on.

241

242 WORMHOLES

: GATEWAY S T O ANOTHE R UNIVERSE ?

the Greeks , which was to have been th e crownin g achievement of mathematics: t o reduc e al l o f mathematic s t o a small , self-consistent set of axioms from whic h everything could b e derived . In a mathematical tou r de force, Godel showed that there will always be theorems i n arithmetic whose correctness or incorrectness can never be demonstrated fro m th e axiom s of arithmetic; tha t is , arithmetic will always be incomplete. Godel' s result was the mos t startling, unexpected development i n mathematical logic in perhaps a thousand years. Mathematics, onc e though t t o b e th e pures t o f all sciences because it was precise an d certain , untarnishe d b y the unpleasan t crudenes s o f our material world, now became uncertain . After Godel , the fundamental basi s fo r mathematic s seeme d t o b e lef t adrift . (Crudel y speaking, Godel's remarkable proo f bega n b y showing that there ar e curiou s paradoxes i n logic. For example, conside r th e statemen t "This sentence is false." If the sentence is true, then it follows that it is false. If the sentence is false , the n th e sentenc e i s true. O r conside r th e statemen t " I a m a liar." Then I am a liar only if I tell the truth. Gode l the n formulated the statement "Thi s sentenc e canno t b e prove d true. " I f th e sentenc e i s correct, the n i t cannot b e prove d t o be correct . B y carefully buildin g a complex web of such paradoxes, Gode l showed that there are true statements that cannot b e proved usin g arithmetic.) After demolishin g one o f the mos t cherished dream s o f all of mathematics, Gode l nex t shattere d th e conventiona l wisdo m surroundin g Einstein's equations . H e showe d tha t Einstein' s theory contain s som e surprising pathologies, includin g time travel. He firs t assumed that the universe was filled wit h gas or dust that was slowly rotating . Thi s seeme d reasonable , sinc e th e fa r reache s o f th e universe do see m t o be fille d wit h gas and dust . However, Godel's solution cause d great concer n fo r tw o reasons. First, his solution violated Mach's principle. He showed that Lwo solutions of Einstein's equations were possible with the sam e distribution of dust an d gas . (Thi s meant tha t Mach' s principle wa s somehow incomplete, tha t hidden assumption s were present.) More important , h e showe d tha t certai n form s o f tim e trave l were permitted. I f one followe d the pat h o f a particl e i n a Gode l universe, eventually it would come back and mee t itself in the past. He wrote, ' 'By making a round tri p o n a rocket shi p i n a sufficientl y wid e curve , it is possible i n thes e world s to trave l into an y region o f th e past , present , and future, and back again."2 Thus Godel found the first CTC in general relativity. Previously, Newto n considere d tim e t o b e movin g lik e a straigh t

To Build a Time Machine 24

3

arrow, which unerringl y flie s forwar d toward it s target . Nothin g coul d deflect o r chang e th e cours e o f thi s arro w once i t wa s shot. Einstein, however, showed that time was more like a mighty river, moving forward but ofte n meandering throug h twistin g valleys and plains. The presenc e of matter o r energ y might momentaril y shift th e directio n o f the river, but overal l th e river' s course wa s smooth: I t neve r abruptl y ended o r jerked backward . However, Godel showe d tha t th e rive r o f tim e coul d be smoothl y bent backwar d int o a circle . Rivers , afte r all , have edd y currents an d whirlpools . In th e main , a rive r may flow forward , bu t a t the edge s ther e ar e alway s sid e pool s wher e wate r flow s i n a circula r motion. Godel's solutio n coul d no t b e dismisse d a s th e wor k of a crackpo t because Gode l ha d vise d Einstein' s own field equation s t o fin d strang e solutions in which time bent into a circle. Because Godel ha d playe d by the rule s and discovere d a legitimate solution to his equations, Einstein was force d t o tak e the evasiv e route an d dismis s it because i t did no t fi t the experimenta l data . The wea k spo t i n Godel' s univers e was the assumptio n tha t th e ga s and dus t in the universe were slowly rotating. Experimentally , we do no t see an y rotation o f th e cosmi c dus t and ga s in space . Our instrument s have verifie d tha t th e univers e is expanding, bu t i t does no t appea r t o be rotating. Thus the Godel universe can be safely ruled out. (Thi s leaves us with th e rathe r disturbing , although plausible , possibilit y that i f ou r universe di d rotate , a s Gode l speculated , the n CTC s an d tim e trave l would b e physicall y possible.) Einstein die d i n 1955 , conten t tha t disturbing solutions to his equations coul d b e swep t unde r th e ru g fo r experimental reason s an d tha t people coul d no t mee t thei r parent s befor e the y were born .

Living in the Twilight Zone Then, i n 1963 , Ezr a Newman , Theodore Unti , an d Loui s Tamburin o discovered a new solution t o Einstein's equation s tha t was even crazier than Godel's . Unlik e the Gode l universe , their solutio n wa s not base d on a rotating dust-fille d universe . On th e surface , i t resembled a typical black hole . As in th e Gode l solution , their univers e allowed for CTC s and tim e travel. Moreover , when goin g 36 0 degrees aroun d th e blac k hole, you would no t win d up wher e you originally started. Instead , like livin g on a universe with a Riemann cut , you would wind up o n anothe r shee t of

244 WORMHOLES

: G A T E W A Y S T O A N O T H E R U N I V E R S E?

the universe . Th e topolog y o f a Newman-Unti-Tamburin o univers e might be compared t o livin g on a spiral staircase. If we move 360 degrees around th e staircase , w e do no t arriv e a t th e sam e poin t a t whic h we started, but on another landing of the staircase. Living in such a universe would surpas s ou r wors t nightmare , wit h commo n sens e bein g com pletely thrown out the window. In fact, this bizarre universe was so pathological tha t i t was quickly coined th e NU T universe, afte r th e initial s of its creators . At first, relativist s dismissed th e NU T solutio n i n th e sam e way they had dismisse d th e Gode l solution ; tha t is , our univers e didn' t see m t o evolve i n th e wa y predicted b y these solutions , so the y were arbitraril y discarded fo r experimenta l reasons . However , as the decade s wen t by, there was a floo d o f such bizarr e solution s t o Einstein's equations tha t allowed fo r tim e travel . I n th e earl y 1970s , Fran k J. Tiple r a t Tulan e University in New Orleans reanalyze d an old solution t o Einstein's equations found by W. J. van Stockum in 1936 , eve n before Godel's solution. This solutio n assume d th e existenc e o f an infinitel y long , rotatin g cylinder. Surprisingl y enough, Tiple r wa s able t o sho w that thi s solution also violated causality. Even the Kerr solution (whic h represents th e most physically realistic description o f black hole s i n oute r space ) wa s shown t o allo w for tim e travel. Rocket ships that pas s throug h th e cente r of the Ker r black hol e (assuming they are not crushe d i n th e process ) coul d violate causality. Soon, physicist s found tha t NUT-type singularities could b e inserted into any black hole o r expandin g universe . I n fact, i t now became pos sible t o coo k u p a n infinit e number o f pathologica l solution s t o Ein stein's equations . Fo r example , ever y wormhol e solutio n t o Einstein's equations coul d b e shown to allow some for m of time travel. According to relativist Frank Tipler, "solutions to the field equations can be found which exhibit virtually any type of bizarre behavior."3 Thus an explosio n o f pathologica l solution s t o Einstein' s equation s wa s discovered tha t certainl y would hav e horrifie d Einstei n ha d h e stil l bee n alive. Einstein's equations, i n some sense , were like a Trojan horse. On th e surface, th e hors e look s lik e a perfectl y acceptabl e gift , givin g us th e observed bendin g of starlight unde r gravit y and a compelling explana tion o f the origin of the universe. However, inside lurk all sorts of strange demons an d goblins , which allow for the possibilit y of interstellar travel through wormhole s and time travel. The price we had to pay for peering into th e darkes t secret s o f th e univers e wa s the potentia l downfal l o f some of our most commonly held beliefs about our world—that its space is simply connected an d it s history is unalterable .

To Build a Time Machine 24

5

But th e questio n stil l remained: Coul d thes e CTC s be dismisse d o n purely experimental grounds , a s Einstein did, o r coul d someon e sho w that the y wer e theoreticall y possibl e an d the n actuall y buil d a tim e machine? To Buil d a Time Machine

In June 1988 , thre e physicist s (Ki p Thome an d Michae l Morri s at th e California Institut e of Technology an d Ulv i Yurtsever at th e Universit y of Michigan) made th e first serious proposal fo r a tim e machine. They convinced th e editor s of Physical Review Letters, one o f th e mos t distin guished publication s i n th e world, that thei r wor k merited seriou s consideration. (Ove r th e decades , score s o f crackpo t proposal s fo r tim e travel have been submitte d to mainstream physics journals, bu t al l have been rejected because the y were not based o n sound physical principles or Einstein' s equations. ) Lik e experience d scientists , the y presente d their argument s i n accepte d fiel d theoretica l languag e an d the n care fully explaine d wher e their weakest assumptions were. To overcome the skepticism of the scientific community, Thorne an d his colleagues realize d that the y would have to overcom e th e standar d objections t o usin g wormhole s as tim e machines . First , as mentione d earlier, Einstei n himself realized that the gravitational forces at the center o f a black hole woul d be s o enormous tha t an y spacecraft would be torn apart . Althoug h wormhole s wer e mathematicall y possible, the y were, in practice, useless. Second, wormhole s migh t b e unstable . On e coul d sho w that small disturbances i n wormhole s would cause th e Einstein-Rose n bridg e t o collapse. Thus a spaceship's presence insid e a black hole would be sufficient to cause a disturbance that would close the entrance to the wormhole. Third, on e woul d have to g o faster tha n th e spee d o f light actually to penetrate th e wormhole t o the other side . Fourth, quantum effect s woul d be so large tha t the wormhole might close b y itself . Fo r example , th e intens e radiatio n emitte d b y th e entrance to the black hole not onl y would kill anyone who tried to enter the blac k hole, but als o might close the entrance . Fifth, tim e slows down in a wormhole and come s t o a complete sto p at the center. Thus wormholes have the undesirable feature that as seen by someon e o n th e earth , a spac e travele r appear s t o slo w dow n an d come t o a total halt a t the cente r o f the blac k hole. Th e spac e traveler looks like he o r sh e is frozen i n time. In other words, it takes an infinit e

246 WORMHOLES

: GATEWAY S T O ANOTHE R UNIVERSE ?

amount o f time fo r a space travele r t o go through a wormhole. Assuming, for the moment, tha t one coul d someho w go through th e center of the wormhol e an d retur n t o earth, th e distortio n o f time would still be so great that million s or eve n billion s of years may have passe d o n th e earth. For al l these reasons , th e wormhol e solution s were never taken seriously. Thorne is a serious cosmologist , one wh o might normall y view time machines with extrem e skepticis m or eve n derision . However , Thorn e was gradually drawn into thi s quest in th e mos t curious way. In the summer o f 1985 , Car l Sagan sen t t o Thorne th e prepublicatio n draf t of his next book , a novel called Contact, which seriously explores th e scientifi c and politica l question s surroundin g a n epoch-makin g event : makin g contact wit h th e firs t extraterrestria l lif e i n oute r space . Ever y scientist pondering th e question of life in outer space must confront the question of how to break th e ligh t barrier. Sinc e Einstein's special theoiy of relativity explicitly forbids travel faster tha n th e spee d o f light, traveling to the distant stars in a conventional spaceship may take thousands of years, thereby makin g interstella r trave l impractical . Sinc e Saga n wante d t o make his book a s scientifically accurat e a s possible, he wrot e to Thorn e asking whether there was any scientifically acceptable way of evading th e light barrier . Sagan's reques t pique d Thome' s intellectual curiosity. Here wa s an honest, scientificall y relevan t reques t mad e b y one scientis t to anothe r that demanded a serious reply. Fortunately, because of the unorthodo x nature o f the request , Thorne and hi s colleagues approached th e question i n a most unusual way: They worked backward. Normally , physicists start with a certain know n astronomical objec t ( a neutron star , a black hole, th e Bi g Bang) and the n solv e Einstein's equations t o find the curvature o f th e surroundin g space . Th e essenc e o f Einstein' s equations, we recall, is that the matte r an d energ y content o f an object determines the amount o f curvature in the surrounding space and time. Proceedin g in thi s way, we are guarantee d t o fin d solution s to Einstein' s equations for astronomicall y relevant objects that we expect to find in outer space. However, becaus e o f Sagan' s strang e request , Thorn e an d hi s colleagues approache d th e questio n backward . They started wit h a roug h idea o f what they wanted t o find . The y wanted a solution t o Einstein's equations in which a space traveler would not b e torn apart by the tida l effects o f the intens e gravitationa l field. They wanted a wormhole tha t would b e stabl e an d no t suddenl y clos e u p i n th e middl e o f th e trip . They wante d a wormhol e i n whic h th e tim e i t take s fo r a roun d tri p

To Build a Time Machine 24

7

would b e measure d i n days , not million s or billion s of earth years , and so on. In fact, their guiding principle was that they wanted a time traveler to have a reasonably comfortable ride back through tim e after enterin g the wormhole. Once the y decided what their wormhole would look like, then, and onl y then, did the y begin t o calculate th e amoun t o f energy necessary t o create suc h a wormhole. From thei r unorthodox poin t of view, they did no t particularl y care if the energ y requirements were well beyond twentieth-centur y science. To them , i t wa s an engineerin g proble m fo r som e futur e civilizatio n actually to construct the time machine. They wanted to prove that it was scientifically feasible, not tha t it was economical o r within the bounds of present-day earth science : Normally, theoretical physicist s ask, "What are th e law s of physics?" and/ or "Wha t d o thos e law s predict abou t th e Universe? " I n thi s Letter , we ask, instead, "What constraints d o the law s of physics place on the activities of an arbitraril y advanced civilization? " This wil l lea d t o som e intriguin g queries about the law s themselves. We begin b y asking whether th e law s of physics permit a n arbitraril y advance d civilizatio n t o construct an d main tain wormhole s for interstella r travel. 4

The ke y phrase, of course, is "arbitrarily advanced civilization. " The laws of physics tell us what is possible, not wha t is practical. Th e law s of physics are independent o f what it might cost to test them. Thus what is theoretically possibl e ma y excee d th e gros s nationa l produc t o f th e planet earth . Thorn e an d hi s colleagues wer e carefu l t o state tha t this mythical civilizatio n that ca n harnes s th e powe r o f wormholes must be "arbitrarily advanced"—tha t is, capable o f performing all experiments that are possibl e (eve n if they are no t practica l for earthlings) . Much t o thei r delight , with remarkabl e eas e the y soon foun d a surprisingly simpl e solutio n tha t satisfie d al l their rigi d constraints . I t was not a typical black hole solution at all, so they didn't hav e to worry about all th e problem s o f being rippe d apar t b y a collapsed star . They christened thei r solution the "transversible wormhole," t o distinguish it from the othe r wormhol e solution s tha t ar e no t transversibl e by spaceship. They wer e s o excited b y their solutio n tha t the y wrote bac k t o Sagan , who the n incorporate d som e o f thei r idea s i n hi s novel . I n fact , the y were so surprised b y the simplicit y of their solution tha t the y were convinced tha t a beginning graduat e studen t i n physic s would b e abl e t o understand thei r solution . In th e autumn o f 1985, o n th e final exam in a cours e o n genera l relativit y given at Caltech , Thorne gav e the worm-

248 WORMHOLES

: GATEWAY S T O ANOTHE R UNIVERSE ?

hole solutio n to the student s without telling them what it was, and they were aske d t o deduc e it s physica l properties . (Mos t student s gav e detailed mathematica l analyses of the solution , but the y failed t o grasp that they were looking at a solution tha t permitte d tim e travel.) If th e student s had bee n a bi t mor e observan t o n tha t fina l exam , they would have been abl e to deduce some rather astonishing properties of the wormhole. In fact, the y would have found tha t a trip through this transversible wormhole would be as comfortable as a trip on an airplane. The maximu m gravitationa l force s experienced b y the traveler s would not exceed 1 g. In other words, their apparent weight would not excee d their weight on th e earth . Furthermore , th e traveler s would never have to worr y about th e entranc e o f th e wormhol e closin g u p durin g th e journey. Thome' s wormhole is , in fact , permanentl y open . Instea d o f taking a million or a billion years, a trip through th e transversible wormhole would be manageable. Morri s and Thorn e writ e that "th e tri p will be fully comfortabl e and wil l require a total of about 20 0 days," or less. 5 So far , Thorn e note s tha t th e tim e paradoxe s tha t on e usuall y encounters i n th e movie s are no t t o b e found: "Fro m exposur e t o science fictio n scenario s (fo r example, thos e i n whic h on e goe s bac k i n time an d kill s oneself ) on e migh t expec t CTC s t o giv e ris e t o initial trajectories with zero multiplicities" (tha t is, trajectories that are impossible).'' However, he ha s shown that th e CTC s that appear i n his wormhole see m t o fulfill th e past , rather tha n chang e i t or initiate time para doxes. Finally, i n presentin g thes e surprisin g result s t o th e scientifi c com munity, Thorn e wrote , " A ne w clas s o f solution s of th e Einstei n field equations i s presented , whic h describ e wormhole s that , i n principle , could b e traversed by human beings. " There is , of course , a catch t o al l this , whic h is one reaso n wh y we do not hav e time machines today. The las t step in Thome's calculation was to deduce th e precise natur e o f the matte r an d energ y necessary to create this marvelous transversible wormhole. Thorne and his colleagues found tha t a t th e cente r o f th e wormhole , ther e mus t b e a n "exotic " form o f matter that has unusual properties. Thorn e is quick to point ou t that thi s "exotic" form o f matter, althoug h unusual , does no t see m t o violate any of the known laws of physics. He cautions that, at some future point, scientist s may prove tha t exoti c matte r doe s not exist. However, at present , exoti c matte r seem s t o b e a perfectl y acceptable for m o f matter i f on e ha s acces s t o sufficientl y advance d technology . Thorn e writes confidently that' 'from a single wormhole an arbitrarily advanced civilization ca n construct a machine fo r backward time travel. "

To Build a Time Machine 24

9

Blueprint fo r a Tim e Machin e Anyone who ha s rea d H . G. Wells's Th e Time Machine, however , may be disappointed with Thome's blueprint for a time machine . You do no t sit in a chai r i n you r livin g room, tur n a fe w dials, se e blinkin g lights, and witnes s the vas t panoram a o f history , including destructiv e world wars, th e ris e an d fal l o f grea t civilizations , or th e fruit s o f futuristi c scientific marvels . One versio n o f Thome's tim e machin e consist s of tw o chambers , each containin g tw o parallel meta l plates . Th e intens e electri c fields created betwee n each pai r o f plates (large r than anythin g possible with today's technology ) rip s th e fabri c o f space-time , creatin g a hol e i n space tha t link s th e tw o chambers. On e chambe r i s then place d i n a rocket shi p an d i s accelerated t o near-ligh t velocities , while th e othe r chamber stay s on th e earth . Sinc e a wormhole can connect tw o regions of spac e wit h differen t times , a cloc k in th e firs t chambe r tick s slower than a cloc k in th e secon d chamber . Becaus e tim e woul d pas s a t different rate s a t th e tw o ends o f th e wormhole , anyone fallin g int o on e end o f th e wormhol e woul d b e instantl y hurled int o th e pas t o r th e future. Another tim e machine might look like the following. If exotic matter can b e foun d an d shape d lik e metal, the n presumabl y th e idea l shap e would be a cylinder. A human stand s i n the cente r o f the cylinder . The exotic matter the n warp s the spac e an d tim e surrounding it , creating a wormhole tha t connect s t o a distant part o f the univers e in a differen t time. At the cente r o f the vorte x i s the human , who the n experience s no mor e tha n 1 g of gravitational stress as he o r she is then sucke d int o the wormhole and finds himself or herself on the other end o f the universe. On the surface, Thome's mathematical reasoning is impeccable. Einstein's equations indeed sho w that wormhole solutions allow for time to pass at different rate s on either side of the wormhole, so that time travel, in principle , is possible. The trick , of course, is to create th e wormhol e in th e firs t place . As Thorne an d hi s collaborator s ar e quic k to poin t out, th e mai n problem i s how to harnes s enoug h energ y t o create an d maintain a wormhole with exoti c matter. Normally, on e o f th e basi c tenet s o f elementar y physic s is tha t al l objects hav e positiv e energy . Vibratin g molecules, movin g cars , flyin g birds, an d soarin g rocket s al l have positiv e energy . (B y definition, th e empty vacuum of space ha s zer o energy. ) However , i f we can produc e objects with "negativ e energies" (tha t is, something that has an energ y

250 WORMHOLES

: GATEWAY S T O ANOTHE R UNIVERSE ?

content les s than th e vacuum), then we might be able to generate exoti c configurations o f space an d tim e in which time is bent into a circle. This rather simple concept goe s by a complicated-sounding title: the averaged weak energy condition (AWEC) . As Thome is careful t o point out, the AWE C mus t b e violated ; energ y mus t becom e temporaril y negative for tim e travel to be successful. However , negative energy has historically been anathem a t o relativists , wh o realiz e tha t negativ e energ y would make possibl e antigravit y and a hos t o f othe r phenomen a tha t hav e never been see n experimentally . But Thorne i s quick to point out tha t there is a way to obtain negative energy, and thi s is through quantu m theory . In 1948 , the Dutch physicist Henrik Casimir demonstrated tha t quantum theor y ca n create negativ e energy: Just tak e tw o large, uncharge d paralle l metal plates. Ordinarily, common sens e tells us that these two plates, because the y are electrically neutral, hav e no forc e betwee n them . Bu t Casimi r proved tha t th e vacuum separatin g these two plates, because of the Heisenberg Uncertainty Principle, is actually teeming wit h activity , with trillions of particles an d antiparticles constantly appearing an d disappearing . The y appear ou t of nowhere and disappea r bac k into the vacuum. Because they are so fleeting, the y are, fo r th e mos t part , unobservable , an d the y d o no t violate any of the law s of physics. These "virtua l particles" create a net attractive force betwee n thes e tw o plates tha t Casimi r predicted wa s measurable. When Casimi r first published hi s paper, i t met wit h extreme skepti cism. After all, how can two electrically neutral objects attract, each other , thereby violating the usua l laws of classical electricity? This was unheard of. However, in 195 8 physicis t M. }. Sparnaay observed thi s effect i n th e laboratory, exactl y as Casimi r ha d predicted . Sinc e then , i t ha s bee n christened th e Casimir effect. One wa y of harnessing th e Casimi r effec t i s to plac e tw o large con ducting paralle l plate s at th e entranc e o f each wormhole , thereb y creating negativ e energy a t eac h end . A s Thorne an d hi s colleagues con clude, "It may turn out that the average weak energy condition can never be violated, in which case there coul d b e no suc h things as transversible wormholes, tim e travel , o r a failure o f causality. It's premature t o tr y to cross a bridge befor e yo u come t o it." 7 At present, th e jury is still out o n Thome' s time machine. Th e deci sive factor , al l agree , i s to hav e a full y quantize d theor y of gravit y settle the matte r once and fo r all. For example, Stephen Hawkin g has pointed out tha t th e radiatio n emitte d a t th e wormhol e entranc e wil l b e quit e large an d wil l contribut e bac k int o th e matter-energ y conten t o f Einstein's equations. Thi s feedback into Einstein's equations will distort th e

To Build a Time. Machine 25

1

entrance t o the wormhole, perhaps even closing it forever. Thorne, however, disagrees tha t the radiatio n wil l be sufficient t o close the entrance . This i s where superstring theor y comes in. Because superstring theory is a fully quantum-mechanica l theor y that includes Einstein's theory of genera l relativit y a s a subset, i t ca n b e use d t o calculat e corrections to th e origina l wormhol e theory . I n principle , i t will allo w us to deter mine whether the AWEC condition i s physically realizable, and whether the wormhol e entranc e stay s ope n fo r tim e traveler s to enjo y a tri p t o the past. Hawking ha s expresse d reservation s abou t Thome' s wormholes . However, thi s i s ironic becaus e Hawkin g himself has propose d a ne w theory of wormholes that is even more fantastic . Instead o f connecting the presen t wit h th e past , Hawkin g proposes t o use wormholes to con nect ou r univers e with an infinit e numbe r o f parallel universes!

12 Colliding Universe s [Nature is ] no t onl y queerer tha n w e suppose, i t i s queerer than we can suppose.

J. B . S. Haldane

C

OSMOLOGIST Stephen Hawkin g is one o f the mos t tragic figures in science . Dyin g o f a n incurable , degenerativ e disease , h e ha s relentlessly pursue d hi s researc h activitie s in th e fac e o f almos t insurmountable obstacles . Althoug h h e ha s los t contro l o f hi s hands , legs , tongue, and finally his vocal cords, he ha s spearheaded ne w avenues of research while confined to a wheelchair. Any lesser physicist would have long ago given up th e struggl e t o tackle the grea t problem s o f science. Unable t o gras p a pencil o r pen, h e perform s al l his calculations in his head, occasionally aided by an assistant. Bereft of vocal cords, he uses mechanical device s to communicate with th e outsid e world. But he no t only maintains a vigorous research program , bu t stil l took tim e to write a best-sellin g book , A Brief History o f Time, an d t o lectur e aroun d th e world. I once visited Hawking in his home just outside Cambridge University when I was invited t o speak a t a physics conference h e wa s organizing. Walking through hi s living room, I was surprised b y the impressive array of ingenious gadgets tha t he uses to continue hi s research. For example, I sa w on hi s des k a devic e muc h lik e thos e use d b y musicians to hol d music sheets. This one, however , was much mor e elaborate an d had th e ability t o gra b eac h pag e an d carefull y tur n i t fo r readin g a book . ( I shivered t o ponder , a s I thin k man y physicists have, whether I would

252

Colliding Universes 25

3

have th e stamin a an d shee r willpowe r t o continu e researc h with out arms , legs , o r a voic e eve n i f I ha d th e fines t mechanica l aid s available.) Hawking is the Lucasia n Professo r of Physic s at Cambridg e Univer sity, th e sam e chair held b y Isaac Newton. And lik e hi s illustrious predecessor, Hawkin g ha s embarke d o n th e greates t ques t o f th e century , the final unification o f Einstein's theory of gravity an d quantu m theory. As a result, he, too , ha s marvele d a t the elegant , self-consistenc y o f th e ten-dimensional theory , an d i n fac t close s hi s best-sellin g boo k wit h a discussion of it. Hawking n o longe r spend s th e bul k o f hi s creativ e energ y o n th e field tha t mad e hi m world-famous—blac k holes—whic h ar e b y no w passe. He i s hunting bigger game—th e unifie d fiel d theory . Strin g the ory, w e recall, began a s a quantum theor y and the n late r absorbed Ein stein's theor y o f gravity . Hawking , startin g a s a pur e classica l relativis t rather than a quantum theorist, approaches the problem from th e other point o f view . H e an d hi s colleagu e James Hartl e star t wit h Einstein' s classical universe, and the n quantiz e the entir e universe!

Wave Functio n o f th e Univers e

Hawking i s on e o f th e founder s o f a ne w scientifi c discipline , calle d quantum cosmology. A t first, this seems like a contradiction i n terms . Th e word quantum applies t o th e infinitesimall y smal l world o f quark s an d neutrinos, while cosmology signifie s th e almos t limitless expanse o f oute r space. However, Hawking and others now believe that the ultimate questions o f cosmology ca n b e answere d only by quantum theory . Hawking takes quantum cosmolog y to its ultimate quantum conclusion , allowing the existenc e o f infinite numbers o f parallel universes. The startin g poin t o f quantum theory , we recall, i s a wave function that describe s al l the variou s possible states of a particle. For example , imagine a large, irregular thundercloud tha t fills up th e sky. The darke r the thundercloud, the greater th e concentration o f water vapor and dust at that point . Thu s b y simply looking at a thundercloud, we can rapidly estimate th e probabilit y o f findin g larg e concentration s o f water an d dust in certain part s of the sky. The thunderclou d ma y be compared t o a single electron's wave function. Like a thundercloud , i t fills up al l space. Likewise , th e greate r it s value at a point, the greater th e probability of finding the electron there . Similarly, wave functions ca n b e associate d wit h large objects , like peo -

254 W O R M H O L E S

: GATEWAY S TO ANOTHE R UNIVERSE ?

pie. As I sit in m y chair i n Princeton , I know that I have a Schodinger probability wave function. If I could somehow see my own wave function, it would resemble a cloud ver y much in the shap e of my body. However, some of the cloud would spread ou t over all space, out t o Mars and eve n beyond th e sola r system , although i t would b e vanishingl y smal l there. This mean s tha t ther e i s very large likelihoo d tha t 1 am, i n fact , sitting in my chair and no t o n th e plane t Mars. Although part o f m y wave func tion has spread eve n beyond th e Milk y Way galaxy, there is only an infin itesimal chanc e tha t I am sitting in another galaxy. Hawking's new idea was to treat the entire universe as though it were a quantum particle . By repeating som e simple steps, we are led t o some eye-opening conclusions . We begi n wit h a wav e functio n describin g th e se t of all possible universes. This mean s tha t th e startin g point o f Hawking's theor y must be an infinite set of parallel universes, the wave functiono f the universe. Hawking's rather simpl e analysis, replacing th e word particle with universe, has led to a conceptual revolutio n i n our thinkin g about cosmology . According to thi s picture, the wav e function of the univers e spreads out over all possible universes. The wav e function i s assumed to be quite large near our ow n universe, so there is a good chanc e that our universe is the correct one , a s we expect. However, the wave function spreads ou t over al l other universes , eve n thos e tha t ar e lifeles s an d incompatibl e with th e familia r laws o f physics. Since th e wav e functio n is supposedly vanishingly smal l for thes e othe r universes , we do no t expec t tha t ou r universe will mak e a quantum leap t o the m i n the nea r future . The goa l facin g quantu m cosmologist s i s t o verif y thi s conjectur e mathematically, t o sho w that th e wav e functio n of th e univers e is large for ou r presen t univers e and vanishingl y small for other universes. This would then prov e that our familiar universe is in some sense unique and also stable . (A t present, quantu m cosmologist s are unabl e t o solv e this important problem. ) If we take Hawking seriously, it means that we must begin our analysis with a n infinit e numbe r o f al l possibl e universes , coexistin g wit h on e another. To put i t bluntly, the definition of the word universe is no longe r "all tha t exists. " I t no w mean s "al l tha t ca n exist. " Fo r example , i n Figure 12. 1 we see how the wave function of the univers e can spread ou t over severa l possible universes , with ou r univers e being th e mos t likely one bu t certainl y not th e onl y one. Hawking' s quantum cosmolog y also assumes that the wav e function of the univers e allow s these universe s to collide. Wormhole s ca n develo p an d lin k thes e universes . However , these wormhole s are no t lik e th e one s w e encountered i n th e previou s

Colliding Universes

255

Figure 12.1. In Hawking's wave function of the universe, the wave function is most likely concentrated around own universe. We live in our universe because it is the most likely, with the largest probability. However, there is a small but nonvanishing probability that the wave function prefers neighboring, parallel universes. Thus transitions between universes may be possible (although with very low probability).

chapters, which connect different part s of three-dimensional space with itself—these wormhole s connect different universe s with one another . Think, for example, of a large collection of soap bubbles, suspended in air . Normally, each soa p bubble i s like a universe unto itself , excep t that periodically it bumps into another bubble, forming a larger one, or splits into two smaller bubbles. The differenc e i s that each soa p bubbl e is now an entire ten-dimensional universe. Since space and tim e can exist only on eac h bubble, there i s no suc h thing as space and tim e between the bubbles. Each universe has its own self-contained "time." It is meaningless to say that time passes at the same rate in all these universes. (We should, however , stress tha t trave l between these universe s is not ope n to u s becaus e o f ou r primitiv e technologica l level . Furthermore ,

Figure 12.2. Our universe may be one of an infinite number of parallel universes, each connected to the others by an infinite series of wormholes. Travel between these wormholes is possible but extremely unlikely.

256

Colliding Universes 25

7

we shoul d als o stres s tha t larg e quantu m transition s o n thi s scal e ar e extremely rare, probably much larger than th e lifetime of our universe.) Most of these universes are dea d universes, devoid o f any life. O n thes e universes, the law s of physics were different, an d henc e the physical conditions tha t mad e lif e possibl e wer e no t satisfied . Perhaps , amon g th e billions of parallel universes, only one (ours ) had the right set of physical laws to allo w life (Figur e 12.2). Hawking's "baby universe" theory , although no t a practical method of transportation, certainl y raises philosophical an d perhap s eve n religions questions . Already , it ha s stimulate d two long-simmering debate s among cosmologists.

Putting God Bac k i n th e Universe?

The firs t debat e concern s th e anthropic principle. Ove r th e centuries , scientists have learned t o view the univers e largely independent o f human bias. We no longer project our human prejudices and whims onto every scientific discovery . Historically, however, early scientists often committed th e fallac y o f anthropomorphism, whic h assumes that object s an d animals hav e humanlik e qualities . This erro r i s committed b y anyone who sees human emotion s and feelings being exhibited by their pets. (It is also committed by Hollywood scriptwriters who regularly assume that beings similar to us must populate planets orbiting the stars in the heavens.) Anthropomorphism i s an age-ol d problem . The Ionia n philosophe r Xenophanes onc e lamented , "Me n imagin e god s t o b e born , an d t o have clothe s an d voice s and shape s lik e theirs. . . . Yea, the god s o f th e Ethiopians are black and flat-nosed , and th e god s of the Thracian s are red-haired an d blue-eyed. " Withi n th e pas t few decades, som e cosmologists have been horrified to find anthropomorphism creepin g back into science, under the guise of the anthropic principle, some of whose advocates openly declare that they would like to put Go d back into science. Actually, there is some scientific merit to this strange debate over the anthropic principle , which revolves around th e indisputabl e fact tha t if the physica l constant s o f th e univers e wer e altere d b y th e smalles t amount, life i n the univers e would be impossible. Is this remarkable fac t just a fortunate coincidence, o r does it show the work of some Supreme Being? There ar e tw o versions of the anthropi c principle . The "weak " version state s tha t th e fac t tha t intelligen t lif e (us ) exists in th e univers e

258 WORMHOLES

: GATEWAY S T O ANOTHE R UNIVERSE ?

should b e take n a s an experimenta l fac t tha t help s u s understan d th e constants o f the universe . As Nobel laureat e Steve n Weinberg explains it, "th e worl d i s the wa y it is , at leas t i n part , becaus e otherwis e ther e would b e n o on e t o as k why it is the wa y it is." 1 State d i n thi s way, the weak version of the anthropi c principl e is hard t o argue with . To hav e lif e i n th e universe , you nee d a rar e conjunctio n o f many coincidences. Life , whic h depends o n a variety of complex biochemical reactions, ca n easil y be rendere d impossibl e i f we change som e o f th e constants of chemistry an d physic s by a small amount . Fo r example , if the constant s tha t gover n nuclea r physic s were change d eve n slightly , then nucleosynthesi s and th e creation of the heav y elements in the stars and supernova e migh t becom e impossible . Then atom s migh t becom e unstable o r impossibl e t o creat e i n supernovae . Lif e depend s o n th e heavy element s (element s beyond iron ) fo r th e creatio n o f DN A and protein molecules . Thus th e smalles t change i n nuclea r physic s would make th e heav y elements of the univers e impossible to manufacture in the stars . We ar e childre n o f th e stars ; however, if the law s o f nuclea r physics chang e i n th e slightest , the n ou r "parents " ar e incapabl e o f having "children " (us) . As another example , i t i s saf e t o sa y that th e creation o f lif e i n th e earl y oceans probabl y took 1 to 2 billio n years. However, if we could somehow shrink the lifetim e o f the proto n t o several millio n years, then lif e woul d be impossible . Ther e woul d not b e enough tim e to create lif e ou t o f random collision s of molecules. In othe r words, the very fact that we exist in the universe to ask these questions abou t i t means tha t a complex sequenc e o f events must necessarily have happened. I t mean s tha t th e physica l constants of nature must have a certain rang e o f values, so that th e star s lived long enoug h to create th e heav y elements in our bodies , so that protons don' t decay too rapidl y before lif e ha s a chance t o germinate , an d s o on. I n othe r words, the existence of humans who can ask questions about the universe places a hug e numbe r o f rigid constraint s o n th e physic s o f th e uni verse—for example , its age, it s chemical composition, it s temperature, its size, and it s physical processes. Remarking on thes e cosmi c coincidences, physicis t Freeman Dyso n once wrote , "A s we look ou t int o th e Univers e and identif y th e man y accidents o f physics and astronom y tha t hav e worked togethe r t o ou r benefit, it almost seems as if the Universe must in some sense have known that w e were coming. " Thi s take s u s t o th e "strong " versio n o f th e anthropic principle , which states that al l th e physica l constants o f th e universe have been precisel y chosen (b y God or som e Supreme Being) so tha t lif e i s possible i n ou r universe . Th e stron g version , because i t

Colliding Universes 25

9

raises question s abou t a deity , is much mor e controversia l amon g sci entists. Conceivably, it might have been blin d luck if only a few constants of nature wer e require d t o assum e certai n value s t o mak e lif e possible . However, it appears tha t a large se t of physical constants mus t assume a narrow ban d o f values in orde r fo r lif e t o for m i n ou r universe . Sinc e accidents o f thi s typ e ar e highl y improbable, perhap s a divin e intelli gence (God ) precisel y chose thos e values in order to create life . When scientists first hear of some version of the anthropic principle , they are immediately taken aback. Physicist Heinz Pagels recalled, "Her e was a form o f reasoning completel y foreign t o th e usua l way that dieoretical physicist s went about thei r business." 2 The anthropi c argumen t i s a more sophisticate d versio n o f the ol d argument tha t God located th e earth a t just the right distanc e fro m th e sun. If God had place d th e eart h to o close , then i t would be to o hot t o support life . I f God ha d place d th e eart h to o far, then i t would be to o cold. The fallac y of this argument i s that millions of planets in the galaxy probably ar e sittin g at the incorrect distance fro m thei r sun, and there fore lif e o n the m i s impossible . However , som e planet s will , b y pur e accident, be a t th e righ t distanc e fro m thei r sun . Ou r plane t i s one o f them, and hence we are here to discuss the question. Eventually, mos t scientist s become disillusione d wit h the anthropi c principle becaus e i t has no predictive power, nor ca n it be tested. Pagels reluctantly conclude d tha t "unlik e th e principle s o f physics, it afford s no way to determine whethe r it is right or wrong; there i s no way to test it. Unlike conventional physical principles, the anthropic principl e is not subject t o experimenta l falsification—th e sur e sig n tha t i t i s not a scientific principle." 3 Physicist Alan Gut h say s bluntly , "Emotionally , th e anthropic principl e kin d o f rubs m e th e wron g way. . . . The anthropi c principle i s something tha t peopl e d o i f the y can' t thin k o f anything better t o do."4 To Richard Feynman, the goa l o f a theoretical physicis t is to "prove yourself wrong as fast a s possible."5 However, the anthropi c principl e is sterile and cannot be disproved. Or, as Weinberg said, "although science is clearly impossible without scientists, it is not clea r tha t th e univers e is impossible withou t science."1' The debat e ove r th e anthropi c principl e (an d hence , abou t God ) was dormant fo r man y years, until i t was recently revived b y Hawking's wave function o f the universe . If Hawking is correct, the n indee d ther e are an infinite number o f parallel universes, many with different physical constants. I n som e o f them, perhap s proton s deca y too rapidly, or stars

260 WORMHOLES

: GATEWAY S T O ANOTHE R UNIVERSE ?

cannot manufacture the heavy elements beyond iron, or the Big Crunch takes place to o rapidl y before lif e ca n begin , and s o on. I n fact , a n infi nite numbe r o f thes e paralle l universe s ar e dead , withou t the physical laws that can mak e life a s we know it possible . On on e suc h parallel univers e (ours) , the law s of physics were compatible wit h lif e a s we know it. Th e proo f i s that w e are her e toda y t o discuss the matter . If this is true, then perhaps Go d does no t hav e to be evoked t o explai n wh y life, preciou s a s it is , is possible in ou r universe . However, this reopens th e possibilit y of the wea k anthropic principle — that is , that w e coexist with many dead universes , and tha t our s i s th e only one compatibl e wit h life. The secon d controvers y stimulated b y Hawking's wav e function o f the univers e is much deepe r an d i n fac t i s still unresolved . It i s called the Schrodinger's ca t problem . Schrodinger's Cat Revisite d Because Hawking' s theor y o f bab y universe s an d wormhole s use s th e power o f quantu m theory , i t inevitabl y reopen s th e stil l unresolve d debates concerning its foundations. Hawking' s wave function of the universe does no t completel y solv e these paradoxe s o f quantum theory ; it only expresses them i n a startling new light. Quantum theory, we recall, states that for every object there is a wave function tha t measures th e probability of finding that object at a certain point in space and time. Quantum theor y also states that you never really know the stat e o f a particle until you have made an observation . Before a measurement i s made, th e particl e ca n be in one o f a variety of states, described b y the Schrodinger wave function. Thus before an observation or measuremen t ca n b e made , you can' t reall y kno w th e stat e o f th e particle. In fact , th e particl e exists in a nether state , a sum of ail possible states, until a measurement is made. When thi s idea was first proposed b y Niels Bohr and Werner Heisenberg, Einstei n revolted agains t thi s concept. "Doe s the moo n exis t just because a mouse look s at it?" h e wa s fond o f asking. According t o th e strict interpretation o f quantum theory , the moon , befor e it is observed, doesn't reall y exist as we know it. The moo n ca n be, i n fact, i n an y on e of an infinit e numbe r o f states, including th e stat e o f bein g i n th e sky, of being blow n up, o r o f not bein g ther e a t all . It is the measuremen t process o f looking a t i t tha t decide s tha t th e moo n i s actually circling the earth .

Colliding Universes 26

1

Einstein ha d man y heated discussion s wit h Niel s Bohr challengin g this unorthodox worl d view. (I n one exchange , Boh r sai d to Einstein in exasperation, "Yo u ar e no t thinking . You are merel y being logical!" 7) Even Erwi n Schrodinge r (wh o initiated th e whol e discussio n wit h hi s celebrated wav e equation ) proteste d thi s reinterpretatio n o f hi s equa tion. H e onc e lamented , " I don' t like it, and I' m sorr y I ever had anything to do with it." 8 To challeng e thi s revisionis t interpretation , th e critic s asked, "I s a cat dead o r aliv e before you look a t it? " To sho w how absurd thi s question is , Schroodinger placed a n imag inary cat in a sealed box . Th e ca t faces a gun, which is connected t o a Geiger counter , which in tur n i s connected t o a piece o f uranium. Th e uranium ato m i s unstable an d wil l underg o radioactiv e decay. If a uranium nucleu s disintegrates, it will be picke d u p b y the Geige r counter , which wil l then trigge r th e gun , whose bullet wil l kill th e cat . To decid e whethe r th e ca t i s dead o r alive , w e must ope n th e bo x and observ e the cat . However, what is the state of the cat before we open the box ? According to quantum theory , we can onl y state tha t the ca t is described b y a wave function that describes the su m of a dead ca t and a live cat. To Schrodinger, th e idea of thinking about cats that are neither dea d nor aliv e was the heigh t o f absurdity, yet nevertheless the experimenta l confirmation o f quantu m mechanic s force s u s t o thi s conclusion . A t present, ever y experiment ha s verified quantu m theory. The parado x o f Schrodinger' s ca t i s s o bizarr e tha t on e i s ofte n reminded o f how Alice reacte d t o th e vanishin g of th e Cheshir e ca t in Lewis Carroll's fable: " 'You'll see me there,' said the Cat, and vanished. Alice wa s not muc h surprise d a t this , sh e wa s getting s o wel l use d t o queer thing s happening. " Ove r th e years , physicists , too , hav e gotte n used t o "queer" things happenin g i n quantum mechanics. There ar e at least three major ways that physicists deal with this complexity. First, we can assum e that God exists . Because all "observations" imply a n observer , the n ther e mus t b e som e "consciousness " i n th e universe. Som e physicists , like Nobe l laureat e Eugen e Wigner , hav e insisted tha t quantu m theor y prove s th e existenc e o f some sor t o f universal cosmi c consciousness in th e universe . The secon d wa y of dealin g wit h th e parado x i s favored b y the vas t majority o f working physicists—to ignore th e problem . Mos t physicists, pointing ou t tha t a camer a withou t an y consciousnes s ca n als o mak e measurements, simpl y wish tha t thi s sticky , bu t unavoidable , proble m would g o away .

262 WORMHOLES

: G A T E W A Y S T O ANOTHE R U N I V E R S E?

The physicis t Richard Feynma n onc e said , " I thin k i t i s safe t o say that n o on e understand s quantu m mechanics . D o no t kee p sayin g to yourself, if you can possibly avoid it, 'But how can it be like that?' because you wil l g o 'dow n th e drain ' int o a blind alle y from which nobody ha s yet escaped. Nobod y knows how it can b e lik e that."9 In fact , i t is often stated tha t o f al l th e theorie s propose d i n thi s century , th e sillies t is quantum theory . Some say that the onl y thing that quantum theor y has going for it , in fact , i s that i t is unquestionably correct . However, there is a third way of dealing with this paradox, calle d the many-worlds theory. Thi s theory (lik e th e anthropi c principle) fel l ou t o f favor i n th e pas t decades, but i s being revive d again b y Hawking's wave function o f the universe .

Many Worlds

In 1957 , physicis t Hugh Everet t raise d th e possibilit y that durin g th e evolution o f the universe , it continually "split" i n half , lik e a fork i n a road. I n one universe , th e uraniu m ato m di d no t disintegrat e an d th e cat was not shot . I n th e other , th e uraniu m ato m di d disintegrat e an d the ca t wa s shot. I f Everett is correct, ther e ar e a n infinit e numbe r o f universes. Each universe is linked to every other through th e network of forks in th e road. Or, a s the Argentinian write r Jorge Luis Borges wrote in Th e Garden o f Forking Paths, "time forks perpetuall y toward innumerable futures. " Physicist Bryc e DeWitt , one o f th e proponent s o f th e many-worlds theory, describe s th e lastin g impac t i t mad e o n him : "Ever y quantu m transition takin g plac e on ever y star, in ever y galaxy, in ever y remot e corner of the univers e is splitting our loca l world on eart h int o myriads of copie s o f itself . I stil l recal l vividly th e shoc k I experience d o n firs t encountering thi s multiworld concept."1" The many-world s theory postulates tha t al l possible quantu m world s exist. In som e worlds, humans exist a s th e dominan t lif e for m o n earth . I n othe r worlds , subatomic events too k plac e tha t prevente d human s fro m eve r evolvin g on thi s planet. As physicist Frank Wilczek noted, It is said that the histor y of the worl d would be entirely different i f Helen of Troy had ha d a wart at th e ti p o f her nose . Well , warts can aris e fro m mutations in single cells, often triggere d by exposure to the ultraviolet rays

Colliding Universes 26

3

of th e sun . Conclusion : ther e ar e many , man y world s i n whic h Hele n of Troy di d have a wart at the ti p of her nose. 11

Actually, the idea that there ma y be multiple universes is an old one . The philosophe r St . Albertus Magnus once wrote, "Do there exis t many worlds, or is there but a single world? This is one o f the mos t noble and exalted question s i n th e stud y of Nature." However , th e ne w twis t o n this ancient ide a is that thes e man y worlds resolve th e Schrodinge r cat paradox. I n on e universe , th e ca t ma y be dead ; i n another , th e ca t is alive. As strange a s Everett's many-worlds theory seems, one ca n show that it i s mathematically equivalent t o the usua l interpretations o f quantum theory. But traditionally, Everett's many-worlds theory has not been popular amon g physicists . Although i t cannot b e rule d out , th e ide a o f an infinite numbe r of equally valid universes, each fissioning in half at every instant in time, poses a philosophical nightmar e fo r physicists, who love simplicity. There i s a principle o f physics called Occam' s razor , which states that we should alway s take the simples t possible pat h an d ignor e more clums y alternatives, especially if the alternative s can never be measured. (Thu s Occam's razo r dismisse s th e ol d "aether " theory , which stated tha t a mysteriou s ga s onc e pervade d th e entir e universe . Th e aether theor y provided a convenien t answe r to a n embarrassin g ques tion: I f light i s a wave , an d ligh t ca n trave l i n a vacuum, the n wha t is waving? The answe r was that aether, lik e a fluid, was vibrating even in a vacuum. Einstein showed that th e aethe r was unnecessary. However, he never said tha t the aethe r didn't exist . He merely said it was irrelevant. Thus by Occam's razor , physicists don't refer to the aethe r anymore.) One ca n sho w that communicatio n betwee n Everett' s many worlds is not possible . Therefore, eac h univers e is unaware of the existenc e of the others . I f experiments canno t tes t for th e existenc e of these worlds, we should, b y Occam's razor , eliminat e them. Somewhat in th e sam e vein, physicists do no t sa y categorically that angels and miracle s cannot exist . Perhaps the y do. But miracles, almost by definition, are not repeatable and therefore not measurable b y experiment. Therefore , b y Occam's razor , w e must dismis s them (unless , of course, we can fin d a reproducible, measurabl e miracl e or angel). On e of th e developer s o f th e many-world s theory, Everett' s mento r John Wheeler, reluctantl y rejected i t becaus e "i t require d to o muc h meta physical baggage t o carry around."12 The unpopularit y of the many-world s theory, however, may subside as Hawking' s wav e functio n o f th e univers e gain s popularity. Everett's

264 WORMHOLES

: GATEWAY S T O ANOTHE R UNIVERSE ?

theory wa s based o n singl e particles , with no possibilit y of communica tion betwee n differen t universes a s they fissioned. However, Hawking' s theory, althoug h related , goe s muc h further : I t i s based o n a n infinite number o f self-contained universe s (an d no t just particles ) an d postu lates the possibilit y of tunneling (vi a wormholes) betwee n them . Hawking has eve n undertake n th e dauntin g tas k o f calculating th e solution t o th e wav e function o f th e universe . H e i s confident that hi s approach i s correct partly becaus e th e theor y i s well defined (if , as we mentioned, th e theory is ultimately defined in ten dimensions). His goal is to sho w that th e wav e function o f the univers e assume s a large value near a universe that looks lik e ours. Thus our univers e i s the most likely universe, bu t certainl y not th e only one . By now, there hav e been a number o f international conference s o n the wav e function o f the universe . However, as before, th e mathematic s involved in the wave function o f the universe is beyond th e calculational ability of any human o n this planet, and we may have to wait years before any enterprisin g individua l ca n fin d a rigorou s solutio n t o Hawking' s equations.

Parallel Worlds

A major difference between Everett's many-worlds theory and Hawking's wave function o f the univers e is that Hawking's theory places wormholes that connec t thes e paralle l universe s a t th e cente r o f his theory . However, ther e i s no nee d t o wonder whethe r you will someda y wal k hom e from work , open th e door , enter a parallel universe , and discove r tha t your famil y neve r hear d o f you. Instea d o f rushing t o mee t yo u after a hard day' s work, your famil y i s thrown int o a panic , screa m abou t a n intruder, an d hav e you throw n i n jai l for illega l entry. This kin d of scenario happen s onl y o n televisio n o r i n th e movies . I n Hawking' s approach, th e wormhole s do , i n fact , constantl y connect ou r univers e with billion s upo n billion s o f paralle l universes , bu t th e siz e o f thes e wormholes, o n th e average , i s extremel y small , abou t th e siz e o f th e Planck lengt h (abou t a 10 0 billion billion times smalle r than a proton , too smal l for huma n travel) . Furthermore, sinc e larg e quantu m transi tions between these universes are infrequent, we may have to wait a long time, longer tha n th e lifetime of the universe, before such an event takes place. Thus it is perfectly consistent with the law s of physics (although highly unlikely) tha t someon e ma y enter a twi n universe tha t i s precisely like

Colliding Universes 26

5

our univers e excep t fo r on e smal l crucia l difference , created a t som e point i n time when the tw o universes split apart. This typ e o f parallel worl d wa s explored b y John Wyndha m in th e story "Rando m Quest. " Coli n Trafford , a Britis h nuclea r physicist , is almost kille d i n 195 4 when a nuclear experimen t blow s up. Instea d o f winding up in the hospital, he wakes up, alone an d unhurt, in a remot e part of London. H e is relieved that everything appears normal , bu t soo n discovers that something is very wrong. The newspape r headlines are all impossible. World War II never took place. The atomi c bomb was never discovered. World histor y has been twisted . Furthermore, h e glance s a t a stor e shelf and notice s his own name, with his picture, a s the author o f a bestselling book . H e i s shocked. A n exac t counterpar t o f himsel f exist s in this parallel world as an autho r instea d o f a nuclear physicist! Is he dreamin g al l this ? Years ago, h e onc e though t o f becoming a writer, but instea d h e chos e t o become a nuclear physicist . Apparently in thi s parallel universe , different choices were made i n the past . Trafford scan s the London telephon e boo k and finds his name listed, but th e addres s is wrong. Shaking, he decides t o visit "his " home . Entering "his " apartment , h e is shocked t o meet "his " wife—some one h e ha s neve r see n before— a beautifu l woman wh o i s bitter an d angry over "his" numerou s affairs with other women. She berates "him " for hi s extramarita l indiscretions , bu t sh e notice s tha t he r husban d seems confused. His counterpart, Trafford finds out, is a cad and a womanizer. However , he find s i t difficult t o argu e wit h a beautiful stranger he has never seen before , eve n if she happens to be "his " wife . Apparently, he an d hi s counterpart hav e switched universes. He graduall y find s himsel f falling in lov e with "his " ow n wife. H e cannot understan d ho w hi s counterpar t coul d eve r hav e treate d hi s lovely wif e i n suc h a despicabl e manner . Th e nex t fe w weeks spen t together ar e the bes t of their lives . He decides t o undo all the har m his counterpart inflicted on his wife over the years. Then, just as the two are rediscovering eac h other , h e i s suddenly wrenched bac k int o hi s own universe, leaving "his" lov e behind. Throw n back into his own universe against hi s will , h e begin s a franti c ques t t o fin d "his " wife . H e ha s discovered tha t most, but no t all , people i n his universe have a counterpart in the other. Surely, he reasons, "his" wif e must have a counterpart in hi s own world. He becomes obsessed, tracking down all the clues that he remember s from th e twi n universe . Using all his knowledge o f history and physics , he concludes that two worlds diverged from each other because of some

266 WORMHOLES

: G A T E W A Y S T O A N O T H E R U N I V E R S E?

pivotal event in 192 6 or 1927 . A single event, he reasons, mus t have split the tw o universes apart . He the n meticulousl y trace s th e birt h an d deat h record s o f several families. H e spend s hi s remaining saving s interviewing scores o f peopl e until he locates "his " wife' s family tree. Eventually, he succeeds in tracking down "his " wif e i n his own universe. In th e end , h e marrie s her .

Attack of th e Giant Wormholes

One Harvar d physicis t who has jumped int o th e fra y concernin g worm holes is Sidney Coleman. Resemblin g a cross between Woody Allen an d Albert Einstein , he shuffle s throug h th e corridor s o f Jefferson Hall , trying t o convinc e th e skeptic s of his latest theory o f wormholes. With hi s Chaplinesque moustache , hi s hai r swep t bac k lik e Einstein's , an d hi s oversize sweatshirt, Coleman stand s ou t i n any crowd. Now he claim s to have solve d th e celebrate d cosmologica l constan t problem , whic h ha s puzzled physicist s for th e pas t 8 0 years. His wor k even mad e th e cove r o f Discover Magazine, wit h a n articl e entitled "Paralle l Universes: The Ne w Reality—From Harvard's Wildest Physicist." He is also wild about science fiction; a serious science-fictio n fan, h e eve n co-founde d Adven t Publishers , which published book s o n science-fiction criticism . At present , Colema n vigorousl y engages th e critic s who say that scientists won't b e abl e t o verify wormhol e theorie s withi n our lifetime . I f we believe in Thome's wormholes, the n w e have to wait, until someon e discovers exoti c matte r o r master s th e Casimi r effect . Unti l then , ou r time machine s hav e n o "engine " capabl e o f shooting u s into th e past . Similarly, if we believe i n Hawking' s wormholes, the n w e have t o trave l in "imaginar y time" in order to travel betwee n wormholes . Either way, it a very sad state of affairs fo r the averag e theoretica l physicist, who feel s frustrated b y the inadequate, feebl e technology of the twentiet h century and who can onl y dream of harnessing th e Planc k energy. This is where Coleman's wor k comes in . He recentl y made th e claim that th e wormhole s might yield a very tangible, very measurable resul t in th e present , an d no t i n som e distant , unforeseeabl e future . A s we pointed ou t earlier , Einstein' s equation s stat e tha t th e matter-energ y content o f an objec t determine s th e curvatur e of space-time surround ing it. Einstein wondered whether the pure vacuum of empty space could contain energy . Is pure emptiness devoid of energy? This vacuum energy is measured b y something calle d th e cosmological constant; in principle ,

Colliding Universes 26

7

there i s nothing t o prevent a cosmological constan t fro m appearin g i n the equations . Einstei n thought thi s term wa s aesthetically ugly, but h e could no t rul e i t out o n physica l or mathematical grounds . In th e 1920s , when Einstein trie d t o solve his equations for th e uni verse, he found , muc h t o his chagrin, tha t th e univers e was expanding. Back then , th e prevailin g wisdo m was that th e univers e wa s static an d unchanging. In order to "fudge" hi s equations to prevent the expansion of the universe , Einstein inserted a tiny cosmological constan t into thi s solution, chose n s o it would just balanc e ou t th e expansion , yieldin g a static univers e b y fiat. In 1929 , whe n Hubbl e conclusivel y proved tha t the univers e i s indeed expanding , Einstei n banished th e cosmologica l constant and sai d it was the "greates t blunder of my life." Today, we know that th e cosmologica l constan t i s very close to zero. If there were a small negative cosmological constant, the n gravit y would be powerfull y attractiv e and th e entir e univers e might be, say, just a few feet across. (B y reaching out with your hand, you should be able to grab the perso n i n front of you, who happens t o be yourself.) If there wer e a small positive cosmological constant, then gravity would be repulsive and everything would be flyin g awa y from you s o fast tha t thei r ligh t would never reach you. Since neither nightmaris h scenario occurs, we are confident tha t the cosmological constant is extremely tin y or eve n zero . But thi s problem resurface d i n th e 1970s , when symmetr y breaking was being intensivel y studied i n th e Standar d Mode l and GU T theory. Whenever a symmetry is broken, a large amoun t o f energy i s dumpe d into th e vacuum. In fact , th e amoun t o f energy floodin g th e vacuum is 10'°° time s large r tha n th e experimentall y observe d amount . I n al l of physics, this discrepancy of 10 100 is unquestionably the largest . Nowhere in physic s do w e see suc h a larg e divergenc e betwee n theor y (whic h predicts a larg e vacuu m energ y wheneve r a symmetr y is broken) an d experiment (whic h measure s zer o cosmologica l constan t i n th e uni verse). This is where Coleman's wormholes comes in; they're needed to cancel th e unwante d contributions t o the cosmologica l constant . According to Hawking, there ma y be a n infinit e numbe r o f alternative universe s coexistin g wit h ours , al l o f whic h ar e connecte d b y a n infinite we b o f interlocking wormholes . Colema n trie d t o ad d u p th e contribution fro m this infinite series. After the su m was performed, h e found a surprising result : The wav e function of the univers e prefer s to have zero cosmological constant, as desired. I f the cosmological constan t was zero, th e wav e functio n became exceptionall y large , meanin g tha t there was a high probability of finding a universe with zero cosmological constant. Moreover , the wav e function o f the univers e quickly vanished

268 WORMHOLES

: GATEWAY S T O ANOTHE R UNIVERSE ?

if th e cosmologica l constan t becam e nonzero , meanin g tha t ther e was zero probabilit y for tha t unwante d universe . Thi s wa s exactly what was needed t o cance l th e cosmologica l constant . I n othe r words , th e cos mological constan t wa s zero becaus e tha t wa s the mos t probabl e out come. Th e onl y effec t o f havin g billion s upon billion s o f paralle l uni verses was to keep th e cosmologica l constan t zer o in our universe . Because thi s wa s such a n importan t result , physicist s immediately began t o lea p int o th e field . "Whe n Sidne y came ou t wit h thi s work, everyone jumped," recalls Stanford physicis t Leonard Susskind. 13 In his typical puckish way, Coleman publishe d thi s potentially important result with a bit o f humor. "I t i s always possible tha t unknow n to mysel f I am up t o my neck in quicksand an d sinkin g fast," h e wrote. 14 Coleman like s t o impres s audience s vividl y with the importanc e o f this problem, tha t the chance s o f canceling out a cosmological constan t to one par t i n I0 100 is fantastically small . "Imagin e tha t ove r a ten-year period yo u spend million s of dollars without looking at your salary, and when you finally compare wha t you earn wit h what you spent, the y balance ou t t o th e penny, " h e notes. 15 Thus hi s calculation, whic h shows that you ca n cance l th e cosmologica l constan t t o on e par t i n 10 10<), i s a highly nontrivia l result. T o ad d frostin g t o th e cake , Colema n empha sizes tha t thes e wormhole s als o solv e anothe r problem : The y hel p t o determine th e values of the fundamental constants of the universe. Coleman adds , "I t wa s a completely differen t mechanis m from any that ha d been considered . I t was Batman swinging in on hi s rope."16 But criticisms also began t o surface; the mos t persistent criticism was that h e assume d tha t th e wormhole s wer e small , o n th e orde r o f th e Planck length, and tha t he forgot to sum over large wormholes. According t o the critics , large wormhole s shoul d als o b e include d i n his sum. But since we don't see large, visibl e wormholes anywhere, it seems tha t his calculation ha s a fatal flaw . Unfazed by this criticism, Coleman sho t back in his usual way: choosing outrageous title s for his papers. T o prove tha t larg e wormholes can be neglected i n his calculation, he wrote a rebuttal to his critics with th e title "Escap e from the Menac e of the Gian t Wormholes." Whe n aske d about his titles, he replied, "I f Nobel Prizes were given for titles, I'd hav e already collected mine." 17 If Coleman's purel y mathematical arguments are correct, they would give hard experimenta l evidenc e that wormholes are an essential feature of all physical processes, an d no t just some pip e dream . I t would mean that wormholes connecting our universe with an infinite number of dead universes are essentia l t o prevent ou r univers e from wrappin g itself up

Colliding Universes 26

9

into a tight , tin y ball, o r fro m explodin g outwar d at fantasti c rates . I t would mea n tha t wormhole s are th e essentia l featur e making ou r uni verse relatively stable. But as with mos t developments tha t occu r a t the Planc k length, th e final solutio n t o thes e wormhol e equation s wil l hav e t o wai t unti l we have a better gras p o f quantu m gravity . Man y o f Coleman' s equation s require a mean s o f eliminating th e infinitie s commo n t o al l quantu m theories o f gravity, and thi s means using superstring theory . In particular, we may have to wait until we can confidently calculate finite quantum corrections t o his theory. Many of these strang e prediction s will have to wait until we can sharpe n ou r calculationa l tools. As we have emphasized, the problem i s mainly theoretical. We simply do no t hav e th e mathematica l brainpowe r t o brea k ope n thes e well defined problems . Th e equation s star e a t u s from th e blackboard , bu t we are helpless to find rigorous, finite solutions to them at present. Onc e physicists hav e a better gras p o f the physic s at th e Planc k energy, the n a whole new universe o f possibilities opens up . Anyone , or an y civilization, tha t trul y master s th e energ y foun d a t th e Planc k lengt h wil l become th e maste r o f all fundamental forces. That i s the nex t topi c t o which w e will turn . Whe n ca n w e expect t o becom e master s of hyperspace?

This page intentionally left blank

PART IV

Masters of Hyperspace

This page intentionally left blank

13 Beyond th e Future What doe s it mean for a civilization to be a million years old? We hav e ha d radi o telescope s an d spaceship s fo r a fe w decades; ou r technica l civilizatio n is a few hundred years ol d . . . an advance d civilizatio n million s o f year s ol d i s a s muc h beyond u s as we are beyon d a bush bab y or a macaque . Carl Saga n

P

HYSICIST Pau l Davie s once commente d o n wha t to expec t onc e we have solved th e mysterie s of the unificatio n o f all forces into a single superforce. He wrote that we coul d chang e th e structur e o f spac e an d time , ti e ou r ow n knot s i n nothingness, an d build matte r t o order. Controlling the superforce woul d enable u s t o construc t an d transmut e particle s a t will , thu s generatin g exotic form s of matter. W e might eve n b e abl e t o manipulate th e dimen sionality of space itself , creating bizarre artificia l worlds with unimaginable properties. Truly we should be lords o f the universe. '

When ca n w e expect t o harnes s th e powe r o f hyperspace ? Experimental verificatio n o f th e hyperspac e theory , a t leas t indirectly , may come in the twenty-firs t century . However, the energ y scale necessary to manipulate (an d not just verify) ten-dimensiona l space-time, t o become "lords o f the universe, " i s many centuries beyond today' s technology. As we have seen, enormous amount s of matter-energy ar e necessar y to

273

274 M A S T E R

S O F HYPERSPAC E

perform near-miraculou s feats , suc h as creating wormholes and alterin g the directio n o f time. To b e master s o f the tent h dimension , eithe r w e encounter intelli gent lif e within the galax y that has already harnessed thes e astronomica l energy levels, or w e struggle fo r several thousand year s before we attain this ability ourselves. For example, our current atom smashers or particle accelerators can boost th e energ y of a particle to over 1 trillion electron volts (th e energ y create d i f a n electro n wer e accelerate d b y 1 trillion volts). Th e larges t accelerato r i s currently locate d i n Geneva , Switzer land, an d operate d b y a consortiu m o f 1 4 European nations . Bu t thi s energy pale s before th e energ y necessar y to probe hyperspace : 10 19 billion electro n volts , o r a quadrillio n time s large r tha n th e energ y tha t might have been produce d b y the SSC . A quadrillion ( 1 with 1 5 zeros after it ) ma y seem lik e an impossibly large number. The technolog y necessary to probe this incredible energy may requir e ato m smasher s billion s o f mile s long, o r a n entirel y new technology altogether . Eve n i f w e wer e t o liquidat e th e entir e gros s national product o f the world and build a super-powerful atom smasher , we would no t b e abl e t o com e clos e to thi s energy. At first, it seems an impossible tas k to harnes s thi s level of energy. However, thi s numbe r doe s no t see m s o ridiculousl y larg e i f we understand tha t technolog y expand s exponentially, whic h is difficult fo r our mind s t o comprehend . T o understan d ho w fas t exponentia l growth is, imagine a bacterium tha t split s in hal f eveiy 30 minutes. If its growth i s unimpeded , the n withi n a fe w weeks thi s singl e bacteriu m will produc e a colon y tha t wil l weig h a s muc h a s th e entir e plane t earth. Although human s hav e existe d o n thi s planet fo r perhap s 2 million years, th e rapi d clim b t o moder n civilizatio n within th e las t 20 0 years was possibl e du e t o th e fac t tha t th e growt h o f scientifi c knowledg e is exponential; tha t is , its rate o f expansio n i s proportional t o ho w much is already known. The mor e w e know, the faste r we can know more. Fo r example, we have amassed mor e knowledg e sinc e World War II than all the knowledg e amasse d i n ou r 2-million-yea r evolutio n o n thi s planet . In fact, the amount o f knowledge that our scientists gain doubles approx imately every 1 0 to 2 0 years. Thus i t becomes importan t t o analyze our ow n development histor ically. To appreciat e ho w technology can grow exponentially, let u s analyze ou r ow n evolution, focusin g strictly o n th e energ y availabl e t o th e average human . Thi s wil l hel p pu t th e energ y necessary t o exploi t th e ten-dimensional theory into prope r historical perspective.

Beyond th e Future 27

5

The Exponentia l Ris e o f Civilizatio n Today, we may think nothing about taking a Sunday drive in the country in a car with a 200-horsepower engine . Bu t th e energ y availabl e to th e average huma n durin g most of our evolutio n on thi s planet was considerably less. During this period, the basic energy source was the power of our own hands, abou t one-eighth o f a horsepower. Human s roamed th e earth in small bands, hunting and foragin g for food i n packs much like animals, using onl y the energ y o f thei r ow n muscles. From a n energ y poin t o f view, this changed onl y within the last 100,000 years. With the invention of han d tools , human s coul d exten d th e powe r o f thei r limbs . Spear s extended th e powe r o f thei r arms , club s th e powe r o f thei r fists , an d knives th e powe r of their jaws. In thi s period, thei r energ y output dou bled, to about one-quarte r o f a horsepower. Within th e pas t 10,00 0 o r s o years, the energ y outpu t o f a huma n doubled onc e again. Th e mai n reason fo r this change was probably th e end o f th e Ic e Age , whic h ha d retarde d huma n developmen t fo r thousands o f years. Human society , which consisted o f small bands of hunters an d gath erers for hundreds o f thousands o f years, changed wit h the discover y of agriculture soon afte r th e ice melted. Roving bands of humans, not having to follow game across the plain s and forests , settled in stable villages where crops coul d be harvested around th e year. Also, with th e melting of th e ic e sheet cam e th e domesticatio n o f animals suc h as horses an d oxen; th e energ y availabl e t o a huma n ros e t o approximatel y 1 horsepower. With th e beginnin g o f a stratified, agrarian lif e cam e th e divisio n of labor, unti l society underwent a n important change : th e transitio n to a slave society . This meant tha t on e person , th e slav e owner, could command th e energ y of hundreds o f slaves. This sudden increas e in energy made possibl e inhuma n brutality ; it als o mad e possibl e th e firs t tru e cities, where kings could comman d thei r slave s to use larg e cranes, levers, an d pulley s t o erec t fortresse s an d monument s t o themselves . Because o f thi s increase i n energy , out o f th e desert s an d forest s ros e temples, towers , pyramids, and cities . From a n energ y point o f view, for abou t 99.99 % o f the existenc e of humanity on thi s planet, th e technologica l leve l of our specie s was only one ste p abov e tha t o f animals . I t ha s onl y bee n withi n th e pas t few hundred years that humans hav e had more tha n 1 horsepower available to them.

276 M A S T E R S O F H Y P E R S P A C

E

A decisive change cam e with the Industrial Revolution. Newton's discovery o f th e universa l la w of gravit y and motio n mad e i t possibl e t o reduce mechanic s to a set of well-defined equations . Thus Newton's classical theor y o f the gravitationa l force, in som e sense , pave d th e wa y for the modern theor y of machines. This helped t o make possible the widespread us e o f steam-powered engine s i n th e nineteent h century ; with steam, the averag e huma n coul d comman d ten s t o hundreds o f horsepowers. For example, th e railroads opened up entire continents to development, and steamships opened up moder n internationa l trade . Bot h were energized b y the powe r of steam, heated b y coal. It took over 10,00 0 years for humanity to create moder n civilizatio n over the fac e o f Europe. Wit h steam-driven and late r oil-fired machines, the Unite d State s was industrialized within a century. Thus th e mastery of just a single fundamental forc e of nature vastly increased th e energ y available to a human bein g and irrevocabl y changed society. By the lat e nineteenth century , Maxwell's mastery of the electromag netic forc e onc e agai n se t off a revolutio n i n energy . Th e electromag netic force made possible the electrification of our cities and our homes, exponentially increasin g th e versatilit y an d powe r o f ou r machines . Steam engine s were now being replace d b y powerful dynamos. Within th e pas t 5 0 years , th e discover y o f th e nuclea r forc e ha s increased th e power available to a single human by a factor of a million. Because the energ y of chemical reactions i s measured i n electron volts , while the energy of fission and fusio n i s measured i n millions of electron volts, we have a millionfold increase in th e powe r available to us. The lesso n fro m analyzin g the historica l energy need s o f humanity shows graphically how for onl y 0.01% o f our existenc e w e have manipulated energ y level s beyond tha t o f animals . Yet within just a few centuries, we have unleashed vast amounts of energy via the electromagneti c and nuclea r forces . Let us now leave the pas t and begi n a discussion of the future , usin g th e sam e methodology , t o understan d th e poin t a t which we may harness th e superforce . Type I , 11 , and il l Civilization s Futurology, o r th e predictio n o f th e futur e fro m reasonabl e scientific judgments, i s a risk y science . Some woul d no t eve n cal l it a science at all, bu t something tha t more resemble s hocus pocus or witchcraft. Futurology ha s deservedl y earned thi s unsavor y reputatio n becaus e ever y "scientific" pol l conducte d b y futurologists about th e nex t decade ha s

Beyond the Future 277

proved t o be wildly off the mark. What makes futurology such a primitive science i s tha t ou r brain s thin k linearly , whil e knowledg e progresse s exponentially. Fo r example , polls of futurologists hav e shown that they take know n technolog y an d simpl y double o r tripl e i t t o predic t th e future. Poll s taken in the 1920 s showed that futurologists predicted tha t we would have , within a few decades, hug e fleet s o f blimps taking passengers acros s the Atlantic. But science also develops in unexpected ways . In the short run, when extrapolating withi n a few years, it is a safe be t tha t science wil l progres s through steady , quantitative improvements on existing technology. However, whe n extrapolatin g ove r a fe w decades, w e fin d tha t qualitativ e breakthroughs i n ne w areas becom e th e dominan t factor , where new industries open up in unexpected places . Perhaps th e mos t famou s exampl e o f futurology gone wron g is th e predictions made by John von Neumann, the father of the modern elec tronic compute r an d on e o f the grea t mathematician s o f th e century . After th e war, he made tw o predictions: first, that in th e future computers would become s o monstrous an d costl y that only large government s would be able to afford them , and second, that computers would be able to predict th e weather accurately. In reality , th e growt h o f computer s wen t i n precisel y th e opposit e direction: W e are floode d with inexpensive, miniature computer s tha t can fit in the palm of our hands . Computer chip s have become s o cheap and plentifu l that they are an integral par t o f some moder n appliances . Already, we have the "smart" typewriter (the word processor), and eventually we will hav e th e "smart " vacuum cleaner , th e "smart " kitchen, the "smart " television , an d th e like . Also, computers, n o matte r ho w powerful, hav e faile d t o predic t th e weather . Althoug h th e classica l motion o f individua l molecule s can , i n principle , b e predicted , th e weather is so complex that even someone sneezing can create distortions that will ripple an d b e magnifie d acros s thousand s o f miles, eventually, perhaps, unleashin g a hurricane . With all these important caveats , let us determine whe n a civilization (either our ow n or on e i n oute r space ) ma y attain th e abilit y to maste r the tent h dimension . Astronome r Nikola i Kardashe v o f th e forme r Soviet Union once categorize d futur e civilization s in th e followin g way. A Type I civilization is one tha t control s th e energ y resources o f an entire planet . Thi s civilizatio n can contro l th e weather , preven t earth quakes, min e dee p i n th e earth' s crust , an d harves t th e oceans . Thi s civilization ha s already completed th e exploratio n o f its solar system. A Type II civilization is one tha t controls th e powe r of the su n itself.

278 M A S T E R

S OF HYPERSPAC E

This doe s no t mea n passivel y harnessing sola r energy; thi s civilization mines th e sun . The energ y needs of this civilization are s o large tha t i t directly consumes th e powe r of the su n t o driv e its machines. This civilization will begin th e colonizatio n o f local star systems. A Typ e II I civilizatio n i s one tha t control s th e powe r o f a n entir e galaxy. Fo r a powe r source , i t harnesse s th e powe r o f billion s of sta r systems. It ha s probably mastere d Einstein' s equations an d ca n manip ulate space-time at will. The basi s of thi s classification is rather simple : Each leve l i s categorized o n th e basi s o f th e powe r sourc e tha t energize s th e civilization. Type I civilizations use the power of an entire planet. Type II civilizations use th e powe r o f an entir e star . Type II I civilization s us e th e powe r of an entir e galaxy. This classification ignores an y predictions concernin g the detailed nature of future civilizations (which are bound to be wrong) and instead focuses on aspects that can be reasonably understood by the laws of physics, such a s energy supply. Our civilization , by contrast, ca n b e categorize d a s a Type 0 civiliza tion, one tha t is just beginning t o tap planetary resources, but doe s no t have the technology and resources to control them. A Type 0 civilization like ours derives its energy from fossil fuels like oil and coa l and, in much of the Third World, from raw human labor . Ou r larges t computers cannot even predict the weather, let alone control it. Viewed from thi s larger perspective, we as a civilization are lik e a newborn infant . Although on e migh t guess that th e slo w march fro m a Type 0 civilization to a Type III civilization might take millions of years, the extraordinary fact abou t thi s classification scheme is that this climb is an expo nential on e an d henc e proceed s muc h faste r tha n anythin g w e ca n readily conceive. With al l thes e qualifications , we ca n stil l mak e educate d guesse s about whe n ou r civilizatio n wil l reac h thes e milestones . Given the rat e at whic h our civilizatio n is growing, we migh t expec t t o reac h Typ e I status within a few centuries. For example , th e larges t energ y source available to ou r Typ e 0 civilization i s the hydroge n bomb . Ou r technolog y is so primitiv e tha t we can unleas h th e powe r of hydrogen fusio n onl y by detonating a bomb, rather tha n controllin g it in a power generator. However , a simple hurricane generate s th e powe r o f hundred s o f hydroge n bombs . Thu s weather control, which is one featur e of Typ e I civilizations, is at least a century away from today' s technology. Similarly, a Type I civilization ha s already colonized most of its solar system. B y contrast, milestone s i n today' s developmen t o f spac e trave l

Beyond th e Future 27

9

are painfully measured o n the scale of decades, and therefore qualitative leaps suc h a s spac e colonizatio n mus t b e measure d i n centuries . Fo r example, th e earlies t dat e fo r NASA' s manned landin g o n th e plane t Mars is 2020. Therefore, th e colonizatio n o f Mars may take place 4 0 to 50 year s afte r that , an d th e colonizatio n o f th e sola r syste m within a century. By contrast, th e transitio n from a Type I to a Type II civilization may take onl y 1,00 0 years. Give n th e exponentia l growt h o f civilization , we may expect tha t within 1,000 years the energ y needs of a civilization will become s o larg e tha t i t mus t begi n t o min e th e su n t o energiz e it s machines. A typical example o f a Type II civilization is the Federatio n o f Planets portrayed i n th e "Sta r Trek " series . This civilizatio n has just begu n t o master the gravitationa l force—that is, the ar t of warping space-time via wormholes—and hence, for the first time, has the capability of reaching nearby stars . I t ha s evade d th e limi t place d b y th e spee d o f ligh t by mastering Einstein' s theor y o f genera l relativity . Small colonie s hav e been establishe d o n som e o f these systems, which the starshi p Enterprise is sworn t o protect . The civilization' s starships are powere d b y the col lision of matter and antimatter. The ability to create large concentration s of antimatter suitabl e for spac e trave l places tha t civilizatio n many centuries to a millennium awa y from ours . Advancing t o a Type III civilization may take several thousand year s or more . Thi s is, in fact , th e tim e scale predicted b y Isaac Asimov in hi s classic Foundatio n Series , whic h describe s th e rise , fall , an d re-emer gence of a galactic civilization. The tim e scale involved in eac h o f these transitions involve s thousands o f years. This civilizatio n ha s harnessed the energ y source containe d withi n the galax y itself. To it , warp drive, instead o f being an exoti c form o f travel to the nearb y stars, is the stan dard mean s of trade and commerc e betwee n sectors o f the galaxy. Thus although i t took 2 million years for our specie s t o leave the safet y of th e forests an d buil d a modern civilization , it ma y take onl y thousands o f years to leave the safet y o f our sola r system and buil d a galactic civilization. One optio n ope n t o a Type II I civilizatio n is harnessing th e powe r of supernovae or black holes. Its starships may even be able to probe the galactic nucleus , whic h i s perhap s th e mos t mysteriou s o f al l energ y sources. Astrophysicist s hav e theorize d tha t becaus e o f th e enormou s size of the galactic nucleus, the center of our galaxy may contain millions of black holes. I f true, this would provide virtually unlimited amount s of energy.

280 M A S T E R

S OF H Y P E R S P A C E

At thi s point , manipulatin g energie s a millio n billio n time s larger than present-da y energies should b e possible . Thus fo r a Type III civilization, with the energ y output of uncountable sta r systems and perhap s the galacti c nucleus at its disposal, th e master y of the tent h dimensio n becomes a real possibility .

Astrochicken

I onc e ha d lunc h wit h physicis t Freeman Dyso n o f th e Institut e fo r Advanced Study. Dyson is a senior figur e i n the world of physics who has tackled some of the mos t intellectually challenging and intriguin g questions facing humanity, such a s new directions in space exploration, th e nature o f extraterrestrial life , an d th e futur e o f civilization. Unlike other physicists, who dwell excessively in narrow, well-defined areas o f specialization , Dyson's fertile imaginatio n ha s roame d acros s the galaxy . "I cannot , a s Bohr an d Feynma n did, si t for year s with my whole min d concentrate d upo n on e dee p question . I am interested i n too man y different directions, " h e confessed. 2 Thin , remarkabl y spry, with th e owlis h expression o f an Oxfor d don , an d speakin g with a trace of his British accent, he engage d i n a long, wide-ranging lunch conversation with me , touching on man y of the ideas that have fascinated him over the years. Viewing the transitio n of our civilizatio n to Type I status, Dyson finds that our primitiv e space program i s headed in the wrong direction. The current tren d i s toward heavier payloads and greate r la g time between space shots, which is severely retarding th e exploratio n o f space. In hi s writings, he has proposed a radical departure fro m thi s trend, base d on what he call s the Aslrochicken. Small, lightweight , and intelligent , Astrochicken is a versatile space probe tha t has a clear advantage over the bulky , exorbitantly expensive space missions of the past, which have been a bottleneck t o space exploration. "Astrochicke n wil l weight a kilogram instea d o f Voyager's ton, " he claims . "Astrochicken wil l no t b e built , it wil l b e grown, " h e adds . "Astrochicken coul d be as agile as a hummingbird with a brain weighing no more tha n a gram."3 It wil l b e par t machin e an d par t animal , usin g th e mos t advance d developments i n bioengineering . I t will b e smal l but powerfu l enoug h to explor e th e oute r planets , such a s Uranus an d Neptune . It wil l no t need huge quantities o f rocket fuel; i t will be bred an d programme d t o "eat" ic e and hydrocarbon s foun d i n th e ring s surrounding th e oute r

Beyond th e Future 28

1

planet. It s genetically engineered stomac h wil l the n diges t these mate rials into chemical fuel. Onc e it s appetite ha s been satisfied, it will the n rocket to the next moo n o r planet . Astrochicken depend s o n technologica l breakthrough s i n geneti c engineering, artificia l intelligence , an d solar-electri c propulsion . Given the remarkabl e progres s i n thes e ares , Dyso n expects tha t th e variou s technologies for Astrochicken may be availabl e by the yea r 2016 . Taking the large r view of the developmen t of civilization, Dyson also believes that, a t th e curren t rat e o f development, w e may attain Type I status within a few centuries. H e doe s not believ e that making the tran sition betwee n the variou s types of civilizations will be very difficult. H e estimates tha t th e differenc e in siz e and powe r separatin g th e variou s types of civilizations is roughly a factor of 1 0 billion. Although this may seem lik e a large number , a civilization growing at the sluggis h rate o f 1 percent per year can expect to make the transition between the various civilizations within 2,500 years. Thus i t is almost guaranteed tha t a civilization ca n steadil y progress towar d Type III status. Dyson ha s written , " A societ y which happen s t o posses s a stron g expansionist drive wil l expan d it s habitat fro m a single planet (Typ e I) to a biosphere exploitin g an entire sta r (Typ e II) within a few thousand years, and fro m a single star t o an entir e galax y (Typ e III) within a few million years. A species which has once passe d beyon d Typ e II status is invulnerable t o extinction b y even th e wors t imaginable natura l or artificial catastrophe."4 However, there i s one problem . Dyson has concluded tha t th e tran sition from a Type II to a Type III civilization may pose formidable physical difficulties , du e mainl y t o th e limitatio n impose d b y the spee d o f light. The expansio n o f a Type II civilization will necessarily proceed a t less than th e spee d o f light, which he fee l place s a severe restriction o n its development . Will a Type I I civilizatio n break th e ligh t barrie r an d th e bond s o f special relativit y by explorin g th e powe r o f hyperspace ? Dyso n i s no t sure. Nothing can be ruled out, but the Planck length, he reminded me , is a fantastically small distance, and th e energies required t o probe dow n to that distance are unimaginable. Perhaps, he mused, the Planck length is a natural barrie r facing all civilizations. Type II I Civilization s i n Outer Space If th e lon g journey t o reac h Typ e II I status seem s remote fo r ou r own civilization, perhaps on e da y we will meet an extraterrestria l civilization

282 M A S T E R

S OF HYPERSPAC F

that has already harnessed hyperspac e for its needs and is willing to share its technology wit h us . The puzzl e facing us , however, is that we do no t see signs of any advanced civilization in th e heavens , a t least not i n ou r solar system or even in our small sector of the galaxy. Our space probes, especially the Viking landing o n Mar s in th e 1970 s and th e Voyager missions t o Jupiter, Saturn , Uranus , an d Neptun e i n th e 1980s , hav e sen t back discouragin g information concernin g th e bleak , lifeless natur e o f our sola r system. The tw o most promisin g planets , Venu s an d Mars , have turne d u p no sign s of life, le t alone advance d civilizations . Venus, named afte r th e goddess o f love, was once envisioned by astronomers as well as romantics to b e a lush , tropica l planet . Instead , ou r spac e probe s hav e foun d a harsh, barren planet , with a suffocating atmosphere o f carbon dioxide , blistering temperatures exceedin g SOOT , and toxi c rains of sulfuric acid . Mars, th e focu s o f speculatio n eve n befor e Orso n Welle s cause d panic i n th e countr y i n 193 8 durin g th e Depressio n wit h hi s fictiona l broadcast abou t a n invasio n fro m tha t planet , ha s bee n equall y disap pointing. W e kno w it t o b e a desolate , deser t plane t withou t traces o f surface water. Ancient riverbeds and long-vanished oceans have left their distinctive mar k o n th e surfac e o f Mars , bu t w e se e n o ruin s o r an y indications of civilization. Going beyon d ou r sola r system , scientists have analyze d th e radi o emissions fro m nearb y star s wit h equall y fruitles s results . Dyso n ha s stressed tha t an y advanced civilization , by the Secon d La w of Thermo dynamics, mus t necessaril y generate larg e quantitie s of waste heat. It s energy consumption shoul d b e enormous , an d a small fraction o f that waste hea t shoul d b e easil y detected b y our instruments . Thus , Dyson claims, by scanning the nearby stars, our instruments should be able fin d the telltal e fingerprin t o f wast e heat bein g generate d b y an advance d civilization. But no matte r wher e we scan th e heavens , we see no trace s of wast e heat o r radi o communication s fro m Typ e I , II , o r II I civiliza tions. On ou r ow n earth, for example, we have mastered th e art of radio and televisio n within the pas t half-century. Thus a n expandin g spher e of radio waves, about 5 0 light-years in radius, surrounds ou r planet . Any star within 50 light-years of earth, if it contains intelligent life , should b e able t o detec t ou r presence . Likewise , any Typ e I I o r II I civilization should b e broadcasting copiou s quantities of electromagnetic radiatio n continuously for th e pas t severa l thousand years, so that any intelligent life within several thousand light-year s of the civilization' s planet shoul d be able t o detect its presence . In 1978 , astronome r Paul Horowitz scanne d al l sunlike star system s

Beyond th e Future 28

3

(185 in all) within 80 light-years of our sola r system, and foun d n o trace s of radio emission s from intelligen t life. Astronomers Donal d Goldsmit h and Tobiu s Owe n reporte d i n 197 9 a search o f more tha n 60 0 star systems, als o wit h negativ e results . Thi s search , calle d SET I (searc h fo r extraterrestrial intelligence) , ha s me t wit h consisten t failure . (Encouragingly, in a rare display of scientific generosity, in 1992 Congress appropriated $10 0 millio n t o b e spen t ove r a 10-yea r perio d fo r th e Hig h Resolution Microwave Survey, which will scan th e nearb y stars for intelligent life . Thes e funds wil l mak e it possible for th e giganti c 305-mete r fixed radio dish at Arecibo, Puerto Rico, to scan select stars systematically within 10 0 light-years of the earth . This will be complemented b y the 34meter movabl e radio antenna a t Goldstone, California, which will sweep broad portion s o f the nigh t sky. After year s of negative results , astrono mer Fran k Drak e o f th e Universit y o f Californi a at Sant a Cru x is cautiously optimistic that they will find some positive signs of intelligent life. He remarks , "Man y huma n societie s developed scienc e independentl y through a combination o f curiosity and tryin g to create a better life, an d I think those sam e motivations would exist in other creatures.") The puzzl e deepens whe n we realize tha t th e probabilit y of intelligent lif e emergin g withi n ou r galax y is surprisingly large. Drak e even derived a simple equation t o calculate the numbe r o f planets with intelligent lif e form s in th e galaxy. Our galaxy , for example , contain s abou t 20 0 billion stars . To ge t a ballpark figure for th e number o f stars with intelligent life forms, we can make the followin g very crude estimate . We can be conservative and say that 10 % of thes e star s are yello w stars much lik e th e sun , tha t 10 % of those have planets orbiting them, that 10% of those hav e earthlike planets, that 10 % of those hav e earthlike planets wit h atmosphere s compat ible with life , tha t 10 % have earthlike atmospheres with lif e forms growing i n them , an d tha t 10 % of thos e hav e some for m o f intelligent life . This means tha t one-milliont h o f the 20 0 billion stars in th e galax y will probably have some intelligen t life form . This implies tha t a staggering 200,000 stars wil l hav e planet s harborin g som e for m o f intelligent life . A slightly more optimisti c set of values for Drake' s equation show s that intelligent lif e migh t be, o n th e average , as close a s 1 5 light-years from our sun. With recent advanced compute r techniques, scientists have been able to refin e Drake's origina l back-of-the-envelop e calculation . Georg e W . Wetherill o f th e Carnegi e Institutio n o f Washington, for example , ha s run compute r simulation s o f th e earl y evolutio n o f ou r sola r system, beginning with a large, swirling disk of gas and dus t around th e sun. He

284 M A S T E R

S O F HYPERSPACE

lets the compute r evolv e the dis k until small, rocky masses begin t o coalesce out of the dust. Much to his pleasant surprise, he found that planets of approximately th e siz e o f th e eart h wer e eas y to evolv e out o f thes e rocky cores. Mos t of the time , i n fact , earth-siz e planet s spontaneousl y coalesced wit h masse s betwee n 80 % and 130 % o f th e earth' s distanc e from th e sun . (Curiously , he als o foun d tha t th e formatio n o f Jupitersize planets far from the sun was important fo r the evolution of the earth size planets. Th e Jupiter-size planet s were essential t o sweep out swarms of comets an d debri s tha t woul d eventuall y strike th e earthlik e planet , extinguishing an y primitive lif e form s o n it . Wetherill's compute r sim ulations show that without a Jupiter-like plane t to clean out these comet s with it s gigantic gravitationa l pull, thes e comet s would hit th e earthlik e planet about 1,00 0 times more frequentl y than the y do in reality, making a life-destroying impac t ever y 100,000 year s or so. ) Thus i t is a compelling (bu t certainly not rigorous ) conclusio n tha t the law s o f probabilit y favor th e presenc e o f other intelligenc e withi n the galaxy. The fac t tha t our galax y is perhaps 1 0 billion years old mean s that there ha s been ampl e tim e for scores of intelligent lif e forms to have flourished within it. Type II and II I civilizations, broadcasting fo r several hundred t o severa l thousan d years , shoul d b e sendin g ou t a n easil y detectable spher e o f electromagnetic radiatio n measurin g severa l hundred t o several thousan d light-year s in diameter . Ye t we see no sign s of intelligent lif e form s in th e heavens . Why? Several speculativ e theorie s hav e been advance d t o explai n wh y we have been unable t o detect sign s of intelligent life out t o 10 0 light-years of our planet . Non e of them i s particularly satisfying, an d th e final truth may be a combination o f all of them . One theor y hold s tha t Drake' s equatio n ma y give us rough proba bilities of how many planets contai n intelligen t life, bu t tell s us nothin g about whe n thes e planet s attai n thi s leve l o f development . Give n th e astronomical tim e scale s involved , perhap s Drake' s equatio n predict s intelligent life forms that existed million s of years before us , or will exist millions of years after us . For example, our sola r syste m is approximately 4. 5 billion years old. Life starte d o n th e eart h abou t 3 to 4 billion years ago, bu t onl y within the pas t million years has intelligent lif e develope d o n th e plane t (an d only within th e pas t fe w decades ha s thi s civilization built radio station s capable o f sending signal s int o oute r space). However, 1 million years, on th e tim e scal e o f billion s o f years , i s bu t a n instan t o f time . I t i s reasonable t o assum e tha t thousand s o f advance d civilization s existe d

Beyond th e Future 28

5

before our distan t ancestors even left the forest and have since perished, or that thousands more civilizations will develop long after ours has died. Either way, we would not b e able t o detect the m via our instruments . The secon d theor y hold s tha t th e galax y is , in fact , teemin g with advanced forms of civilizations, but they are advanced enough to conceal their existenc e fro m ou r pryin g instruments. We would mean nothin g to the m becaus e the y ar e s o man y millions o f years ahead o f us . Fo r example, i f we stumble o n a n an t colon y while walking in a field , ou r first impulse i s certainly not t o mak e contac t wit h th e ants , as k to se e their leader, wave trinkets before their eyes, and offer the m unparallele d prosperity an d th e fruit s o f our advance d technology . Mor e likely , ou r first temptatio n is to ignor e the m (or perhap s eve n ste p on a few of them). Puzzled by these long-standing questions, I asked Dyson if he though t we would soon b e makin g contac t wit h extraterrestrial lif e forms . Hi s answer rathe r surprise d me . H e said , " I hop e not. " I though t i t was strange tha t someon e wh o had spen t decade s speculatin g about intelligent civilization s in outer space should hav e reservations about actually meeting them. Knowing British history, however, he must have had good reasons fo r no t rushin g i n t o embrac e othe r civilizations . British civilization wa s probably onl y severa l hundre d year s mor e advance d tha n many of the civilizations , such as the India n an d th e African, conquere d by the Britis h army and navy . Although mos t science-fiction writers bewail the limitations on space exploration place d b y the spee d o f light, Dyson take s th e unorthodo x view that perhaps this is a good thing. Viewing the ofte n blood y history of colonialism throughout ou r ow n world history, perhaps it is a blessing in disguise, h e muses , that various Type II civilizations will be separated by large distances and tha t the Planc k energy is inaccessible. Looking at the bright side, he quipped, "A t least, one can evade the tax collector." Unfortunately, the meeting of two unequal civilizations has often had catastrophic implication s fo r th e weake r one . Fo r example , th e Aztec civilization ha d rise n ove r thousand s o f years to grea t prominenc e i n central Mexico. In some areas, its mastery of science, art, and technology rivaled the achievements of Europe. However, in the area of gunpowder and warships , th e Aztec s were perhap s severa l centurie s behin d th e Spanish. Th e sudde n clas h betwee n a small , ragge d ban d o f 400 con quistadors and th e advance d civilization s of the Aztecs ended i n tragedy in 1521 . Within a brief perio d o f time, the Azte c people, wit h a popu lation numberin g i n th e millions , wer e systematicall y crushed an d enslaved to work in the mines. Their treasuries were looted, their history

286 MASTER

S O F HYPERSPAC E

was erased, an d eve n th e faintes t memor y of the grea t Aztec civilization was obliterated b y waves of missionaries. When w e think o f how we might react t o visitor s from oute r space , it is sobering t o read how the Aztec s reacted t o th e visitor s from Spain : "They seized upon th e gol d a s if they were monkeys, their faces gleam ing. Fo r clearl y their thirs t fo r gol d wa s insatiable; the y starve d fo r it ; they lusted fo r it ; they wanted t o stuf f themselve s with i t as if they were pigs. S o the y wen t abou t fingering , takin g u p th e streamer s o f gold , moving the m bac k an d forth , grabbin g the m t o themselves , babbling , talking gibberish amon g themselves."* 3 On a cosmi c scale , th e sudde n interaction s betwee n civilizations could be even more dramatic. Because we are talking about astronomical time scales, it is likely that a civilization that i s a millio n years ahead o f us will find u s totally uninteresting. Furthermore, ther e i s probably little that our plane t can offe r thes e alien s in term s of natural resource s that isn't simultaneously available in numerou s othe r star systems. In the "Star Trek" series, however, the Federation o f Planets encounters othe r hostil e civilizations , the Klingon s an d Romulans , whic h ar e precisely a t th e sam e stag e o f technologica l developmen t a s the Federa tion. This may increase the drama and tension of the series, but the odds of this happening are trul y astronomical. Mor e likely , as we venture off into th e galax y in starships , we will encounte r civilization s a t vastly different level s o f technologica l development , som e perhap s million s of years ahead o f us.

The Ris e an d Fal l o f Civilization s In addition t o the possibilities that we may have missed other civilizations by millions of years and tha t other civilizations may not conside r u s worthy o f notice , a thir d theory , whic h i s mor e interesting , hold s tha t thousands o f intelligent lif e form s di d aris e fro m th e swamp , but the y were unable t o negotiate a series of catastrophes, bot h natural and self *So perhaps w e shouldn't be s o enthusiasti c abou t makin g contact with intelligent , extraterrestrials. Scientist s poin t ou t tha t o n th e earth , ther e ar e tw o type s o f animals : predators like cats, dogs, and tiger s (which have eyes to the fron t o f their face, so they can stereoscopically zero i n o n thei r target) and pre y like rabbit s and dee r (whic h hav e eye s to the sid e of their face i n order to look around 36 0 degrees for the predators). Typically, predators ar e mor e intelligen t the n prey . Tests show that cat s are mor e intelligent than mice, and foxe s are more intelligent than rabbits. Humans, with eyes to the front , ar e also predators. I n ou r searc h for intelligen t life i n th e heavens , we should keep in min d that the alien s we meet will probably also have evolved from predators .

Beyond th e Future 28

7

inflicted. I f this theory i s correct, the n perhap s someda y ou r starship s will fin d th e ruin s o f ancien t civilization s o n far-of f planets , or , mor e likely, our ow n civilization will be face d with thes e catastrophes. Instea d of becomin g "lord s o f th e universe, " w e ma y follo w th e roa d t o self destruction. Thu s th e questio n w e ask is: What is the fat e o f advance d civilizations? Wil l w e (they ) surviv e long enoug h t o maste r th e physics of the tent h dimension? The ris e of civilizations is not marke d by a steady and sur e growth in technology an d knowledge . Histor y show s u s tha t civilization s rise, mature, an d the n disappear , sometime s without a trace . I n th e future , perhaps humanit y will unleash a Pandora's bo x of technological horror s that threaten ou r very existence, from atomi c bombs t o carbon dioxide . Far from trumpetin g th e comin g o f the Ag e of Aquarius, some futurol ogists predic t tha t w e ma y be facin g technological an d ecologica l col lapse. For the future, the y conjure up the frightening image of humanity reduced to a pathetic, terrified Scrooge in Charles Dickens's fable, groveling on the ground o f his own grave and pleadin g for a second chance . Unfortunately, th e bul k of humanity is largely uncaring, or unaware, of th e potentia l disaster s facin g us . Som e scientist s have argue d tha t perhaps humanity , considered a s a single entity , can be compare d t o a teenager careening out of control. For example, psychologists tell us that teenagers ac t a s if they ar e invulnerable . Thei r driving , drinking, an d drug habits are graphic proof, they say, of the devil-may-care recklessness that pervade s thei r life-styl e an d outlook . Th e mai n caus e o f deat h among teenager s i n thi s countr y i s n o longe r disease , bu t accidents , probably caused by the fac t that they think the y will live forever . If that is true, then we are abusin g technology and th e environmen t as if we will live forever, unaware of the catastrophes that lie in the future . Society a s a whole may have a "Pete r Pa n complex, " neve r wantin g to grow up an d fac e th e consequence s o f its own irresponsibility. To concretiz e ou r discussion , using the knowledg e at ou r disposal , we can identify severa l important hurdles that must be crossed over during th e nex t severa l aeons befor e w e can becom e master s of the tent h dimension: the uranium barrier , ecological collapse, a new ice age, astronomical close encounters, Nemesis arid extinction, and th e death of the sun an d th e Milk y Way galaxy. The Uraniu m Barrier

Jonathan Schell , in hi s watershed boo k Th e Fate o f the Earth, points ou t how perilously close we have come to mutual annihilation. Although the

288 MASTER

S O F HYPERSPACE

recent collaps e o f th e Sovie t Union ha s mad e possibl e sweepin g arms cuts, ther e ar e stil l clos e t o 50,00 0 nuclea r weapons , bot h tactica l an d strategic, in the world today, and wit h deadly accurate rocket s to deliver them. Humanity has finally mastered th e possibility of total annihilation . If th e missile s do no t destro y everyon e i n th e openin g shot s o f a nuclear war, we can stil l look forward to th e agonizin g death cause d by nuclear winter, during which the soot and as h from burning cities slowly chokes off all the life-givin g sunlight. Computer studie s have shown that as few as 10 0 megatons o f explosive s may generate enoug h fir e storm s in th e citie s t o clou d th e atmospher e significantly . A s temperature s plummet, crops fail, and citie s freeze over, the las t vestiges of civilization will b e snuffe d ou t lik e a candle . Finally, ther e i s th e increasin g dange r o f nuclea r proliferation . United State s intelligence estimates that India, which detonated it s first bomb i n 1974 , no w ha s a stockpil e o f abou t 2 0 atomi c bombs . Arch enemy Pakistan, these source s claim , has built four atomi c bombs, on e of which weighs no mor e than 40 0 pounds, a t its secret Kahut a nuclear facility. An atomic worke r at Israel's Dimona nuclea r installatio n in th e Negev desert claimed tha t h e sa w enough materia l t o build 20 0 atomic bombs there . And South Afric a admitte d tha t it had mad e seve n atomic bombs an d apparentl y tested tw o atomic bombs i n th e lat e 1970 s off its coast. Th e U.S . sp y satellit e Vela picke d u p th e "fingerprint " o f th e atomic bomb, a characteristic, unmistakabl e double-flash, on tw o occasions of f th e coas t o f Sout h Afric a i n th e presenc e o f Israel i warships. Nations lik e North Korea , Sout h Korea , an d Taiwa n ar e poise d a t th e brink o f goin g nuclear . It' s highl y probable, give n recen t U.S . intelligence disclosures, that 20 nations will possess the bomb by the year 2000. The bomb will have proliferated int o the hottest spots around the world, including th e Middl e East. This situation is highly unstable, an d wil l continu e t o become mor e so a s nations compet e fo r diminishin g resource s an d sphere s o f influ ence. No t just our society , but ever y intelligent civilization in th e galaxy building a n industria l society , will discove r elemen t 9 2 (uranium ) an d with it the abilit y for mass destruction. Elemen t 92 has the curiou s property o f sustainin g a chai n reactio n an d releasin g th e vas t amoun t o f energy stored withi n it s nucleus. With the abilit y to maste r elemen t 9 2 comes th e abilit y either t o liberat e ou r specie s fro m want , ignorance , and hunger , o r t o consum e th e plane t i n nuclea r fire . Th e powe r o f element 92 , however, can be unleashed onl y when an intelligent species reaches a certai n poin t o f developmen t a s a Typ e 0 civilization . I t depends on th e siz e of its cohesive socia l uni t and it s state of industria l development.

Beyond th e Future 28

9

Fire, for example, ca n be harnesse d b y isolated groups of intelligent individuals (suc h as a tribe) . Smeltin g and primitiv e metallurgy, necessary for th e manufactur e o f weapons, requires a larger socia l unit, perhaps numberin g i n th e thousand s (suc h as a small village). The devel opment o f the internal-combustio n engin e (fo r example, a car engine) requires th e developmen t o f a comple x chemica l an d industria l base , which can be accomplishe d b y only a cohesive social unit numbering in the million s (fo r example, a nation-state). The discover y of element 9 2 upset s thi s balanc e betwee n th e slow , steady rise of the cohesive social unit and its technological development. The releasin g o f nuclear energ y dwarf s chemica l explosive s by a factor of a million , bu t th e sam e nation-stat e tha t ca n harnes s th e internal combustion engin e can also refine element 92 . Thus a severe mismatch occurs, especiall y when th e socia l development o f this hypothetical civilization i s still locked i n th e for m of hostile nation-states . The technol ogy for mayhe m an d destructio n abruptl y outpace s th e slo w develop ment o f social relations with the discover y of element 92 . It is natural t o conclude, therefore , that Type 0 civilizations arose on numerous occasion s withi n the pas t 5 - to 10-billion-yea r histor y of ou r galaxy, bu t tha t the y all eventually discovered elemen t 92 . If a civiliza tion's technologica l capabilit y outrace d it s socia l development , then , with th e ris e o f hostile nation-states , there was a large chanc e tha t th e civilization destroyed itself long ago in an atomic war.6 Regrettably, if we live lon g enoug h t o reac h nearb y star s i n ou r secto r o f th e galaxy , we may see th e ashe s o f numerous, dea d civilization s that settle d nationa l passions, personal jealousies, an d racia l hatreds with nuclear bombs . As Heinz Pagels has said , The challeng e to our civilizatio n which ha s come from ou r knowledg e o f the cosmic energies that fuel th e stars, the movement o f light and electrons through matter, the intricate molecular order which is the biological basi s of life, mus t b e me t b y the creatio n o f a mora l an d politica l orde r which will accommodat e thes e force s o r w e shal l b e destroyed . I t wil l tr y ou r deepest resources of reason and compassion. 7

It seem s likely , therefore , tha t advance d civilization s sprang u p o n numerous occasion s within our galaxy , but tha t few of them negotiate d the uranium barrier , especially if their technolog y outpaced thei r socia l development. If we plot, for example , the ris e of radio technolog y on a graph, we see that ou r plane t evolve d for 5 billion years before a n intelligen t species discovere d ho w t o manipulat e th e electromagneti c an d nuclea r

290 M A S T E R

S OF H Y P E R S P A C E

forces. However , i f we annihilate ourselve s i n a nuclea r war , then thi s curve wil l becom e a spike an d retur n t o zero . Thu s i n orde r t o com municate wit h an advance d civilization , we must sca n a t precisel y th e right era , t o a n accurac y of a few decades, befor e th e civilizatio n blows itself up . Ther e is a vanishingly small "window " throug h whic h we may make contac t wit h another living civilization, before i t destroys itself. In Figure 13.1 , we see th e ris e o f alien civilization s throughou t th e galaxy represented a s a serie s o f peaks , eac h representin g th e rapi d ris e o f a civilization an d th e eve n mor e rapi d fal l du e t o nuclea r war . Scannin g the heavens for intelligent life, therefore, ma y be a difficult task . Perhaps there hav e bee n man y thousand s o f peak s withi n th e pas t fe w billion years, wit h thousand s o f planet s briefl y masterin g radi o technolog y before blowin g themselve s up . Eac h brie f peak , unfortunately , take s place a t different cosmic times . Ecological Collaps e

Assuming that a Type 0 civilization can master uranium without destroying itself in a nuclear war, the nex t barrier i s the possibilit y of ecological collapse. We recall the earlie r exampl e o f a single bacterium, which divides so frequently tha t i t eventuall y outweigh s th e plane t earth . However , i n reality we do no t se e gigantic masse s o f bacteria o n th e earth—i n fact , bacterial colonie s usually do no t eve n gro w to th e siz e o f a penny. Laboratory bacteri a place d i n a dis h fille d wit h nutrients wil l indee d gro w exponentially, bu t eventuall y die because the y produce to o much waste and exhaus t th e foo d supply . These bacteria l colonie s essentiall y suffo cate in thei r ow n waste products . Like bacteria l colonies , w e ma y als o b e exhaustin g ou r resource s while drowning in th e waste products tha t we relentlessly produce. Ou r oceans an d th e atmospher e ar e not limitless, but ultrathi n films on th e surface o f th e earth . Th e populatio n o f a Type 0 civilization, before i t reaches Typ e I status , ma y soa r t o th e billions , creatin g a strai n o n resources an d exacerbatin g th e problems o f pollution. On e o f the mos t immediate danger s i s the poisonin g o f the atmosphere , i n th e for m o f carbon dioxide , whic h trap s sunlight an d raise s the averag e worl d tem perature, possibl y initiating a runaway greenhouse effect . Since 1958 , carbo n dioxid e concentration s i n the air have increase d 25%, mostl y from oi l and coa l burnin g (45 % of carbon dioxid e come s from th e United States and th e former Sovie t Union). This, in turn, may have accelerated th e mean temperatur e ris e of the earth . It took almos t

Beyond the Future

291

Figure 13.1. Why don't we see other intelligent life in the galaxy? Perhaps intelligent life forms that could build radio telescopes flourished millions of years in the past, but perished in a nuclear war. Our galaxy could have been teeming with intelligent life, but perhaps most are dead now. Will our civilization be any different ?

a century, from 1880, to raise the mean world temperature 1°F . However, the mea n temperatur e i s now rising a t almost 0.6°F per decade . B y the year 2050, this translates into a rise of coastal waters by 1 to 4 feet, which could swam p nations like Bangladesh an d floo d area s like Los Angeles and Manhattan . Eve n mor e seriou s woul d b e a devastatio n o f th e nation's foo d baske t i n th e Midwest , the acceleratio n o f the sprea d o f deserts, and destruction of tropical rain forests, which in turn accelerates the greenhous e effect . Famin e an d economi c rui n coul d sprea d o n a global scale. The faul t lie s in a n uncoordinate d planetar y policy . Pollution take s place in millions of individual factories all over the planet, but the power to curb this unbridled pollutio n resides with a planetary policy, which is difficult, i f no t impossible , t o enforc e i f th e dominan t cohesiv e socia l unit is the nation-state , numbering onl y in the hundred s o f millions. In the shor t term , thi s may mean emergenc y policie s an d th e shar p cur tailment o f the internal-combustio n engin e an d coa l an d oi l burning . The standard o f living could also drop. It means additional hardship s in

292 M A S T E R

S OF HYPERSPAC E

developing nations , whic h need acces s t o chea p source s o f energy. I n the lon g term , however , our societ y may be force d t o resor t t o on e o f three possibl e solution s tha t d o no t giv e of f carbo n dioxid e an d ar e essentially inexhaustible: solar energy , fusion plants , and breede r reac tors. O f these , sola r an d fusio n hol d th e mos t promise . Fusio n power (which fuse s th e hydroge n atom s foun d i n se a water) and sola r energ y are stil l several decades away , but shoul d provide ampl e energy supplies into th e nex t few centuries, until society makes the transitio n t o a Type I civilization. The faul t once again lies in the fact that the technology has outpaced social development . A s lon g a s pollutio n i s produce d b y individual nation-states, while the measure s necessary to correct thi s are planetary, there wil l be a fatal mismatc h tha t invite s disaster. The uraniu m barrie r and ecologica l collapse wil l exist as life-threatening disasters for Type 0 civilizations until this mismatch is bridged . Once a civilization passes Type 0 status, however, there is much more room for optimism. To reach Type I status requires a remarkable degre e of socia l cooperation o n a planetary scale. Aggregate s o n th e orde r of tens t o hundreds o f millions of individuals are necessar y to exploit th e resources o f uranium , interna l combustion , an d chemicals . However, aggregates o n th e orde r of billions are probabl y necessar y truly to har ness planetary resources . Thu s th e socia l organizatio n o f a Type I civi lization must be very complex and very advanced, or else the technology cannot b e developed . By definition, a Type I civilization requires a cohesive social unit that is the entir e planet' s population . A Type I civilization by its very natur e must be a planetary civilization . It cannot function on a smaller scale. This can , in some sense , be compare d t o childbirth. Th e mos t dan gerous perio d for a child i s the firs t fe w months o f life, whe n the tran sition t o an external , potentiall y hostile environmen t places enormou s biological strain s on th e baby . After th e first year of life, th e deat h rat e plunges dramatically . Similarly, the mos t dangerous perio d fo r a civili zation is the first few centuries afte r i t has reached nuclea r capability. It may turn ou t tha t once a civilization has achieved a planetary political system, the wors t is over.

A Ne w Ice Age No on e know s what causes an ic e age, which has a duratio n measure d in tens to hundreds of thousands of years. One theor y is that it is caused by minute variation s in th e earth' s rotation , which ar e to o smal l t o b e

Beyond th e Future 29

3

noticed eve n ove r a perio d o f centuries . Thes e tin y effects , ove r hun dreds o f thousand s o f years , apparentl y accumulat e t o caus e sligh t changes in the jet strea m over the poles. Eventually, the jet stream s are diverted, sendin g freezin g pola r ai r masse s farther an d farthe r south, causing temperature s t o plumme t around th e globe , unti l a n ic e age begins. Th e ic e age s di d considerabl e damag e t o th e ecolog y o f th e earth, wiping out score s of mammalian life forms and perhap s even isolating bands of humans on different continents , perhaps even giving rise to the various races , which is a relatively recent phenomenon . Unfortunately, ou r computer s ar e to o primitiv e eve n t o predic t tomorrow's weather , le t alon e whe n th e nex t ic e ag e wil l strike . For example, computers ar e no w entering thei r fifth generation. We sometimes forget that n o matte r ho w large o r comple x a fourth-generation computer is, it can only add tw o numbers at a time. This is an enormou s bottleneck that is just beginnin g to be solved with fifth-generatio n com puters, whic h hav e paralle l processor s tha t ca n perfor m severa l opera tions simultaneously. It is highly likely that our civilizatio n (if it successfully negotiate s th e uranium barrie r an d ecologica l collapse ) wil l attai n Typ e I status, an d with i t the abilit y to control th e weather , within a few hundred years . If humanity reaches Type I status or higher before the next ice age occurs, then ther e i s ample reaso n t o believ e tha t a n ic e ag e wil l no t destro y humanity. Humans either will chang e th e weathe r and preven t th e ice age or wil l leave the earth . Astronomical Clos e Encounter s On a time scale of several thousand t o several million years, Types 0 and I civilization s have to worry about asteroi d collision s and nearb y supernovas. Only within thi s century , wit h refined astronomica l measurements , has i t becom e apparen t tha t th e earth' s orbi t cut s across th e orbit s of many asteroids , makin g th e possibilit y o f nea r misse s uncomfortably large. (On e way for a Type 0 or I civilization to prevent a direct collision is t o sen d rocket s wit h hydroge n bomb s t o intercep t an d deflec t th e asteroid while it is still tens of millions of miles away from th e earth . This method has, in fact, been proposed b y international bodies of scientists.) These nea r misse s are more frequent than mos t people realize . The last on e too k place o n January 3 , 1993, an d wa s actually photographed using radar b y NASA astronomers. Photos of the asteroi d Toutatis show that i t consist s o f tw o rocky cores, eac h 2 mile s i n diameter . I t cam e

294 M A S T E R

S O F H Y P E R S P A CE

within 2. 2 millio n mile s o f th e plane t earth . O n Marc h 23 , 1989 , a n asteroid abou t hal f a mile across drifte d even closer to the earth , about 0.7 million miles (roughly three times the distance from th e earth to the moon). In fact, it was also announced i n late 1992 that a gigantic comet would hit th e eart h o n exactl y August 14, 2126, perhaps endin g all life o n th e planet. Astronomer Brian Marsden of the Harvard-Smithsonian Cente r for Astrophysics estimated the chances of a direct hit as 1 in 10,000. The Swift—Tuttle come t (name d afte r di e tw o American astronomer s wh o first spotted i t durin g th e Civi l War ) wa s soon dubbe d th e Doomsday Rock by the media . Soon-to-be-unemployed nuclea r weapons physicists argued, perhap s i n a self-servin g way, that the y should b e allowe d t o build massive hydrogen bomb s to blow it to smithereens when the time comes. Bits and piece s of the Swift-Tuttl e come t hav e already impacted o n the earth. Making a complete revolution around the sun every 130 years, it sheds a considerable amount of debris, creating a river of meteors and particles in oute r space . When th e eart h crosse s this river, we have th e annual Perseid meteo r shower, which rarely fails t o light up th e sky with celestial fireworks. (W e should also point out dial predicting near misses of comet s i s a risk y business. Becaus e th e hea t o f th e sun' s radiatio n causes th e comet' s ic y surface t o vaporiz e irregularly and sputte r lik e thousands o f smal l firecrackers , there ar e sligh t bu t importan t distor tions in its trajectory. Not surprisingly, Marsden retracted hi s prediction a fe w weeks later a s being incorrect . "We'r e saf e fo r th e nex t millennium," admitte d Marsden.) A NASA panel i n January 199 1 estimate d that there ar e abou t 1,000 to 4,000 asteroids tha t cross the earth's orbit and are bigger tha n a half mile across , sufficient t o pose a threat t o human civilization . However , only about 150 of these large asteroids have been adequately tracked by radar. Furthermore , ther e are estimate d t o be abou t 300,00 0 asteroids that cross the eardi's orbit that are at least 300 feet across. Unfortunately, scientists hardly know the orbit s of any of these smalle r asteroids. My own personal clos e encounter with an extraterrestrial object came when I was a senior a t Harvar d i n th e winte r of 1967 . A close friend of mine in my dormitory, who had a part-time job a t the universit y observatory, told me a closely held secret: The astronomers there had detecte d a gigantic asteroid, severa l miles across, heading directly for th e plane t earth. Furthermore , althoug h i t was too earl y to tell , h e informe d m e that thei r computer s calculate d i t might strike the eart h i n June 1968 ,

Beyond th e Future 29

5

the tim e o f our graduation . A n object that size would crack th e earth' s crust, spew open billions of tons of molten magma , and send huge earth quakes and tida l waves around th e world. As the months went by, I would get periodi c update s o n th e cours e o f th e Doomsda y asteroid . Th e astronomers a t the observatory were obviously being careful not to cause any undue pani c with thi s information. Twenty years later, I had forgotte n all about th e asteroid , until I was browsing through a n articl e on asteroi d nea r misses . Sure enough , th e article mad e referenc e to th e asteroi d o f 1968. Apparently, the asteroi d came within about 1 million miles of a direct impact with th e earth . More rare , bu t mor e spectacula r than asteroi d collision s are supernova bursts in th e vicinit y of the earth . A supernova releases enormou s quantities of energy, greater tha n th e outpu t o f hundreds o f billions of stars, unti l eventuall y it outshine s th e entir e galax y itself. I t create s a burst of x-rays, which would be sufficient t o cause severe disturbances in any nearby star system. At the very minimum, a nearby supernova would create a gigantic EMP (electromagnetic pulse), similar to th e on e tha t would be unleashed b y a hydrogen bomb detonated in outer space. Th e x-ray burst would eventually hit our atmosphere , smashin g electrons out of atoms; the electron s would then spira l through th e earth' s magneti c field, creatin g enormou s electri c fields . Thes e field s ar e sufficien t t o black ou t al l electrica l an d communicatio n device s fo r hundred s o f miles, creatin g confusio n an d panic . I n a large-scal e nuclear war , th e EMP would be sufficien t t o wipe out o r damag e an y form o f electronics over a wide area o f the earth' s population. At worst, in fact, a supernova burst in the vicinity of a star system might be sufficient t o destroy all life. Astronomer Carl Sagan speculates that such an event may have wiped out the dinosaurs : If there were by chance a supernova within ten o r twenty light-years of the solar syste m som e sixty-fiv e millio n year s ago , it woul d hav e spraye d an intense flu x o f cosmi c ray s int o space , an d som e o f these , enterin g th e Earth's envelope of air, would have burned the atmospheric nitrogen. The oxides of nitrogen thus generated would have removed the protective layer of ozone from th e atmosphere, increasing the flux of solar ultraviolet radiation a t th e surfac e an d fryin g an d mutatin g the man y organisms imperfectly protected against intense ultraviolet light. Unfortunately, th e supernov a would give littl e warning of its explosion. A supernova eruption take s place quit e rapidly , and it s radiation

296 MASTER

S O F HYPERSPAC E

travels at the spee d of light, so a Type I civilization would hav e t o mak e a speedy escap e int o outer space . The onl y precaution tha t a civilization can tak e i s to monitor carefull y thos e nearb y star s that ar e o n th e verg e of going supernova .

The Nemesi s Extinctio n Facto r In 1980, the late Luis Alvarez, his son Walter, and Frank Asaro and Hele n Michel of the Universit y of California at Berkeley proposed tha t a comet or a n asteroi d hi t th e eart h 6 5 million years ago, thereb y initiating vast atmospheric disturbance s tha t led to the sudde n extinctio n o f the dinosaurs. B y examining th e rock y strata lai d dow n b y river bed s 6 5 million years ago, the y were abl e t o determin e th e presenc e o f unusuall y high amounts o f iridium, which is rarely found o n earth bu t commonly found in extraterrestria l objects , lik e meteors . Th e theor y i s quite plausible , since a comet 5 miles in diameter hittin g th e earth a t about 2 0 miles per second (te n time s faste r tha n a speedin g bullet ) woul d hav e th e forc e of 10 0 millio n megaton s o f TN T (o r 10,00 0 time s th e world' s tota l nuclear arsenal) . I t would creat e a crate r 6 0 miles acros s an d 2 0 miles deep, sending up enough debris to cut off all sunlight fo r an extended period o f time . A s temperature s fal l dramatically , th e vas t majorit y o f the specie s o n thi s plane t woul d b e eithe r kille d of f o r seriousl y depleted. In fact , i t wa s announced i n 199 2 tha t a stron g candidat e fo r th e dinosaur-killing come t o r asteroi d ha d bee n identified . I t wa s already known tha t ther e i s a large impac t crater , measurin g 11 0 mile s across , in Mexico, in the Yucatan, near th e village of Chicxulub Puerto. I n 1981 , geophysicists wit h th e Mexica n nationa l petroleu m company , Pemex , told geologists that they had picked up gravitational and magnetic anom alies that were circular in shape a t the site . However, only after Alvarez's theory becam e popula r di d geologist s activel y analyze the remnant s o f that cataclysmi c impact . Radioactive-datin g method s usin g argon-3 9 have show n tha t th e Yucata n crate r i s 64.98 ± 0.0 5 millio n year s old . More impressively, it was shown that Mexico, Haiti, and eve n Florida ar e littered wit h small, glassy debris calle d tektites, which were probably sili cates tha t wer e glassifie d b y the impac t o f this larg e asteroi d o r comet . These glass y tektite s ca n b e foun d i n sedimen t tha t wa s lai d down betwee n th e Tertiar y an d Cretaceou s periods . Analyse s o f fiv e different tektit e sample s sho w a n averag e ag e o f 65.0 7 ± 0.1 0 millio n years. Give n th e accurac y o f thes e independen t measurements ,

Beyond the Future 297

geologists no w hav e th e "smokin g gun " fo r th e dinosaur-killin g asteroid o r comet . But one o f the astonishin g features of life on eart h i s that the extinction of the dinosaur s i s but on e o f several well-documented mass extinctions. Other mass extinctions were much worse than th e one that ende d the Cretaceou s perio d 6 5 million year s ago . Th e mas s extinctio n tha t ended the Permian period , for example, destroyed full y 96% of all plant and animal specie s 250 million years ago. The trilobites, which ruled th e oceans a s one o f earth's dominant lif e forms, mysteriously and abruptl y perished durin g this great mas s extinction. In fact, ther e hav e been five mass extinction s o f anima l an d plan t life . I f one include s mas s extinctions tha t ar e les s well documented , a pattern become s evident : Every 26 million years or so , there i s a mass extinction. Paleontologist s Davi d Raup andJohn Sepkoski have shown that if we plot the number of known species on the earth at any given time, then the chart shows a sharp drop in th e numbe r o f lif e form s o n th e eart h ever y 2 6 millio n years , like clockwork. This can be show n to extend ove r ten cycle s going bac k 260 million years (excludin g two cycles). In one extinctio n cycle , at the en d o f the Cretaceou s period, 6 5 million year s ago, most of the dinosaur s were killed off. In anothe r extinc tion cycle, at the en d o f the Eocen e period , 3 5 million years ago, man y species o f land mammal s wer e extinguished . Bu t th e centra l puzzl e to this is : What i n heaven' s nam e ha s a cycl e tim e o f 2 6 million years ? A search through biological, geological, o r even astronomical dat a suggests that nothin g ha s a cycle time of 26 million years. Richard Muller of Berkeley has theorized tha t our sun is actually part of a double-star system , arid tha t ou r siste r star (calle d Nemesi s or th e Death Star ) i s responsible fo r periodi c extinction s o f life o n th e earth . The conjectur e i s that our su n has a massive unseen partne r tha t circles it ever y 26 million years. As it passes throug h th e Oor t clou d ( a clou d of comet s tha t supposedl y exist s beyon d th e orbi t o f Pluto) , i t bring s with i t a n unwelcom e avalanch e o f comets , som e o f whic h strik e th e earth, causin g enough debri s that the sunlight is blocked fro m reachin g the earth' s surface . Experimental evidenc e fo r thi s unusual theor y comes fro m th e fac t that th e geologica l layer s from th e past , correspondin g t o th e en d o f each extinctio n cycle , contain unusuall y large quantities of the elemen t iridium. Since iridium i s naturally found i n extraterrestrial meteors , i t is possible tha t thes e trace s o f iridiu m ar e remnant s o f th e comet s sen t down by Nemesis. At present, w e are half-wa y between extinction cycles, meaning tha t Nemesis, if it exists, is at its farthest point in its orbit (prob -

298 M A S T E R

S OF HYPERSPAC E

ably several light-year s away). Thi s woul d give us ove r 1 0 million years or s o until its next arrival. * Fortunately, by the tim e comets from th e Oor t clou d streak through the sola r syste m again, w e will hav e reache d Typ e II I status , meaning that we will have conquered no t just the nearby stars, but trave l through space-time. The Deat h o f th e Su n Scientists sometime s wonder wha t will eventuall y happen t o th e atom s of our bodie s lon g after we are dead . The mos t likel y possibility i s that our molecule s will eventually return t o th e sun. Our su n is a middle-aged star. It is approximately 5 billion years old, and wil l probably remain a yellow star for another 5 billion years. When our sun exhausts its supply of hydrogen fuel , however, it will burn helium and becom e vastly inflated—a red giant . Its atmosphere wil l expand rapidly, eventually extending out t o the orbit of Mars, and th e earth's orbit will b e entirel y within the sun' s atmosphere , s o tha t th e eart h wil l b e fried b y the sun' s enormous temperatures . Th e molecule s making u p our bodies , an d i n fac t th e eart h itself , wil l b e consume d b y the sola r atmosphere. Sagan paint s th e followin g picture: Billions o f year s from now , ther e wil l b e a las t perfect da y on Earth . . . . The Arctic and Antarctic icecaps will melt, flooding the coasts of the world. The hig h oceani c temperature s wil l release more water vapor into the air, increasing cloudiness, shielding the Eart h fro m sunligh t and delayin g the end a little . Bu t sola r evolutio n i s inexorable. Eventuall y the ocean s wil l boil, the atmospher e wil l evaporate awa y to space an d a catastrophe o f the most immense proportion s imaginable will overtake ou r planet. 8

Thus, for those who wish to know whether the earth will be consumed in ice or fire, physics actually gives a definite answer. It will be consumed in fire . However , it is highly likely that humans, if we have survived that *Another theory that might explain periodi c extinctions on thi s vast tim e scale is the orbit of our sola r system around the Milk y Way galaxy. The sola r system actually dips below and above the galactic plane in its orbit around the galaxy, much lik e carousel horses move up an d dow n as a merry-go-round turns. As it dips periodically through the galacti c plane, the solar system may encounter large quantities of dust that disturb the Oor t cloud, bringing dow n a hail of comets.

Beyond th e Future 29

9

long, will have long departed from the sola r system. Unlike a supernova, there is ample warning of the demis e o f our sun .

The Deat h o f the Galax y On a tim e scal e o f severa l billion s of years , w e must confron t th e fac t that the Milk y Way galaxy i n whic h we live, will die. Mor e precisely, we live on the Orion spiral arm of the Milk y Way. When we gaze at the nigh t sky and fee l dwarfed by the immensit y of the celestia l lights dotting th e heavens, w e are actually looking a t a tiny portion o f the stars located o n the Orio n arm. Th e million s of stars that hav e inspired bot h lover s and poets fo r generations occup y only a tiny part of the Orion arm. The res t of the 20 0 billion stars within the Milk y Way are s o distant tha t they can barely be see n a s a hazy ribbon tha t cuts across the nigh t sky. About 2 million light-years from the Milk y Way is our neares t galacti c neighbor, th e great Andromeda galaxy , which is two to three times larger than ou r ow n galaxy. The tw o galaxies ar e hurtlin g towar d eac h othe r at 12 5 kilometers pe r second , an d shoul d collid e withi n 5 to 1 0 billion years. As astronomer Lar s Hernquis t a t th e Universit y of Californi a a t Santa Cru z ha s said , thi s collisio n wil l b e "analogou s t o a hostil e take over. Ou r galax y will be consume d an d destroyed." 9 As seen fro m oute r space, th e Andromed a galax y will appear t o collide wit h and the n slowl y absorb th e Milk y Way galaxy. Computer sim ulations of colliding galaxies show that the gravitational pull of the large r galaxy will slowl y overwhelm th e gravit y of the smalle r galaxy , and afte r several rotation s th e smalle r galax y will b e eate n up . Bu t becaus e th e stars within the Milk y Way galaxy are s o widely separated b y the vacuum of space, th e numbe r o f collisions between star s will be quite low, on th e order o f severa l collision s pe r century . S o our su n ma y avoid a direc t collision for a n extende d perio d o f time. Ultimately, o n thi s tim e scal e o f billion s o f years, we have a muc h more deadl y fate , th e deat h of the univers e itself. Clever forms of intelligent lif e ma y find ways to buil d spac e ark s to avoid mos t natural catas trophes, bu t ho w can w e avoid th e deat h o f th e universe , when spac e itself is our wors t enemy? The Aztecs believed tha t the en d o f the world would come when th e sun one day falls from the sky. They foretold tha t this would come ' 'when the Eart h ha s becom e tire d . . . , when th e see d o f Earth ha s ended. " The star s would be shaken fro m th e heavens . Perhaps the y were close to the truth . One ca n hope that by the time our sun begins t o flicker out , human -

300 M A S T E R

S OF HYPERSPAC E

ity will hav e lon g sinc e lef t th e sola r syste m an d reache d fo r th e stars . (In fact , i n Asimov's Foundation series , th e locatio n o f our origina l star system has been los t for thousands of years.) However, inevitably, all the stars i n th e heaven s wil l flicke r ou t a s thei r nuclea r fue l i s exhausted . On a scal e o f ten s t o hundred s o f billion s of years, w e are facin g th e death of the univers e itself . Either the univers e is open, in which case it will expan d foreve r unti l temperature s graduall y reac h nea r absolut e zero, o r th e univers e i s closed , i n whic h cas e th e expansio n wil l b e reversed an d th e univers e will die in a fiery Big Crunch. Even for a Type III civilization, this is a daunting threa t t o it s existence. Ca n master y of hyperspace sav e civilization from it s ultimate catastrophe , th e deat h o f the universe?

14 The Fat e o f the Univers e Some sa y the worl d will end i n fire . Some sa y in ice. From what I've tasted o f desire I hold wit h those who favor fire .

Robert Frost

It ain't over 'til it's over. Yogi Berr a

W

HETHER a civilization , either o n eart h o r i n oute r space , ca n reach a poin t i n it s technological developmen t t o harnes s th e power o f hyperspace depend s partly , as we have seen, on negotiatin g a series o f disaster s typica l of Typ e 0 civilizations . Th e dange r perio d i s the first several hundred year s after th e daw n of the nuclea r age , when a civilization's technological developmen t has far outpaced it s social and political maturity in handling regiona l conflicts. By th e tim e a civilizatio n has attaine d Typ e H I status , i t wil l hav e achieved a planetary social structure advanced enough t o avoid self-annihilation an d a technology powerfu l enough t o avoid an ecologica l or a natural disaster , suc h a s an ic e ag e o r sola r collapse . However , even a Type III civilization will have difficulty avoiding the ultimate catastrophe: the deat h o f the univers e itself. Even th e mighties t an d mos t sophisti cated o f the Typ e II I civilization's starships will b e unabl e t o escape th e final destiny of the universe . That th e univers e itsel f mus t di e wa s known t o nineteenth-century

301

302 M A S T E R

S O F H Y P E R S P A CE

scientists. Charles Darwin, in his Autobiography, wrote of his anguish when he realize d thi s profoun d bu t depressin g fact : "Believin g as I d o tha t man i n th e distan t futur e wil l b e a fa r mor e perfec t creatur e tha n h e now is, it is an intolerable though t tha t he an d al l other sentien t beings are doome d t o complet e annihilatio n afte r suc h long-continue d slo w progress."1 The mathematician and philosopher Bertrand Russell wrote that the ultimate extinctio n o f humanity is a cause o f "unyieldin g despair. " I n what mus t b e on e o f th e mos t depressin g passage s eve r writte n b y a scientist, Russell noted: That man is the produc t of causes which had n o previsio n of the end the y were achieving ; tha t hi s origin, hi s growth, hi s hopes and fears , hi s loves and hi s beliefs, ar e bu t th e outcom e o f accidental collocation s of atoms ; that n o fire, no heroism , n o intensit y of thought o r feeling , can preserv e a lif e beyon d the grave ; tha t all the labor s o f the ages , all the devotion , all the inspiration , all the noonda y brightnes s o f human genius , ar e destine d to extinctio n i n th e vas t death o f the sola r system ; and th e whol e templ e of Man's achievemen t mus t inevitabl y be burie d beneat h th e debri s o f a universe i n ruins—al l these things , if not quit e beyon d dispute , ar e yet so nearly certain , tha t n o philosoph y which rejects the m ca n hop e t o stand . Only withi n the scaffoldin g o f these truths , onl y on th e fir m foundatio n of unyielding despair, ca n th e soul' s habitatio n b e safel y built. 2

Russell wrote this passage in 1923, decades before the advent of space travel. The death of the solar system loomed larg e in his mind, a rigorous conclusion o f th e law s o f physics . Within th e confine s o f th e limite d technology of his time, this depressing conclusion seeme d inescapable. Since that time, we have learned enoug h abou t stellar evolution to know that our sun will eventually become a red gian t and consum e the eart h in nuclea r fire . However , we also understand th e basic s of space travel. In Russell' s time, th e ver y though t o f larg e ship s capabl e o f placin g humans o n th e moo n o r th e planet s was universally considere d t o b e the thinkin g of a madman . However , with th e exponentia l growt h of technology, th e prospec t o f the deat h o f the sola r system i s not suc h a fearsome event for humanity, as we have seen. By the tim e our su n turns into a re d giant , humanit y either wil l hav e long perishe d int o nuclear dust or, hopefully , wil l have found its rightful plac e among th e stars. Still, it is a simple matter to generalize Russell's "unyielding despair" from th e deat h o f our sola r syste m to th e deat h o f the entir e universe. In tha t event , it appears tha t n o spac e ar k can transpor t humanity out

The Fate o f th e Universe 30

3

of harm's way. The conclusio n seem s irrefutable ; physic s predicts tha t all intelligent life forms, no matte r ho w advanced, wil l eventually perish when th e univers e itself dies. According t o Einstein' s genera l theor y o f relativity , th e univers e either will continu e t o expand foreve r i n a Cosmic Whimper, i n which case th e univers e reache s nea r absolut e zer o temperatures , o r wil l con tract int o a fiery collapse, th e Bi g Crunch. The univers e wil l di e eithe r in "ice, " wit h a n ope n universe , o r i n "fire, " wit h a close d universe . Either way , a Type II I civilizatio n is doomed becaus e temperature s wil l approach eithe r absolut e zer o or infinity . To tel l which fate awaits us, cosmologists use Einstein's equations t o calculate the total amount of matter-energy i n the universe. Because the matter i n Einstein's equation determine s th e amount of space-time curvature, we must know the average matter density of the universe in order to determin e i f ther e i s enough matte r an d energ y fo r gravitatio n t o reverse th e cosmi c expansion o f the origina l Bi g Bang. A critica l value fo r th e averag e matte r densit y determines th e ulti mate fat e o f the univers e an d al l intelligent life within it. If the averag e density o f th e univers e i s les s tha n 10~ 29 gra m pe r cubi c centimeter , which amounts t o 10 milligrams of matter sprea d ove r the volume of the earth, the n the universe will continue t o expand forever , until it becomes a uniformly cold, lifeless space. However , if the averag e density is larger than thi s value, then ther e i s enough matte r fo r th e gravitationa l force of the universe to reverse the Big Bang, and suffe r th e fiery temperature s of the Bi g Crunch. At present, th e experimental situatio n is confused. Astronomers have several way s o f measuring th e mas s of a galaxy , and henc e th e mas s of the universe . The first is to count th e numbe r o f stars in a galaxy, and multiply tha t numbe r b y the averag e weigh t of eac h star . Calculation s performed i n thi s tediou s fashio n sho w that th e averag e densit y is less than th e critica l amount, an d tha t th e univers e will continue to expan d forever. Th e proble m wit h this calculation i s that i t omits matter that is not luminou s (fo r example, dus t clouds , black holes, cold dwarf stars). There i s also a secon d wa y to perfor m thi s calculation , which is t o use Newton' s laws . B y calculating th e tim e i t take s fo r star s t o mov e around a galaxy, astronomers ca n use Newton's laws to estimate the tota l mass o f the galaxy , in th e sam e wa y that Newto n used th e tim e i t too k for th e moo n t o orbit th e eart h t o estimate th e mas s of the moo n an d earth. The proble m i s the mismatch between these two calculations. In fact , astronomers kno w that up t o 90% of the mas s of a galaxy is in the for m

304 M A S T E R

S O F HYPERSPACE

of hidden, undetectabl e "missin g mass " o r "dark matter," which is not luminous bu t ha s weight. Even if we include a n approximat e valu e for the mas s of nonluminous interstella r gas, Newton's laws predict that th e galaxy is far heavier tha n th e valu e calculated by counting stars. Until astronomers resolv e the questio n o f this missing mass or dar k matter, we cannot resolve the questio n of whether the univers e will contract and collaps e into a fiery bal l or wil l expand forever. Entropy Deat h

Assume, for the moment , tha t th e averag e density of the univers e is less than th e critical value. Since the matter-energy conten t determines the curvature of space-time, we find tha t there is not enough matter-energ y to make th e univers e recollapse. I t will the n expan d limitlessl y until its temperature reache s almos t absolute zero . This increases entropy (whic h measures th e tota l amoun t o f chao s o r randomnes s i n th e universe) . Eventually, the univers e dies in an entropy death . The Englis h physicist and astronome r Si r James Jeans wrot e about the ultimate death o f the universe , which he called the "hea t death," as early a s the tur n o f th e century : "The secon d la w of thermodynamics predicts that ther e ca n be but on e en d t o the universe— a 'hea t death' in which [the] temperatur e i s so low as to mak e lif e impossible." 3 To understan d ho w entropy death occurs , it i s important t o under stand th e thre e law s of thermodynamics, which govern all chemical and nuclear processes on the earth an d in the stars. The British scientist and author C . P. Snow ha d a n elegan t way of remembering th e thre e laws : 1. Yo u cannot win (tha t is , you canno t ge t somethin g fo r nothing , because matte r an d energ y are conserved). 2. Yo u cannot break even (yo u cannot retur n t o th e sam e energ y state, becaus e ther e i s always a n increas e i n disorder ; entrop y always increases). 3. Yo u cannot get out oj th e game (because absolute zero is unattainable). For the death o f the universe , the most important is the Second Law, which state s tha t an y process create s a ne t increas e i n th e amoun t of disorder (entropy ) in th e universe . The Secon d La w is actually an integral par t o f ou r everyda y lives. Fo r example , conside r pourin g crea m into a cu p o f coffee . Orde r (separat e cup s o f crea m an d coffee ) ha s

The Fate o f th e Universe 30

5

naturally change d int o disorde r ( a random mixtur e o f cream an d cof fee). However , reversin g entropy , extractin g orde r fro m disorder , i s exceedingly difficult. "Unmixing " th e liqui d back into separate cup s of cream an d coffe e i s impossible withou t an elaborat e chemistr y laboratory. Also, a lighted cigarett e can fill an empty room wit h wisps of smoke, increasing entrop y i n that room . Orde r (tobacc o an d paper ) ha s again turned int o disorde r (smok e and charcoal) . Reversing entropy—that is, forcing the smoke back into the cigarette an d turnin g the charcoal back into unburne d tobacco—i s impossibl e eve n wit h th e fines t chemistr y laboratory o n the planet . Similarly, everyone know s that it's easier t o destroy tha n t o build. I t may take a year to construct a house, bu t onl y an hour or s o to destroy it in a fire. It took almost 5,000 years to transform roving bands of hunters into th e grea t Azte c civilization, which flourished ove r Mexic o and Central Americ a and buil t towering monuments t o its gods. However, it only too k a few months fo r Corte z an d th e conquistador s t o demolis h that civilization. Entropy is relentlessly increasing in the star s as well as on our planet . Eventually, thi s mean s that th e star s will exhaust thei r nuclea r fue l an d die, turnin g int o dea d masse s o f nuclea r matter . Th e univers e wil l darken as the stars , on e b y one, ceas e to twinkle. Given ou r understandin g o f stellar evolution , we can pain t a rathe r dismal pictur e o f how th e univers e will die. Al l stars will becom e blac k holes, neutro n stars , o r col d dwar f star s (dependin g o n thei r mass ) within 10 24 years as their nuclea r furnace s shut down. Entropy increases as stars slide dow n th e curv e o f binding energy , until n o mor e energ y can b e extracte d b y fusing thei r nuclea r fuel . Withi n 10 s2 years, all protons an d neutron s i n the universe will probably decay. According to the GUTs, th e proton s an d neutron s ar e unstabl e ove r tha t vast time scale. This means that eventually all matter as we know it, including the earth and th e solar system, will dissolve into smaller particles, such as electrons and neutrinos . Thu s intelligen t beings wil l hav e t o face th e unpleasan t possibility that the protons and neutrons i n their bodies will disintegrate. The bodie s of intelligent organisms will no longer b e made o f the famil iar 100 chemical elements, which are unstable over that immense period of time. Intelligent life will have to find ways of creating new bodies mad e of energy, electrons, and neutrinos . After a fantastic 10'° ° ( a googol ) years , th e universe' s temperatur e will reac h nea r absolut e zero . Intelligen t lif e i n thi s disma l future wil l face th e prospect o f extinction. Unabl e to huddle nex t t o stars, they will freeze t o death . Bu t eve n i n a desolate , col d univers e at temperature s

306 M A S T E R

S OF HYPERSPAC E

near absolut e zero , ther e i s on e las t remainin g flickerin g source o f energy: black holes. Accordin g t o cosmologist Stephen Hawking , black holes are no t completel y black, but slowl y leak energy into oute r spac e over an extende d perio d o f time. In thi s distan t future , blac k hole s ma y becom e "lif e preservers " because the y slowly evaporat e energy . Intelligen t lif e woul d necessarily congregate nex t t o thes e blac k holes an d extrac t energ y fro m the m t o keep thei r machine s functioning . Intelligent civilizations, like shivering homeless peopl e huddle d nex t t o a fadin g fire , woul d b e reduce d t o pathetic outpost s o f misery clinging to a black hole. 4 But what, we may ask, happens afte r 10'°° years, when the evaporatin g black holes will have exhausted mos t of their ow n energy? Astronomers John D . Barrow of th e Universit y of Susse x arid Joseph Sil k o f th e Uni versity of California at Berkeley caution tha t this question may ultimately have n o answe r with present-da y knowledge . O n tha t tim e scale , quantum theory , fo r example , leave s open th e possibilit y that ou r univers e may "tunnel" into anothe r universe . The probabilitie s for these kinds of events are exceedingly small; one would have to wait a time interval larger tha n th e lifetim e of our presen t universe, so we need no t worr y that reality will suddenly collapse in ou r lifetime, bringing with it a new set of physical laws. However, on th e scale of 10 lo° years, these kind s of rare cosmi c quantum events can no longe r be ruled out . Barrow and Sil k add, "Wher e there is quantum theory there is hope. We ca n neve r b e completel y sur e thi s cosmi c hea t deat h wil l occu r because w e can neve r predict th e futur e o f a quantum mechanica l universe with complete certainty; for in an infinite quantum future anything that can happen, eventuall y will."5 Escape Through a Higher Dimensio n

The Cosmic Whimper i s indeed a dismal fate awaiting us if the averag e density of the univers e is too low . Now assume that th e averag e density is larger tha n th e critica l value. This mean s tha t th e expansio n proces s will contract within tens of billions of years, and th e univers e will en d i n fire, not ice . In thi s scenario, ther e i s enough matter an d henc e a strong enoug h gravitational pul l i n th e univers e t o hal t th e expansion , an d the n th e universe wil l begi n t o slowl y recollapse , bringin g th e distan t galaxie s together again . Starligh t wil l becom e "blu e shifted, " instea d o f re d

The Fate o f the Universe 30

7

shifted, indicatin g tha t th e star s are rapidl y approaching on e another . The temperature s onc e agai n will rise to astronomical limits. Eventually, the hea t wil l become sufficientl y grea t t o vaporize all matter into a gas. Intelligent beings will find that their planets' oceans have boiled away and tha t thei r atmosphere s hav e turned int o a searing furnace. As their planets begin t o disintegrate, the y will be forced to flee int o outer spac e in gian t rockets. Even the sanctuary of outer space may prove to be inhospitable, however. Temperatures wil l eventuall y rise past th e poin t wher e atoms ar e stable, an d electron s wil l b e rippe d of f their nuclei , creating a plasma (like tha t foun d i n ou r sun) . At this point, intelligen t life ma y have t o build giganti c shield s aroun d thei r ship s an d us e thei r entir e energ y output t o keep thei r shield s from disintegrating from th e intens e heat. As temperatures continu e t o rise , th e proton s an d neutron s i n th e nucleus will be ripped apart . Eventually, the protons and neutrons themselves will be tor n apart into quarks. As in a black hole, th e Bi g Crunch devours everything . Nothing survive s it . Thus i t seem s impossibl e tha t ordinary matter, let alone intelligen t life, ca n survive the violent disruption. However, there is one possibl e escape. If all of space—time is collapsing into a fiery cataclysm, then th e onl y way to escape th e Big Crunch is to leave space and time—escap e via hyperspace. This may not b e a s farfetched a s i t sounds . Compute r calculation s performe d wit h Kaluza Klein and superstrin g theories have shown that moments after Creation , the four-dimensiona l univers e expande d a t th e expens e o f th e six dimensional universe . Thus th e ultimat e fat e o f th e four - an d th e six dimensional universe s are linked. Assuming that thi s basic picture is correct, ou r six-dimensiona l twin universe ma y gradually expand, a s our ow n four-dimensional universe collapses. Moment s before ou r univers e shrinks to nothing, intelligent life may realize that the six-dimensional universe is opening up, and fin d a means to exploit that fact . Interdimensional trave l is impossible toda y because ou r siste r universe has shrunk dow n t o th e Planc k scale. However,ain th e final staare s of a collapse, the sister universe may open up , making dimensional travel possible once again . If the siste r universe expands enough , then matte r and energ y may escape int o it, making an escap e hatc h possible for any intelligent beings smart enough t o calculate the dynamics of space-time . The lat e Columbi a Universit y physicis t Gerald Feinber g speculated on thi s long shot o f escaping the ultimat e compression o f the universe through extra dimensions: 7

308 M A S T E R

S OF HYPERSPAC E

At present , thi s is no mor e tha n a scienc e fiction plot. However , if there are mor e dimension s tha n thos e w e know , o r four-dimensiona l space times in addition t o the one we inhabit, then I think it very likely that ther e are physical phenomena tha t provide connections between them. It seems plausible tha t i f intelligence persist s in th e universe , it will, i n muc h less time tha n th e man y billion s o f year s befor e th e Bi g Crunch , fin d ou t whether there i s anything to this speculation, and i f so how to take advantage o f it.' 1

Colonizing the Universe

Almost al l scientist s wh o hav e investigate d th e deat h o f th e universe , from Bertran d Russell to current cosmologists , have assumed tha t intelligent life will be almost helpless in the fac e of the inevitable , final death throes of the universe. Even the theory that intelligent beings can tunnel through hyperspac e and avoid the Big Crunch assume s that these beings are passiv e victims until th e final moments of the collapse . However, physicists John D . Barrow o f the Universit y o f Sussex an d Frank J. Tipler o f Tulane University , in thei r boo k Th e Anthropic Cosmological Principle, have departed from conventiona l wisdom and conclude d just the opposite : that intelligen t life, ove r billions of years of evolution, will pla y an activ e role i n th e fina l moment s o f our universe . They tak e the rathe r unorthodo x vie w tha t technolog y wil l continu e t o rise expo nentially over billions of years, constantly accelerating i n proportio n t o existing technology . Th e mor e star system s that intelligen t beings hav e colonized, th e mor e sta r system s they can colonize . Barro w and Tiple r argue tha t ove r severa l billio n years , intelligen t being s wil l hav e com pletely colonized vast portions o f the visible universe. But the y are con servative; they do no t assum e that intelligen t lif e wil l have mastered th e art o f hyperspace travel . They assume onl y that thei r rocket s wil l travel at near-ligh t velocities. This scenari o shoul d b e take n seriousl y fo r severa l reasons . First , rockets travelin g a t near-ligh t velocitie s (propelled , say , b y photo n engines usin g th e powe r o f larg e lase r beams ) ma y tak e hundred s o f years t o reac h distan t sta r systems . But Barro w an d Tiple r believ e tha t intelligent beings will thrive for billions of years, which is sufficient tim e to colonize their own and neighboring galaxie s even with sub-light-speed rockets. Without assumin g hyperspac e travel , Barro w an d Tiple r argu e tha t intelligent being s wil l sen d million s o f smal l "vo n Neuman n probes "

The Fate o f th e Universe 30

9

into th e galax y at near-ligh t speeds t o fin d suitabl e star systems for colonization. John von Neumann, the mathematical genius who developed the first electronic computer a t Princeton Universit y during World War II, proved rigorousl y that robots or automatons coul d b e built with th e ability to program themselves , repair themselves, and even create carbon copies of themselves. Thus Barrow and Tiple r suggest that the von Neumann probes will function largely independently of their creators. These small probes will be vastly different from the current generation o f Viking and Pioneer probes, whic h are little more tha n passive , preprogramme d machines obeying orders from thei r human masters . The von Neumann probes will be simila r to Dyson's Astrochicken, except vastly more powerful an d intelligent . They will enter ne w star systems, land o n planets , and min e th e roc k fo r suitabl e chemical s an d metals . The y wil l the n create a small industrial complex capabl e o f manufacturing numerous robotic copie s o f themselves . Fro m thes e bases , mor e vo n Neuman n probes wil l be launche d t o explor e eve n more sta r systems. Being self-programmin g automatons , thes e probe s wil l no t nee d instructions from their mothe r planet ; they will explore million s of star systems entirel y on thei r own , pausing onl y to periodicall y radio bac k their findings . Wit h million s of thes e vo n Neuman n probe s scattere d throughout th e galaxy , creating millions of copies of themselves as they "eat" an d "digest " th e chemical s on eac h planet , a n intelligen t civilization will be able t o cut down the tim e wasted exploring uninterestin g star systems . (Barrow and Tiple r eve n conside r th e possibilit y that von Neumann probes from distan t civilizations have already entered ou r own solar system . Perhaps th e monolit h feature d s o mysteriously in 2001: A Space Odyssey wa s a von Neuman n probe.) In th e "Sta r Trek" series, for example, the exploratio n o f other star systems b y the Federatio n i s rather primitive . The exploratio n proces s depends totall y on th e skill s o f humans aboar d a small number o f starships. Although thi s scenario ma y make fo r intriguin g human-interes t dramas, it is a highly inefficient metho d o f stellar exploration, given th e large number o f planetary systems that ar e probabl y unsuitable for life . Von Neuman n probes , althoug h the y ma y no t hav e th e interestin g adventures of Captain Kirk or Captain Picard and thei r crews , would be more suitabl e for galactic exploration . Barrow and Tiple r make a second assumption that is crucial to their argument: The expansio n o f the univers e will eventually slow down and reverse itsel f over tens of billions of years. During the contractio n phas e of th e universe , th e distanc e betwee n galaxie s will decrease , makin g it vastly easie r fo r intelligen t being s t o continu e th e colonizatio n o f th e

310 M A S T E R

S O F HYPERSPAC E

galaxies. As the contractio n o f the univers e accelerates, th e rat e o f colonization o f neighborin g galaxie s wil l als o accelerate , unti l the entir e universe is eventually colonized. Even though Barro w and Tiple r assum e that intelligent life wil l pop ulate th e entir e universe , they are stil l a t a loss t o explain ho w any lif e form wil l b e able t o withstand th e unbelievabl y large temperature s an d pressures create d b y the fina l collaps e o f th e universe . They conced e that th e hea t create d b y the contractio n phas e wil l be grea t enoug h t o vaporize any living being, bu t perhaps th e robots that they have created will be sufficientl y hea t resistan t t o withstand the fina l moment s o f th e collapse. Re-Creating the Bi g Ban g

Along thes e lines , Isaac Asimov has conjectured ho w intelligent beings might reac t to the final death o f the universe. In "Th e Las t Question," Asirnov asks the ancient question of whether the universe must inevitably die, and what will happen to all intelligent life when we reach Doomsday. Asimov, however, assumes that th e univers e will di e in ice , rather tha n in fire, as the star s cease t o burn hydroge n an d temperature s plumme t to absolute zero . The stor y begin s i n th e yea r 2061 , whe n a colossa l compute r ha s solved the earth' s energy problems by designing a massive solar satellite in spac e that can beam th e sun' s energ y back to earth. Th e AC (analog computer) i s so large an d advance d tha t it s technician s hav e onl y th e vaguest ide a o f how it operates. O n a $ 5 bet, tw o drunken technician s ask the compute r whethe r th e sun' s eventual death ca n be avoide d or , for tha t matter , whethe r th e univers e must inevitably die. Afte r quietl y mulling ove r thi s question, th e A C responds : INSUFFICIEN T DAT A FOR A MEANINGFUL ANSWER .

Centuries into th e future , th e A C has solved the proble m o f hyperspace travel , an d human s begi n colonizin g thousand s o f sta r systems . The AC is so large tha t it occupies several hundred square miles on eac h planet an d s o comple x tha t i t maintain s an d service s itself . A young family is rocketing through hyperspace , unerringly guided b y the AC, in search o f a new star system to colonize . When th e fathe r casuall y mentions that th e star s must eventually die, the childre n become hysterical. "Don't let the stars die," plea d th e children. To calm the children , h e asks the AC if entropy can be reversed. "See," reassures the father, reading th e AC' s response, th e A C can solv e everything. He comfort s them

The Fate o f th e Universe 31

1

by saying, "It wil l take care o f everything when the tim e comes, so don't worry." H e never tells the children tha t the AC actually prints out: INSUFFICIENT DATA FOR A MEANINGFUL ANSWER. Thousands of years into th e future , th e Galax y itself has been colo nized. The AC has solved the problem o f immortality and harnesse s th e energy o f the Galaxy , but mus t fin d ne w galaxies fo r colonization . Th e AC is so complex that it is long past the point where anyone understand s how it works. It continually redesigns and improve s its own circuits. Two members o f th e Galacti c Council , eac h hundred s o f years old , debat e the urgen t question o f finding new galactic energy sources, and wonder if the univers e itself is running down. Can entropy be reversed? they ask. The A C responds : INSUFFICIENT DAT A FOR A MEANINGFUL ANSWER. Millions o f year s int o th e future , humanit y ha s sprea d acros s th e uncountable galaxie s o f th e universe . The A C has solve d th e proble m of releasin g th e min d fro m th e body , an d huma n mind s ar e fre e t o explore the vastness of millions of galaxies, with their bodies safely stored on some long forgotten planet. Two minds accidentally meet each other in oute r space, and casuall y wonder where among th e uncountabl e gal axies humans originated . Th e AC , which is now so large tha t mos t o f it has to be housed i n hyperspace, responds by instandy transporting the m to an obscur e galaxy . They are disappointed . Th e galax y is so ordinary, like millions of other galaxies , and th e origina l sta r has long since died. The tw o minds become anxiou s becaus e billion s of stars in the heaven s are slowl y meeting th e sam e fate. The tw o minds ask, can th e deat h o f the universe itself be avoided? From hyperspace, the AC responds: INSUFFICIENT DATA FOR A MEANINGFUL ANSWER. Billions o f years into th e future , humanit y consists of a trillion, trillion, trillion immortal bodies, each cared for by automatons. Humanity's collective mind , which is free t o roa m anywher e in th e univers e a t will, eventually fuses into a single mind, which in turn fuses with the AC itself. It n o longe r make s sens e t o as k what th e A C is made of , o r wher e i n hyperspace it really is. "The univers e is dying," think s Man, collectively. One b y one, a s the star s and galaxie s cease t o generate energy , temper atures throughout th e universe approach absolut e zero. Man desperately asks if the col d and darknes s slowl y engulfing the galaxie s mean its eventual death . Fro m hyperspace , th e A C answers: INSUFFICIENT DATA FOR A MEANINGFUL ANSWER.

When Ma n ask s th e A C to collec t th e necessar y data, i t responds: I WILL DO SO . I HAVE BEEN DOING SO FOR A HUNDRED BILLION YEARS. M Y PREDECESSORS HAV E BEEN ASKED THIS QUESTIO N MAN Y TIMES. AL L TH E DAT A I HAVE REMAINS INSUFFICIENT .

312 M A S T E R

S O F HYPERSPAC E

A timeles s interva l passes , an d th e univers e ha s finall y reache d it s ultimate death . Fro m hyperspace , th e AC spends a n eternit y collecting data an d contemplatin g th e final question. At last, the AC discovers the solution, eve n thoug h ther e i s no longer anyone t o give the answer. The AC carefull y formulate s a program , an d the n begin s th e proces s o f reversing Chaos . I t collect s cold , interstella r gas , brings togethe r th e dead stars , until a gigantic bal l is created. Then, when it s labors ar e done , fro m hyperspac e th e A C thunders: LET THERE BE LIGHT !

And ther e was light— And o n th e sevent h day, He rested .

15 Conclusion The know n i s finite , th e unknow n infinite ; intellectuall y we stand on a n islet in the mids t of an illimitable ocean of inexplicability. Our busines s i n every generation is to reclaim a little more land. Thomas H. Huxley

P

ERHAPS the mos t profound discover y of the past century in physics has been th e realizatio n tha t nature , a t its most fundamental level, is simpler tha n anyon e thought . Althoug h th e mathematica l complexity of th e ten-dimensiona l theor y ha s soared t o dizzyin g heights , openin g up ne w areas o f mathematics in th e process , th e basi c concept s driving unification forward , such a s higher-dimensional spac e an d strings , ar e basically simple and geometric . Although i t i s to o earl y t o tell , futur e historian s o f science , whe n looking back at the tumultuou s twentieth century , may view one o f th e great conceptua l revolution s t o b e th e introductio n o f higher-dimensional space-tim e theories , suc h a s superstring an d Kaluza-Klein-typ e theories. A s Copernicus simplifie d th e sola r syste m wit h hi s serie s o f concentric circle s an d dethrone d th e centra l rol e o f th e eart h i n th e heavens, the ten-dimensiona l theory promises t o vastly simplify th e law s of nature an d dethron e th e familia r world o f thre e dimensions . A s we have seen, th e crucial realization is that a three-dimensional descriptio n of the world, such as the Standar d Model , is "too small " t o unite al l the fundamental force s of nature int o one comprehensive theory . Jamming

313

314

M A S T E R S OF H Y P E R S P A C E

the four fundamental forces into a three-dimensional theor y creates an ugly, contrived, and ultimatel y incorrect descriptio n o f nature. Thus th e mai n curren t dominatin g theoretica l physic s in th e pas t decade ha s bee n th e realizatio n tha t th e fundamenta l law s of physics appear simpler in highe r dimensions , an d tha t al l physical laws appea r to b e unifie d i n te n dimensions . These theorie s allo w u s t o reduc e a n enormous amoun t o f information into a concise , elegan t fashio n tha t unites the two greatest theories of the twentieth century: quantum theory and genera l relativity . Perhap s i t i s time t o explor e som e o f th e man y implications that the ten-dimensional theory has for the future o f physics and science , th e debat e betwee n reductionis m an d holis m i n nature , and th e aestheti c relatio n amon g physics , mathematics, religion , an d philosophy. Ten Dimension s an d Experimen t

When caught up in the excitement, and turmoil accompanying the birth of any great theory , there i s a tendency to forget that ultimately all theories must be tested against the bedroc k of experiment. N o matter how elegant o r beautifu l a theor y ma y appear , i t i s doomed i f i t disagree s with reality . Goethe onc e wrote, "Gray is the dogma, but green is the tree of life." History has repeatedly borne ou t th e correctnes s o f his pungent obser vation. Ther e ar e man y example s o f old , incorrec t theorie s tha t stub bornly persisted fo r years, sustained onl y by the prestig e o f foolish bu t well-connected scientists . At times , i t eve n becam e politicall y risky t o oppose th e powe r o f ossified , senio r scientists . Many o f thes e theorie s have been kille d off only when some decisive experiment expose d thei r incorrectness. For example , because o f Hermann vo n Helmholtz' s fam e and con siderable influenc e in nineteenth-centur y Germany, hi s theory o f electromagnetism was much more popula r amon g scientist s than Maxwell's relatively obscure theory . But no matte r how well known Helmholtz was, ultimately experimen t confirme d th e theor y of Maxwell an d relegate d Helmholtz's theor y to obscurity . Similarly, when Einstei n proposed hi s theory of relativity, many politically powerful scientist s in Nazi Germany, like Nobel laureate Philip Lenard, hounded hi m until he was driven ou t of Berlin in 1933 . Thus the yeoman's work in any science, and especially physics, is done by the experimentalist, who must keep the theoretician s honest.

Conclusion 31

5

Victor Weisskopf , a theoretica l physicis t a t MIT , once summarize d the relationship betwee n theoretical an d experimental scienc e when he observed tha t ther e ar e thre e kind s of physicists: th e machin e builder s (who buil d th e ato m smasher s that mak e the experimen t possible) , the experimentalists (wh o plan an d execut e th e experiment) , and th e the oreticians (wh o devise the theor y t o explain th e experiment) . H e the n compared thes e thre e classe s t o Columbus' s voyag e t o America . H e observed tha t the machin e builder s correspon d t o th e captain s an d shi p builder s who really develope d th e technique s a t tha t time . Th e experimentalist s wer e those fellow s o n th e ship s tha t saile d t o th e othe r sid e o f th e worl d an d then jumpe d upo n th e ne w island s an d just wrot e dow n wha t the y saw. The theoretica l physicist s are thos e fellow s who stayed back in Madrid an d told Columbu s tha t he wa s going to land i n India. 1

If, however , the law s o f physic s become unite d i n te n dimension s only a t energie s fa r beyon d anythin g available wit h ou r presen t tech nology, the n th e futur e o f experimental physics i s in jeopardy. I n th e past, ever y new generation o f atom smasher s has brough t forth a new generation o f theories. This period ma y be coming to a close. Although everyone expected new surprises if the SS C became operational b y about th e yea r 2000, some were betting that it would simply reconfirm th e correctnes s o f ou r present-da y Standard Model . Mos t likely, th e decisiv e experiments that wil l prov e or disprov e the correct ness of the ten-dimensional theory cannot be performed anytime in th e near future. W e may be entering a long dry spell where research in tendimensional theorie s will become a n exercis e in pur e mathematics . Al l theories derive their power and strengt h from experiment , which is like fertile soi l that can nourish an d sustai n a field of flowering plant s once they take root. I f the soi l becomes barren an d dry , then th e plant s will wither along with it. David Gross, one of the originators of the heterotic string theory, has compared th e developmen t of physics to th e relationshi p between two mountain climbers: It used to be that a s we were climbin g th e mountai n of nature, the exper imentalists would lea d th e way . We laz y theorists woul d la g behind. Every once in a while the y would kic k down a n experimental ston e which woul d bounce of f ou r heads . Eventuall y we woul d ge t th e ide a an d w e woul d follow th e pat h tha t was broken b y the experimentalists . . . . But no w we theorists migh t hav e t o tak e th e lead . This is a muc h mor e lonel y enter -

316 M A S T E R

S OF HYPERSPAC E

prise. I n the past we always knew where the experimentalists wer e and thu s what we should ai m for. Now we have n o ide a ho w large th e mountai n is, nor wher e the summi t is.

Although experimentalists have traditionally taken the lead in breaking ope n ne w territory, the nex t er a i n physic s may be an exceptionally difficult one , forcin g theoretician s t o assum e th e lead , a s Gros s notes. The SS C probably would have foun d ne w particles. Th e Higg s particles may have been discovered , or "super" partners o f the quark s may have shown up, or mayb e a sublayer beneath th e quarks may have bee n revealed. However , th e basi c force s bindin g thes e particle s will , i f th e theory holds up , b e th e same . We may have seen mor e comple x YangMills fields and gluon s coming forth from the SSC , but thes e fields may represent onl y larger an d large r symmetr y groups, representin g frag ments o f th e eve n large r E(8 ) X E(8 ) symmetr y coming fro m strin g theory. In some sense, the origi n o f this uneasy relation between theor y an d experiment i s due t o th e fac t tha t this theory represents , a s Witten has noted, "21s t century physics tha t fel l accidentall y int o th e 20t h century."2 Because th e natural dialectic between theor y and experiment was disrupted b y the fortuitou s accidental discover y of the theor y i n 1968, perhaps w e must wait until the twenty-firs t century , when we expect th e arrival of new technologies that will hopefully open u p a new generation of atom smashers , cosmic-ray counters, and dee p spac e probes. Perhap s this i s the pric e w e must pa y for havin g a forbidde n "snea k preview " into th e physic s of the nex t century . Perhaps by then, throug h indirec t means, we may experimentally see the glimme r of the tent h dimensio n in ou r laboratories . Ten Dimension s an d Philosophy: Reductionis m versus Holis m Any great theor y has equally great repercussions on technolog y and th e foundations o f philosophy . Th e birt h o f genera l relativit y opene d u p new areas o f research i n astronom y an d practicall y created th e scienc e of cosmology. The philosophica l implications of the Bi g Bang have sent reverberations throughou t th e philosophica l an d theologica l commu nities. A few years ago , thi s eve n le d t o leadin g cosmologist s havin g a special audience with the pope at the Vatican to discuss the implications of th e Bi g Bang theory on th e Bibl e an d Genesis . Similarly, quantu m theor y gav e birt h t o th e scienc e o f subatomi c particles and helpe d fue l th e current revolution in electronics. The tran-

Conclusion 31

7

sistor—the linchpin o f modern technologica l society—is a purely quantum-mechanical device . Equally profound wa s the impac t tha t th e Hei senberg Uncertainly Principle has had o n th e debat e ove r free wil l an d determinism, affectin g religiou s dogma on th e rol e o f sin and redemp tion fo r th e church . Bot h th e Catholi c Churc h an d th e Presbyteria n Church, with a large ideological stake in the outcome of this controversy over predestination , hav e bee n affecte d b y thi s debat e ove r quantu m mechanics. Although th e implications of the ten-dimensional theory are still unclear, w e ultimately expect tha t th e revolutio n no w germinatin g in th e worl d of physics will have a similar far-reachin g impact onc e th e theory becomes accessibl e t o the averag e person . In general, however, most physicists feel uncomfortable talking about philosophy. They are supreme pragmatists. They stumble across physical laws not b y design o r ideology , bu t largel y through tria l and erro r an d shrewd guesses . Th e younge r physicists , who d o th e lion' s shar e o f research, ar e to o bus y discovering new theorie s t o waste time philoso phizing. Younger physicists , i n fact , loo k askanc e a t olde r physicist s if they spend to o much tim e sitting on distinguishe d policy committees or pontificating o n th e philosoph y of science. Most physicist s feel that , outsid e o f vagu e notion s o f "truth " an d "beauty," philosophy has no business intruding on their private domain. In general, they argue, reality has always proved to be much more sophisticated an d subtl e tha n an y preconceived philosophy . The y remin d u s of some well-know n figure s i n scienc e who, in thei r waning years, took up embarrassingl y eccentri c philosophica l idea s tha t le d dow n blin d alleys. When confronte d wit h stick y philosophica l questions , suc h a s th e role o f "consciousness" in performin g a quantum measurement , mos t physicists shru g thei r shoulders . A s long a s they can calculat e th e out come o f an experiment , the y reall y don't car e abou t it s philosophica l implications. I n fact , Richar d Feynma n almos t mad e a caree r tryin g to expose th e pompou s pretense s o f certai n philosophers . Th e greate r their puffed-u p rhetori c an d erudite vocabulary, he thought, the weaker the scientifi c foundatio n o f their arguments . (Whe n debatin g th e rela tive merits of physics and philosophy , I am sometime s reminded o f th e note writte n b y a n anonymou s universit y president wh o analyze d th e differences between them. He wrote, ' 'Why is it that you physicists always require so much expensiv e equipment ? No w the Departmen t o f Mathematics requires nothin g but money for paper, pencils, and waste paper baskets and th e Department of Philosophy is still better. I t doesn't even ask for waste paper baskets." 3) Nevertheless, although th e average physicist is not bothered by philo-

318 M A S T E R

S O F H Y P E R S P A CE

sophical questions, the greatest of them were. Einstein, Heisenberg, an d Bohr spent long hours in heated discussions , wrestling late into the night with th e meanin g o f measurement, th e problem s o f consciousness, and the meanin g of probability in their work. Thus it is legitimate to ask how higher-dimensional theories reflec t on thi s philosophical conflict , espe cially regarding th e debat e betwee n "reductionism " and "holism. " Heinz Pagels once said , "We are passionate about our experienc e of reality, and mos t of us project our hope s an d fear s onto th e universe.'" 1 Thus i t i s inevitabl e tha t philosophical , eve n persona l question s wil l intrude int o th e discussio n on higher-dimensiona l theories . Inevitably, the reviva l o f highe r dimension s i n physic s wil l rekindl e th e debat e between "reductionism " an d "holism " tha t ha s flared, on an d off , for the past decade . Webster's Collegiate Dictionary define s reductionism a s a "procedur e o r theory that reduces complex dat a or phenomena t o simple terms." Thi s has bee n on e o f th e guidin g philosophie s o f subatomi c physics—t o reduce atom s an d nucle i t o thei r basi c components . Th e phenomena l experimental success, for example, o f the Standar d Mode l in explaining the propertie s o f hundred s o f subatomi c particle s show s that ther e i s merit in looking for th e basi c building blocks of matter. Webster's Collegiate Dictionary define s holism a s th e "theor y tha t th e determining factor s esp . i n livin g natur e ar e irreducibl e wholes. " Thi s philosophy maintain s tha t th e Wester n philosoph y o f breakin g thing s down int o thei r component s i s overl y simplistic , tha t on e misse s th e larger picture , whic h ma y contai n vitall y importan t information . Fo r example, think of an ant colon y containing thousand s o f ants that obeys complex, dynamic rules of social behavior. Th e questio n is : What is the best way to understand th e behavio r of an an t colony ? The reductionis t would brea k th e ant s int o thei r constituents : organi c molecules . However, on e ma y spend hundred s o f years dissectin g ant s ari d analyzing their molecula r makeu p without finding the simples t clues as to how an ant colon y behaves. The obviou s way is to analyze the behavio r of an an t colony as an integra l whole , without breaking it down. Similarly, thi s debat e ha s sparke d considerabl e controvers y within the are a o f brain researc h an d artificia l intelligence . Th e reductionis t approach i s to reduce th e brain t o its ultimate units, the brai n cells , an d try t o reassembl e th e brai n fro m them . A whole schoo l o f researc h i n artificial intelligenc e hel d tha t b y creating elementa l digita l circuit s we could buil d u p increasingl y complex circuits , until we created artificia l intelligence. Although thi s school o f though t ha d initia l success in th e 1950s by modeling "intelligence" along the lines of modern digita l com-

Conclusion 31

9

puters, i t prove d disappointin g becaus e i t coul d no t mimi c eve n th e simplest o f brai n functions , such a s recognizin g patterns i n a photo graph. The secon d schoo l o f though t ha s trie d t o tak e a mor e holisti c approach t o th e brain . I t attempts t o defin e th e function s of the brai n and creat e model s tha t trea t th e brai n a s a whole . Although thi s ha s proved mor e difficul t t o initiate, it holds grea t promis e becaus e certai n brain functions that we take for granted (fo r example, toleranc e of"error , weighing o f uncertainty, and makin g creative associations between different objects ) ar e buil t into th e syste m from th e start . Neural network theory, for example , uses aspects of this organic approach . Each side of this reductionist-holistic debat e take s a di m vie w of th e other. In their strenuous attempts to debunk each other, they sometimes only diminish themselves. They often tal k past eac h other , not address ing each other' s mai n points. The lates t twis t i n th e debat e i s that th e reductionist s have, for th e past few years, declared victor y over holism. Recently , there ha s been a flurry o f claim s in th e popula r pres s b y the reductionist s tha t th e suc cesses o f th e Standar d Mode l an d th e GU T theor y are vindication s of reducing natur e t o smalle r an d mor e basi c constituents . B y probin g down t o the elementa l quarks , leptons, and Yang-Mills fields, physicists have finally isolated th e basi c constituent s o f al l matter . Fo r example , physicist James S . Trefi l o f th e Universit y of Virgini a takes a swip e a t holism when he write s about th e "Triump h of Reductionism": During th e 1960 s an d 1970s , when th e complexit y of th e particl e worl d was being made manifest in one experiment after another, some physicists broke fait h with the reductionis t philosophy and bega n t o look outside of the Wester n traditio n fo r guidance . I n hi s boo k Th e Too o f Physics, fo r example, Fritjho f Capr a argue d tha t th e philosoph y of reductionism ha d failed an d tha t i t wa s tim e t o tak e a mor e holistic , mystica l vie w o f nature. . . . [T]h e 1970 s [however ] ca n b e though t o f a s th e perio d i n which th e grea t tradition s of Western scientifi c thought, seemingl y imperiled b y the advance s o f twentieth-centur y science, hav e bee n thoroughl y vindicated. Presumably, it will take a while for this realization to percolate away from a small group of theoretical physicists and becom e incorporate d into our genera l worl d view. 5

The disciple s o f holism , however , tur n thi s debat e around . The y claim tha t th e ide a o f unification, perhap s th e greates t them e i n al l of physics, i s holistic , no t reductionist . The y poin t t o ho w reductionist s

320 M A S T E R

S O F HYPERSPACE

would sometimes snicker behind Einstein' s back in th e las t years of his life, saying that he was getting senile trying to unite all the force s of the world. Th e discover y o f unifyin g pattern s i n physic s was an ide a pio neered b y Einstein, not th e reductionists . Furthermore, th e inabilit y of the reductionist s to offe r a convincing resolution o f the Schrodinger' s cat paradox show s that the y have simply chosen t o ignore th e deeper , philosophical questions . The reductionist s may have had grea t success with quantu m field theory and th e Standar d Model , but ultimately that success is based o n sand , because quantum theory , in th e final analysis, is an incomplet e theory . Both sides , o f course , hav e merit . Eac h sid e i s merel y addressin g different aspects of a difficult problem. However, taken to extremes, this debate sometime s degenerates int o a battle betwee n what I call belligerent scienc e versus know-nothing science. Belligerent science club s the oppositio n wit h a heavy , rigi d vie w of science tha t alienate s rathe r tha n persuades . Belligeren t science seek s to win points in a debate, rathe r tha n win over the audience . Instead of appealing t o th e fine r instinct s of the la y audience b y presenting itsel f as the defende r o f enlightened reaso n an d soun d experiment, it comes off a s a ne w Spanis h Inquisition . Belligeren t scienc e i s science with a chip on its shoulder. Its scientists accuse the holists of being soft-headed , of getting their physics confused, o f throwing pseudoscientific gibberis h to cover their ignorance . Thu s belligeren t science ma y be winning the individual battles, bu t i s ultimately losing the war . In ever y one-on-one skirmish, belligeren t scienc e ma y trounce th e oppositio n b y parading out mountain s o f data an d learne d Ph.D.s . However , in th e lon g run , arrogance an d concei t ma y eventuall y backfire b y alienatin g th e ver y audience tha t it is trying to persuade . Know-nothing science goes to the opposite extreme, rejecting experiment an d embracin g whateve r faddis h philosoph y happen s t o com e along. Know-nothin g science see s unpleasant facts a s mere details , an d the overal l philosophy as everything. If the fact s d o no t see m t o fi t the philosophy, the n obviousl y something i s wrong with th e facts . Know nothing scienc e come s in with a preformed agenda , base d o n persona l fulfillment rathe r tha n objectiv e observation, an d trie s t o fit in th e science as an afterthought . This split between these tw o factions first appeared durin g th e Vietnam War, when the flowe r generation was appalled by the massive, excessive use of deadly technology against a peasant nation. Bu t perhaps th e area in which this legitimate debate has flared u p mos t recently is personal health. For example, well-paid lobbyists for the powerfu l agri-busi-

Conclusion 32

1

ness an d foo d industr y i n th e 1950 s an d 1960 s exerte d considerabl e influence o n Congres s an d th e medica l establishment , preventin g a thorough examinatio n o f th e harmfu l effect s o f cholesterol , tobacco , animal fats , pesticides , and certai n foo d additive s on hear t diseas e an d cancer, which have now been thoroughl y documented . A recent exampl e is the scandal that surrounded th e uproar over the pesticide Ala r i n apples . Whe n th e environmentalist s at th e Nationa l Resources Defense Council announced tha t curren t level s of pesticides in apple s coul d kil l upwar d o f 5,00 0 children , the y sparke d concer n among consumer s an d indignatio n withi n th e foo d industry , which denounced the m a s alarmists. Then i t was revealed tha t the repor t use d figures an d dat a from th e federal governmen t t o arrive at these conclu sions. This, in turn, implied that th e Food an d Drug Administration was sacrificing 5,00 0 children i n th e interest s of "acceptabl e risk. " In addition , th e revelation s about th e widesprea d possibl e contamination o f our drinkin g water by lead, which can cause serious neurolog ical problem s i n children , onl y served t o lowe r th e prestig e o f science in the minds of most Americans. The medical profession, the food industry, an d th e chemica l industr y have begu n t o ear n th e distrus t of wide portions o f society . These an d othe r scandal s hav e als o contribute d t o the national flareup of faddish healt h diets, most of which are well intentioned, but som e of which are no t scientificall y sound .

Higher Synthesi s i n Highe r Dimension s These tw o philosophical viewpoints , apparently irreconcilable , must be viewed fro m th e large r perspective . The y ar e antagonisti c onl y when viewed in thei r extreme form . Perhaps a highe r synthesi s of both viewpoint s lies in highe r dimen sions. Geometry , almost by definition, cannot fi t th e usua l reductionis t mode. B y studying a tiny strand o f fiber, we cannot possibl y understand an entir e tapestry . Similarly, by isolating a microscopic region o f a surface, w e cannot determin e th e overal l structure o f th e surface . Highe r dimensions, b y definition , imply tha t w e mus t tak e th e larger , globa l viewpoint. Similarly, geometr y i s no t purel y holistic , either . Simpl y observing that a higher-dimensional surface is spherical does not provide the infor mation necessar y t o calculat e th e propertie s o f th e quark s containe d within it . Th e precis e wa y in whic h a dimensio n curl s u p int o a bal l determines th e nature of the symmetries of the quarks and gluons living

322 M A S T E R

S O F HYPERSPAC E

on tha t surface. Thus holism by itself does not giv e us the dat a necessary to turn th e ten-dimensiona l theory into a physically relevant theory. The geometr y of higher dimensions, in some sense, forces us to realize the unit y between the holistic and reductionist approaches. The y are simply two ways of approaching th e sam e thing: geometry. They are two sides of the sam e coin. From the vantage point of geometry, it makes no difference whethe r we approach i t fro m th e reductionis t point of view (assembling quarks and gluon s in a Kaluza-Klein space ) o r th e holistic approach (takin g a Kaluza—Klein surface and discovering the symmetries of the quark s and gluons) . We ma y prefe r on e approac h ove r th e other , bu t thi s i s onl y fo r historical or pedagogical purposes . For historical reasons, we may stress the reductionis t root s o f subatomi c physics , emphasizing how particle physicists over a period o f 40 years pieced togethe r thre e o f the funda mental force s b y smashin g atoms , o r w e ma y tak e a mor e holistic approach an d clai m tha t th e fina l unificatio n o f quantu m force s wit h gravity implie s a dee p understandin g o f geometry . Thi s lead s u s t o approach particl e physics through Kaluza-Klei n and string theories and to vie w th e Standar d Mode l a s a consequenc e o f curlin g u p higher dimensional space . The tw o approaches ar e equall y valid. In ou r boo k Beyond Einstein: The Cosmic Quest for th e Theory o f the Universe, Jennifer Traine r an d I too k a more reductionist approach an d described how the discoveries of phenomena in the visible universe eventually led to a geometric description of matter. In thi s book, we took the opposit e approach , beginnin g with the invisibl e universe and takin g the concep t o f how the law s of nature simplify i n highe r dimension s a s ou r basi c theme . However , bot h approaches yield th e sam e result. By analogy, we can discuss the controvers y ove r th e "left " brai n an d "right" brain . Th e neurologist s who originally made th e experimenta l discovery that th e lef t an d righ t hemisphere s o f our brai n perfor m distinctly different function s became distresse d that their data were grossly misrepresented i n th e popula r press . Experimentally , they found tha t when someone i s shown a picture, the left eye (or right brain) pays more attention t o particula r details , while the righ t ey e (o r lef t brain ) mor e easily grasp s th e entir e photo . However , they became disturbe d when popularizers began to say that the lef t brain was the "holistic brain" and the righ t brai n wa s the "reductionis t brain. " Thi s too k th e distinction between th e tw o brains ou t o f context, resultin g in man y bizarre inter pretations o f how one shoul d organize one' s thought s in dail y life . A more correct approach t o brain function, they found, was that th e

Conclusion 32

3

brain necessaril y use s bot h halve s i n synchrony , tha t th e dialecti c between bot h halve s of th e brai n i s more importan t tha n th e specifi c function o f eac h hal f individually . The trul y interesting dynamic s take place when both halve s of the brai n interac t in harmony. Similarly, anyon e wh o see s th e victor y of on e philosoph y ove r th e other i n recen t advance s i n physic s is perhaps readin g to o muc h int o the experimenta l data . Perhap s the safes t conclusio n tha t we can reach is that science benefits most from the intens e interaction between these two philosophies. Let us see concretely how this takes place, analyzing how the theor y of highe r dimension s give s u s a resolutio n betwee n diametricall y opposed philosophies , usin g two examples, Schrodinger's ca t and th e S matrix theory.

Schrodinger's Cat The disciple s of holism sometimes attack reductionism b y hitting quantum theor y where i t i s weakest, on th e questio n o f Schrodinger' s cat . The reductionists cannot give a reasonable explanatio n of the paradoxe s of quantum mechanics . The mos t embarrassing feature o f quantum theory , we recall, is that an observer is necessary to make a measurement. Thus before the obser vation i s made, cat s can b e eithe r dead o r aliv e and th e moo n ma y or may not be in the sky. Usually, this would be considered crazy , but quantum mechanics has been verifie d repeatedl y in the laboratory. Since the process o f makin g a n observatio n require s a n observer , an d sinc e a n observer require s consciousness , then th e disciple s of holism claim that a cosmi c consciousnes s mus t exist in orde r t o explai n th e existenc e of any object. Higher-dimensional theorie s d o no t resolv e thi s difficul t questio n completely, but the y certainly put i t in a new light. The proble m lie s in the distinctio n betwee n th e observe r an d th e observed . However , i n quantum gravity we write down the wav e function of the entir e universe. There i s no mor e distinctio n betwee n th e observe r an d th e observed ; quantum gravit y allow s for th e existenc e o f onl y the wav e functio n of everything. In the past, such statements were meaningless because quantum gravity did not reall y exist as a theory. Divergences would crop up every time someone wante d to do a physically relevant calculation. So the concep t of a wave function for the entire universe, although appealing, was mean-

324 M A S T E R

S O F HYPERSPACE

ingless. However , wit h th e comin g o f th e ten-dimensiona l theory , th e meaning of the wave function of the entir e universe becomes a relevant concept once again. Calculations with the wave function of the universe can appea l t o th e fac t tha t th e theor y i s ultimately a ten-dimensional theory, and i s hence renormalizable . This partial solution t o the questio n of observation once again takes the bes t o f both philosophies . O n th e on e hand , thi s picture i s reductionist becaus e i t adheres closel y to th e standar d quantum-mechanica l explanation o f reality, without recourse to consciousness. O n th e othe r hand, i t is also holisti c because i t begins with th e wav e functio n o f th e entire universe , which is the ultimat e holistic expression! Thi s pictur e does not mak e th e distinctio n between th e observe r an d th e observed . In thi s picture, everything , including all objects and thei r observers , is included i n th e wav e function. This is still only a partial solution because th e cosmi c wave functio n itself, which describes th e entir e universe , does no t liv e i n an y definit e state, but is actually a composite o f all possible universes. Thus the prob lem o f indeterminacy, first discovered b y Heisenberg, i s now extende d to the entir e universe. The smalles t uni t tha t on e ca n manipulat e i n thes e theorie s i s the universe itself , an d th e smalles t unit that on e ca n quantiz e is the spac e of al l possibl e universes , which include s bot h dea d cat s an d liv e cats . Thus i n on e universe , the ca t is indeed dead; bu t i n another , th e ca t is alive. However, both universe s reside i n th e sam e home: th e wav e func tion of the universe . A Chil d of 5-Matrix Theor y

Ironically, in th e 1960s , th e reductionist approach looke d lik e a failure; the quantu m theor y o f field s wa s hopelessly riddle d wit h divergence s found i n the perturbation expansion . With quantum physics in disarray, a branch o f physics called 5-matri x (scatterin g matrix ) theor y broke off from th e mainstrea m an d bega n t o germinate . Originall y founded b y Heisenberg, i t was further developed b y Geoffrey Che w at the Universit y of California at Berkeley. S-matrix theory, unlike reductionism, tried t o look at the scatterin g of particles as an inseparable , irreducible whole. In principle , i f w e kno w th e S matrix , w e kno w everything abou t particle interaction s an d ho w they scatter. In thi s approach, ho w particles bump into one another is everything; the individual particle is nothing. S-matrix theory said that the self-consistency of the scattering matrix,

325 and self-consistency alone, was sufficient t o determin e th e S matrix. Thu s fundamental particle s an d fields were banished foreve r from th e Ede n of S-matrix theory. In th e fina l analysis , only the S matrix ha d an y physical meaning . As an analogy, let us say that we are give n a complex, strange-looking machine an d ar e aske d t o explai n wha t i t does . Th e reductionis t wil l immediately get a screw driver an d tak e the machin e apart. B y breaking down th e machin e t o thousand s o f tiny pieces, th e reductionis t hope s to fin d ou t ho w the machin e functions . However, if the machin e i s too complicated, takin g it apart onl y makes matters worse. The holists , however , d o no t wan t t o tak e th e machin e apar t fo r several reasons . First, analyzin g thousand s of gears and screw s may not give us the slightest hint of what the overall machine does. Second, trying to explain how each tiny gear works may send u s on a wild-goose chase. The correc t way, they feel, i s to loo k a t th e machin e a s a whole. They turn th e machine on an d as k how the part s move and interac t with one another. I n moder n language , thi s machin e i s th e S matrix, an d thi s philosophy became th e 5-matri x theory. In 1971 , however, the tid e shifte d dramaticall y in favo r o f reductionism with Gerard ' t Hooft's discovery that the Yang—Mills field can provide a self-consisten t theor y o f subatomic forces. Suddenly , eac h of the par ticle interaction s cam e tumblin g down lik e huge tree s i n a forest. Th e Yang-Mills fiel d gav e uncann y agreemen t wit h th e experimenta l dat a from ato m smashers , leadin g t o th e establishmen t o f th e Standar d Model, whil e 5-matri x theor y becam e entangle d i n mor e an d mor e obscure mathematics . B y the lat e 1970s , i t seemed lik e a total , irreversible victor y of rcductionis m ove r holis m an d th e 5-matri x theory . Th e reductionists bega n t o declar e victor y over th e prostrat e bod y o f th e holists an d th e S matrix. The tide , however, shifted once again i n th e 1980s . With the failur e of the GUT s to yield any insight into gravitatio n o r yield any experimentally verifiabl e results , physicist s bega n t o loo k fo r ne w avenue s o f research. Thi s departur e fro m GUT s bega n wit h a ne w theory , which owed its existence t o th e 5-matri x theory. In 1968 , whe n 5-matri x theor y wa s i n it s heyday , Venezian o an d Suzuki were deepl y influence d b y the philosoph y o f determining th e S matrix in its entirety. They hit o n th e Eule r bet a functio n because the y were searching for a mathematical representation o f the entire S matrix. If they had looke d fo r reductionist Feynman diagrams, they never would have stumbled o n one of the great discoveries of the past several decades. Twenty years later, w e see th e flowerin g of th e see d plante d b y th e

326 M A S T E R

S O F HYPERSPACE

S-matrix theory. Th e Veneziano-Suzuk i theory gave birth t o string theory, whic h i n tur n ha s bee n reinterprete d vi a Kaluza-Klei n a s a ten dimensional theor y of the universe . Thus w e se e tha t th e ten-dimensiona l theor y straddle s bot h traditions. It was born as a child of a holistic S-matrix theory, but contains the reductionis t Yang-Mill s an d quar k theories . I n essence , i t ha s matured enoug h t o absorb bot h philosophies . Ten Dimensions and Mathematics

One o f the intriguing features of superstring theor y is the leve l to which the mathematics has soared. No other theory known to science uses such powerful mathematic s a t such a fundamental level. In hindsight , this is necessarily so , becaus e an y unifie d fiel d theor y firs t mus t absor b th e Riemannian geometr y o f Einstein's theory and th e Li e groups comin g from quantu m field theory, and the n mus t incorporate a n eve n highe r mathematics to make them compatible . This new mathematics, which is responsible fo r th e merge r o f thes e tw o theories , i s topology, an d i t i s responsible fo r accomplishin g th e seemingl y impossible tas k of abolishing th e infinitie s o f a quantum theor y of gravity. The abrup t introductio n o f advance d mathematic s int o physic s via string theory has caught man y physicists off guard. Mor e than on e physicists has secretly gone to the library to check out huge volumes of mathematical literatur e t o understan d th e ten-dimensiona l theory . CER N physicist John Ellis admits, "I find myself touring through th e bookshops trying to find encyclopedias o f mathematics so that I can mu g up o n all these mathematica l concept s lik e homology and homotop y and al l this sort o f stuf f whic h I neve r bothere d t o lear n before!" 6 T o thos e wh o have worrie d abou t th e ever-widenin g split betwee n mathematic s an d physics in thi s century, this is a gratifying, histori c event in itself . Traditionally, mathematic s an d physic s have been inseparabl e sinc e the tim e o f the Greeks . Newto n and hi s contemporaries neve r mad e a sharp distinctio n betwee n mathematic s an d physics ; they calle d them selves natural philosophers , and fel t a t home i n th e disparat e world s of mathematics, physics , and philosophy . Gauss, Riemann , an d Poincar e al l considere d physic s t o b e o f th e utmost importanc e a s a sourc e o f ne w mathematics . Throughou t th e eighteenth an d nineteent h centuries , ther e was extensive cross-pollination betwee n mathematics and physics . But after Einstei n and Poincare , the development o f mathematics an d physics took a sharp turn . For the

Conclusion 32

7

past 70 years, there ha s been little, if any, real communicatio n between mathematicians an d physicists . Mathematicians explore d th e topolog y of ^dimensiona l space , developin g ne w discipline s suc h a s algebrai c topology. Furtherin g th e work of Gauss, Riemann, and Poincare , mathematicians in the past century developed a n arsenal of abstract theorem s and corollarie s tha t hav e n o connectio n t o th e wea k or stron g forces . Physics, however, began t o probe th e real m o f the nuclea r force, using three-dimensional mathematic s known in th e nineteenth century . All thi s change d wit h th e introductio n o f th e tent h dimension . Rather abruptly, the arsena l of the pas t century of mathematics i s being incorporated int o th e worl d of physics. Enormously powerful theorem s in mathematics , lon g cherishe d onl y by mathematicians, no w tak e o n physical significance . A t last , i t seem s a s thoug h th e divergin g ga p between mathematics and physic s will be closed. I n fact, eve n the math ematicians have been startle d a t the floo d o f new mathematics that th e theory has introduced. Som e distinguished mathematicians , such as Isadore A . Singe r o f MIT , hav e state d tha t perhap s superstrin g theor y should be treate d a s a branch o f mathematics, independent o f whether it is physically relevant. No one ha s the slightes t inkling why mathematics and physic s are so intertwined. The physicist Paul A. M. Dirac, one o f the founders of quantum theory, stated that "mathematics can lead us in a direction we would not tak e if we only followed up physica l ideas by themselves."7 Alfred Nort h Whitehead, one o f the greates t mathematician s o f the past century, once sai d tha t mathematics , at the deepes t level , is inseparable from physics at the deepest level . However, the precise reason for the miraculou s convergence seem s totall y obscure . N o on e ha s even a reasonable theor y to explai n why the tw o disciplines should shar e con cepts. It i s often sai d tha t "mathematic s i s the languag e o f physics. " Fo r example, Galileo once said, "No on e will be able to read th e great book of the Univers e if he doe s not understan d it s language, which is that of mathematics."8 But thi s begs th e questio n o f why. Furthermore, math ematicians would be insulted to think that their entire discipline is being reduced t o mere semantics . Einstein, notin g thi s relationship, remarke d tha t pur e mathematic s might be on e avenu e to solve the mysterie s of physics: "It i s my conviction that pure mathematical constructio n enables us to discover the concepts and th e laws connecting them, which gives us the key to the understanding of nature. . .. In a certain sense , therefore, I hold i t true tha t pure though t ca n grasp reality , as the ancient s dreamed."9 Heisenberg

328 M A S T E R

S OF H Y P E R S P A C E

echoed thi s belief : "I f natur e lead s u s t o mathematica l form s o f grea t simplicity an d beaut y . . . tha t n o on e ha s previousl y encountered, w e cannot hel p thinkin g tha t the y ar e 'true, ' tha t the y revea l a genuin e feature o f nature." Nobel laureat e Eugen e Wigne r onc e even penned a n essa y with th e candid titl e "The Unreasonabl e Effectivenes s o f Mathematics in the Natural Sciences. "

Physical Principle s versus Logica l Structure s

Over th e years , I hav e observe d tha t mathematic s an d physic s hav e obeyed a certai n dialectica l relationship . Physic s is not just a n aimless , random sequenc e o f Feynma n diagram s an d symmetries , and mathe matics is not just a set o f mess y equations, bu t rathe r physic s and math ematics obe y a definite symbiotic relationship . Physics, I believe, i s ultimately based o n a small set o f physical principles. These principles ca n usuall y be expresse d i n plai n Englis h without reference t o mathematics . Fro m th e Copernica n theory , t o Newton' s laws o f motion, an d eve n Einstein' s relativity , the basi c physica l principles can be expressed in just a few sentences, largel y independent o f any mathematics. Remarkably , only a handful of fundamental physica l principles are sufficien t t o summarize most o f modern physics . Mathematics, b y contrast, i s the se t o f al l possibl e self-consistent structures, an d ther e ar e vastl y man y mor e logica l structure s tha n physica l principles. The hallmar k o f any mathematical syste m (for example, arithmetic, algebra, o r geometry) i s that it s axioms and theorem s ar e consis tent with one another . Mathematician s are mainly concerned tha t thes e systems neve r resul t i n a contradiction , an d ar e les s intereste d i n dis cussing the relative merits of one system over another. Any self-consistent structure, o f which there are many , is worthy of study. As a result, mathematicians ar e muc h mor e fragmente d tha n physicists ; mathematicians in one area usually work in isolation from mathematicians in other areas . The relationshi p between physic s (based on physical principles) an d mathematics (base d o n self-consisten t structures) i s no w evident : T o solve a physica l principle , physicist s ma y requir e man y self-consistent structures. Thu s physics automatically unites many diverse branches o f mathematics. Viewed i n thi s light , w e can understan d ho w the grea t idea s i n theoretical physic s evolved. For example, both mathematician s and physicists claim Isaa c Newto n a s one o f the giant s of their respectiv e profes sions. However , Newto n did no t begi n th e stud y of gravitation startin g with mathematics . B y analyzing the motio n o f falling bodies , h e wa s led

Conclusion 32

9

to believ e tha t th e moo n wa s continually fallin g toward th e earth , bu t never collided with it because th e earth curved beneath it ; the curvature of the eart h compensate d fo r th e fallin g of the moon . H e was therefore led t o postulate a physical principle: th e universa l law of gravitation. However, because h e wa s at a loss t o solve the equation s fo r gravity , Newton began a 30-year quest to construct fro m scratc h a mathematics powerful enoug h t o calculate them. In the process, he discovere d many self-consistent structures, which are collectively called calculus. From this viewpoint, the physical principle came first (law of gravitation), and the n came th e constructio n o f diverse self-consistent structures necessar y to solve i t (suc h a s analyti c geometry, differentia l equations , derivatives , and integrals). In the process, the physical principle united thes e diverse self-consistent structure s into a coherent bod y o f mathematics (th e cal culus). The sam e relationshi p applie s t o Einstein' s theory o f relativity. Einstein began wit h physical principles (suc h as the constanc y of the spee d of ligh t an d th e equivalenc e principl e fo r gravitation ) an d then , b y searching through th e mathematical literature, found the self-consisten t structures (Li e groups, Riemann' s tenso r calculus , differential geome try) tha t allowe d hi m t o solv e these principles . I n th e process , Einstein discovered ho w to link thes e branche s o f mathematics int o a coherent picture. String theory also demonstrate s thi s pattern, bu t i n a startlingly different fashion. Because of its mathematical complexity, string theory has linked vastl y differen t branche s o f mathematic s (suc h a s Riemann sur faces, Kac-Mood y algebras , supe r Li e algebras, finit e groups , modula r functions, an d algebrai c topology ) i n a way that has surprised the mathematicians. A s with other physical theories, i t automaticall y reveals th e relationship amon g man y different self-consistent structures. However, the underlyin g physical principle behind string theory is unknown. Physicists hope that once thi s principle i s revealed, new branches o f mathematics will be discovered i n the process . I n othe r words , the reason why the strin g theory cannot b e solved is that twenty-first-centur y mathemat ics has not ye t been discovered . One consequenc e o f this formulation is that a physical principle that unites man y smalle r physica l theorie s mus t automaticall y unit e man y seemingly unrelate d branche s o f mathematics . Thi s i s precisel y what string theory accomplishes. In fact, o f all physical theories, string theory unites by far the largest number of branches of mathematics into a single coherent picture . Perhaps one of the by-products of the physicists' quest for unificatio n will be th e unificatio n of mathematics as well. Of course , th e se t o f logically consistent mathematical structure s i s

330 M A S T E R

S O F HYPERSPACE

many time s larger tha n th e se t of physical principles. Therefore, som e mathematical structures , such a s numbe r theor y (whic h some mathe maticians clai m t o b e th e pures t branc h o f mathematics) , hav e neve r been incorporated int o an y physical theory. Som e argue tha t this situation ma y always exist : Perhap s th e huma n min d wil l alway s b e abl e t o conceive o f logicall y consisten t structure s tha t canno t b e expresse d through an y physica l principle . However , ther e ar e indication s that string theor y may soon incorporat e numbe r theor y into its structure as well. Science and Religio n

Because th e hyperspac e theor y ha s opene d u p new , profoun d link s between physic s and abstrac t mathematics, som e peopl e hav e accused scientists of creating a new theology based o n mathematics ; that is , we have rejecte d th e mytholog y o f religion , onl y t o embrac e a n eve n stranger religio n base d o n curve d space—time, particle symmetries, and cosmic expansions . Whil e priests ma y chant incantation s in Lati n tha t hardly anyon e understands , physicist s chant arcan e superstrin g equa tions that even fewer understand . Th e "faith " i n a n all-powerfu l Go d is now replaced b y "faith" i n quantum theory and general relativity . When scientists protest tha t ou r mathematica l incantations can be checked in the laboratory, th e response is that Creation canno t be measured i n the laboratory, an d henc e thes e abstrac t theorie s lik e th e superstrin g ca n never be tested . This debate i s not new . Historically, scientists have often been asked to debat e th e law s o f natur e wit h theologians . Fo r example , th e grea t British biologist Thomas Huxle y was the foremos t defender of Darwin's theory o f natura l selectio n agains t th e church' s criticism s in th e lat e nineteenth century . Similarly , quantu m physicist s hav e appeare d o n radio debate s wit h representatives o f th e Catholi c Churc h concernin g whether the Heisenber g Uncertaint y Principle negate s fre e will , a question tha t ma y determine whethe r our soul s will enter heave n or hell. But scientist s usually are reluctan t t o engage i n theologica l debate s about Go d an d Creation . On e problem , I hav e found , i s that "God " means many things to many people, an d th e us e of loaded word s full of unspoken, hidde n symbolis m only clouds the issue. To clarif y thi s problem somewhat , I hav e foun d i t usefu l t o distinguis h carefull y betwee n two types of meanings for th e wor d God. It i s sometimes helpfu l t o differentiate betwee n th e Go d of Miracles and th e Go d of Order.

Conclusion 33

1

When scientist s us e th e wor d God, they usuall y mean th e Go d o f Order. For example, on e of the most important revelation s in Einstein's early childhood too k place when he rea d hi s first books on science . He immediately realize d tha t mos t of what he ha d bee n taugh t abou t reli gion coul d no t possibl y be true . Throughou t hi s career , however , h e clung t o th e belie f tha t a mysterious , divine Orde r existe d i n th e uni verse. His life's calling , h e woul d say, was to ferret ou t hi s thoughts, to determine whethe r he had an y choice i n creating th e universe. Einstein repeatedly referre d t o thi s God in hi s writings, fondly callin g him "th e Old Man. " Whe n stumpe d wit h a n intractabl e mathematica l problem , he woul d ofte n say, "God i s subtle, bu t no t malicious. " Mos t scientists, it is safe t o say , believe tha t there i s some for m o f cosmic Order i n th e universe. However, to th e nonscientist , the wor d Go d almost universally refers t o th e Go d o f Miracles, and thi s is the sourc e o f miscommunication between scientists and nonscientists. The God of Miracles intervenes in ou r affairs , perform s miracles , destroy s wicked cities, smites enem y armies, drowns the Pharaoh's troops , an d avenges the pure and noble . If scientist s and nonscientist s fai l t o communicat e wit h each othe r over religiou s questions, i t i s because the y are talkin g past eac h other , referring t o entirel y differen t Gods . Thi s i s because th e foundatio n of science is based o n observin g reproducible events , but miracles , by definition, are not reproducible . The y happen only once i n a lifetime, if at all. Therefore, th e Go d o f Miracles is, in som e sense , beyon d wha t we know a s science. Thi s i s not t o sa y that miracle s canno t happen , onl y that they are outsid e what is commonly called science. Biologist Edwar d O . Wilson of Harvard Universit y has puzzle d ove r this questio n an d aske d whethe r ther e i s an y scientifi c reaso n wh y humans clin g s o fiercel y t o thei r religion . Eve n traine d scientists , h e found, wh o ar e usuall y perfectl y rational abou t thei r scientifi c specialization, laps e int o irrationa l argument s t o defen d thei r religion . Fur thermore, he observes , religion ha s been use d historicall y as a cover to wage hideou s war s and perfor m unspeakabl e atrocitie s against infidel s and heathens . Th e shee r ferocit y o f religious or hol y wars, in fact, rival s the wors t crime tha t an y human ha s ever committed against any other. Religion, note s Wilson , is universally found i n ever y human cultur e ever studie d o n earth . Anthropologist s hav e foun d tha t al l primitive tribes have an "origin " myth that explains where they came from. Fur thermore, thi s mythology sharply separates "us" fro m "them," provides a cohesive (an d often irrational) forc e that preserves the tribe , and sup presses divisive criticism of the leader . This is not a n aberration, but th e norm of human society. Religion ,

332 M A S T E R

S OF HYPERSPAC E

Wilson theorizes , i s so prevalen t becaus e i t provide d a definit e evolutionary advantage fo r thos e earl y humans wh o adopted it . Wilson notes that animals that hunt in packs obey the leader becaus e a pecking order based o n strengt h an d dominanc e ha s been established . Bu t roughly 1 million year s ago, whe n ou r apelik e ancestor s graduall y became mor e intelligent, individual s could rationall y begin t o questio n th e powe r of their leader . Intelligence , by its very nature, questions authority by reason, an d henc e coul d b e a dangerous , dissipativ e forc e o n th e tribe . Unless ther e was a force t o counterac t thi s spreading chaos , intelligent individuals would leave the tribe , the trib e would fall apart , and al l individuals would eventually die. Thus, according to Wilson, a selection pressure was placed o n intelligen t apes t o suspend reaso n an d blindl y obey the leade r an d hi s myths , since doin g otherwis e woul d challeng e th e tribe's cohesion . Surviva l favored th e intelligen t ape wh o coul d reaso n rationally about tool s and foo d gathering , but als o favored the one who could suspen d tha t reaso n whe n i t threatene d th e tribe' s integrity . A mythology was needed to define and preserv e th e tribe . To Wilson, religion was a very powerful, life-preservin g force for apes gradually becomin g mor e intelligent , an d forme d a "glue " tha t hel d them together . I f correct, thi s theory would explain why so many religions rely on "faith " ove r commo n sense , and wh y the floc k i s asked t o suspend reason . I t would also help t o explai n th e inhuma n ferocit y o f religious wars, and wh y the Go d o f Miracle s always seem s t o favo r th e victor in a bloody war. The Go d of Miracles has one powerfu l advantag e over th e Go d o f Order. The Go d o f Miracles explains the mytholog y of our purpose i n the universe; on thi s question, the God of Order is silent.

Our Rol e i n Natur e Although th e Go d o f Orde r canno t giv e humanity a shared destin y or purpose, wha t I fin d personall y most astonishin g abou t thi s discussion is that we humans, who are just beginning ou r ascen t up th e technolog ical scale, should be capabl e o f making such audacious claims concerning th e origi n an d fat e o f the universe. Technologically, w e are just beginnin g t o leav e th e earth' s gravita tional pull; we have only begun t o send crud e probe s t o the oute r plan ets. Yet imprisoned o n ou r smal l planet, with onl y our mind s and a few instruments, we have been abl e to decipher th e law s that govern matte r billions of light-years away. With infinitesimally smal l resources, without even leavin g th e sola r system , w e hav e bee n abl e t o determin e wha t

Conclusion 33

3

happens deep inside the nuclear furnaces of a star or inside the nucleus itself. According t o evolution , w e ar e intelligen t ape s wh o hav e onl y recently left th e trees , living on th e thir d planet from a minor star, in a minor spira l ar m o f a minor galaxy , in a minor grou p o f galaxies near the Virgo supercluster. If the inflatio n theor y is correct, then ou r entir e visible universe is but a n infinitesima l bubbl e in a much large r cosmos . Even then , give n the almos t insignificant role tha t we play in the large r universe, i t seem s amazin g that w e should b e capabl e o f makin g th e claim t o have discovered th e theor y of everything. Nobel laureat e Isido r I . Rab i was once aske d wha t event i n hi s lif e first set him o n th e lon g journey to discover th e secret s of nature. H e replied tha t it was when he checked out some books on the planets from the library . What fascinated him was that th e huma n min d is capable of knowing such cosmic truths. The planets and the stars are so much larger than th e earth , s o muc h mor e distan t tha n anythin g eve r visite d by humans, yet the huma n min d i s able t o understand them . Physicist Hein z Pagel s recounte d hi s pivota l experience when , as a child, h e visited the Hayde n Planetarium in Ne w York. He recalled, The drama and power of the dynamic universe overwhelmed me. I learned that single galaxies contain more stars than all the human beings who have ever lived . . . . The realit y of the immensit y and duratio n o f th e universe caused a kind of 'existential shock' that shook the foundations of my being. Everything that I had experience d or know n seemed insignifican t placed in tha t vast ocean o f existence. 10

Instead o f being overwhelme d by the universe , I think that perhap s one of the deepest experience s a scientist can have, almost approaching a religious awakening, is to realize that we are childre n o f the stars , and that our minds are capable of understanding the universal laws that they obey. The atom s within our bodie s wer e forged on th e anvi l o f nucleosynthesis withi n an explodin g sta r aeon s befor e th e birt h o f th e sola r system. Our atom s are olde r tha n th e mountains . We are literally made of sta r dust . No w these atoms , in turn , hav e coalesced int o intelligent beings capabl e o f understandin g th e universa l law s governin g tha t event. What I find fascinatin g is that the law s of physics that we have found on ou r tiny , insignificant planet ar e th e sam e a s the law s foun d every where els e i n th e universe , yet thes e law s were discovered withou t ou r ever having left th e earth . Without mighty starships or dimensional win-

334

M A S T E R S OF H Y P E R S P A C E

dows, we have been abl e t o determine th e chemica l nature o f the star s and decod e th e nuclear processes tha t tak e place deep i n their cores . Finally, i f ten-dimensional superstring theor y i s correct, the n a civi lization thriving on the farthest star will discover precisely the same truth about ou r universe. It, too, will wonder about th e relatio n between marble an d wood , an d com e t o th e conclusio n tha t th e traditiona l threedimensional world i s "too small " t o accommodate the known force s in its world. Our curiosit y is part of the natural order. Perhaps we as humans want to understand th e univers e in the sam e way that a bird wants to sing. As the grea t seventeenth-centur y astronomer Johanne s Keple r onc e said , "We d o no t as k for wha t usefu l purpos e th e bird s do sing , for son g is their pleasur e sinc e the y were create d fo r singing . Similarly, we ought not t o as k why the huma n min d trouble s t o fatho m th e secret s of th e heavens." Or , a s the biologis t Thoma s H . Huxle y said i n 1863 , "Th e question o f all questions for humanity , the proble m whic h lie s behind all others an d i s more interestin g than an y of them i s that o f the deter mination o f man's plac e in Nature and hi s relation t o the Cosmos. " Cosmologist Stephe n Hawking , who has spoken of solving the problem o f unification within this century, has written eloquently about th e need t o explai n t o th e wides t possible audienc e th e essentia l physical picture underlying physics: [If] w e do discover a. complete theory , it should i n time be understandabl e in broad principle b y everyone, not just a lew scientists. Then we shall all, philosophers, scientists , and jus t ordinary people , be abl e t o tak e part in the discussio n o f th e questio n o f why it is that w e and th e univers e exist. If we fin d th e answe r t o that , it . would be th e ultimat e triumph o f human reason—for the n w e would kno w the min d o f God. "

On a cosmic scale, we are stil l awakening to the large r world around us. Yet the power of even our limited intellect is such that we can abstract the deepes t secret s of nature. Does this give meaning o r purpose t o life ? Some peopl e see k meanin g i n lif e throug h persona l gain , through personal relationships , o r throug h persona l experiences . However , it seems to me tha t being blessed with the intellec t to divine the ultimate secrets of nature give s meaning enough t o life .

Notes

Preface 1. Th e subjec t is so new that ther e is yet no universall y accepted ter m use d by theoretica l physicist s when referrin g t o higher-dimensiona l theories . Tech nically speaking, when physicist s address th e theory , they refer t o a specific theory, suc h as Kaluza-Klein theory, supergravity, or superstring, although hyperspace is the ter m popularly used when referring to higher dimensions, and hyper- is the correct scientifi c prefi x fo r higher-dimensiona l geometri c objects . I hav e adhered t o popula r custo m an d use d th e wor d hyperspace t o refe r t o highe r dimensions.

Chapter 1 1. Hein z Pagels, Perfect Symmetry: Th e Search far th e Beginning of rime (New York: Bantam, 1985) , 324 . 2. Pete r Freund, intervie w with author, 1990 . 3. Quote d i n Abraham Pais , Subtle h the Lord: Th e Science and th e Life o f Albert Einstein (Oxford : Oxford Universit y Press, 1982), 235. 4. Thi s incredibl y small distanc e wil l continuall y reappear throughou t thi s book. I t i s th e fundamenta l lengt h scal e tha t typifie s an y quantu m theor y o f gravity. The reaso n fo r thi s is quite simple. In an y theory of gravity, the strengt h of the gravitationa l forc e i s measured b y Newton's constant. However, physicists use a simplified set o f units where th e spee d o f light c is set equa l t o one . Thi s means tha t 1 secon d i s equivalen t t o 186,00 0 miles . Also , Planck' s constan t divided by 2ir is also set equal to one, which sets a numerical relationship between seconds an d erg s o f energy. I n thes e strang e bu t convenien t units, everything, including Newton's constant, ca n be reduced t o centimeters. Whe n we calculate the lengt h associate d wit h Newton's constant , i t i s precisely the Planc k length, or 10~ 33 centimeter, or 10 19 billion electron volts . Thus all quantum gravitational

335

336 Notes effects ar e measure d i n term s of this tiny distance. I n particular, th e siz e of these unseen highe r dimension s i s the Planc k length . 5. Lind a Dalrymple Henderson, Th e Fourth Dimension and Non-Euclidean Geometry i n Modern Ar t (Princeton , N.J. : Princeton Universit y Press, 1983), xix.

Chapter 2 1. E . T. Bell, Men o f Mathematics (Ne w York: Simon an d Schuster , 1937) , 484 . 2. Ibid. , 487 . Thi s inciden t mos t likel y sparke d Riemann' s earl y interest i n number theory . Years later, he would make a famous speculation about a certain formula involvin g the zet a function in numbe r theory . After 10 0 years of grap pling wit h "Riemann' s hypothesis, " th e world' s greates t mathematician s have failed t o offer an y proof. Ou r mos t advanced computer s have failed to give us a clue, and Riemann' s hypothesi s has now gone dow n in history as one of the most famous unprove n theorem s i n numbe r theory , perhap s i n al l o f mathematics. Bell notes, "Whoeve r prove s o r disprove s it will cover himself with glory" (ibid. , 488). 3. John Wallis, Der Barycentrische Calcul (Leipzig , 1827), 184 . 4. Althoug h Rieman n i s credited a s havin g bee n th e drivin g creative force who finally shattered th e confines of Euclidean geometry, by rights, the man who should hav e discovered th e geometr y of higher dimensions was Riemann's aging mentor, Gaus s himself. In 1817 , almos t a decade befor e Riemann' s birth, Gaus s privatel y expressed his deep frustration with Euclidean geometry. In a prophetic lette r to his friend the astronome r Heinric h Olbers , h e clearl y stated tha t Euclidea n geometr y is mathematically incomplete . In 1869 , mathematicia n James J. Sylvester recorded tha t Gaus s had seriously considered th e possibilit y of higher-dimensiona l spaces . Gaus s imagine d th e properties o f beings, which he calle d "bookworms, " tha t coul d liv e entirel y on two-dimensional sheet s o f paper . H e the n generalize d thi s concept t o include "beings capabl e o f realizing space o f four or a greater numbe r o f dimensions " (quoted i n Lind a Dalrymple Henderson, The. Fourth Dimension and Non-Euclidean Geometry i n Modern Ar t [Princeton , N.J.: Princeton Universit y Press, 1983] , 19) . But i f Gauss was 40 years ahead o f anyone else i n formulatin g the theor y of higher dimensions, then why did h e mis s this historic opportunity to shatter th e bonds o f three-dimensional Euclidea n geometry? Historians hav e noted Gauss' s tendency to be conservative in his work, his politics, and hi s personal life . In fact , he neve r onc e lef t Germany , and spen t almos t hi s entire lif e i n on e city . Thi s also affected his professional life . In a revealing letter writte n in 1829 , Gaus s confessed t o hi s friend Friedric h Bessel that he would never publish his work on non-Euclidea n geometry for fea r of the controvers y it would raise among th e "Boeotians. " Mathematician Morris Kline wrote , "[Gauss ] sai d i n a lette r t o Besse l o f Januar y 27 , 1829 , tha t h e

Notes 33

7

probably would neve r publish hi s findings i n thi s subject because h e feare d rid icule, or as he put it, he feared the clamor of the Boeotians, a figurative reference to a dull-witted Greek tribe" (Mathematics an d the Physical World [Ne w York : Crowell, 1959] , 449). Gauss was so intimidated b y the ol d guard , th e narrow-minde d "Boeotians" who believed in the sacred natur e of three dimensions, that he kept secret som e o f his finest work. In 1869 , Sylvester , in a n intervie w with Gauss' s biographe r Sartoriou s vo n Waltershausen, wrot e tha t "thi s grea t ma n use d t o sa y that h e ha d lai d asid e several questions which he ha d treate d analytically , and hope d t o apply to them geometrical method s in a future state of existence, when his conceptions of space should hav e become amplified and extended ; for as we can conceive beings (lik e infinitely attenuated book-worm s in an infinitel y thin sheet of paper) whic h possess only the notio n of space of two dimensions, so we may imagine beings capable o f realizin g space o f four o r a greate r numbe r o f dimensions " (quote d i n Henderson, Fourth Dimension and Non-Euclidean Geometry i n Modern Art, 19) . Gauss wrote t o Olbers, " I a m becoming mor e an d mor e convince d tha t th e (physical) necessit y of our (Euclidean ) geometr y cannot b e proved , at least no t by human reaso n no r fo r human reason . Perhap s i n another lif e we will be able to obtain insight into the nature of space, which is now unattainable. Until then, we must place geometr y no t i n th e sam e clas s with arithmetic , which i s purely a priori, but wit h mechanics" (quote d i n Morri s Kline, Mathematical Thought from Ancient t o Modem Times [Ne w York: Oxford Universit y Press, 1972] , 872). In fact, Gauss was so suspicious of Euclidean geometry that he even conducted an ingeniou s experiment t o test it. He an d hi s assistants scaled thre e mountai n peaks: Rocken , Hohehagen , an d Inselsberg . Fro m eac h mountai n peak , th e other tw o peak s wer e clearl y visible . B y drawing a triangl e betwee n th e thre e peaks, Gauss was able to experimentally measure the interior angles. If Euclidean geometry is correct, the n th e angl e shoul d hav e summed t o 18 0 degrees. T o his disappointment, h e foun d tha t th e su m was exactly 180 degrees (plu s or minus 15 minutes). The crudenes s o f his measuring equipment di d no t allo w him t o conclusively show that Euclid was wrong. (Today, we realize that this experimen t would hav e t o b e performe d betwee n thre e differen t sta r system s t o detec t a sizable deviation from Euclid's result.) We shoul d also poin t ou t tha t th e mathematician s Nikolau s 1 . Lobachevski and Jano s Bolya i independentl y discovere d th e non-Euclidea n mathematic s defined o n curve d surfaces . However, their construction wa s limited to the usual lower dimensions. 5. Quote d i n Bell , Me n o f Mathematics, 497. 6. Th e Britis h mathematicia n Willia m Clifford , wh o translate d Riemann' s famous speec h fo r Nature in 1873 , amplifie d many of Riemann' s semina l ideas and was perhaps th e first to expand on Riemann's idea that the bending of space is responsible fo r th e forc e o f electricit y and magnetism , thu s crystallizin g Riemann's work. Clifford speculate d tha t th e tw o mysterious discoveries in mathe matics (higher-dimensiona l spaces ) an d physic s (electricity and magnetism ) ar e

338 Notes really th e sam e thing , that th e forc e o f electricit y and magnetis m i s caused b y the bendin g o f higher-dimensional space. This is the first time tha t anyone had speculated that a "force" is nothing bu t the bendin g of space itself , precedin g Einstei n by 50 years. Clifford' s ide a tha t electromagnetism wa s caused b y vibrations i n th e fourt h dimensio n als o pre ceded th e wor k o f Theodr Kaluza , wh o woul d also attemp t t o explai n electro magnetism with a higher dimension. Clifford an d Rieman n thus anticipated th e discoveries of the pioneer s o f the twentiet h eentury, that the meaning of higherdimensional space i s i n it s abilit y t o giv e a simpl e an d elegan t descriptio n o f forces. Fo r th e first time, someone correctl y isolated th e tru e physica l meaning of higher dimensions, that a theory about space actually gives us a unifying picture oi forces. These prophetic view s were recorded by mathematician James Sylvester, who wrote in 1869 , "Mr. W. K. Clifford ha s indulged i n some remarkable speculations as t o th e possibilit y of ou r bein g abl e t o infer , fro m certai n unexplaine d phe nomena o f light and magnetism , the fac t o f our leve l space of three dimensions being i n th e ac t o f undergoin g i n spac e o f fou r dimension s . . . a distortio n analogous t o th e rumplin g of a page" (quote d i n Henderson , Fourth Dimension and Non-Kudidean Geometry i n Modem Art, 19) . In 1870 , in a paper with the intriguing title "On th e Space-Theory of Matter," he say s explicitl y tha t "thi s variatio n o f th e curvatur e o f spac e i s what reall y happens i n tha t phenomeno n whic h we call th e motion o f matter, whether pon derable o r ethereal" (Willia m Clifford, "On th e Space-Theor y o f Matter," Proceedings-of the Cambridge Philosophical Society 2 [1876] : 157-158). 7. Mor e precisely , in N dimensions the Rieman n metric tensor g^ i s an ./V X TVmatrix, which determines th e distance between two points, such that the infin itesimal distance betwee n tw o points i s given b y d.f = X/ix f g ^ dx". I n th e limi t of flat space, the Rieman n metric tensor becomes diagonal, that is, g^ = 8^, and hence the formalism reduces back to the Pythagorean Theorem i n JVdimensions. The deviatio n o f th e metri c tenso r fro m 8^,, , roughl y speaking, measure s th e deviation of the spac e from flat space. From th e metri c tensor, we can construc t the Rieman n curvatur e tensor , represente d b y fl p(tl,CI. The curvatur e of space at any given point can be measured b y drawing a circle at tha t poin t an d measurin g th e are a insid e that circle. In fla t two-dimensional space, th e are a insid e the circl e is irr2. However, if the curvatur e i s positive, as in a sphere , th e are a i s less than TT? 2. I f th e curvatur e is negative, as in a saddl e o r trumpet, the are a i s greater tha n irr 2. Strictly speaking , by thi s convention , the curvatur e o f a crumple d shee t o f paper i s zero. This is because th e area s o f circles drawn o n thi s crumpled shee t of paper stil l equal irr 2. In Riemann's example o f force created by the crumplin g of a sheet of paper, we implicitly assume that the paper is distorted and stretche d as well a s folded, so that th e curvatur e is nonzero. 8. Quote d in Bell , Me n o f Mathematics, 501 . 9. Ibid. , 14.

Notes 33

9

10. Ibid . 11. I n 1917 , physicist Paul F>hrenfest, a friend of Einstein, wrote a paper entitled "I n Wha t Way Does It Become Manifes t in the Fundamenta l Law s of Physics that Spac e ha s Three Dimensions? " Ehrenfes t asked himsel f whether th e star s and planet s are possible in higher dimensions. For example, th e light of a candle gets dimmer a s we move farthe r away from it . Similarly, the gravitationa l pull of a star gets weaker as we go farther away. According to Newton, gravity gets weaker by an invers e square law . If we double th e distanc e awa y from a candl e o r star , the ligh t or gravitationa l pul l gets fou r time s weaker. I f we triple the distance , it gets nine time s weaker. If space were four dimensional , then candleligh t or gravit y would get weaker much mor e rapidly, as the invers e cube . Doublin g the distanc e from a candle o r star woul d weaken th e candleligh t o r gravit y by a factor of eight. Can sola r system s exist i n suc h a four-dimensiona l world? In principle , yes, but the planets' orbits would not be stable. The slightest vibration would collapse the orbit s of the planets. Ove r time , all the planet s would wobble away from thei r usual orbit s an d plung e int o th e sun . Similarly, the sun would no t be able to exist i n higher dimensions. The forc e of gravity tends to crush th e sun . It balances ou t th e forc e of fusion, which tends to blow the su n apart . Thu s th e su n i s a delicate balancing ac t between nuclea r forces tha t woul d caus e i t t o explod e an d gravitationa l force s tha t woul d con dense i t down t o a point. I n a higher-dimensional universe , this delicate balanc e would b e disrupted , an d star s might spontaneously collapse . 12. Henderson , Fourth Dimension and Non-EucKdean Geometry i n Modern Art, 22.

13. Zollne r ha d bee n converte d t o spiritualis m i n 187 5 whe n h e visite d th e laboratory o f Crookes , th e discovere r o f th e elemen t thalium , invento r o f th e cathode ray tube, an d edito r of the learne d Quarterly Journal of Science. Crookes's cathode ra y tub e revolutionize d science ; anyon e wh o watche s television , use s a compute r monitor , play s a vide o game , o r ha s bee n x-raye d owe s a deb t t o Crookes's famou s invention. Crookes, i n turn , wa s no crank . I n fact , h e wa s a lio n o f Britis h scientifi c society, wit h a wal l ful l o f professiona l honors . H e wa s knighte d i n 189 7 an d received th e Order of Merit in 1910 . His deep interest in spiritualism was sparked by th e tragi c deat h o f hi s brothe r Phili p o f yellow fever i n 1867 . H e becam e a prominent membe r (an d late r president ) o f the Societ y for Psychica l Research, which include d a n astonishin g number o f important scientist s in th e lat e nine teenth century . 14. Quote d in Rudy Rucker, Th e Fourth Dimension (Boston: Houghton Mifflin , 1984), 54. 15. T o imagin e ho w knot s ca n b e unravele d i n dimension s beyon d three , imagine two rings that are intertwined. Now take a two-dimensional cross section of this configuration, such tha t on e rin g lie s on thi s plane whil e the othe r rin g becomes a poin t (becaus e i t lie s perpendicula r t o th e plane) . W e no w hav e a point inside a circle. In highe r dimensions , w e have th e freedo m o f moving this

340 Notes dot completel y outsid e th e circl e without cutting any of the rings . This mean s that th e tw o rings hav e no w completely separated, a s desired. Thi s means tha t knots i n dimension s highe r tha n thre e ca n alway s b e untie d becaus e ther e i s "enough room. " But also notice tha t we cannot remov e th e do t fro m th e rin g if we are i n three-dimensional space, which is the reaso n wh y knots stay knotted only in the thir d dimension .

Chapter 3 1. A . T. Schofield wrote , "W e conclude , therefore , tha t a higher worl d than ours i s not onl y conceivably possible, but probable ; secondl y that suc h a world may be considere d as a world of four dimensions; and thirdly , that th e spiritual world agrees largel y in it s mysterious laws . . . with what by analogy would be th e laws, language, and claim s of a fourth dimension" (quote d i n Rudy Rucker, The Fourth Dimension [Boston : Houghton Mifflin , 1984] , 56) . 2. Arthu r Willink wrote, "When w e have recognize d th e existenc e of Space of Four Dimensions there is no greater strain called for in the recognitio n of the existence o f Spac e o f Fiv e Dimensions , an d s o o n u p t o Spac e o f a n infinit e number o f Dimensions" (quote d i n ibid. , 200). 3. H . G . Wells , Th e Time Machine: A n Invention (London : Heincmarm , 1895), 3 . 4. Lind a Dalrymple Henderson, Th e Fourth Dimension and Non-Euclidean Geometry i n Modem Art (Princeton , N.J. : Princeton Universit y Press, 1983), xxi. 5. Ibid . Accordin g t o Henderson , "[T]h c fourt h dimensio n attracte d th e notice o f such literar y figures a s H. G . Wells, Oscar Wilde, Joseph Conrad , Ford Madox Ford, Marce l Proust , and Gertrud e Stein . Among musicians , Alexander Scriabin, Edga r Varese , an d Georg e Anthei l wer e activel y concerne d wit h th e fourth dimension , and wer e encouraged t o make bold innovations in the nam e of a higher reality " (ibid. , xix—xx). 6. Lenin' s Materialism and Empiro-Criticismis important today because it deeply affected moder n Sovie t and Easter n Europea n science . For example , Lenin's celebrated phras e "th e inexhaustibilit y of the electron " signifie d th e dialectical notion tha t we find ne w sublayers and contradiction s whenever we probe deepl y into th e hear t o f matter . Fo r example , galaxie s are compose d o f smalle r star systems, which in tur n contai n planets, which are composed o f molecules, which are made of atoms, which contain electrons, which, in turn, are "inexhaustible." This is a variation of the "world s withi n worlds" theory . 7. Vladimi r Lenin, Materialism an d Empiro-Crittcism, i n Kar l Marx, Friedrich Engels, and Vladimi r Lenin, On Dialectical Materialism (Moscow : Progress, 1977) , 305-306. 8. Ibid . 9. Quote d in Rucker , Fourth Dimension, 64 . 10. Imagin e a Flatlander building a sequence o f six adjacent squares , i n the

Notes 34

1

shape o f a cross. To a Flatlander, th e square s ar e rigid . They cannot b e twisted or rotated along any of the sides connecting the squares. Now imagine, however, that we grab the square s and decid e t o fold u p th e serie s of squares, forming a cube. Th e joints connectin g th e squares , whic h were rigi d i n tw o dimensions, can b e easil y folded i n thre e dimensions . I n fact , th e foldin g operatio n ca n b e performed smoothly without a Flatlander even noticing that the folding is taking place. Now, if a Flatlander wer e inside the cube , he would notice a surprising thing. Each squar e lead s to another square . Ther e is no "outside " t o th e cube . Eac h time a Flatlander move s from on e squar e t o the next , he smoothl y (without his knowledge) bends 90 degrees in the third dimension and enters the next square. From th e outside , th e hous e i s just a n ordinar y square. However , to someon e entering th e square , h e woul d fin d a bizarre sequenc e o f squares, each squar e leading impossibly to the nex t square. To him, it would seem impossible that the interior o f a single square coul d hous e a series of six squares.

Chapter 4 1. Jaco b Bronowski, Th e Ascent o f Man (Boston : Little, Brown, 1974), 247 2. Quote d i n Abraha m Pais , Subtle Is th e Lord: Th e Science and th e Life o f Albert Einstein (Oxford : Oxford Universit y Press, 1982) , 131 . 3. Normally , it is absurd t o think that tw o people ca n each b e taller than th e other. However , in thi s situatio n w e hav e tw o people, eac h correctl y thinking that th e othe r has been compressed . Thi s is not a true contradictio n because it takes time in whic h t o perfor m a measurement , an d tim e a s well as space ha s been distorted . In particular , events that appear simultaneous in on e fram e are not simultaneou s when viewed in another frame . For example, let's say that people on the platform take out a ruler and, as the train passes by, drop the measurin g stick onto the platform . As the trai n goes by, they dro p th e tw o ends o f th e stic k s o tha t th e end s hi t th e platfor m simultaneously. I n thi s way , they can prov e tha t th e entir e lengt h o f th e compresse d train, from fron t to back, is only 1 foot long . Now consider th e sam e measuring process from th e poin t of view of the passengers on th e train . They think they are at rest and se e the compresse d subwa y station comin g towar d them , wit h compresse d peopl e abou t t o dro p a com pressed rule r onto the platform. At first it seems impossible that such a tiny ruler would be able to measure the entire length of the train . However, when the ruler is dropped, th e end s o f the rule r d o no t hit th e floo r simultaneously . One en d of the rule r hit s th e floo r jus t as the statio n goes b y the fron t en d o f th e train . Only when th e statio n ha s move d completel y by the lengt h o f the entir e trai n does the second en d o f the rule r finally hit th e floor . I n thi s way, the same ruler has measured th e entir e length o f the trai n in either frame. The essenc e o f thi s "paradox, " an d man y others tha t appea r i n relativit y

342 Notes theory, is thai th e measurin g proces s take s time, and tha t bot h space an d tim e become distorted i n differen t way s in differen t frames. 4. Maxwell' s equations look like this (w e set c = 1) :

The second and las t lines are actually vector equations representing three equa tions each. Therefore, ther e are eigh t equations in Maxwell's equations. We ca n rewrit e these equation s relativistically . I f we introduce th e Maxwel l tensor F vv = d^A f — drAl±, the n thes e equation s reduc e t o on e equation :

which i s the relativisti c version of Maxwell's equations. 5. Quote d i n Pais , Subtle Is the Lord, 239 . 6. Ibid. , 179 . 7. Einstein' s equations look lik e this:

where T^ i s th e energy-momentu m tenso r tha t measure s th e matter-energ y content, whil e JK^,, i s th e contracte d Riernar m curvatur e tensor . Thi s equatio n says tha t th e energy-momentu m tenso r determine s th e amoun t o f curvatur e present i n hyperspace . 8. Quote d i n Pais , Subtle Is the Lord, 212. 9. Quote d i n K . C . Cole , Sympathetic Vibrations: Reflections o n Physics a s a Wa y of Life (Ne w York: Bantam, 1985) , 29 . 10. A hyperspherc can be defined in much the same way as a circle or sphere . A circle is defined as the set of points that satisfy th e equatio n xf + f = f i n th e x-y plane . A sphere i s defined a s the se t o f points that satisf y x* + f + z 2 = r 1 in x-y-z space. A four-dimensional hypersphere is defined as the se t of points that satisfy y? + f + z? + u 2 = f i n x-y-z-u space . Thi s procedur e ca n easil y b e extended t o JV-dimensiona l space . 11. Quote d i n Abdus Salam, "Overview of Particle Physics," in Th e New Physics, ed. Pau l Davies (Cambridge: Cambridge Universit y Press, 1989) , 487 . 12. Theod r Kaluza, "Zu m Unitatsproblem der Physik, " Silzungsberichle Preussische Akademie de r Wissenschaften 9 6 (1921) : 69 . 13. I n 1914 , even befor e Einstei n propose d hi s theor y o f genera l relativity ,

Notes 34

3

physicist Gunna r Nordstro m trie d t o unif y electromagnetis m wit h gravit y b y introducing a five-dimensional Maxwell theory . If one examine s his theory, one finds that i t correctly contains Maxwell's theory of light in four dimensions, but it is a scalar theory of gravity, which is known to be incorrect. A s a consequence , Nordstrom's ideas were largely forgotten. In some sense, he published too soon. His pape r wa s written 1 year before Einstein' s theor y of gravit y was published, and henc e i t was impossible for hi m t o write down a five-dimensional Einsteintype theor y of gravity. Kaluza's theory , i n contras t t o Nordstrom's , bega n wit h a metri c tenso r g^ v defined i n five-dimensional space. Then Kaluz a identified g^ with th e Maxwel l tensor A^. The ol d four-dimensional Einstein metric was then identifie d by Kaluza's new metric only if u , and v di d no t equa l 5 . In thi s simple but elegan t way, both th e Einstei n field and th e Maxwel l field were place d insid e Kaluza' s five dimensional metri c tensor . Also, apparently Heinrich Mandel and Gusta v Mie proposed five-dimensional theories. Thu s th e fac t tha t higher dimension s were such a dominant aspec t of popular cultur e probably helped t o cross-pollinate the worl d o f physics. In thi s sense, th e wor k of Riemann was coming full circle . 14. Pete r Freund, inter-vie w with author , 1990 . 15. Ibid .

Chapter 5 1. Quote d i n K . C. Cole , Sympathetic Vibrations: Reflections o n Physics a s a Wa y of Life (Ne w York: Bantam, 1985) , 204. 2. Quote d in Nige l Calder, Th e Key to Ihe Universe (Ne w York: Penguin, 1977), 69. 3. Quote d i n R . P . Creas e an d C . C . Mann , Th e Second Creation (Ne w York: Macmillan, 1986) , 326 . 4. Ibid. , 293. 5. Willia m Blake, "Tyger ! Tyger ! burnin g bright, " fro m "Song s o f Experience," in Th e Poems o f William Blake, ed. W . B. Yeats (London: Routledge, 1905) . 6. Quote d in Hein z Pagels, Perfect Symmetry: Th e Search for th e Beginning of Time (New York : Bantam, 1985) , 177 . 7. Quote d in Cole , Sympathetic Vibrations, 229 . 8. Quote d in John Gribben, In Search o f Schrodinger's Cat (Ne w York: Bantam, 1984), 79.

Chapter 6 I. Quote d i n R . P . Creas e an d C . C . Mann , Th e Second Creation (Ne w York: Macmillan, 1986) , 411 .

344 Notes 15.

2. Quote d in Nige l Calder, Th e Key to the Universe (Ne w York: Penguin, 1977) ,

3. Quote d in Creas e an d Mann , Second Creation, 418. 4. Hein z Pagels, Perfect Symmetry: Th e Search for th e Beginning ofFime (Ne w York: Bantam, 1985) , 327 . 5. Quote d i n Creas e an d Mann , Second Creation, 417 . 6. Pete r va n Nieuwenhuizen , "Supergravity," in Supersymmelry an d Supergravity, ed . M.Jaco b (Amsterdam : North Holland, 1986) , 794 . 7. Quote d i n Creas e an d Mann , Second Creation, 419 .

Chapter 7 1. Quote d i n K. C. Cole, "A Theory o f Everything," New York Times Magazine, 18 October 1987 , 20 . 2. Joh n Horgan, "Th e Pie d Piper of Superstrings," Scientific American, November 1991 , 42 , 44. 3. Quote d in Cole , "Theor y o f Everything," 25. 4. Edwar d Witten, Interview , in Superstrings: A Theory o f Everything 1? ed . Pau l Davies and J. Brow n (Cambridge : Cambridge Universit y Press, 1988) , 90-91. 5. Davi d Gross, Interview, in Superstrings, ed . Davie s and Brown , 150. 6. Witten , Interview, in Superslrings, ed . Davie s and Brown , 95. Witten stresse s tha t Einstei n was led t o postulat e th e genera l theor y of rela tivity starting from a physical principle, the equivalenc e principle (tha t the gravitational mass and inertia l mass of an objec t are th e same , so that all bodies, n o matter ho w large, fal l a t the sam e rat e o n th e earth) . However, the counterpar t of the equivalenc e principle for string theory has not ye t been found. As Witten point s out , "It' s bee n clea r tha t string theor y does, i n fact , giv e a logically consisten t framework , encompassin g bot h gravit y an d quantu m mechanics. At the sam e time, the conceptua l framewor k in which this should b e properly understood , analogou s t o th e principl e o f equivalenc e tha t Einstein found i n hi s theory of gravity, hasn't yet emerged" (ibid. , 97). This i s why, at present , Witte n i s formulating wha t are calle d topological field theories—that is , theorie s tha t ar e totall y independen t o f th e wa y we measur e distances. Th e hop e i s that thes e topologica l fiel d theorie s ma y correspond t o some "unbroke n phas e o f strin g theory"—tha t is , string theor y beyon d th e Planck length . 7. Gross , Interview , in Superstrings, ed . Davie s and Brown , 150. 8. Horgan , "Pie d Pipe r of Superstrings," 42 . 9. Le t us examine compactification in terms of the ful l heterotic string, which has two kinds of vibrations: one vibratin g in th e ful l 26-dimensiona l space-time, and th e othe r in th e usua l ten-dimensiona l space time . Since 26 — 10 = 16 , we now assum e tha t 1 6 o f th e 2 6 dimension s hav e curle d up—tha t is , "com -

Notes 34

5

pactified" int o som e manifold—leavin g u s with a ten-dimensiona l theory . Anyone walkin g along an y of these 1 6 directions wil l wind up precisel y at th e sam e spot. It was Peter Freun d wh o suggested tha t th e symmetry group of this 16-dimensional compactified spac e wa s the grou p E(8) X E(8). A quick check show s that this symmetry is vastly larger an d include s th e symmetr y group of th e Standar d Model, give n by SU(3) X SU(2 ) X TJ(1) . In summary , th e ke y relation i s 26 — 10 = 16 , which mean s tha t i f we compactify 1 6 of th e origina l 2 6 dimensions o f the heteroti c siring, w e are lef t with a 16-dimensiona l compac t spac e wit h a leftove r symmetr y called K(8 ) X E(8). However, in Kalu/.a-Klei n theory , when a particle i s forced t o liv e on a compac tified space , i t must necessaril y inherit th e symmetr y of that space . Thi s mean s that th e vibration s o f th e strin g mus t rearrang e themselve s accordin g t o th e symmetry group E(8) X E(8). As a result, we can conclud e tha t group theory reveals to us that this group is much large r tha n th e symmetr y group appearin g i n th e Standar d Model , an d can thu s includ e th e Standar d Mode l a s a smal l subse t o f th e ten-dimensiona l theory. 10. Althoug h th e supergravity theory is defined in 1 1 dimensions, the theor y is still to o smal l to accommodat e al l particle interactions. The larges t symmetry group for supergravity is O(8), which is too small to accommodate th e Standar d Model's symmetries . At first, it appears that the 11-dimensiona l supergravity has more dimensions, and henc e mor e symmetry, than th e ten-dimensiona l superstring . This is an illusion becaus e th e heteroti c strin g begins b y compactifying 26-dimensiona l spac e down t o ten-dimensiona l space , leavin g u s wit h 1 6 compactifie d dimensions , which yields the grou p E(8) X E(8). This is more than enoug h t o accommodat e the Standar d Model . 11. Witten , Interview, in Superstrings, ed . Davie s and Brown , 102 . 12. Not e tha t othe r alternative nonperturbativ e approaches t o string theory have bee n proposed , bu t the y ar e no t a s advanced a s string fiel d theory . Th e most ambitious i s "universal moduli space, " which tries to analyze the properties of string surfaces with an infinit e number of holes in them . (Unfortunately , no one know s how t o calculat e with this kind of surface.) Another i s the renormal ization grou p method , whic h ca n s o fa r reproduc e onl y surface s withou t an y holes (tree-typ e diagrams). There is also th e matri x models , which so far can b e defined onl y in tw o dimensions o r less . 13. T o understan d thi s mysteriou s factor o f two , consider a ligh t beam tha t has tw o physical modes of vibration. Polarized ligh t can vibrate , say, either hor izontally o r vertically . However , a rclativisti c Maxwel l fiel d A (i ha s fou r compo nents, wher e | x = 1,2,3,4 . We are allowe d t o subtrac t tw o of thes e fou r compo nents usin g th e gauge symmetr y of Maxwell' s equations. Sinc e 4 — 2 = 2 , th e original fou r Maxwel l field s hav e bee n reduce d b y two . Similarly , a relativistic string vibrates in 2 6 dimensions. However, tw o of these vibrator y modes ca n b e

346 Notes removed when we break th e symmetr y of the string , leaving us with 2 4 vibratory modes, whic h ar e th e one s tha t appear i n the Ramanuja n function. 14. Quote d i n Godfre y H. Hardy , Ramanujan (Cambridge : Cambridg e University Press, 1940), 3. 15. Quote d i n James Newman , Th e World o f Mathematics (Redmond , Wash.: Tempus Books , 1988) , 1 : 363 . 16. Hardy , Ramanujan, 9. 17. Ibid. , 10. 18. Ibid. , 11. 19. Ibid. , 12. 20. Jonatha n Borwei n an d Pete r Borwein , "Ramanuja n an d Pi, " Scientific American, February 1988 , 112 .

Chapter 8 1. Davi d Gross, Interview, in Superstrings: A Theory oj'Everything? ed. Pau l Davies and f . Brown (Cambridge : Cambridge Universit y Press, 1988) , 147 . 2. Sheldo n Glashow , Interactions (Ne w York: Warner, 1988) , 335 . 3. Ibid. , 333 . 4. Ibid. , 330 . 5. Steve n Weinberg , Dreams o f a Final Theory (Ne w York: Pantheon, 1992) , 218-219. 6. Quote d i n John D . Barrow and Fran k J. Tipler , Th e Anthropic Gosmological Principle (Oxford : Oxford Universit y Press, 1986), 327. 7. Quote d i n F . Wilczek an d B . Devine, Longing for th e Harmonies (Ne w York: Norton, 1988) , 65. 8. Joh n Updike, "Cosmic Gall, " in Telephone Poles and Other Poems (New York: Knopf, 1960) . 9. Quote d in K . C. Cole, "A Theory o f Everything," New York Times Magazine, 18 October 1987 , 28 . 10. Quote d i n Hein z Pagels, Perfect Symmetry: Th e Search for th e Beginning of Time (Ne w York: Bantam, 1985) , 11 . 11. Quote d in K . C. Cole , Sympathetic Vibrations: Reflections o n Physics a s a Way of Life (Ne w York: Bantam, 1985) , 225.

Chapter 9 1. Quote d i n E . Harrison, Masks o f the Universe (Ne w York: Macmillan, 1985), 211. 2. Quote d i n Core y S . Powell , "The Golde n Ag e o f Cosmology, " Scientific American, July 1992 , 17 .

Notes 34

7

3. Th e orbifol d theory is actually the creation of several individuals, including L. Dixon, J. Harvey , and Edwar d Witten of Princeton. 4. Year s ago, mathematician s asked themselve s a simpl e question : Give n a curved surfac e in A^dimensiona l space, ho w many kinds of vibrations can exis t on it? For example, think of pouring sand on a drum. When the drum is vibrated at a certai n frequency , th e particle s o f sand s danc e o n th e dru m surfac e an d form beautifu l symmetrical patterns. Differen t pattern s o f sand particle s corre spond t o differen t frequencie s allowed o n th e dru m surface . Similarly, mathe maticians have calculated th e numbe r an d kin d of resonating vibrations allowed on th e surfac e o f a curve d A?-dimensiona l surface . They eve n calculate d th e number and kind of vibrations that an electron could have on such a hypothetical surface. T o th e mathematicians , thi s wa s a cut e intellectua l exercise . N o on e thought it could possibly have any physical consequence. After all, electrons, they thought, don't vibrate on A^-dimensiona l surfaces. This larg e bod y of mathematica l theorems ca n no w be brough t t o bea r o n the proble m of GUT families. Each GU T family , i f string theory is correct, must be a reflection of some vibration on an orbifold. Since the various kinds of vibrations have been cataloge d b y mathematicians, all physicists have to do i s look in a math book t o tell them how many identical families ther e are ! Thus the origin of th e famil y proble m i s topology. I f string theory is correct, th e origi n o f thes e three duplicat e familie s o f GU T particle s canno t b e understoo d unles s w e expand ou r consciousnes s to ten dimensions. Once w e have curled u p th e unwante d dimension s into a tin y ball , we can then compar e th e theor y with experimenta l data . For example, th e lowes t excitation o f the strin g corresponds to a closed strin g with a very small radius . Th e particles tha t occu r i n th e vibratio n of a small closed strin g are precisel y those found i n supergravity. Thus we retrieve all the goo d results of supergravity, without th e ba d results . The symmetr y group of this new supergravity is E(8) X E(8), which is much larger than th e symmetry of the Standard Model or even the GUT theory. Therefore, th e superstrin g contains both th e GU T and th e supergravity theory (withou t many of th e ba d feature s of eithe r theory) . Instea d o f wiping out it s rivals, the superstrin g simply eats them up . The proble m with these orbifolds, however, is that we can construct hundreds of thousands o f them. We have an embarrassmen t of riches! Each one o f them, in principle , describes a consisten t universe. How d o w e tel l whic h univers e is the correc t one? Among these thousand s of solutions, we find many that predict exactly three generation s o r families o f quarks and leptons . We can also predict thousands of solutions where there are many more than three generations. Thus while GUTs consider thre e generations t o be to o many, many solutions of string theory consider three generation s t o be to o few! 5. Davi d Gross, Interview, in Superslrings: A Theory o f Everything'? ed. Paul Davies andj. Brow n (Cambridge : Gambridge University Press, 1988), 142-143 . 6. Ibid .

348 Notes Chapter 1 0 1. Mor e precisely , the Paul i exclusio n principl e state s tha t n o tw o electrons can occup y th e sam e quantu m stat e wit h th e sam e quantu m numbers . Thi s means tha t a white dwar f can b e approximate d a s a Fermi sea , o r a gas of electrons obeying the Paul i principle. Since electron s canno t b e i n th e sam e quantu m state , a ne t repulsiv e forc e prevents them from being compresse d dow n t o a point. In a white dwarf star, it is this repulsive force tha t ultimatel y counteracts th e gravitationa l force. The sam e logic applies to neutrons in a neutron star, since neutrons also obey the Paul i exclusio n principle , althoug h th e calculatio n i s mor e complicate d because o f other nuclea r and genera l rclativisti c effects . 2. John Michel! , in Philosophical Transactions of the Royal Society 74 (1784) : 35. 3. Quote d in Hein z Pagels, Perfect Symmetry: Th e Search for the Beginning of Time (New York : Bantam, 1985) , 57.

Chapter 1 ! 1. Quote d in Anthony Zee, Fearful Symmetry (Ne w York: Macmillan, 1986), 68. 2. K . Godel, "A n Exampl e o f a Ne w Type o f Cosmologica l Solution o f Einstein's Field Equations of Gravitation, " Reviews o f Modem Physics 21 (1949) : 447. 3. F. Tipler, "Causality Violation in Asymptotically Flat Space-Times," Physical Review Letters 37 (1976) : 979. 4. M . S. Morris, K. S. Thorne, and U.Yurtsever , "Wormholes, Time Machines, and th e Wea k Energy Condition," Physical Review Letters 61 (J988) : 1446 . 5. M . S. Morris and K . S. Thorne, "Wormhole s in Spacetim e an d Thei r Us e for Interstella r Travel: A Tool for Teaching Genera l Relativity, " American Journal of Physics 56 (1988) : 411 . 6. Fernand o Echeverria , Gunnar Klinkhammer, and Ki p S. Thorne, "Billiar d Balls in Wormhol e Spacetimes with Closed Timelike Curves : Classical Theory," Physical Review D 44 (1991) : 1079 . 7. Morris , Thorne, an d Yurtsever, "Wormholes, " 1447 .

Chapter 1 2 1. Steve n Weinberg, "Th e Cosmologica l Constant. Problem," Reviews o f Modem Physics Ql (1989) : 6. 2. Hein/ . Pagels, Perfect Symmetry: Th e Search for the Beginning of Time (New York: Bantam, 1985) , 377 . 3. Ibid. , 378 . 4. Quote d i n Ala n Lightrnar i an d Robert a Brawer , Origins: Th e Lives an d

Notes 34

9

Worlds o f Modern Co.smologists (Cambridge, Mass.: Harvard University Press, ]990), 479. 5. Richar d Feynman, Interview, in Superstrings: A Theory of Everything?ed. Paul Davies andj. Brow n (Cambridge : Cambridge University Press, 1988), 196 . 6. Weinberg , "Cosmologica l Constant Problem," 7 . 7. Quote d i n K . C . Cole , Sympathetic Vibrations: Reflections o n Physics as a Way of Life (Ne w York: Bantam, 1985), 204. 8. Quote d i n Joh n Gribben, In Search of Schrodinger's Ca t (New York: Bantam, 1984), vi. 9. Quote d in Hein z Pagels, Th e Cosmic Code (Ne w York: Bantam, 1982), 113 . 10. Quote d in E. Harrison, Masks o f the Universe (New York: Macmillan, 1985), 246. 11. F . Wilczek an d B . Devine, Longing for th e Harmonies (Ne w York: Norton , 1988), 129 . 12. Pagels , Cosmic Code, 155. 13. Quote d in David Freedman, "Paralle l Universes : The Ne w Reality—From Harvard's Wildest Physicist," Discover Magazine, July 1990, 52. 14. Ibid. , 48. 15. Ibid. , 49. 16. Ibid. , 51. 17. Ibid. , 48.

Chapter 1 3 1. Pau l Davies, Superforce: Th e Search for a Grand Unified Theory o f Nature (New York: Simon and Schuster , 1984), 168 . 2. Freema n Dyson , Disturbing the Universe (Ne w York: Harper & Row, 1979) , 76. 3. Freema n Dyson , Infinite i n All Directions (New York: Harper & Row, 1988), 196-197. 4. Dyson , Disturbing the Universe, 212. 5. Car l Sagan , Cosmos (New York: Random House , 1980) , 306-307. 6. I n fact , aeon s ag o it was even easier t o self-destruct . In orde r t o make a n atomic bomb , th e fundamenta l problem facin g any species i s to separat e ura nium-235 fro m it s more abundan t twin , uranium-238 , whic h cannot sustai n a chain reaction. Only the uranium-235 will sustain a chain reaction. But uranium235 i s only 0.3% o f naturall y occurrin g uranium . To sustai n a runawa y chain reaction, you nee d a n enrichmen t leve l of at least 20%. In fact , weapons-grad e uranium ha s a 90% or mor e enrichmen t rate . (Thi s is the reaso n wh y uranium mines d o no t suffe r fro m spontaneou s nuclea r detonations . Becaus e naturall y occurring uranium i n a uranium mine is only 0.3% enriched, i t contains far too low a concentration o f U-235 to sustain a runaway nuclear chain reaction.)

350 Notes Because uranium-23 5 i s relatively short-live d compared wit h it s mor e abun dant twin , uranium-238, aeons ago , th e naturall y occurring enrichmen t rat e i n our univers e was much large r tha n 0.3% . In other words, it was far easier the n for any civilization to fabricate an atomi c bomb becaus e th e naturall y occurring enrichment rat e wa s much large r tha n i t is today. 7. Hein z Pagels, Th e Cosmic Code (Ne w York: Bantam, 1982), 309. 8. Sagan , Cosmos, 231. 9. Quote d i n Melind a Beck an d Danie l Click , "An d I f th e Come t Misses, " Newsweek, 2 3 November 1992 , 61.

Chapter 1 4 1. Quote d i n John D . Barrow and Fran k J. Tipler , Th e Anthropic Gosmological Principle (Oxford : Oxford Universit y Press, 1986) , 167 . 2. Quote d in Hein z Pagels, Perfect Symmetry: Th e Search for th e Beginning of Time (New York: Bantam, 1985) , 382 . 3. Ibid. , 234. 4. Astronomer s John D . Barrow of th e Universit y of Susse x in Englan d an d Joseph Sil k o f th e Universit y o f Californi a at Berkele y see som e hop e i n thi s dismal scenario . The y write , "I f life , i n an y shap e o r form , i s t o surviv e thi s ultimate environmental crisis, then th e univers e must satisfy certai n basi c requirements. The basi c prerequisite for intelligenc e to survive is a source o f energy. "The anisotropie s in th e cosmi c expansion, th e evaporatin g blac k holes, the remnant nake d singularitie s are al l lif e preserver s o f a sort . . . . A n infinit e amount of information is potentially available in an open universe, and it s assimilation woul d be th e principa l goa l o f any surviving noncorporeal intelligence " (The Left Hand o f Creation [Ne w York: Basic Books, 1983] , 226). 5. Ibid . 6. Geral d Feinberg , Solid Clues (Ne w York: Simon and Schuster , 1985) , 95.

Chapter I S 1. Quote d in Heinz Pagels, Th e Cosmic Code (Ne w York: Bantam Books, 1982), 173-174. 2. Edwar d Witten , Interview , in Superstrings: A Theory o f Everything? ed . Pau l Davies and J. Brow n (Cambridge : Cambridg e Universit y Press, 1988) , 102 . 3. Quote d i n John D . Barrow and Fran k J. Tipler , Th e Anthropic Cosmological Principle (Oxford : Oxford Universit y Press, 1986) , 185 . 4. Pagels , Cosmic Code, 382. 5. Jame s Trefil , Th e Moment o f Creation (Ne w York: Macmillan, 1983), 220.

Notes 35

1

6. John Ellis , Interview, in Superstrings, ed . Davie s and Brown , 161 . 7. Quote d i n R . P . Creas e an d C . C . Mann , Th e Second Creation (Ne w York: Macmillan, 1986) , 77. 8. Quote d i n Anthon y Zee , Fearful Symmetry (Ne w York: Macmillan , 1986) , 122. 9. Ibid. , 274 . 10. Hein z Pagels , Perfect Symmetry: Th e Search for th e Beginning of Time (Ne w York: Bantam, 1985) , xiii. 11. Stephe n Hawking , A Brief History of Time (New York: Bantam, 1988) , 175 .

This page intentionally left blank

References an d Suggeste d Readin g

Abbot, E . A. Flatland: A Romance o f Many Dimensions. New York: Ne w American Library, 1984 . Barrow, J. D., and F . J. Tipler. TheAnthropic Cosmological Principle. Oxford: Oxfor d University Press, 1986 . Bell, E . T. Me n o f Mathematics. Ne w York: Simon and Schuster , 1937 . Calder, N. Th e Key to the Universe. Ne w York: Penguin, 1977 . Chester, M. Particles. New York: Macmillan, 1978 . Crease, R., and C . Mann. Th e Second Creation. New York: Macmillan , 1986 . Davies, P . Th e Forces o f Nature. Cambridge : Cambridge University Press, 1979 . Davies, P . Superforce: Th e Search for a Grand Unified Theory o f Nature. Ne w York : Simon an d Schuster , 1984 . Davies, P. , an d J . Brown , eds . Superslrings: A Theory o f Everything? Cambridge: Cambridge University Press , 1988 . Dyson, F . Disturbing the Universe. Ne w York: Harper & Row, 1979 . Dyson F . Infinite i n All Directions. New York: Harper & Row, 1988 . Feinberg, G. Solid Clues. New York: Simo n an d Schuster , 1985 . Feinberg, G . What I s the World Made Of ? New York: Doubleday, 1977 . French, A. P. Einstein: A Centenary Volume. Cambridge, Mass.: Harvard Universit y Press, 1979 . Gamow, G. Th e Birth an d Death of Our Sun. Ne w York: Viking, 1952 . Glashow, S. L. Interactions. New York: Warner, 1988 . Gribben,J. I n Search o f Schrodinger's Cat. New York: Bantam, 1984 . Hawking, S . W. A Brief History o f Time. New York: Bantam, 1988 . Heisenberg, W. Physics an d Beyond. Ne w York: Harper Torchbooks, 1971 . Henderson, L. D. Th e Fourth Dimension and Non-Eudidean Geometry i n Modem Art, Princeton, N.J.: Princeton Universit y Press, 1983 . Kaku, M. Introduction to Superstrings. Ne w York: Springer-Verlag, 1988 . Kaku, M. , an d J . Trainer . Beyond Einstein: Th e Cosmic Quest for th e Theory o f th e Universe. Ne w York: Bantam, 1987 . Kaufmann, W . J. Black Holes an d Warped Space—Time. Sa n Francisco : Freeman , 1979.

353

354 References

an d Suggested Reading

Lenin, V . Materialism an d Empiro-Criticism. I n K . Marx , F. Engels, an d V . Lenin , On Dialectical Materialism. Moscow : Progress, 1977 . Pagels, H . Th e Cosmic Code. Ne w York: Bantam, 1982 . Pagels, H . Perfect Symmetry: Th e Search for th e Beginning oj Time. New York: Bantam, 1985. Pais, A. Subtle Is th e Lord: Th e Science and th e Life o f Albert Einstein. Oxford: Oxfor d University Press, 1982 . Penrose, R. Th e Emperor's Ne w Mind. Oxford : Oxfor d Universit y Press, 1989 . Polkinghorne, J. C . Th e Quantum World. Princeton , N.J. : Princeto n Universit y Press, 1984 . Rucker, R. Geometry, Relativity, an d th e Fourth Dimension. New York: Dover, 1977 . Rucker, R. Th e Fourth Dimension. Boston: Houghto n Mifflin , 1984 . Sagan, C . Cosmos. New York: Random House , 1980 . Silk, J. Th e Big Bang: The Creation and Evolution o f the Universe. 2n d ed . Sa n Fran cisco: Freeman, 1988 . Trefil, J . S. From Atoms to Quarks. Ne w York: Scribner, 1980 . Trefil,J. S . Th e Moment o f Creation. New York: Macmillan, 1983. Weinberg, S . Th e First Three Minutes: A Modern View o f the Origin o f th e Universe. New York: Basic Books, 1988 . Wilczek, F., and B . Devine. Longing for th e Harmonies. Ne w York: Norton, 1988 . Zee, A. Fearful Symmetry. Ne w York: Macmillan, 1986 .

Index

Abbot, Edwin , 55-58 Alvarez, Luis, 296 Alvarez, Walter, 296 Antheil, George, 2 2 Anthropic principle, 257-259 Antimatter, 122-123 , 12 6 Aristotle, 34 Asimov, Isaac, 5, 279 , 31 0 Askey, Richard, 17 6 Astrochicken, 280-281, 309 Averaged weak energy conditio n (AWEC) , 250-251 Aztecs, 285-286, 299, 305 Banchoff, Thomas , 1 1 Barrett, Sir W. F., 53 Barrow, John D. , 306, 308-310, 350n. 4 Bayeux Tapestry , 63-6 4 Bell, E. T., 31 Big Bang theory, x , 27, 180 , 195-197 , 213, 218, 303 , 31 0 Big Crunch, 28 , 303, 30 7 Binding energ y curve , 218-219 Blackbody radiation, 19 7 Black holes, 22, 217-218, 222-227, 245 , 253, 30 6 Blake, William, 124 Bohr, Niels , 137, 26 0 Bolsheviks, 65, 67-68 Bolyai, Janos, 377n.4 Bond, Nelson s 75 Borgcs, Jorge Luis, 26 2 Borwein, Jonathan, 176 Borwein, Peter, 17 6 Bose, Satyendra , 14 4 Boson, 14 4

Bronowski, Jacob, 81 Buller, A. H. R. , 233 Bush, Ia n D. , 18 6 Capra, Fritjhof, 31 9 Carroll, Lewi s (Charle s Dodgson), 22, 42, 62, 12 4 Casimir, Hcnrik, 250 Casimir effect, 25 0 Causality, 234-235 Chandrasekhar, Subrahmanyan, 94 , 226 Chew, Geoffrey, 32 4 Clifford, William , 337n.6 Closed time-like curv e (CTC) , 240, 24 8 Coleman, Sidney, 266-268 Compactified dimension , 105, 158-15 9 Compte, Auguste, 186 Conrad, Joseph, 22 Cosmic Background Explorer (COBE), 199 202 Cosmic rays , 184-185 Cosmological constant , 267-26 8 Cosmological proof of God, 192-19 4 Crookes, William, 50, 339n.l3 Curvature, 4 0 Dali, Salvadore, 7 0 Dark matter , 304 Darwin, Charles, 28 , 131 , 30 2 Davies, Paul, 273 DeWitt, Bryce, 144 , 26 2 Dirac, P. A. M., 112 , 147 , 189 , 32 7 Dirkson, Everett , 18 2 Dixon, L. , 347n.3 Dostoyevsky, Fyodor, 22 , 65-67 Doyle, Sir Arthur Conan , 167

355

356

Index

Drake, Frank, 283-28 4 Ouchamp, Marcel , 22 Dyson, Freeman , 258, 280-281 , 285 Ehrenfest, Paul , 339n.ll Einstein, Albert, 6, 10 , 13 , 15 , 79 , 80-107, 112-113, 133 , 138 , 142 , 154, 157 , 177, 201, 233 , 243-246 , 266 , 303 , 314 , 327-328, 342nn.7 , 13 Einstein-Rosen bridge, 224-226 Electromagnetic interactions, 13, 101 , 122 , 125, 338n. 6 Ellis, John, 189 , 32 6 Entropy death, 304-305 Equivalence principle , 8 9 Erikson, Erik, 209 Escape velocity , 22 3 Euclidean geometry , 33 , 38 Everett, Hugh , 262 Family problem, 127 , 206 Faraday, Michael , 25, 79, 100-101, 168 , 189 Faraday's Law, 35 Feinbcrg, Gerald, 28 , 307-308 Fermi, Enrico , 118 , 14 4 Fcrmions, 144 Ferrara, Sergio , 145 Feyiiman, Richard, 130 , 259 Feymnan diagrams , 119-120 , 138-139 , 166, 32 5 Field theory , 23, 25, 39, 79, 93-94, 156 , 166-168 Flatland, 46-48, 70-74, 106 , 180-181 , 340n.lO Frecdman, Daniel , 145 Freund, Peter, 11-12 , 104-105 , 345n.9 Gamow, George, 197-198 , 238 Gauss, Carl Friedrich, 32, 62, 336n.4 Gcller, Uri , 5 3 (Jell-Mann, Murray , 17 9 General relativity , 91-95, 100-101 , 138 150, 251 Generation problem , 127-128 , 206 Georgi, Howard , 140 Gladsone, William, 25 Glashow, Sheldon, 121 , 17 9 Gluons, 15, 122-123 God, 191-193 , 330-332 cosmological proof of, 192-194 ontological proo f of , 193—19 4 ideological proof of , 192-19 4

Godel, Kurt, 240, 242-24 3 Goldsmith, Donald, 283 Grand Unifie d Theories (GUI's) , 131-134 , 143, 157 , 159 , 206 , 213, 267, 305, 319 , 325, 347n. 4 Gravitino, 145 , 18 3 Graviton, 138-139 , 154 , 183 Gravity, 14-45 , 90-93, 95, 100-101, 126, 138-139, 146-148 , 154 , 183 , 253 , 33511.4 Green, Michael , 16, 155 , 169 Gross, David , 157 , 178 , 206 , 315-316 Grossman, Mareel, 93 Guth, Alan, 20, 26, 201, 259 Half-life, 13 4 Hardy, Godfrey , 174-175 Hartlc, James, 25 3 Harvey, Jeffrey, 157 , 347n. 3 Hawking, Stephen, 147 , 235, 252-254 , 267, 33 4 Hcinlein, Robert, 77 , 236-237 Heisenberg, Werner, 111 , 136 , 166 , 260, 324 Heisenberg Uncertaint y Principle, 114 , 187 Henderson, Linda Dalrymple , 22 , 62 Hernquist, Lars, 299 Heterotic string, 158-159 , 345n.l() Higgs boson, 127 , 183 Hinton, Charles , 54, 68-79, 84 , 88 Hinton's cubes, 69-70 Holism, 318-321 Horowitz, Paul , 282 Hubble, Edwin , 19 6 Hubble's Law, 196 Hurnc, David, 181 Huxley, Thomas H., 330 Hypcrcubc, 70, 77-78 Hyperdoughnut, 96-97 Hypersphere, 95, 342n.l() Inflation, 20 1 James, William , 22 Jeans, Sir James, 304 Johnson, Lyndon , 164 , 182 Kaluza, Theodr, 99-100, 105-107, 338n.6, 343n.l3 Kalu/a-Klein theory, vii , 8, 16, 99-103, 140-144, 146 , 154-155 , 169 , 207 , 313, 322, 335n.l , 345n. 9

Index Kardashev, Nikolai, 277 Kepler, Johannes, 334 Kerr, Roy, 226 Kikkawa, Keiji , 162 , 166, 20 7 Klein, Oskar, 106-107 , 144 , 207 Lawrence, Ernest, 184 Lenard, Philip, 314 Lenin, Vladimir, 22 , 67-68, 87, 340n.6 Leonardo d a Vinci, 64 Leptons, 123 , 127 , 142 , 143 , 146 , 18 3 Littlewood.John, 175 Lobachevski, Nikolaus I., 337n.4 Lodge, Sir Oliver, 53 Lovelace, Claude, 168 Mach, Ernst, 6 7 Mach's principle, 91, 242 Mandel, Heinrich, 343n.l3 Mandelstam, Stanley, 165 Many-worlds theory, 262 Marsden, Brian, 294 Martinec, Ernil, 15 7 Marx, Karl, 32 Maxwell, James Clerk, xi, 7 , 86, 101 , 189 , 314 Maxwell's equations, 101-103, 123 , 130 , 137, 142 , 143 , 276 , 342n.4, 345n.l 3 McDonald, George, 6 2 McGovern, George, 152 Michel, Helen, 296 Michell,John, 223 Microwave background, 197-200 Mie, Gustav, 343n.l3 Mills, R. L., 26 , 11 8 Missing mass, 304 Mobius, August, 51 Mobius strip, 60-61, 96 Modular functions , 172-173 , 176-177 More, Henry, 21 Morris, Michael , 24 5 Muller, Richard, 297 Multiply connected spaces , 1 8 Muon, 12 8 Nambu, Yoichiro, 161 Nanopoulous, D. V., 15 5 Nappi, Chiara, 151 Nemesis theory, 296 Neutrino, 125 , 128 , 187-188 Neutron star , 220-221, 348n. l Newman, Ezra, 243

357

Newton, Isaac, xi, 85, 115 , 147 , 242, 329, 339n.ll Newton's constant, 335n. 4 Non-Euclidean geometry, 34-36 Nonrenormalizable theory, 126 , 150 Norstrom, Gunnar, 104 , 343n.l 3 NUT solution, 244 Ontological proof of God, 193-19 4 Oort cloud, 297 Oppenheimer, J. Robert , 11 2 Orbifolds, 202-204 , 206 , 211, 347nn.3, 4 Ostriker, Jeremiah P. , 199 Ouspensky, P. D., 65 Owen, Tobius, 283 Pagels, Heinz, 9, 140 , 259, 289, 333 Pauli, Wolfgang, 106-107, 137 , 18 7 Pauli exclusion principle, 348n.l Pcnzias, Arno , 19 7 Perturbation theory , 119 Phase transition, 210-214 Photon, 11 3 Piaget.Jean, 210 Picasso, Pablo, 6 5 Planck, Max, 88 Planck energy, 107, 138 , 177 , 185, 26 9 Planck length, 16, 269, 335n.l Planck's constant, 113 , 335n. l Poincare, Henri, 130 , 327 Proton decay , 133-134 Proust, Marcel, 22 Ptolemy, 3 4 Pulsar, 220 Pythagorean Theorem, 37, 338n.7 Quanta, 11 3 Quantum chromodynamic s (QCD), 122 Quantum electrodynamic s (QED) , 123 Quantum theory , 112-115 Quarks, 15,122-123,125,142,143,183,213 bottom quark , 12 8 charmed quark , 128 colored quarks, 122 , 128 flavored quarks , 122 , 128 strange quark, 12 8 super quarks, 183 top quark , 128 Rabi, Isidor I., xii, 333 Ramanujan, Srinivasa , 172-177 Ramanujan function , 346n.l3 Raup, David, 297

358

Index

Red giant, 218 Red shift , 19 6 Reductionism, 318-321 Reissner-Nordstrom solution, 225 Resonance, 141 , 15 3 Riemann, Georg Bernhard, 22-23, 30-45, 62, 79 , 90-91, 107 , 243, 326 , 329 , 336nn.2, 4, 337n.6, 343n.l3 Riemann's metric tensor , 39-41 , 79, 93, 101, 143-144 , 146 , 147 , 148 , 338n. 7 Rohm, Ryan , 15 7 Russell, Bertrand , 28, 302 Rutherford, Ernest , 13 1 Sagan, Carl, 246, 295, 29 8 Sakita, Bunji, 16 2 Salam, Abdus, 145 , 211 Schapiro, Meyer, 65 Schell, Jonathan, 287 Scherk,Joel, 168 Schofield, A. T., 55, 340n.l Schrödinger, Erwin, 111 Schrödinger's cat, 260-261 Schwarz, John, 16, 155, 157 , 168-16 9 Schwarzschild, Karl, 164 Schwinger, Julian, 13 7 Scriabin, Alexander, 2 2 Search fo r extraterrestria l intelligenc e (SETI), 283 Sepkoski.John, 297 Sheehy, Gail, 209 Silk, Joseph, 306, 350n. 4 Singer, Isadore A., 327 Slade, Henry, 49, 52 Slepton, 183 S-matrix theory, 324-326 Smoot, George, 199-20 0 Snow, C. P., 30 4 Space warp, 90-92 Sparnaay, M.J. , 250 Special relativity , 82-85 Spielberg, Steven , 1 8 Spin, 144 , 15 0 Standard Model , 121-127, 131-134, 137 , 150, 153 , 155 , 170-171 , 211, 267 , 313 , 319, 345n.9 , 347n. 4 Stefan-Bollzmann law , 197 Stein, Gertrude, 2 2 Strong interactions , 14, 114, 121 , 21 3 Superconducting supercollider (SSC) , 16, 182-185, 187 , 274 , 316 Supergravity, vu, 16 , 144-148, 150 , 183 , 335n.l,345n.lO, 347n. 4

Supernova, 220 , 295 Superstrings, viii, 16 , 152-183, 335n.l, 345n.lO Supersymmetry, 145 , 18 3 Susskind, Leonard, 268 Suzuki, Mahiko, 160-161, 167 , 32 5 Swift-Tuttle comet , 294 Symmetry, 86 , 124-130, 209-213 Symmetry breaking, 209-21 3 Tamburino, Louis , 243 Tau lepton, 127-12 8 Teleological proof of God, 192-19 4 Tesseract, 70-71,77-78 Thermodynamics, secon d la w of, 304 Thomas Aquinas, 192 Thompson, J. J., 50 't Hooft , Gerard , 118-119, 121 , 148 , 32 5 Thome, Kip, 20, 245-249 Time travel , 18-20 , 232-251 Tipler, Frank , 244, 308-310 Townsend, Paul, 149 Trainer, Jennifer , xi , xii, 322 Trefil, James S. , 319 Treiman, Samuel , 151 Tunneling, 116 , 208 Type I , II, III civilizations, 277-279, 290 292,301-303 Unified fiel d theory , 6, 98, 11 2 Unti, Theodore, 243 Updike,John, 187 Vacuum, false , 209 , 21 1 Vafa, Curnrum , 202 van Nieuwenhuizen , Peter, 145 , 147-15 0 van Stockum, W. J., 244 Veltman, Martinus, 119, 148 Vene/iano, Gabriel, 160-161, 167 , 170 , 325 Virasoro, Miguel , 16 2 von Fraunhofer , Joseph, 18 6 von Helmholtz , Hermann, 10 , 44-45, 314 von Neumann , John, 309 Vranceanu, George , 104-10 5 Wave function o f the universe, 254-255, 264-265 W bosons, 114 , 12 2 Weak interactions , 14 , 114, 122 , 196 , 21 3 Weber, Wilhelm , 35 , 50 Weinberg, Steven , 9, 121 , 124 , 140, 148, 179, 25 9

Index Weisskopf, Victor, 94, 315 Welles, H . G. , 20, 22, 59-61, 84, 96, 232, 249 Wetherill, George W. , 283 White dwarf , 220 Whitehead, Alfred North, 327 Wigner, Eugene, 328 Wilczek, Frank, 262-263 Wilde, Oscar, 22 , 59 Willink, Arthur, 21 , 55, 340n. 2 Wilson, Edward O. , 331 Wilson, Robert, 19 7 Witten, Edward , 151-152, 161 , 179 , 188 , 207, 316 , 344n.6 , 347n.3 Witten, Leonard, 15 1 World line , 237-239

359

Wormholes, x, xi, 17 , 24, 213, 224-226, 228-231, 246-247, 256, 266-268 Wulf, Theodor, 184 Wyndham, John, 265 Xenophanes, 25 7 Yang, C. N., 26, 118 , 12 9 Yang-Mills field , 26 , 118 , 121-123, 132 , 134, 140 , 142 , 143 , 32 5 Yu, Loh-ping , 16 5 Yukawa, Hideki, 166 Yurtsever, Ulvi , 24 5 Z boson, 12 2 Zollnerjohann, 49-53 , 84, 339n.l3