Theory and Experiment in Gravitational Physics

This is a revised edition of a classic and highly regarded book, first published in 1981, giving a comprehensive survey...

Author: Clifford M. Will

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This is a revised edition of a classic and highly regarded book, first published in 1981, giving a comprehensive survey of the intensive research and testing of general relativity that has been conducted over the last three decades. As a foundation for this survey, the book first introduces the important principles of gravitation theory, developing the mathematical formalism that is necessary to carry out specific computations so that theoretical predictions can be compared with experimental findings. A completely up-to-date survey of experimental results is included, not only discussing Einstein's "classical" tests, such as the deflection of light and the perihelion shift of Mercury, but also new solar system tests, never envisioned by Einstein, that make use of the high precision space and laboratory technologies of today. The book goes on to explore new arenas for testing gravitation theory in black holes, neutron stars, gravitational waves and cosmology. Included is a systematic account of the remarkable "binary pulsar" PSR 1913+16, which has yielded precise confirmation of the existence of gravitational waves. The volume is designed to be both a working tool for the researcher in gravitation theory and experiment, as well as an introduction to the subject for the scientist interested in the empirical underpinnings of one of the greatest theories of the twentieth century. Comments on the previous edition: "consolidates much of the literature on experimental gravity and should be invaluable to researchers in gravitation" Science "a c»ncise and meaty book . . . and a most useful reference work . . . researchers and serious students of gravitation should be pleased with it" Nature

Theory and Experiment in Gravitational Physics Revised Edition

THEORY AND EXPERIMENT IN GRAVITATIONAL PHYSICS CLIFFORD M.WILL McDonnell Center for the Space Sciences, Department of Physics Washington University, St Louis

Revised Edition

[CAMBRIDGE UNIVERSITY PRESS

CAMBRIDGE u n i v e r s i t y p r e s s

Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi, Dubai, Tokyo, Mexico City Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521439732 © Cambridge University Press 1981, 1993 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1981 First paperback edition 1985 Revised edition 1993 A catalogue recordfor this publication is available from the British Library Library of Congress Cataloguing in Publication Data

Will, Clifford M. Theory and experiment in gravitational physics / Clifford M. Will. Rev. ed. p. cm. Includes bibliographical references and index. ISBN 0 521 43973 6 1. Gravitation. I. Title. QC178.W47 1993 53i'.4dc20 92-29555 CIP ISBN 978-0-521-43973-2 Paperback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. Information regarding prices, travel timetables, and other factual information given in this work is correct at the time of first printing but Cambridge University Press does not guarantee the accuracy of such information thereafter.

To Leslie

Contents

1 2 2.1 2.2 2.3 2.4 2.5 2.6 3 3.1 3.2 3.3 4 4.1 4.2 4.3 4.4 5 5.1 5.2 5.3 5.4 5.5 5.6 5.7

Preface to Revised Edition Preface to First Edition Introduction The Einstein Equivalence Principle and the Foundations of Gravitation Theory The Dicke Framework Basic Criteria for the Viability of a Gravitation Theory The Einstein Equivalence Principle Experimental Tests of the Einstein Equivalence Principle Schiff 's Conjecture The THsu Formalism Gravitation as a Geometric Phenomenon Universal Coupling Nongravitational Physics in Curved Spacetime Long-Range Gravitational Fields and the Strong Equivalence Principle The Parametrized Post-Newtonian Formalism The Post-Newtonian Limit The Standard Post-Newtonian Gauge Lorentz Transformations and the PPN Metric Conservation Laws in the PPN Formalism Post-Newtonian Limits of Alternative Metric Theories of Gravity Method of Calculation General Relativity Scalar-Tensor Theories Vector-Tensor Theories Bimetric Theories with Prior Geometry Stratified Theories Nonviable Theories

page xiii xv 1 13 16 18 22 24 38 45 67 67 68 79 86 87 96 99 105 116 116 121 123 126 130 135 138

ix

Contents 6 6.1 6.2 6.3 6.4 6.5 7 7.1 7.2 7.3 8 8.1 8.2 8.3 8.4 8.5 9 9.1 9.2 9.3 10 10.1 10.2 10.3 11 11.1 11.2 11.3 12 12.1 12.2 12.3 13 13.1 13.2

Equations of Motion in the PPN Formalism Equations of Motion for Photons Equations of Motion for Massive Bodies The Locally Measured Gravitational Constant N-Body Lagrangians, Energy Conservation, and the Strong Equivalence Principle Equations of Motion for Spinning Bodies The Classical Tests The Deflection of Light The Time-Delay of Light The Perihelion Shift of Mercury Tests of the Strong Equivalence Principle The Nordtvedt Effect and the Lunar Eotvos Experiment Preferred-Frame and Preferred-Location Effects: Geophysical Tests Preferred-Frame and Preferred-Location Effects: Orbital Tests Constancy of the Newtonian Gravitational Constant Experimental Limits on the PPN Parameters Other Tests of Post-Newtonian Gravity The Gyroscope Experiment Laboratory Tests of Post-Newtonian Gravity Tests of Post-Newtonian Conservation Laws Gravitational Radiation as a Tool for Testing Relativistic Gravity Speed of Gravitational Waves Polarization of Gravitational Waves Multipole Generation of Gravitational Waves and Gravitational Radiation Damping Structure and Motion of Compact Objects in Alternative Theories of Gravity Structure of Neutron Stars Structure and Existence of Black Holes The Motion of Compact Objects: A Modified EIH Formalism The Binary Pulsar Arrival-Time Analysis for the Binary Pulsar The Binary Pulsar According to General Relativity The Binary Pulsar in Other Theories of Gravity Cosmological Tests Cosmological Models in Alternative Theories of Gravity Cosmological Tests of Alternative Theories

x 142 143 144 153 158 163 166 167 173 176 184 185 190 200 202 204 207 208 213 215 221 223 227 238 255 257 264 266 283 287 303 306 310 312 316

Contents 14 An Update 14.1 The Einstein Equivalence Principle 14.2 The PPN Framework and Alternative Metric Theories of Gravity 14.3 Tests of Post-Newtonian Gravity 14.4 Experimental Gravitation: Is there a Future? 14.5 The Rise and Fall of the Fifth Force 14.6 Stellar-System Tests of Gravitational Theory 14.7 Conclusions References References to Chapter 14 Index

xi 320 320 331 332 338 341 343 352 353 371 375

Preface to the Revised Edition

Since the publication of thefirstedition of this book in 1981, experimental gravitation has continued to be an active and challenging field. However, in some sense, the field has entered what might be termed an Era of Opportunism. Many of the remaining interesting predictions of general relativity are extremely small effects and difficult to check, in some cases requiring further technological development to bring them into detectable range. The sense of a systematic assault on the predictions of general relativity that characterized the "decades for testing relativity" has been supplanted to some extent by an opportunistic approach in which novel and unexpected (and sometimes inexpensive) tests of gravity have arisen from new theoretical ideas or experimental techniques, often from unlikely sources. Examples include the use of laser-cooled atom and ion traps to perform ultra-precise tests of special relativity, and the startling proposal of a "fifth" force, which led to a host of new tests of gravity at short ranges. Several major ongoing efforts continued nonetheless, including the Stanford Gyroscope experiment, analysis of data from the Binary Pulsar, and the program to develop sensitive detectors for gravitational radiation observatories. For this edition I have added chapter 14, which presents a brief update of the past decade of testing relativity. This work was supported in part by the National Science Foundation (PHY 89-22140). Clifford M. Will 1992

xm

Preface to First Edition

For over half a century, the general theory of relativity has stood as a monument to the genius of Albert Einstein. It has altered forever our view of the nature of space and time, and has forced us to grapple with the question of the birth and fate of the universe. Yet, despite its subsequently great influence on scientific thought, general relativity was supported initially by very meager observational evidence. It has only been in the last two decades that a technological revolution has brought about a confrontation between general relativity and experiment at unprecedented levels of accuracy. It is not unusual to attain precise measurements within a fraction of a percent (and better) of the minuscule effects predicted by general relativity for the solar system. To keep pace with these technological advances, gravitation theorists have developed a variety of mathematical tools to analyze the new highprecision results, and to develop new suggestions for future experiments made possible by further technological advances. The same tools are used to compare and contrast general relativity with its many competing theories of gravitation, to classify gravitational theories, and to understand the physical and observable consequences of such theories. The first such mathematical tool to be thoroughly developed was a "theory of metric theories of gravity" known as the Parametrized PostNewtonian (PPN) formalism, which was suited ideally to analyzing solar system tests of gravitational theories. In a series of lectures delivered in 1972 at the International School of Physics "Enrico Fermi" (Will, 1974, referred to as TTEG), I gave a detailed exposition of the PPN formalism. However, since 1972, significant progress has been made, on both the experimental and theoretical sides. The PPN formalism has been refined, and new formalisms have been developed to deal with other aspects of xv

Preface to First Edition

xvi

gravity, such as nonmetric theories of gravity, gravitational radiation, and the motion of condensed objects. A irecent review article (Will, 1979)1 summarizes the principal results of these new developments, but gives none of the physical or mathematical details. Since 1972, there has been a need for a complete treatment of techniques for analyzing gravitation theory and experiment. To fill this need I have designed this study. It analyzes in detail gravitational theories, the theoretical formalisms developed to study them, and the contact between these theories and experiments. I have made no attempt to analyze every theory of gravity or calculate every possible effect; instead I have tried to present systematically the methods for performing such calculations together with relevant examples. I hope such a presentation will make this book useful as a working tool for researchers both in general relativity and in experimental gravitation. It is written at a level suitable for use as either a reference text in a standard graduate-level course on general relativity or, possibly, as a main text in a more specialized course. Not the least of my motivations for writing such a book is the fact that it was my "centennial project" for 1979 - the 100th anniversary of Einstein's birth. It is a pleasure to thank Bob Wagoner, Martin Walker, Mark Haugan, and Francis Everitt for helpful discussions and critical readings of portions of the manuscript. Ultimate responsibility for errors or omissions rests, of course, with the author. For his constant support and encouragement, I am grateful to Kip Thorne. Victoria LaBrie performed her usual feats of speedy and accurate typing of the manuscript. Thanks also go to Rose Aleman for help with the typing. Preparation of this book took place while the author was in the Physics Department at Stanford University, and was supported in part by the National Aeronautics and Space Administration (NSG 7204), the National Science Foundation (PHY 76-21454, PHY 79-20123), the Alfred P. Sloan Foundation (BR 1700), and by a grant from the Mellon Foundation. 1

See also Will (1984).

Introduction

On September 14,1959,12 days after passing through her point of closest approach to the Earth, the planet Venus was bombarded by pulses of radio waves sent from Earth. Anxious scientists at Lincoln Laboratories in Massachusetts waited to detect the echo of the reflected waves. To their initial disappointment, neither the data from this day, nor from any of the days during that month-long observation, showed any detectable echo near inferior conjunction of Venus. However, a later, improved reanalysis of the data showed a bona fide echo in the data from one day: September 14. Thus occurred the first recorded radar echo from a planet. On March 9, 1960, the editorial office of Physical Review Letters received a paper by R. V. Pound and G. A. Rebka, Jr., entitled "Apparent Weight of Photons." The paper reported the first successful laboratory measurement of the gravitational red shift of light. The paper was accepted and published in the April 1 issue. In June, 1960, there appeared in volume 10 of the Annals of Physics a paper on "A Spinor Approach to General Relativity" by Roger Penrose. It outlined a streamlined calculus for general relativity based upon "spinors" rather than upon tensors. Later that summer, Carl H. Brans, a young Princeton graduate student working with Robert H. Dicke, began putting the finishing touches on his Ph.D. thesis, entitled "Mach's Principle and a Varying Gravitational Constant." Part of that thesis was devoted to the development of a "scalartensor" alternative to the general theory of relativity. Although its authors never referred to it this way, it came to be known as the Brans-Dicke theory. On September 26,1960, just over a year after the recorded Venus radar echo, astronomers Thomas Matthews and Allan Sandage and co-workers at Mount Palomar used the 200-in. telescope to make a photographic

Theory and Experiment in Gravitational Physics plate of the star field around the location of the radio source 3C48. Although they expected to find a cluster of galaxies, what they saw at the precise location of the radio source was an object that had a decidedly stellar appearance, an unusual spectrum, and a luminosity that varied on a timescale as short as 15 min. The name quasistellar radio source or "quasar" was soon applied to this object and to others like it. These disparate and seemingly unrelated events of the academic year 1959-60, in fields ranging from experimental physics to abstract theory to astronomy, signaled a new era for general relativity. This era was to be one in which general relativity not only would become an important theoretical tool of the astrophysicist, but would have its validity challenged as never before. Yet it was also to be a time in which experimental tools would become available to test the theory in unheard-of ways and to unheard-of levels of precision. The optical identification of 3C48 (Matthews and Sandage, 1963) and the subsequent discovery of the large red shifts in its spectral lines and in those of 3C273 (Schmidt, 1963; Greenstein and Matthews, 1963),presented theorists with the problem of understanding the enormous outpourings of energy (1047 erg s"1) from a region of space compact enough to permit the luminosity to vary systematically over timescales as short as days or hours. Many theorists turned to general relativity and to the strong relativistic gravitationalfieldsit predicts, to provide the mechanism underlying such violent events. This was the first use of the theory's strong-field aspect (outside of cosmology), in an attempt to interpret and understand observations. The subsequent discovery of pulsars and the possible identification of black holes showed that it would not be the last. However, the use of relativistic gravitation in astrophysical model building forced theorists and experimentalists to address the question: Is general relativity the correct relativistic theory of gravitation? It would be difficult to place much confidence in models for such phenomena as quasars and pulsars if there were serious doubt about one of the basic underlying physical theories. Thus, the growth of "relativistic astrophysics" intensified the need to strengthen the empirical evidence for or against general relativity. The publication of Penrose's spinor approach to general relativity (Penrose, 1960) was one of the products of a new school of relativity theorists that came to the fore in the late 1950s. These relativists applied the elegant, abstract techniques of pure mathematics to physical problems in general relativity, and demonstrated that these techniques could also aid in the work of their more astrophysically oriented colleagues. The

2

Introduction

3

bridging of the gaps between mathematics and physics and mathematics and astrophysics by such workers as Bondi, Dicke, Sciama, Pirani, Penrose, Sachs, Ehlers, Misner, and others changed the way that research (and teaching) in relativity was carried out, and helped make it an active and exciting field of physics. Yet again the question had to be addressed: Is general relativity the correct basis for this research? The other three events of 1959-60 contributed to the rebirth of a program to answer that question, a program of experimental gravitation that had been semidormant for 40 years. The Pound-Rebka (1960) experiment, besides verifying the principle of equivalence and the gravitational red shift, demonstrated the powerful use of quantum technology in gravitational experiments of high precision. The next two decades would see further uses of quantum technology in such high-precision tools as atomic clocks, laser ranging, superconducting gravimeters, and gravitational-wave detectors, to name only a few. Recording radar echos from Venus (Smith, 1963) opened up the solar system as a laboratory for testing relativistic gravity. The rapid development during the early 1960s of the interplanetary space program made radar ranging to both planets and artificial satellites a vital new tool for probing relativistic gravitational effects. Coupled with the theoretical discovery in 1964 of the relativistic time-delay effect (Shapiro, 1964), it provided new and accurate tests of general relativity. For the next decade and a half, until the summer of 1974, the solar system would be the sole arena for high-precision tests of general relativity. Finally, the development of the Brans-Dicke (1961) theory provided a viable alternative to general relativity. Its very existence and agreement with experimental results demonstrated that general relativity was not a unique theory of gravity. Many even preferred it over general relativity on aesthetic and" theoretical grounds. At the very least, it showed that discussions of experimental tests of relativistic gravitational effects should be carried on using a broader theoretical framework than that provided by general relativity alone. It also heightened the need for high-precision experiments because it showed that the mere detection of a small general relativistic effect was not enough. What was now required was measurements of these effects to accuracy within 10%, 1%, or fractions of a percent and better, to distinguish between competing theories of gravitation. To appreciate more fully the regenerative effect that these events had on gravitational theory and its experimental tests, it is useful to review briefly the history of experimental gravitation in the 45 years following the publication of the general theory of relativity.

Theory and Experiment in Gravitational Physics In deriving general relativity, Einstein was not particularly motivated by a desire to account for unexplained experimental or observational results. Instead, he was driven by theoretical criteria of elegance and simplicity. His primary goal was to produce a gravitation theory that incorporated the principle of equivalence and special relativity in a natural way. In the end, however, he had to confront the theory with experiment. This confrontation was based on what came to be known as the "three classical tests." One of these tests was an immediate success - the ability of the theory to account for the anomalous perihelion shift of Mercury. This had been an unsolved problem in celestial mechanics for over half a century, since the discovery by Leverrier in 1845 that, after the perturbing effects of the planets on Mercury's orbit had been accounted for, and after the effect of the precession of the equinoxes on the astronomical coordinate system had been subtracted, there remained in the data an unexplained advance in the perihelion of Mercury. The modern value for this discrepancy is 43 arc seconds per century (Table 1.1). A number of ad hoc proposals were made in an attempt to account for this excess, including, among others, the existence of a new planet, Vulcan, near the Sun; a ring of planetoids; a solar quadrupole moment; and a deviation from the inversesquare law of gravitation (for a review, see Chazy, 1928). Although these proposals could account for the perihelion advance of Mercury, they either involved objects that were detectable by direct optical observation, or predicted perturbations on the other planets (for example, regressions of nodes, changes in orbital inclinations) that were inconsistent with observations. Thus, they were doomed to failure. General relativity accounted Table 1.1. Perihelion advance of Mercury Cause of advance

Rate (arc s/century)

General precession (epoch 1900) Venus Earth Mars Jupiter Saturn Others

5025'.'6 211".% 9070 275 15376 773 072

Sum Observed Advance Discrepancy

555770 559977 4277

4

Introduction

5

for the anomalous shift in a natural way without disturbing the agreement with other planetary observations. This result would go unchallenged until 1967. The next classical test, the deflection of light by the Sun, was not only a success, it was a sensation. Shortly after the end of World War I, two expeditions set out from England: one for Sobral, in Brazil; and one for the island of Principe off the coast of Africa. Their goal was to measure the deflection of light as predicted by general relativity -1.75 arc seconds for a ray that grazes the Sun. The observations had to be made in the path of totality of a solar eclipse, during which the Moon would block the light from the Sun and reveal thefieldof stars behind it. Photographic plates taken of the star field during the eclipse were compared with plates of the same field taken when the Sun was not present, and the angular displacement of each star was determined. The results were 1.13 + 0.07 times the Einstein prediction for the Sobral expedition, and 0.92 ±0.17 for the Principe expedition (Dyson et al., 1920). The announcement of these results confirming the theory caught the attention of a war-weary public and helped make Einstein a celebrity. But Einstein was so convinced of the "correctness" of the theory because of its elegance and internal consistency that he is said to have remarked that he would have felt sorry for the Almighty if the results had disagreed with the theory (see Bernstein, 1973). Nevertheless, the experiments were plagued by possible systematic errors, and subsequent independent analyses of the Sobral plates yielded values ranging from 1.0 to 1.3 times the general relativity value. Later eclipse expeditions made very little improvement (Table 1.2). The main sources of error in such optical deflection experiments are unknown scale changes between eclipse and comparison photographic plates, and the precarious conditions, primarily associated with bad weather and exotic locales, under which such expeditions are carried out. By 1960, the best that could be said about the deflection of light was that it was definitely more than 0'.'83, or half the Einstein value. This was the amount predicted from a simple Newtonian argument, by Soldner in 1801 (Lenard, 1921),1 or from an extension of the principle of equivalence, by Einstein (1911). Beyond that, "the subject [was] still a live one" (Bertotti et al., 1962). The third classical test was actually thefirstproposed by Einstein (1907): the gravitational red shift of light. But by contrast with the other two 1

In 1921, the physicist Philipp Lenard, an avowed Nazi, reprinted Soldner's paper in the Annalen der Physik in an effort to discredit Einstein's "Jewish" science by showing the precedence of Soldner's "Aryan" work.

Theory and Experiment in Gravitational Physics

6

tests, there was no reliable confirmation of it until the 1960 Pound-Rebka experiment. One possible test was a measurement of the red shift of spectral lines from the Sun. However, 30 years of such measurements revealed that the observed shifts in solar spectral lines are affected strongly by Doppler shifts due to radial mass motions in the solar photosphere. For example, the frequency shift was observed to vary between the center of the Sun and the limb, and to depend on the line strength. For the gravitational red shift the results were inconclusive, and it would be 1962 before a reliable solar red-shift measurement would be made. Similarly inconclusive were attempts to measure the gravitational red shift of spectral lines from white dwarfs, primarily from Sirius B and 40 Eridani B, both members of binary systems. Because of uncertainties in the determination of the masses and radii of these stars, and because of possible complications in their spectra due to scattered light from their companions, reliable, precise measurements were not possible [see Bertotti et al. (1962) for a review]. Furthermore, by the late 1950s, it was being suggested that the gravitational red shift was not a true test of general relativity after all. According to Leonard I. Schiff and Robert H. Dicke, the gravitational red shift was a consequence purely of the principle of equivalence, and did not test the field equations of gravitational theory. Schiff took the argument one step Table 1.2. Optical measurements of light deflection by the Suri*

Eclipse

Approximate number of stars

Minimum distance from center of Sun in solar radii

Result in units of Einstein prediction

1919 1919 1922 1922 1922 1922 1929 1936 1936 1947 1952 1973"

7 5 92 145 14 18 17 25 8 51 10 39

2 2 2.1 2.1 2 2 1.5 2 4 3.3 2.1 2

1.13 + 0.07 0.92 + 0.17 0.98 ± 0.06 1.04 + 0.09 0.7 to 1.3 0.8 to 1.2 1.28 ±0.06 1.55 + 0.15 0.7 to 1.2 1.15 ±0.15 0.97 + 0.06 0.95 + 0.11

a b

See Bertotti et al. (1962) for details. Texas Mauritanian Eclipse Team (1976), Jones (1976).

Results from different analyses 1.0 to 1.3 1.3 to 0.9 1.2 0.9 to 1.2 1.55 ± 0.2 1.0 to 1.4 0.82 ± 0.09

Introduction

7

further and suggested that the gravitational red-shift experiment was superseded in importance by the more accurate Eotvos experiment, which verified that bodies of different composition fall with the same acceleration (Schiff, 1960a; Dicke, 1960). Other potential tests of general relativity were proposed, such as the Lense-Thirring effect, an orbital perturbation due to the rotation of a body, and the de Sitter effect, a secular motion of the perigee and node of the lunar orbit (Lense and Thirring, 1918; de Sitter, 1916), but the prospects for ever detecting them were dim. Cosmology was the other area where general relativity could be confronted with observation. Initially the theory met with success in its ability to account for the observed expansion of the universe, yet by the 1940s there was considerable doubt about its applicability. According to pure general relativity, the expansion of the universe originated in a dense primordial explosion called the "big bang." The age of the universe since the big bang could be determined by extrapolating the expansion of the universe backward in time using the field equations of general relativity. However, the observed values of the present expansion rate were so high that the inferred age of the universe was shorter than that of the Earth. One result of this doubt was the rise in popularity during the 1950s of the steady-state cosmology of Herman Bondi, Thomas Gold, and Fred Hoyle. This model avoided the big bang altogether, and allowed for the expansion of the universe by the continuous creation of matter. By this means, the universe would present the same appearance to all observers for all time. But by the late 1950s, revisions in the cosmic distance scale had reduced the expansion rate by a factor of five, and had thereby increased the age of the universe in the big bang model to a more acceptable level. Nevertheless, cosmological observations were still in no position to distinguish among different theories of gravitation or of cosmology [for a detailed technical and historical review, see Weinberg (1972), Chapter 14]. Meanwhile, a small "cottage industry" had sprung up, devoted to the construction of alternative theories of gravitation. Some of these theories were produced by such luminaries as Poincare, Whitehead, Milne, Birkhoff, and Belinfante. Many of these authors expressed an uneasiness with the notions of general covariance and curved spacetime, which were built into general relativity, and responded by producing "special relativistic" theories of gravitation. These theories considered spacetime to be "special relativistic" at least at a background level, and treated gravitation as a Lorentz-invariant field on that background. As of 1960, it was possible

Theory and Experiment in Gravitational Physics

8

to enumerate at least 25 such alternative theories, as found in the primary research literature between 1905 and 1960 [for a partial list, see Whitrow and Morduch (1965)]. Thus, by 1960, it could be argued that the validity of general relativity rested on the following empirical foundation: one test of moderate precision (the perihelion shift, approximately 1%), one test of low precision (the deflection of light, approximately 50%), one inconclusive test that was not a real test anyway (the gravitational red shift), and cosmological observations that could not distinguish between general relativity and the steady-state theory. Furthermore, a variety of alternative theories laid claim to viability. In addition, the attitude toward the theory seemed to be that, whereas it was undoubtedly of importance as a fundamental theory of nature, its observational contacts were limited to the classical tests and cosmology. This view was present for example in the standard textbooks on general relativity of this period, such as those by Mcller (1952), Synge (1960), and Landau and Lifshitz (1962). As a consequence, general relativity was cut off from the mainstream of physics. It was during this period that one young, beginning graduate student was advised not to enter this field, because general relativity "had so little connection with the rest of physics and astronomy" (his name: Kip S. Thorne). However, the events of 1959-60 changed all that. The pace of research in general relativity and relativistic astrophysics began to quicken and, associated with this renewed effort, the systematic high-precision testing of gravitational theory became an active and challengingfield,with many new experimental and theoretical possibilities. These included new versions of old tests, such as the gravitational red shift and deflection of light, with accuracies that were unthinkable before 1960. They also included brand new tests of gravitational theory, such as the gyroscope precession, the time delay of light, and the "Nordtvedt effect" in lunar motion, that were discovered theoretically after 1959. Table 1.3 presents a chronology of some of the significant theoretical and experimental events that occurred in the two decades following 1959. In many ways, the years 19601980 were the decades for testing relativity. Because many of the experiments involved the resources of programs for interplanetary space exploration and observational astronomy, their cost in terms of money and manpower was high and their dependence upon increasingly constrained government funding agencies was strong. Thus, it became crucial to have as good a theoretical framework as possible for comparing the relative merits of various experiments, and for pro-

Introduction Table 1.3. A chronology: 1960-80 Time

Experimental or observational events

Theoretical events

1960

Hughes-Drever mass-anisotropy experiments Pound-Rebka gravitational red-shift experiment Discovery of nonsolar x-ray sources Discovery of quasar red shifts Princeton Eotvos experiment

Penrose paper on spinors Gyroscope precession (Schiff)

1962

1964

Brans-Dicke theory Bondi mass-loss formula Kerr metric discovery Time-delay of light (Shapiro)

Pound-Snider red-shift experiment Discovery of 3K microwave background

Singularity theorems in general relativity

1966

1968

Reported detection of solar oblateness Discovery of pulsars Planetary radar measurement of time delay Launch of Mariners 6 and 7 Acquisition of lunar laser echo First radio deflection measurements

Element production in the big bang Nordtvedt effect and early PPN framework

1970 CygXl: a black hole candidate Mariners 6 and 7 time-delay measurements 1972 1974

Moscow Eotvos experiment Discovery of binary pulsar

1976

1978

1980

Rocket gravitational red-shift experiment Lunar test of Nordtvedt effect Time delay results from Mariner 9 and Viking Measurement of orbit period decrease in binary pulsar SS433 Discovery of gravitational lens

Preferred-frame effects Refined PPN framework Area increase of black holes in general relativity Quantum evaporation of black holes Dipole gravitational radiation in alternative theories

Theory and Experiment in Gravitational Physics

10

posing new ones that might have been overlooked. Another reason that such a theoretical framework was necessary was to make some sense of the large (and still growing) number of alternative theories of gravitation. Such a framework could be used to classify theories, elucidate their similarities and differences, and compare their predictions with the results of experiments in a systematic way. It would have to be powerful enough to be used to design and assess experimental tests in detail, yet general enough not to be biased in favor of general relativity. A leading exponent of this viewpoint was Robert Dicke (1964a). It led him and others to perform several high-precision null experiments which greatly strengthened our faith in the foundations of gravitation theory. Within this viewpoint one asks general questions about the nature of gravity and devises experiments to test them. The most important dividend of the Dicke framework is the understanding that gravitational experiments can be divided into two classes. The first consists of experiments that test the foundations of gravitation theory, one of these foundations being the principle of equivalence. These experiments (Eotvos experiment, Hughes-Drever experiment, gravitational red-shift experiment, and others, many performed by Dicke and his students) accurately verify that gravitation is a phenomenon of curved spacetime, that is, it must be described by a "metric theory" of gravity. General relativity and Brans-Dicke theory are examples of metric theories of gravity. The second class of experiments consists of those that test metric theories of gravity. Here another theoretical framework was developed that takes up where the Dicke framework leaves off. Known as the "Parametrized Post-Newtonian" or PPN formalism, it was pioneered by Kenneth Nordtvedt, Jr. (1968b), and later extended and improved by Will (1971a), Will and Nordtvedt (1972), and Will (1973). The PPN framework takes the slow motion, weak field, or post-Newtonian limit of metric theories of gravity, and characterizes that limit by a set of 10 real-valued parameters. Each metric theory of gravity has particular values for the PPN parameters. The PPN framework was ideally suited to the analysis of solar system gravitational experiments, whose task then became one of measuring the values of the PPN parameters and thereby delineating which theory of gravity is correct. A second powerful use of the PPN framework was in the discovery and analysis of new tests of gravitation theory, examples being the Nordtvedt effect (Nordtvedt 1968a), preferredframe effects (Will, 1971b) and preferred-location effects (Will, 1971b, 1973). The Nordtvedt effect, for instance, is a violation of the equality of acceleration of massive bodies, such as the Earth and Moon, in an

Introduction

11

external field; the effect is absent in general relativity but present in many alternative theories, including the Brans-Dicke theory. The third use of the PPN formalism was in the analysis and classification of alternative metric theories of gravitation. After 1960, the invention of alternative gravitation theories did not abate, but changed character. The crude attempts to derive Lorentz-invariant field theories described previously were mostly abandoned in favor of metric theories of gravity, whose development and motivation were often patterned after that of the BransDicke theory. A "theory of gravitation theories" was developed around the PPN formalism to aid in their systematic study. The PPN formalism thus became the standard theoretical tool for analyzing solar system experiments, looking for new tests, and studying alternative metric theories of gravity. One of the central conclusions of the two decades of testing relativistic gravity in the solar system is that general relativity passes every experimental test with flying colors. But by the middle 1970s it became apparent that the solar system could no longer be the sole testing ground for gravitation theories. One reason was that many alternative theories of gravity agreed with general relativity in their post-Newtonian limits, and thereby also agreed with all solar system experiments. But they did not necessarily agree in other predictions, such as cosmology, gravitational radiation, neutron stars, or black holes. The second reason was the possibility that experimental tools, such as gravitational radiation detectors, would ultimately be available to perform such extra-solar system tests. This suspicion was confirmed in the summer of 1974 with the discovery by Joseph Taylor and Russell Hulse of the binary pulsar (Hulse and Taylor, 1975). Here was a system that combined large post-Newtonian gravitational effects, highly relativistic gravitational fields associated with the pulsar, and the possibility of the emission of gravitational radiation by the binary system, with ultrahigh precision data obtained by radiotelescope monitoring of the extremely stable pulsar clock. It was also a system where relativistic gravity and astrophysics became even more intertwined than in the case, say, of quasars. In the binary pulsar, relativistic gravitational effects provided a means for accurate measurement of astrophysical parameters, such as the mass of a neutron star. The role of the binary pulsar as a new arena for testing relativistic gravity was cemented in the winter of 1978 with the announcement (Taylor et al., 1979) that the rate of change of the orbital period of the system had been measured. The result agreed with the prediction of general relativity for the rate of orbital energy loss due to the emission of gravitational radiation. But it

Theory and Experiment in Gravitational Physics

12

disagreed violently with the predictions of most alternative theories, even those with post-Newtonian limits identical to general relativity. As a young student of 17 at the Poly technical Institute of Zurich, Einstein studied closely the work of Helmholtz, Maxwell, and Hertz, and ultimately used his deep understanding of electromagnetic theory as a foundation for special and general relativity. He appears to have been especially impressed by Hertz's confirmation that light and electromagnetic waves are one and the same (Schilpp, 1949). The electromagnetic waves that Hertz studied were in the radio part of the spectrum, at 30 MHz. It is amusing to note that, 60 years later, the decades for testing relativistic gravity began with radio waves, the 440 MHz waves reflected from Venus, and ended with radio waves, the signals from the binary pulsar, observed at 430 MHz. During these two decades, that closed on the centenary of Einstein's birth, the empirical foundations of general relativity were strengthened as never before. But this does not end the story. The confrontation between general relativity and experiment will proceed, using new tools, in new arenas. Whether or not general relativity will continue to survive is a matter of speculation for some, pious hope for another group, and supreme confidence for others. Regardless of one's theoretical prejudices, it can certainly be agreed that gravitation, the oldest known, and in many ways most fundamental interaction, deserves an empirical foundation second to none. Throughout this book, we shall adopt the units and conventions of Misner, Thorne, and Wheeler, 1973 (hereafter referred to as MTW). Although we have attempted to produce a reasonably self-contained account of gravitation theory and gravitational experiments, the reader's path will be greatly smoothed by a familiarity with at least the equivalent of "track 1" of MTW. A portion of the present book (Chapters 4-9) is patterned after the author's 1972 Varenna lectures "The Theoretical Tools of Experimental Gravitation" (Will, 1974a, hereafter referred to as TTEG), with suitable modification and updating. An overview of this book without the mathematical details is provided by the author's "The Confrontation between Gravitation Theory and Experiment" (Will, 1979). Other useful reviews of this subject are of three types: (i) semipopular: Nordtvedt (1972), Will (1972, 1974b); (ii) technical: Richard (1975), Brill (1973), Rudenko (1978); (iii) "early": Dicke (1964a,b), Bertotti et al. (1962). The reader is referred to these works for background or for different points of view.

The Einstein Equivalence Principle and the Foundations of Gravitation Theory

The Principle of Equivalence has played an important role in the development of gravitation theory. Newton regarded this principle as such a cornerstone of mechanics that he devoted the opening paragraphs of the Principia to a detailed discussion of it (Figure 2.1). He also reported there the results of pendulum experiments he performed to verify the principle. To Newton, the Principle of Equivalence demanded that the "mass" of any body, namely that property of a body (inertia) that regulates its response to an applied force, be equal to its "weight," that property that regulates its response to gravitation. Bondi (1957) coined the terms "inertial mass" mb and "passive gravitational mass" mP, to refer to these quantities, so that Newton's second law and the law of gravitation take the forms F = m,a,

F = mPg

where g is the gravitational field. The Principle of Equivalence can then be stated succinctly: for any body mP = m1 An alternative statement of this principle is that all bodies fall in a gravitational field with the same acceleration regardless of their mass or internal structure. Newton's equivalence principle is now generally referred to as the "Weak Equivalence Principle" (WEP). It was Einstein who added the key element to WEP that revealed the path to general relativity. If all bodies fall with the same acceleration in an external gravitational field, then to an observer in a freely falling elevator in the same gravitational field, the bodies should be unaccelerated (except for possible tidal effects due to inhomogeneities in the gravitational field, which can be made as small as one pleases by working in a sufficiently small elevator). Thus insofar as their mechanical motions are

Figure 2.1. Title page and first page of Newton's Principia.

PHILOSOPHISE NATURALIS

PRINCIPIA MATHEMATICA Autore JS. UEfFTON, Trin. CM. Cantab. Soc. Mathefeos Profeflbre Lucafuoto, & Sodetatis Regalis Sodali.

IMPRIMATUR S. P E P Y S, Reg. Soc. P R R S E S. Jutii 5. 1686.

L 0 N D I N /, Juflii Societatis Regia ac Typis Jofepbi Streater. Proftat apud plures Bibliopolas. Anno MDCLXXXVIl.

14

Figure 2.1 (continued)

MATHEMATICAL PRINCIPLES OF

NATURAL PHILOSOPHY1 D eft

nitions

DEFINITION I The quantity of matter is the measure of the same, arising from its density and bulk, conjointly.2

T

HUS AIR of a double density, in a double space, is quadruple in quantity; in a triple space, sextuple in quantity. The same thing is to be understood of snow, and fine dust or powders, that are condensed by compression or liquefaction, and of all bodies that are by any causes whatever differently condensed. I have no regard in this place to a medium, if any such there is, that freely pervades the interstices between the parts of bodies. It is this quantity that I mean hereafter everywhere under the name of body or mass. And the same is known by the weight of each body, for it is proportional to the weight, as I have found by experiments on pendulums, very accurately made, which shall be shown hereafter. DEFINITION

IIs

The quantity of motion is the measure of the same, arising from the velocity and quantity of matter conjointly. The motion of the whole is the sum of the motions of all the parts; and therefore in a body double in quantity, with equal velocity, the motion is double; with twice the velocity, it is quadruple. t l Appendix, Note 10.] [ 2 Appendix, Note 11.]

[ 3 Appendix, Note 12.]

CO

15

Theory and Experiment in Gravitational Physics

16

concerned, the bodies will behave as if gravity were absent. Einstein went one step further. He proposed that not only should mechanical laws behave in such an elevator as if gravity were absent but so should all the laws of physics, including, for example, the laws of electrodynamics. This new principle led Einstein to general relativity. It is now called the "Einstein Equivalence Principle" (EEP). Yet, it is only relatively recently that we have gained a deeper understanding of the significance of these principles of equivalence for gravitation and experiment. Largely through the work of Robert H. Dicke, we have come to view principles of equivalence, along with experiments such as the Eotvos experiment, the gravitational red-shift experiment, and so on, as probes more of the foundations of gravitation theory, than of general relativity itself. This viewpoint is part of what has come to be known as the Dicke Framework described in Section 2.1, allowing one to discuss at a very fundamental level the nature of space-time and gravity. Within it one asks questions such as: Do all bodies respond to gravity with the same acceleration? Does energy conservation imply anything about gravitational effects? What types of fields, if any, are associated with gravitation-scalar fields, vector fields, tensor fields... ? As one product of this viewpoint, we present in Section 2.2 a set of fundamental criteria that any potentially viable theory should satisfy, and as another, we show in Section 2.3 that the Einstein Equivalence Principle is the foundation for all gravitation theories that describe gravity as a manifestation of curved spacetime, the so-called metric theories of gravity. In Section 2.4 we describe the empirical support for EEP from a variety of experiments. Einstein's generalization of the Weak Equivalence Principle may not have been a generalization at all, according to a conjecture based on the work of Leonard Schiff. In Section 2.5, we discuss Schiif 's conjecture, which states that any complete and self-consistent theory of gravity that satisfies WEP necessarily satisfies EEP. Schiff's conjecture and the Dicke Framework have spawned a number of concrete theoretical formalisms, one of which is known as the THsu formalism, presented in Section 2.6, for comparing and contrasting metric theories of gravity with nonmetric theories, analyzing experiments that test EEP and WEP, and proving Schiff's conjecture. 2.1

The Dicke Framework

The Dicke Framework for analyzing experimental tests of gravitation was spelled out in Appendix 4 of Dicke's Les Houches lectures

Einstein Equivalence Principle and Gravitation Theory (1964a). It makes two main assumptions about the type of mathematical formalism to be used in discussing gravity: (i) Spacetime is a four-dimensional differentiable manifold, with each point in the manifold corresponding to a physical event. The manifold need not a priori have either a metric or an affine connection. The hope is that experiment will force us to conclude that it has both. (ii) The equations of gravity and the mathematical entities in them are to be expressed in a form that is independent of the particular coordinates used, i.e., in covariant form. Notice that even if there is some physically preferred coordinate system in spacetime, the theory can still be put into covariant form. For example, if a theory has a preferred cosmic time coordinate, one can introduce a scalar field T{0>) whose numerical values are equal to the values of the preferred time t: T(0>) = t{0>),

0> a point in spacetime

If spacetime is endowed with a metric, one might also demand that VT be a timelike vector field and be consistently oriented toward the future (or the past) throughout spacetime by imposing the covariant constraints VT-VT<0,

VVT = 0

where V is a covariant derivative with respect to the metric. Other types of theories have "flat background metrics" IJ; these can also be written covariantly by defining i; to be a second-rank tensor field whose Riemann tensor vanishes everywhere, i.e., Riem(>r) = 0

and by defining covariant derivatives and contractions with respect to i\. In most cases, this covariance is achieved at the price of the introduction into the theory of "absolute" or "prior geometric" elements (T, i/), that are not determined by the dynamical equations of the theory. Some authors regard the introduction of absolute elements as a failure of general covariance (Einstein would be one example), however we shall adopt the weaker assumption of coordinate invariance alone. (For further discussion of prior geometry, see Section 3.3.) Having laid down this mathematical viewpoint [statements (i) and (ii) above] Dicke then imposes two constraints on all acceptable theories of gravity. They are: (1) Gravity must be associated with one or more fields of tensorial character (scalars, vectors, and tensors of various ranks).

17

Theory and Experiment in Gravitational Physics

18

(2) The dynamical equations that govern gravity must be derivable from an invariant action principle. These constraints strongly confine acceptable theories. For this reason we should accept them only if they are fundamental to our subsequent arguments. For most applications of the Dicke Framework only the first constraint is often needed. It is a fact, however, that the most successful gravitation theories are those that satisfy both constraints. The Dicke Framework is particularly useful for designing and interpreting experiments that ask what types of fields are associated with gravity. For example, there is strong evidence from elementary particle physics for at least one symmetric second-rank tensorfieldthat is approximated by the Minkowski metric i\ when gravitational effects can be ignored. The Hughes-Drever experiment rules out the existence of more than one second-rank tensor field, each coupling directly to matter, and various ether-drift experiments rule out a long-range vectorfieldcoupling directly to matter. No experiment has been able to rule out or reveal the existence of a scalar field, although several experiments have placed limits on specific scalar-tensor theories (Chapters 7 and 8). However, this is not the only powerful use of the Dicke Framework. 2.2

Basic Criteria for the Viability of a Gravitation Theory

The general unbiased viewpoint embodied in the Dicke Framework has allowed theorists to formulate a set of fundamental criteria that any gravitation theory should satisfy if it is to be viable [we do not impose constraints (1) and (2) above]. Two of these criteria are purely theoretical, whereas two are based on experimental evidence. (i) It must be complete, i.e., it must be capable of analyzing from "first principles" the outcome of any experiment of interest. It is not enough for the theory to postulate that bodies made of different material fall with the same acceleration. The theory must incorporate a complete set of electrodynamic and quantum mechanical laws, which can be used to calculate the detailed behavior of bodies in gravitational fields. This demand should not be extended too far, however. In areas such as weak and strong interaction theory, quantum gravity, unified field theories, spacetime singularities, and cosmic initial conditions, even special and general relativity are not regarded as being complete or fully developed. We also do not regard the presence of "absolute elements" and arbitrary parameters in gravitational theories as a sign of incompleteness, even though they are generally not derivable from "first principles," rather we

Einstein Equivalence Principle and Gravitation Theory

19

view them as part of the class of cosmic boundary conditions. Fortunately, so simple a demand as one that the theory contain a set of gravitationally modified Maxwell equations is sufficiently telling that many theories fail this test. Examples are given in Table 2.1. (ii) It must be self-consistent, i.e., its prediction for the outcome of every experiment must be unique, i.e., when one calculates the predictions by two different, though equivalent methods, one always gets the same

Table 2.1. Basically nonviable theories of gravitation - a partial list Theory and references

Comments"

Newtonian gravitation theory Milne's kinematical relativity (Milne, 1948)

Is not relativistic Was devised originally to handle certain cosmological problems. Is incomplete: makes no gravitational red-shift prediction Contain a vector gravitational field in flat spacetime. Are incomplete: do not mesh with the other nongravitational laws of physics (viz. Maxwell's equations) except by imposing them on the flat background spacetime. Are then inconsistent: give different results for light propagation for light viewed as particles and light viewed as waves. Action-at-a-distance theory in flat spacetime. Is incomplete or inconsistent in the same manner as Kustaanheimo's theories Contains a vector gravitational field in flat spacetime. Is incomplete or inconsistent in the same manner as Kustaanheimo's theories. Contains a tensor gravitational field used to construct a metric. Violates the Newtonian limit by demanding that p = pc2, i.e.

Kustaanheimo's various vector theories (Kustaanheimo and Nuotio, 1967; Whitrow and Morduch, 1965)

Poincare's theory (as generalized by Whitrow and Morduch, 1965) Whitrow-Morduch (1965) vector theory Birkhoff's (1943) theory

''sound

Yilmaz's (1971,1973) theory

=

"light-

Contains a tensor gravitational field used to construct a metric. Is mathematically inconsistent: functional dependence of metric on tensor field is not well defined.

° These theories are nonviable in their present form. Future modifications or specializations might make some of them viable. If I have misinterpreted any theory here I apologize to its proponents, and urge them to demonstrate explicitly its completeness, self-consistency, and compatibility with special relativity and Newtonian gravitation theory.

Theory and Experiment in Gravitational Physics

20

results. An example is the bending of light computed either in the geometrical optics limit of Maxwell's equations or in the zero-rest-mass limit of the motion of test particles. Furthermore, the system of mathematical equations it proposes should be well posed and self-consistent. Table 2.1 shows some theories that fail this criterion. (iii) It must be relativistic, i.e., in the limit as gravity is "turned off" compared to other physical interactions, the nongravitational laws of physics must reduce to the laws of special relativity. The evidence for this comes largely from high-energy physics and from a variety of optical ether-drift experiments. Since these experiments are performed at high energies and velocities and over very small regions of space and time, the effects of gravity on their outcome are negligible. Thus we may treat such experiments as if they were being performed far from all gravitating matter. The evidence provided by these experiments is of two types. First are experiments that measure space and time intervals directly, e.g., measurements of the time dilation of systems ranging from atomic clocks to unstable elementary particles, experiments that verify the velocity of light is independent of the velocity of the source for sources ranging from pions at 99.98% of the speed of light to pulsating binary x-ray sources at 10" 3 of the speed of light [for a thorough review and reference list, see Newman et al. (1978)] and Michelson-Morley-type experiments [for recent high-precision results, see Trimmer et al. (1973) and Brillet and Hall (1979); see also Mansouri and Sexl (1977a,b,c) for theoretical discussion]. Second are experiments which reveal the fundamental role played by the Lorentz group in particle physics, including verifications of fourmomentum conservation and of the relativistic laws of kinematics, electron and muon "g-2" experiments, and tests of esoteric predictions of Lorentz-in variant quantumfieldtheories [Lichtenberg (1965), Blokhintsev (1966), Newman et al. (1978), Combley et al. (1979), and Cooper et al. (1979)]. The fundamental theoretical object that enters these laws is the Minkowski metric i\, with a signature of + 2, which has orthonormal tetrads related by Lorentz transformations, and which determines the ticking rates of atomic clocks and the lengths of laboratory rods. If we view q as a field [Dicke statement (ii)], then we conclude that there must exist at least one second-rank tensor field in the Universe, a symmetric tensor ^, which reduces to r\ when gravitational effects can be ignored. Let us examine what particle physics experiments do and do not tell us about the tensor field V- First, they do not guarantee the existence of global Lorentz frames, i.e., coordinate systems extending throughout

Einstein Equivalence Principle and Gravitation Theory

21

spacetime in which (-1,1,1,1) Nor do they demand that at each event 2P, there exist local frames related by Lorentz transformations, in which the laws of elementary-particle physics take on their special form. They only demand that, in the limit as gravity is "turned off," the nongravitational laws of physics reduce to the laws of special relativity. Second, elementary-particle experiments do tell us that the times measured by atomic clocks in the limit as gravity is turned off depend only on velocity, not upon acceleration. The measured squared interval, ds2 = i^dx^dx", is independent of acceleration. Equivalently, but more physically, the time interval measured by a clock moving with velocity vJ relative to a coordinate system in the absence of gravity is ds = (-q^tordx*)112

= dt(l - |v|2)1/2

independent of the clock's acceleration d2xi/dt2. (For a review of experimental tests, see Newman et al., 1978.) We shall henceforth assume the existence of the tensor field $. (iv) It must have the correct Newtonian limit, i.e., in the limit of weak gravitational fields and slow motions, it must reproduce Newton's laws. Massive amounts of empirical data support the validity of Newtonian gravitation theory (NGT), at least as an approximation to the "true" relativistic theory of gravity. Observations of the motions of planets and spacecraft agree with NGT down to the level (parts in 108) at which post-Newtonian effects can be observed. Observations of planetary, solar, and stellar structure support NGT as applied to bulk matter. Laboratory Cavendish experiments provide support for NGT for small separations between gravitating bodies. One feature of NGT that has recently come under experimental scrutiny is the inverse-square force law. Despite one claim to the contrary (Long, 1976), there seems to be no hard evidence for a deviation from this law (other than those produced by post-Newtonian effects) over distances ranging from a few centimeters to several astronomical units (see Mikkelson and Newman, 1977; Spero et al., 1979; Paik, 1979; Yu et al., 1979; Panov and Frontov, 1979; and, Hirakawa et al., 1980). Thus, to at least be viable, a gravitation theory must be complete, self-consistent, relativistic, and compatible with NGT. Table 2.1 shows examples of theories that violate one or more of these criteria.

Theory and Experiment in Gravitational Physics 2.3

22

The Einstein Equivalence Principle The Einstein Equivalence Principle is the foundation of all curved spacetime or "metric" theories of gravity, including general relativity. It is a powerful tool for dividing gravitational theories into two distinct classes: metric theories, those that embody EEP, and nonmetric theories, those that do not embody EEP. For this reason, we shall discuss it in some detail and devote the next section (Section 2.4) to the supporting experimental evidence. We begin by stating the Weak Equivalence Principle in more precise terms than those used before. WEP states that if an uncharged test body is placed at an initial event in spacetime and given an initial velocity there, then its subsequent trajectory will be independent of its internal structure and composition. By "uncharged test body" we mean an electrically neutral body that has negligible self-gravitational energy (as estimated using Newtonian theory) and that is small enough in size so that its coupling to inhomogeneities in external fields can be ignored. In the same spirit, it is also useful to define "local nongravitational test experiment" to be any experiment: (i) performed in a freely falling laboratory that is shielded and is sufficiently small that inhomogeneities in the external fields can be ignored throughout its volume, and (ii) in which self-gravitational effects are negligible. For example, a measurement of the fine structure constant is a local nongravitational test experiment; a Cavendish experiment is not. The Einstein Equivalence Principle then states: (i) WEP is valid, (ii) the outcome of any local nongravitational test experiment is independent of the velocity of the (freely falling) apparatus, and (iii) the outcome of any local nongravitational test experiment is independent of where and when in the universe it is performed. This principle is at the heart of gravitation theory, for it is possible to argue convincingly that if EEP is valid, then gravitation must be a curvedspacetime phenomenon, i.e., must satisfy the postulates of Metric Theories of Gravity. These postulates state: (i) spacetime is endowed with a metric g, (ii) the world lines of test bodies are geodesies of that metric, and (iii) in local freely falling frames, called local Lorentz frames, the nongravitational laws of physics are those of special relativity. General relativity, BransDicke theory, and the Rosen bimetric theory are metric theories of gravity (Chapter 5); the Belinfante-Swihart theory (Section 2.6) is not. The argument proceeds as follows. The validity of WEP endows spacetime with a family of preferred trajectories, the world lines of freely falling test bodies. In a local frame that follows one of these trajectories,

Einstein Equivalence Principle and Gravitation Theory

23

test bodies have unaccelerated motions. Furthermore, the results of local nongravitational test experiments are independent of the velocity of the frame. In two such frames located at the same event, 9, in spacetime but moving relative to each other, all the nongravitational laws of physics must make the same predictions for identical experiments, that is, they must be Lorentz invariant. We call this aspect of EEP Local Lorentz Invariance (LLI). Therefore, there must exist in the universe one or more second-rank tensor fields i/t(1), ij/(2\ . . . , that reduce in a local freely falling frame to fields that are proportional to the Minkowski metric, (j)(1\^)tl, 0 (2) (^)«J,..., where 4>(A\0>) are scalar fields that can vary from event to event. Different members of this set of fields may couple to different nongravitationalfields,such as bosonfields,fermionfields,electromagnetic fields, etc. However, the results of local nongravitational test experiments must also be independent of the spacetime location of the frame. We call this Local Position Invariance (LPI). There are then two possibilities, (i) The local versions of ijf{A) must have constant coefficients, that is, the scalarfields4>(A\^) must be constants. It is therefore possible by a simple universal rescaling of coordinates and coupling constants (such as the unit of electric charge) to set each scalar field equal to unity in every local frame, (ii) The scalarfieldsiA)({A\0>) = cA4>(0>). If this is true, then physically measurable quantities, being dimensionless ratios, will be location independent (essentially, the scalar field will cancel out). One example is a measurement of the fine structure constant; another is a measurement of the length of a rigid rod in centimeters, since such a measurement is a ratio between the length of the rod and that of a standard rod whose length is defined to be one centimeter. Thus, a combination of a rescaling of coupling constants to set the cA's equal to unity (redefinition of units), together with a "conformal" transformation to a new field ij/ = cj>~ V. guarantees that the local version of if/ will be ij. In either case, we conclude that there exist fields that reduce to r\ in every local freely falling frame. Elementary differential geometry then shows that thesefieldsare one and the same: a unique, symmetric secondrank tensor field that we now denote g. This g has the property that it possesses a family of preferred worldlines called geodesies, and that at each event $* there exist local frames, called local Lorentz frames, that follow these geodesies, in which

<W^) = 1** + 0(Y |X« - x\0>)\\

dgjdx* = 0,

at 0>

Theory and Experiment in Gravitational Physics

24

However, geodesies are straight lines in local Lorentz frames, as are the trajectories of test bodies in local freely falling frames, hence the test bodies move on geodesies of g and the Local Lorentz frames coincide with the freely falling frames. We shall discuss the implications of the postulates of metric theories of gravity in more detail in Chapter 3. Because EEP is so crucial to this conclusion about the nature of gravity, we turn now to the supporting experimental evidence. 2.4

Experimental Tests of the Einstein Equivalence Principle (a) Tests of the Weak Equivalence Principle

A direct test of WEP is the Eotvos experiment, the comparison of the acceleration from rest of two laboratory-sized bodies of different composition in an external gravitational field. If WEP were invalid, then the accelerations of different bodies would differ. The simplest way to quantify such possible violations of WEP in a form suitable for comparison with experiment is to suppose that for a body of inertial mass m,, the passive mass mP is no longer equal to mv Now the inertial mass of a typical laboratory body is made up of several types of mass energy: rest energy, electromagnetic energy, weak-interaction energy, and so on. If one of these forms of energy contributes to mP differently than it does to m,, a violation of WEP would result. One could then write p = m, + I r]AEA/c2

(2.1)

A

where EA is the internal energy of the body generated by interaction A, and nA is a dimensionless parameter that measures the strength of the violation of WEP induced by that interaction, and c is the speed of light.1 For two bodies, the acceleration is then given by

^

( + S r,AEA/m2Ag

(2.2)

where we have dropped the subscript I on mj and m2. 1 Throughout this chapter we shall avoid units in which c = 1. The reason for this is that if EEP is not valid then the speed of light may depend on the nature of the devices used to measure it. Thus, to be precise we should denote c as the speed of light as measured by some standard experiment. Once we accept the validity of EEP in Chapter 3 and beyond, then c has the same value in every local Lorentz frame, independently of the method used to measure it, and thus can be set equal to unity by appropriate choice of units.

Einstein Equivalence Principle and Gravitation Theory

25

A measurement or limit on the relative difference in acceleration then yields a quantity called the "Eotvos ratio" given by

K + a\

t

\mC2

m2c2j

v

'

Thus, experimental limits on r\ place limits on the WEP-violation parameters rjA. Many high-precision Eotvos-type experiments have been performed, from the pendulum experiments of Newton, Bessel, and Potter to the classic torsion-balance measurements of Eotvos, Dicke, and Braginsky and their collaborators. The latter experiments can be described heuristically. Two objects of different composition are connected by a rod of length r, and suspended in a horizontal orientation by afinewire ("torsion balance"). If the gravitational acceleration of the bodies differs, there will be a torque N induced on the suspension wire, given by N = tjr(g x ew) er where g is the gravitational acceleration, and ew and er are unit vectors along the wire and rod, respectively (see Figure 2.2). If the entire apparatus is rotated about a direction
(

'

where the limits are \a formal standard deviations. For further discussion of the experiments, see Dicke (1964a) and Braginsky (1974). The primary sources of error in these experiments are seismic noise and coupling of the torsion balance to gradients in the external gravitational field (produced, for example, by the experimenters). Attempts to improve these results have centered on different forms of suspension of the masses,

Theory and Experiment in Gravitational Physics

26

Figure 2.2. Schematic arrangement of a torsion-balance Eotvos experiment; g is the external gravitational acceleration, and to is the angular velocity vector about which the apparatus is rotated. The unit vectors e w and er are parallel to the wire and rod, respectively. In the Eotvos experiments, g was the acceleration toward the Earth, and to was parallel to e w ; in the Princeton and Moscow experiments g was that of the Sun, and co was parallel to the Earth's rotation axis. \\W\\\\\

including magnetic levitation (Worden and Everitt, 1974), flotation on liquids (Keiser and Faller, 1979), and free fall in orbit. Experiments to test WEP for individual atoms and elementary particles have been inconclusive or inaccurate, with the exception of neutrons (Fairbank et al, 1974 and Koester, 1976). Table 2.2 discusses various experiments and quotes the limits they set on Y) for different pairs of materials. Future improved tests of WEP must reduce noise due to thermal, seismic, and gravity-gradient effects, and may have to be performed in space using cryogenic techniques. Anticipated limits on r\ in such experiments range between 10""15 and 10" 18 (Worden, 1978). To determine the limits placed on individual parameters r\K by, say, the best of the torsion-balance experiments, we must estimate the co-

Table 2.2. Tests of the weak equivalence principle Experiment

Reference

Method

Substances tested

Newton Bessel Eotvos Potter Renner Princeton Moscow Munich Stanford Boulder Orbital-

Newton (1686) Bessel (1832) Eotvos, Pekar, and Fekete (1922) Potter (1923) Renner (1935) Roll, Krotkov, and Dicke (1964) Braginsky and Panov (1972) Koester (1976) Worden (1978) Keiser and Faller (1979) Worden (1978)

Pendula Pendula Torsion balance Pendula Torsion balance Torsion balance Torsion balance Free fall Magnetic suspension Flotation on water Free fall in orbit

Various Various Various Various Various Aluminum and gold Aluminum and platinum Neutrons Niobium, Earth Copper, tungsten Various

" Experiments yet to be performed.

Limit on \tj\

2 5 2 2

x x x x

10" 3 10" 5 10" 9 10" 5 lO" 9

io-»

lO" 1 2 3 x 10" 4

io- 4u

4 x 10" 10~15 - lO" 18 "

Theory and Experiment in Gravitational Physics

28

efficients EA/m for the different interactions and for different materials. For laboratory-sized bodies, the dominant contribution to £A comes from the atomic nucleus. We begin with the strong interactions. The semiempirical mass formula (see, for example, Leighton, 1959) gives Es = -15.74 + 17.SA213 + 23.604 - 2Z)2A~l + 132/1"lS MeV

(2.5)

where Z and A are the atomic number and mass number, respectively, of the nucleus, and where S = 1 if {Z,A} = {odd, even}, 5 1 if {Z,A} = {even, even} and 8 = 0 if A is odd. Then, Es/mc2 = -1.7 x 1(T2 + 1.9 x 10" 2 ,4" 1/3 + 2.5 x 10~2(l - 2Z/A)2 + 1.41 x 10"M-2<5

(2.6)

For platinum (Z = 78, A = 195), and aluminum (13,27) the difference in Es/mc2 is approximately 2 x 10"3, so from the limit \n\ < 10" 12 , we obtain the limit \ns\ < 5 x 10" 10 . In the case of electromagnetic interactions, we can distinguish among a number of different internal energy contributions, each potentially having its own n* parameter. For the electrostatic nuclear energy, the semiempirical mass formula yields the estimate £ ES = 0.71Z(Z - X)A ~1/3 MeV

(2.7)

Thus, EES/mc2 = 7.6 x 10"4Z(Z - \)A"4/3

(2.8) 3

with the difference for platinum and aluminum being 2.5 x 10" . The resulting limit on nES is |T/ES| < 4 X 10" 10 . Another form of electromagnetic energy is magnetostatic, resulting from the nuclear magnetic fields generated by the proton currents. To estimate the nuclear magnetostatic energy requires a detailed shell model computation. For example, the net proton current in any closed angular momentum shell vanishes, hence there is no energy associated with the magnetostatic interaction between such a closed shell and any particle outside the shell. For aluminum and platinum, Haugan and Will (1977) have shown (£MS/mc2)A1 = 4.1 x 10~7, us

6

(£M7mc2)pt = 2.4 x 10"7

(2.9)

thus \r\ \ < 6 x 10 " . A third form of electromagnetic energy that has been studied is hyperfine, the energy of interaction between the spins of the nucleons and the magnetic fields generated by the proton and neutron magnetic moments. Computations by Haugan (1978) have yielded the

Einstein Equivalence Principle and Gravitation Theory

29

estimate

£ H F = (2n/V)ntig2pZ2 + g2(A - Z) 2 ]

(2.10)

where V is the nuclear volume, (iN is the nuclear magneton, and gp = 2.79 and ga = 1.91 are gyromagnetic ratios for the proton and neutron, respectively. Then, Em/mc2 = 2.1 x 10"5[>2Z2 + g2(A - Zf^A'2

(2.11)

with the difference between aluminum and platinum being 4 x 10~ 6 ; thus|>7 HF |<2 x 10" 7. For some time, it was believed that the contribution of the weak interactions to nuclear energy was of the order of a part in 1012, and that the Eotvos experiment was not yet sufficiently accurate to test WEP for weak interactions (see for example, Chiu and Hoffman, 1964; Dicke, 1964a). However, these estimates took into account only the parity nonconserving parts of the weak interactions, which make no contribution to the energy of a nucleus in its ground state, to first order in the weak-interaction coupling constant G w . On the other hand, the parityconserving parts of the weak interactions do contribute at first order in Gw and yield a value E/me2 ~ 10" 8 (Haugan and Will, 1976). Specifically, in the Weinberg-Salam model for weak and electromagnetic interactions, the result is E^/mc2 = 2.2 x 10"8(iVZM2)[l + g(N,Z)], g(N,Z) = 0.295[i(iV - Z)2/ATZ + 4 sin2 0W + (Z/N) sin2 0W(2 sin2 0W - 1)]

(2.12)

where N = (A Z) is the neutron number, and where 0W ~ 20° is the "Weinberg" angle. For aluminum and platinum, the difference is 2 x 10~ 10 , yielding |>?w| < 10~2. Gravitational interactions are specifically excluded from WEP and EEP. In Chapter 3, we shall extend these two principles to incorporate local gravitational effects, thereby defining the Gravitational Weak Equivalence Principle (GWEP) and the Strong Equivalence Principle (SEP). These two principles will be useful in classifying alternative metric theories of gravity. In any case, for laboratory Eotvos experiments, gravitational interactions are totally irrelevant, since for an atomic nucleus Ea/mc2 ~ Gmp/Raucleusc2 ~ 10" 3 9 To test for gravitational effects in GWEP, it will be necessary to employ planetary objects and planetary Eotvos experiments (Section 8.1).

Theory and Experiment in Gravitational Physics

30

(b) Tests of Local Lorentz Invariance Any experiment that purports to test special relativity (Section 2.2) also tests some aspect of Local Lorentz Invariance, since every Earthbound laboratory resides in a gravitational field (although it is only partially in free fall). However, very few of these experiments have been used to make quantitative tests of LLI in the same way that Eotvos experiments have been used to test WEP. For example, although elementary-particle experimental results are consistent with the validity of Lorentz invariance in the description of high-energy phenomena, they are not "clean" tests because in many cases it is unlikely that a violation of Lorentz invariance could be distinguished from effects due to the complicated strong and weak interactions. For instance, the observed violation of conservation of four momentum in beta decay was found to be due not to a violation of LLI, but to the emission of a hitherto unknown particle, the neutrino. However, there is one experiment that can be interpreted as a "clean" test of Local Lorentz invariance, and an ultrahigh precision one at that. This is the Hughes-Drever experiment, performed in 1959-60 independently by Hughes and collaborators at Yale University and by Drever at Glasgow University (Hughes et al., 1960; Drever, 1961). In the Glasgow version, the experiment examined the J = § ground state of the 7Li nucleus in an external magnetic field. The state is split into four levels by the magneticfield,with equal spacing in the absence of external perturbations, so the transition line is a singlet. Any external perturbation associated with a preferred direction in space (the velocity of the Earth relative to the mean rest frame of the universe, for example) that has a quadrupole (/ = 2) component will destroy the equality of the energy spacing and split the transition lines. Using NMR techniques, the experiment set a limit of 0.04 Hz (1.7 x 10" 16 eV) on the separation in frequency (energy) of the lines. One interpretation of this result is that it sets a limit on a possible anisotropy 3m\j in the inertial mass of the 7Li nucleus: |5mjJc2| ;$ 1.7 x 10" 16 eV. If any of the forms of internal energy of the 7Li nucleus suffered a breakdown of Local Lorentz Invariance, one would expect a contribution to 5m{J of the form dmij ~ X 5AEx/c2

(2.13)

A

where <5A is a dimensionless parameter that measures the strength of anisotropy induced by interaction A. Using formulae from Section 2.4(a), we can then make estimates of EA for 7Li and obtain the following limits

Einstein Equivalence Principle and Gravitation Theory

31

on<5\ |<5S| < 1(T 23 , |£ES| < 1 0 - 2 2 )

|<5HF| < 5 x 1(T 2 2 , |gW| < 5 x 1 Q -18

(2 .14)

Notice that the magnetostatic energy for 7Li is zero, since the proton shell structure is ls1/2lp3/2 and there is no magnetostatic interaction either within the closed s-shell (/ = 0) or between that shell and the valence proton. Because of the remarkably small size of these limits, the HughesDrever experiment has been called the most precise null experiment ever performed. If Local Lorentz Invariance is violated, then there must be a preferred rest frame, presumably that of the mean rest frame of the universe, or, equivalently of the cosmic microwave background, in which the local laws of physics take on their special form. Deviations from this form would then depend on the velocity of the laboratory relative to the preferred frame. Since the anisotropy is a quadrupole effect, one would expect it to be proportional to the square of the velocity w of the laboratory. If <>o is a parameter that measures the "bare" strength of LLI violation, then one would expect

For the motion of the Earth relative to the universe rest frame, w2 ~ 10 ~6. Limits on the <5Q can then be inferred from Equation (2.14). As a special case of this general argument, the Hughes-Drever experiment has also been interpreted as a test of the existence of additional long-range tensor fields that couple directly to matter (Peebles and Dicke, 1962; Peebles, 1962). Other experiments that can be interpreted as tests of LLI include various ether-drift experiments, such as the Turner-Hill experiment (Dicke, 1964a; Haugan, 1979). (c) Tests of Local Position Invariance

The two principal tests of Local Position Invariance are gravitational red-shift experiments that test the existence of spatial dependence on the outcomes of local experiments, and measurements of the constancy of the fundamental nongravitational constants that test for temporal dependence.

Theory and Experiment in Gravitational Physics

32

Gravitational Red-Shift Experiments A typical gravitational red-shift experiment measures the frequency or wavelength shift Z = Av/v = AA/A between two identical frequency standards (clocks) placed at rest at different heights in a static gravitational field. To illustrate how such an experiment tests LPI, we shall assume that the remaining parts of EEP, namely WEP and Local Lorentz Invariance, are valid. (In Sections 2.5 and 2.6, we shall discuss this question under somewhat different assumptions.) WEP guarantees that there exist local freely falling frames whose acceleration g relative to the static gravitational field is the same as that of test bodies. Local Lorentz Invariance guarantees that in these frames, the proper time measured by an atomic clock is related to the Minkowski metric by c2dx2 oc - r\^dx% dx\ oc c2dt\ - dx\ - dy\ - dz%

(2.15)

where x% are coordinates attached to the freely falling frame. However, in a local freely falling frame that is momentarily at rest with respect to the atomic clock, we permit its rate to depend on its location (violation of Local Position Invariance), that is, relative to an arbitrarily chosen atomic time standard based on a clock whose fundamental structure is different than the one being analyzed, the proper time between ticks is given by T = T(O)

(2.16)

where O is a gravitational potential whose gradient is related to the testbody acceleration by g = £V. Now the emitter, receiver, and gravitational field are assumed to be static, therefore in a static coordinate system (ts,xs), the trajectories of successive wave crests of emitted signal are identical except for a time translation Ats from one crest to the next. Thus, the interval of time Afs between ticks (passage of wave crests) of the emitter and of the receiver must be equal (otherwise there would be a build up or depletion of wave crests between the two clocks, in violation of our assumption that the situation is static). The static coordinates are not freely falling coordinates, but are accelerated upward (in the +z direction) relative to the freely falling frame, with acceleration g. Thus, for \gts/c\ ~ |#zs/c2| « 1 (i.e., for g uniform over the distance between the clocks), a sequence of Lorentz transformations yields (MTW, Section 6.6) ctF = (zs + c2/g)sinh{gts/c), zF = (zs + c2/g)cosh(gts/c), xF=xs, yF = y s

(217)

Einstein Equivalence Principle and Gravitation Theory

33

Thus, the time measured by the atomic clocks (relative to the standard clock) is given by c2dx2 = T2(
dz2)

= T2(<J>)[(1 + gzs/c2)2c2 dti - dx2 - dy2 - dz2}

(2.18)

Since the emission and reception rates are the same (1/Afj) when measured in static coordinate time, and since dxs = dys = dzs = 0 for both clocks, the measured rates (v = AT" ') are related by 7

Vrec

Vem_,

~^T~

[T(g>rec)(l +

2 gZ)~\ rJc rJc)\

Wc 22))JJ U<"U(i +flWc

(

}

For small separations, Az = zrec z em , we can expand T(rec) = T0 + ^ r ^ A z

(2.20)

where T 0 = T(
(2.21)

where a =C 2 C~ 1 TJ ) /T 0 and where AC/ = g Az = g(zrec zem). If there is no location dependence in the clock rate, then a = 0, and the red shift is the standard prediction, i.e.,

Z = AU/c2

(2.22)

An alternative version of this argument assumes the validity of both LLI and LPI and shows that, if the red shift is given by Equation (2.22), then the acceleration of the local frames in which Lorentz and Position Invariance hold is the same as that of test bodies, i.e., the local frames are freely falling frames (Thorne and Will, 1971). Although there were several attempts following the publication of the general theory of relativity to measure the gravitational red shift of spectral lines from white dwarf stars, the results were inconclusive (see Bertotti et al., 1962 for a review). The first successful, high-precision red-shift measurement was the series of Pound-Rebka-Snider experiments of 1960-65, which measured the frequency shift of y-ray photons from Fe 57 as they ascended or descended the Jefferson Physical Laboratory tower at Harvard University. The high accuracy achieved (1%) was obtained by making use of the Mossbauer effect to produce a narrow resonance line whose shift could be accurately determined. Other experiments since 1960 measured the shift of spectral lines in the Sun's gravitational field and the change in rate of atomic clocks transported aloft on aircraft, rockets,

Table 2.3. Gravitational red-shift experiments Experiment

Reference

Method

Pound-Rebka-Snider Brault Jenkins

Pound and Rebka (1960) Pound and Snider (1965) Brault (1962) Jenkins (1969)

Snider Jet-Lagged Clocks (A)

Snider (1972,1974) Hafele and Keating (1972a,b)

Jet-Lagged Clocks (B)

Alley (1979)

Vessot-Levine Rocket Red-shift Experiment Null Red-shift Experiment Close Solar Probe"

Vessot and Levine (1979) Vessot et al. (1980) Turneaure et al. (1983) Nordtvedt (1977)

Fall of photons from Mossbauer emitters Solar spectral lines Crystal oscillator clocks on GEOS-1 satellite Solar spectral lines Cesium beam clocks on jet aircraft Rubidium clocks on jet aircraft Hydrogen maser on rocket

' Experiments yet to be performed.

Hydrogen maser vs. SCSO Hydrogen maser or SCSO on satellite

Limit on |a|

io- 2 5 x 10"2 9 x 10"2 6 x 10"2 10"1 2 x 10"2 2 x 10"4

io- 2 10 -6«

Einstein Equivalence Principle and Gravitation Theory

35

and satellites. Table 2.3 summarizes the important red-shift experiments that have been performed since 1960. Recently, however, a new era in red-shift experiments has been ushered in with the development of frequency standards of ultrahigh stability parts in 1015 to 1016 over averaging times of 10 to 100 s and longer. Examples are hydrogen-maser clocks (Vessot, 1974), superconducting-cavity stabilized oscillator (SCSO) clocks (Stein, 1974; Stein and Turneaure, 1975), and cryogenically cooled monocrystals of dielectric materials such as silicon and sapphire (McGuigan et al., 1978). The first such experiment was the Vessot-Levine Rocket Red-shift Experiment that took place in June, 1976. A hydrogen-maser clock was flown on a rocket to an altitude of about 10,000 km and its frequency compared to a similar clock on the ground. The experiment took advantage of the high frequency stability of hydrogen-maser clocks (parts in 1015 over 100 s averaging times) by monitoring the frequency shift as a function of altitude. A sophisticated data acquisition scheme accurately eliminated all effects of the first-order Doppler shift due to the rocket's motion, while tracking data were used to determine the payload's location and velocity (to evaluate the potential difference AU, and the second-order Doppler shift). Analysis of the data yielded a limit (Vessot and Levine, 1979; Vessot et al., 1980) |a| < 2 x 10" 4

(2.23)

Coincidentally, the Scout rocket that carried the maser aloft stood 22.6 m in its gantry, almost exactly the height of the Harvard Tower. In an interplanetary version of this experiment, a stable clock (H-maser or SCSO clock) would be flown on a spacecraft in a very eccentric solar orbit (closest approach ~ 4 solar radii); such an experiment could test a to a part in 106 (Nordtvedt, 1977) and could conceivably look for "secondorder" red-shift effects of O(AC/)2 (Jaffe and Vessot, 1976). Advances in stable clocks have also made possible a new type of redshift experiment that is a direct test of Local Position Invariance (LPI): a "null" gravitational red-shift experiment that compares two different types of clocks, side by side, in the same laboratory. If LPI is violated, then not only can the proper ticking rate of an atomic clock vary with position, but the variation must depend on the structure and composition of the clock, otherwise all clocks would vary with position in a universal way and there would be no operational way to detect the effect (since one clock must be selected as a standard and ratios taken relative to that clock).

Theory and Experiment in Gravitational Physics

36

Thus, we must write for a given clock type A, T

= t\U) = TA(1 - aA AU/c2)

(2.24)

Then a comparison of two different clocks at the same location would measure TA/TB =

( T A/ t B )o [ 1 _ (aA _ aB) A [// C 2]

£.25)

where (TA/TB)0 i s t n e constant ratio between the two clock times observed at a chosen initial location. A null red-shift experiment of this type was performed in April, 1978 at Stanford University. The rates of two hydrogen maser clocks and of an ensemble of three SCSO clocks were compared over a 10 day period (Turneaure et al., 1983). During this time, the solar potential U changed sinusoidally with a 24 hour period by 3 x 10~13 because of the Earth's rotation, and changed linearly at 3 x 10""12 per day because the Earth is 90° from perihelion in April. However, analysis of the data set an upper limit on both effects, leading to a limit on the LPI violation parameter |aH_ascso|< 10-2 (2.26) The art of atomic timekeeping has advanced to such a state that it may soon be necessary to take red-shift and Doppler-shift corrections into account in making comparisons between timekeeping installations at different altitudes and latitudes. Constancy of the constants The other key test of Local Position Invariance is the constancy of the nongravitational constants over cosmological timescales (we delay discussion of the gravitational "constant" until Section 8.5). We shall not review here the various theories and proposals, originating with Dirac, that permit variable fundamental constants [for detailed review and references, see Dyson (1972)], rather we shall cite the most recent observational evidence (Table 2.4). The observations range from comparisons of spectral lines in distant galaxies and quasars, to measurements of isotopic abundances of elements in the solar system, to laboratory comparisons of atomic clocks. Recently, Shlyakhter (1976a,b) has made significant improvements in the limits on variations in the electromagnetic, weak, and strong coupling constants by studying isotopic abundances in the "Oklo Natural Reactor," a sustained U 235 fission reactor that evidently occurred in Gabon, Africa nearly two billion years ago (Maurette, 1976). Measurements of ore samples yielded an abnormally low value for the ratio of two isotopes of samarium (Sm149/

Table 2.4. Limits on cosmological variation of nongravitational constants

Constant k

Limit on kjk per Hubble time 2 x 1010 yr (H 0 = 5 5 k m s " 1 M p c " 1 )

Fine structure Constant: a = e2/hc

4 x 10" 4 8 x 10" 2 8 x 10" 2

Weak Interaction Constant:

Method

Reference

Re 187 ft decay rate over geological time Mgll fine structure and 21 cm line in radio source at Z = 0.5 SCSO clock vs. cesium beam clock

Dyson (1972) Wolfe, Brown, and Roberts (1976)

2

Re 187 , K 4 0 decay rates

Dyson (1972)

Electron-Proton Mass Ratio: mjmp

1

Mass shift in quasar spectral lines (Z ~ 2)

Pagel (1977)

10" 1

Mgll, 21 cm line

Wolfe, Brown, and Roberts (1976)

8 x 10" 2

Nuclear stability

Davies (1972)

g me/mp

10" 7

Turneaure and Stein (1976)

P = g(mlc/h3

Proton Gyromagnetic Factor:

Limit from Oklo reactor (Shlyakhter, 1976a,b)

2 x 10~ 2

Strong Interactions:

gl

8 x 10" 9

Theory and Experiment in Gravitational Physics

38

Sm147). Neither of these isotopes is a fission product, but Sm149 can be depleted by a dose of neutrons. Estimates of the neutron fluence (integrated dose) during the reactor's "on" phase, combined with the measured abundance anomaly yielded a value for the neutron capture cross section for Sm149 two billion years ago which agrees with the modern value. However, the capture cross section is extremely sensitive to the energy of a low-lying level (E ~ 0.1 eV) of Sm149, so that a variation of only 20 x 10 ~3 eV in this energy over 109 years would change the capture cross section from its present value by more than the observed amount. By estimating the contributions of strong, electromagnetic, and weak interactions to this energy, Shlyakhter obtained the limits on the rate of variation of the corresponding coupling constants shown in Table 2.4, column 5 (see also Dyson, 1978). 2.5

Schiff 's Conjecture

Because the three parts of the Einstein Equivalence Principle discussed above are so very different in their empirical consequences, it is tempting to regard them as independent theoretical principles. However, any complete and self-consistent gravitation theory must possess sufficient mathematical machinery to make predictions for the outcomes of experiments that test each principle, and because there are limits to the number of ways that gravitation can be meshed with the special relativistic laws of physics, one might not be surprised if there were theoretical connections between the three subprinciples. For instance, the same mathematical formalism that produces equations describing the free fall of a hydrogen atom must also produce equations that determine the energy levels of hydrogen in a gravitational field, and thereby determine the ticking rate of a hydrogen maser clock. Hence a violation of EEP in the fundamental machinery of a theory that manifests itself as a violation of WEP might also be expected to emerge as a violation of Local Position Invariance. Around 1960, Leonard I. Schiff conjectured that this kind of connection was a necessary feature of any self-consistent theory of gravity. More precisely, Schiff's conjecture states that any complete, self-consistent theory of gravity that embodies WEP necessarily embodies EEP. In other words, the validity of WEP alone guarantees the validity of Local Lorentz and Position Invariance, and thereby of EEP. This form of Schiff's conjecture is an embellished classical version of his original 1960 quantum mechanical conjecture (Schiff, 1960a). His interest in this conjecture was rekindled in November, 1970 by a vigorous argument with Kip S. Thorne at a conference on experimental gravitation held at the California Insti-

Einstein Equivalence Principle and Gravitation Theory

39

tute of Technology. Unfortunately, his untimely death in January, 1971 cut short his renewed effort. If Schiff's conjecture is correct, then the Eotvos experiments may be seen as the direct empirical foundation for EEP, and for the interpretation of gravity as a curved-spacetime phenomenon. Some authors, notably Schiff, have gone further to argue that if the conjecture is correct, then gravitational red-shift experiments are weak tests of gravitation theory, compared to the more accurate Eotvos experiment. For these reasons, much effort has gone into "proving" Schiff's conjecture. Of course, a rigorous proof of such a conjecture is impossible, yet a number of powerful "plausibility" arguments using a variety of assumptions can be formulated. The most general and elegant of these arguments is based upon the assumption of energy conservation. This assumption allows one to perform very simple cyclic gedanken experiments in which the energy at the end of the cycle must equal that at the beginning of the cycle. This approach was pioneered by Dicke (1964a), and subsequently generalized by Nordtvedt (1975) and Haugan (1979). Specifically, we restrict attention to theories of gravity in which there is a conservation law of energy for nongravitating "test" systems that reside in given static and external gravitational fields. To guarantee the existence of such a law, it is sufficient for the theory to be based on an invariant action principle [cf. Dicke's constraint (2)], but it is not necessary. We consider an idealized composite body made up of structureless test particles that interact by some nongravitational force to form a bound system. For a system that moves sufficiently slowly in a weak, static gravitationalfield,the laws governing its motion can be put into a quasiNewtonian form (we assume the theory has a Newtonian limit); in particular, the conserved energy function Ec associated with the conservation law can be assumed to have the general form Ec = MKc2 - MRU(X) + | M R F 2 + O(MRU2, MRV\MRUV2)

(2.27)

where X and V are quasi-Newtonian coordinates of the center of mass of the body, MR is the "rest" energy of the body, U is the external gravitational potential, and c is a fundamental speed used to convert units of mass into units of energy. If EEP is violated, we must allow for the possibility that the speed of light and the limiting speed of material particles may differ in the presence of gravity; to maintain this possibility we do not set c = 1 automatically in Equation (2.27) (see also footnote, p. 24). Note that V is the velocity relative to some preferred frame. In

Theory and Experiment in Gravitational Physics

40

problems involving external, static gravitational potentials, the preferred frame is generally the rest frame of the external potential, while in problems involving cosmological gravitational effects where localized potentials can be ignored, the preferred frame is that of the universe rest frame. [In problems involving both kinds of effects, the simple form of Equation (2.27) no longer holds.] The possible occurrence of EEP violations arises when we write the rest energy MRc2 in the form MRc2 = Moc2 - £B(X, V)

(2.28)

where M o is the sum of the rest masses of the structureless constituent particles, and £ B is the binding energy of the body. It is the position and velocity dependence of £ B , a dependence that in general is a function of the structure of the system, which signals the breakdown of EEP. Roughly speaking, an observer in a freely falling frame can monitor the binding energy of the system, thereby detecting the effects of his location and velocity in local nongravitational experiments. Haugan (1979) has made this more precise by showing that in fact it is the possible functional difference in £B(X, V) between the system under study and a "standard" system arbitrarily chosen as the basis for the units of measurement that leads to measurable effects. Because the location and velocity dependence in £ B is a result of the external gravitational environment, it is useful to expand it in powers of U and V2. To an order consistent with the quasi-Newtonian approximation in Equation (2.27), we write

£B(X,V) = El + 8myUiJ(X) - tfntfV'V1 + O(E%U2,...)

(2.29)

iJ

where U is the external gravitational potential tensor [cf. Equation (4.28)]; it is of the same order as U and satisfies U" = U. The quantities <5w# and bm\' are called the anomalous passive gravitational and inertial mass tensors, respectively. They are expected to be of order t\E%, where r\ is a dimensionless parameter that characterizes the strength of EEP-violating effects; they depend upon the detailed internal structure of the composite body. Summation over repeated spatial indices i, j is assumed. The conserved energy can thus be written, to quasi-Newtonian order, Ec = (M0c2 - £g) - [(Mo -

E°c-2)S» ^ ' J + O(M0U2,...)

(2.30)

We first give examples of violations of Local Position and Lorentz Invariance generated by £B(X, V). Consider two different systems at rest in the gravitational potential. Each system makes a transition from one

Einstein Equivalence Principle and Gravitation Theory

41

quantum energy level to another, and emits a quantum of frequency v = AEc/h. The ratio of the two frequencies is given from Equation (2.30) by

L (AEE)x

(AEg) 2 Jc 2

In the case where <5mj/ oc diJ, the quantity in square brackets can be identified as the coefficient a. 1 a 2 in Equation (2.25). Thus the anomalous passive gravitational mass tensor dmtf produces preferred-location effects in a null gravitational red-shift experiment. Consider the same two systems far from gravitating matter, but moving relative to the universe rest frame with velocity V. Then the ratio of the two frequencies is given by

(A£g)21

1

jAQSmiV

\ (A£°)x

J A(<5m{V2l V'V 2

(A£°)2 J c

Thus, the anomalous inertial mass tensor produces preferred-frame effects in an experiment such as the Hughes-Drever experiment, where the two systems in question are two excited states of 7 Li nuclei of different azimuthal quantum numbers in an external magnetic field. In this case, because the Zeeman splitting is the same for all levels in the absence of a preferred-frame effect, (A£B)X = (A£ B ) 2 , however because of the possible anisotropy in dm?, one would expect A(dm\J) to differ for transitions between different pairs of levels [for further details, see Section 2.6]. Thus dmy is responsible for violations of Local Position Invariance and <5m{J is responsible for violations of Local Lorentz Invariance. In order to verify Schiff s conjecture, it remains only to show that 5mj,J and dm\3 also produce violations of WEP. To do this we make use of a cyclic gedanken experiment first used by Dicke (1964a). We begin with a set of n free particles of mass m0 at rest at X = h. From Equation (2.27), the conserved energy is simply nm 0 c 2 [l t/(h)/c 2 ]. We then form a composite body and release the binding energy £B(h,0), in the form of free particles of rest mass m 0 , stored in a massless reservoir. The conserved energy of the composite body is [nmoc2 £ B (h,0)][l - [7(h)/c2] and that of the reservoir is £ B (h,0)[l t/(h)/c 2 ]. The composite body falls freely to X = 0 with an acceleration assumed to be A = g 4- <5A while the stored test particles fall with acceleration g = Vt/ (by definition). At X = 0 we bring both systems to rest, and place the energies thus gained, - [nm0 - £B(0, V)/c 2 ]A h - dniJgihi,

and

- EB(h, 0)g h/c 2

into the reservoir (we have assumed g, h, and V are parallel). Dropping terms of order (g h) 2 , we see that the reservoir now contains conserved

Theory and Experiment in Gravitational Physics

42

energy £B(h,O)[l - C/(0)/c2] - £jjg h/c 2 - (nmo - E°/c2)A h - &nitfh> From this we extract enough energy £ B (0,0)[l t/(0)/c 2 ] to disassemble the composite system into its n constituents, and enough energy nmog h to give the particles sufficient kinetic energy to return to their initial state of rest at X = h. The cycle is now closed, and if energy is to be conserved, the reservoir must be empty. To quasi-Newtonian order, this implies £B(h, 0) - £B(0,0) - (nm0 - E$/c2)SA h - <5m{W = 0

(2.33)

£B(h, 0) - £B(0,0) = dmy\UlJ h

(2.34)

Since

we obtain A' = g> + (5mik/MR)U{f - (<5mp/MRV

(2.35)

where M R s nm0 £ B /c 2 . The first term is the universal gravitational acceleration that would be expected in a theory satisfying WEP. The remaining terms depend upon the body's structure through the anomalous mass tensors in £B(X, V). Hence a violation of Local Lorentz or Position Invariance implies a violation of WEP. Equivalently, WEP {dm^ = dm{k = 0) implies Local Lorentz and Position Invariance. Equivalently, WEP implies EEP. The gravitational red-shift experiment can also be studied within this framework, using a cyclic gedanken experiment suggested by Nordtvedt (1975). The cycle begins as before with a set of n free particles of mass m0 at rest at X = h. We form a composite body and release the binding energy £ B (h,0)[l L/(h)/c2] in the form of a massless quantum which propagates to X = 0. Its energy there, compared to the energy £ B (0,0)[l l/(0)/c 2 ] of a quantum emitted from an identical system at X = 0, is assumed to be given by (1 - Z)£ B (0,0)[l - t/(0)/c 2 ]. This energy is stored in a reservoir. Our goal is to evaluate the red shift Z. The body is then allowed to fall freely to X = 0, where it is brought to rest, with the kinetic energy of motion, -{nm0

- £ B (0,V)/c 2 ]A h - <5mJW

added to the reservoir. If we substitute for A from Equation (2.35), we see that the reservoir now contains energy (to quasi-Newtonian order) (1 - Z)£ B (0,0)[l - t/(0)/c 2 ] - [rnno - £ B (0,0)/c 2 ]g h -

Einstein Equivalence Principle and Gravitation Theory

43

We extract from the reservoir enough energy £B(0,0)[l - t/(O)/c2] to disassemble the system, and enough energy nmog h to return the n free particles to the starting point. Again, conservation of energy requires that the reservoir be empty and therefore that Z must satisfy (to first order in g h) - ZE% + Elg h/c 2 - dmyVUiJ h = 0

(2.36)

or Z = [At/ - (8r4Jc2/E$) AUir\/c2

(2.37)

where A C / s g h and AUiJ = Vt/° h. By a similar analysis one can show that the second-order Doppler shift between an emitter moving at velocity V and a receiver at rest, relative to a preferred universe rest frame, is given by (Haugan, 1979) Z D = - i V2/c2 +

ttSnf/E&VV'

(2.38)

Thus, the simple assumption of energy conservation has allowed us to "prove" Schiff's conjecture, as well as elucidate the empirical consequences of possible violations of the three aspects of the Einstein Equivalence Principle. Thorne, Lee, and Lightman (1973) have proposed a more qualitative "proof" of Schiff's conjecture for that class of gravitation theories that are based on an invariant action principle, so-called Lagrangian-based theories of gravity. They begin by defining the concept of "universal coupling": a generally covariant Lagrangian-based theory is universally coupled if it can be put into a mathematical form (representation) in which the action for matter and nongravitational fields / NG contains precisely one gravitational field: a symmetric, second-rank tensor # with signature + 2 that reduces to J/ when gravity is turned off; and when ^ is replaced by if, 7NG becomes the action of special relativity. Clearly, among all Lagrangian-based theories, one is universally coupled if and only if it is a metric theory (for details see Thorne, Lee, and Lightman, 1973). Let us illustrate this point with a simple example. Consider a Lagrangian-based theory of gravity that possesses a globally flat background metric i\ and a symmetric, second-rank tensor gravitational field h. The nongravitational action for charged point particles of rest mass m0 and charge e, and for electromagnetic fields has the form /NG = h + /in, + hm

(2-39)

Theory and Experiment in Gravitational Physics

44

where Io= -m0 jdr,

dx2 = -(rj^ + h^)dx"dx\

^ ^

d4x

(2.40)

where F^ = AVfll A ^ , and where

IMNIWI 1

'

(2.41)

We work in a coordinate system in which if = diag( 1,1,1,1). To see whether this theory is universally coupled, the obvious step is to assume that the single gravitational field i/^v is given by "/V = V + V

(2.42)

This would make J o and Jinl appear universally coupled. However, in the electromagnetic Lagrangian, we obtain, for example, tf* _ W = yj,** _ }f% + O(h3)

(2.43)

1

where ||^""|| = | | ^ | | " . Thus, there is no way to combine */ and h^ into a single gravitational field in ING, hence the theory is not universally coupled. To see that the theory is also not a metric theory, we transform to a frame in which at an event 2P,

Note that in this frame, h^ into the form

^ 0 in general, thus the action can be put

/NG = ^SRT + A/

(2.44)

where JSRT

= -m 0 Jdx + e JAndx* - (len)'1

j^F^F^-fj)1!2d*x,

AI = + O(h3F2) flC

9

(2.45)

wherefc = h* ]? ¥= 0. So in a local Lorentz frame, the laws of physics are not those of special relativity, so the theory is not a metric theory. Notice that in this particular case, for weak gravitationalfields(|fy,v| « 1), the theory is metric to first order in h^, while the deviations from metric form occur at second order in h^. In the next section, we shall present a

Einstein Equivalence Principle and Gravitation Theory

45

mathematical framework for examining a class of theories with nonuniversal coupling and for making quantitative computations of its empirical consequences. Consider now all Lagrangian-based theories of gravity, and assume that WEP is valid. WEP forces 7NG to involve one and only one gravitational field (which must be a second-rank tensor ty which reduces to r\ far from gravitating matter). If / NG were to involve some other gravitational fields <j>, Kp, h^,... they would all have to conspire to produce exactly the same acceleration for a body made largely of electromagnetic energy as for one made largely of nuclear energy, etc. This is unlikely unless i/^v and the other fields appear everywhere in JNG in the same form, for example, /((p)^^ if a scalar field is present, i//^ + ah^ if a tensor field is present, and so on. In this case, one can absorb these fields into a new field g^ and end up with only one gravitational field in JNG. This means that the theory must be universally coupled, and therefore a metric theory, and must satisfy EEP. One possible counterexample to Schiff 's conjecture has been proposed by Ni (1977): a pseudoscalar field <j> that couples to electromagnetism in a Lagrangian term of the form ^""F^F^, where s*11 is the completely antisymmetric Levi-Civita symbol. Ni has argued that such a term, while violating EEP, does not violate WEP, although it does have the observable effect of producing an anomalous torque on systems of electromagnetically bound charged particles. Whether this torque then can lead to observable WEP violations is an open question at present. 2.6

The THs/i Formalism

The discussion of Schiff's conjecture presented in the previous section was very general, and perhaps gives compelling evidence for the validity of the conjecture. However, because of the generality of those arguments, there was little quantitative information. For example, no means was presented to compute explicitly the anomalous mass tensors (5mj/ and 8m\J for various systems. In order to make these ideas more concrete, we need a model theory of the nongravitational laws of physics in the presence of gravity that incorporates the possibility of both nonmetric (nonuniversal) and metric coupling. This theory should be simple, yet capable of making quantitative predictions for the outcomes of experiments. One such "model" theory is the THe/x formalism, devised by Lightman and Lee (1973a). It restricts attention to the motions and electromagnetic interactions of charged structureless test particles in an external, static, and spherically symmetric (SSS) gravitational field. It

Theory and Experiment in Gravitational Physics

46

assumes that the nongravitational laws of physics can be derived from an action / N G given by f NG

=

Jo + hat + Iem> \{T -

E2 - li-^d'x

(2.46)

(we use units here in which x and t both have units of length) where mOa, ea, and x£(t) are the rest mass, charge, and world line of particle a, x° = t, v»a = dxljdt, E = \A0 - A o, B = (V x A), and where scalar products between 3-vectors are taken with respect to the Cartesian metric 8ij. The functions T, H, e, and n are assumed to be functions of a single external gravitational potential 4>, but are otherwise arbitrary. For an SSS field in a given theory, T, H, e, and /x will be particular functions of O. It turns out that, for SSS fields, equations (2.46) are general enough to encompass all metric theories of gravitation and a wide class of nonmetric theories, such as the Belinfante-Swihart (1957) theory and the nonmetric theory discussed in Section 2.5. In many cases, the form of / N G in equation (2.46) is valid only in special coordinate systems ("isotropic" coordinates in the case of metric theories of gravity). An example of a theory that does not fit the THsfi form of / N G is the Naida-Capella nonmetric theory (see Lightman and Lee, 1973a for discussion). Cases such as this must then be analyzed on an individual basis. For an "en" formalism, see Dicke (1962). (a) Einstein Equivalence Principle in the THe/x formalism We begin by exploring in some detail the properties of the formalism as presented in equations (2.46). Later, we shall discuss the physical restrictions built into it, and shall apply it to the interpretation of experiment. In order to examine the Einstein Equivalence Principle in this formalism we must work in a local freely falling frame. But we do not yet know whether WEP is satisfied by the THsn theory (and suspect that it is not, in general), so we do not know to which freely falling trajectories local frames should be attached. We must therefore arbitrarily choose a set of trajectories: the most convenient choice is the set of trajectories of neutral test particles, i.e., particles governed only by the action l0, since

Einstein Equivalence Principle and Gravitation Theory

47

their trajectories are universal and independent of the mass mOa. We make a transformation to a coordinate system x" = (?, x) chosen according to the following criteria: (i) the origins of both coordinate systems coincide, that is, for a selected event 3P, xx{@) - x\0>) = 0, (ii) at 0>, a neutral test body has zero acceleration in the new coordinates, i.e., d2xJ'/dt2^ = 0, and in the neighborhood of 9 the deviations from zero acceleration are quadratic in the quantities Ax* = x* x%0>), and (iii) the motion of the neutral-test body is derivable from an action Jo. The required transformation, correct to first order in the quantities g0? and gj, x, assumed small, is x = Hy\x

+ |tf 0-»Togof2 + ±Ho ' H^2xg 0 x - gox2)]

(2.47)

where the subscript (0) and superscript (') on the functions T, H, E, and fi denote To = T(x* = xs = 0),

r 0 = ^r/a
(2.48)

and where go = V*

(2.49)

The action Io in the new coordinates then has the form 'o = - I > o a fd - v2a)il2 dt{\ + O[(xa)2]}

(2.50)

where va = dxjdf. Note that our choice of the multiplicative factors Tj / 2 and HQ12 resulted in unit coefficients in / 0 , making it look exactly like that of special relativity. Similarly the actions /int and Iem can be rewritten in the new coordinates, with the result /in, = 2 X UfV? dt{\ + O[(XS)2]}, Im =

(2.51)

(Snr1eoTlo'2Ho1 - To

'HOEE V O

+ H0TE

1/2

'^(l -

JT'OTO

l

HE

1/2

A o go x)

r 0 f g 0 -(E x S)(l - To 'HoEE V O 1 ) } ^

+ [corrections of order (x*)2]

(2.52)

where A% =

(dx*/dxii)Aa

E = *At - A o,

6 = Vx A

(2.53)

Theory and Experiment in Gravitational Physics

48

and Ao = (2r o /T' o )(^o * + iT'oTo ' - iH'oHo *)

(2.54)

Let us now examine the consequences for EEP of physics governed by I NG . Focus on the form of JNG at the event 0>(xj = t = 0), since local test experiments are assumed to take place in vanishingly small regions surrounding 3". Because such experiments are designed to be electrically neutral overall, we can assume that the E and B fields do not extend outside this region. Then at 9

/NG= - X > o a f ( l -v2a) + (8TT)- hoT^Ho

1/2

J [ £ 2 - (To 'Hoeo Vo X)B2] d*x

(2.55)

We first see that, in general, /NG violates Local Lorentz Invariance. A simple Lorentz transformation of particle coordinates and fields in 7NG shows that JNG is a Lorentz invariant if and only if To 1 /f o £oVo 1 = l

or eofio=TolHo

(2.56)

Since we have not specified the event 0>, this condition must hold throughout the SSS spacetime. Notice that the quantity (TQ lHtfo Vo x ) 1/2 plays the role of the speed of light in the local frame, or more precisely, of the ratio of the speed of light clight to the limiting speed c0 of neutral test particles, i.e., To 'Hoeo Vo l = (clight/c0)2

(2.57)

Our units were chosen in such a way that, in the local freely falling frame, c0 = 1; equivalently, in the original THsfi coordinate system [cf. Equation (2.46)] c0 = (To/Ho)1'2,

clight = (EoAio)-1/2

(2.58)

These speeds will be the same only if Equation (2.56) is satisfied. If not, then the rest frame of the SSS field is a preferred frame in which / NG takes its THe/j. form, and one can expect observable effects in experiments that move relative to this frame. Thus, the quantity 1 ToHo lfioMo plays the role of a preferred-frame parameter: if it is zero everywhere, the formalism is locally Lorentz invariant; if it is nonzero anywhere, there will be preferred-frame effects there. As we shall see, the Hughes-Drever experiment provides the most stringent limits on this preferred-frame parameter.

Einstein Equivalence Principle and Gravitation Theory

49

Next, we observe that / NG is locally position invariant if and only if o 1/2 = [constant, independent of 9\ o 1/2 = [constant, independent of ^>]

(2.59)

Even if the theory is locally Lorentz invariant (TQ 1H0EQ VO * = 1> independent of &) there may still be location-dependent effects if the quantities in Equation (2.59) are not constant. This would correspond, for example, to the situation discussed in Section 2.3, in which different parts of the local physical laws in a freely falling frame couple to different multiples of the Minkowski metric; in this case, free particle motion coupling to 7 itself, electrodynamics coupling to the position-dependent tensor i\* =. eTi/2H'i/2ti in the manner given by the field Lagrangian ri*'"'rivfFllvFltf. The nonuniversality of this coupling violates EEP and leads to position-dependent effects, for example, in gravitational redshift experiments (also see Section 2.4). An alternative way to characterize these effects in the case where Local Lorentz Invariance is satisfied is to renormalize the unit of charge and the vector potential at each event & according to e*a = ettso 1/2To 1/4 H S/4,

Af = A^T^H^

(2.60)

then the action, (2.55), takes the form

+ (8TT)- * j(E*2 - B*2) d4x

(2.61)

This action has the special relativistic form, except that the physically measured charge e* now depends on location via Equation (2.60), unless E0TII2HQXI2 is independent of 9. In the latter case, the units of charge can be effectively chosen so that everywhere in spacetime, soTlol2Ho112

= 1

(2.62)

Note, however, that if LPI alone is satisfied, one can renormalize the charge and vector potential to make either £0TQI2HQ 1/2 = 1 or fioTll2Ho 1/2 = 1, but not both, thus in general LLI need not be satisfied. Combining Equations (2.56), (2.59), and (2.62), we see that a necessary and sufficient condition for both Local Lorentz and Position Invariance to be valid is e0 = n0 = (Ho/T0)112, for all events 9 (2.63)

Theory and Experiment in Gravitational Physics

50

Consider now the terms in / NG , in Equations (2.50)-(2.52) that depend on the first-order displacements x, t from the event 9. These occur only in / em , and presumably produce polarizations of the electromagnetic fields of charged bodies proportional to the external "acceleration" g0 = V4>. One would expect these polarizations to result in accelerations of composite bodies made up of charged test particles relative to the local freely falling frame (i.e., relative to neutral test particles), in other words, to result in violations of WEP. These terms are absent if F o = Ao = 0, and U i.e., eoTlol2Ho

1/2

= const, noTl'2Ho EoHo^HoTo1

m

= const, (2.64)

Again, the units of charge can be normalized so that e0 = Ho = (H0/T0)1'2, for all 9

(2.65)

But this condition also guarantees Local Lorentz and Position Invariance. Thus, within the THe/j. formalism, for SSS fields [Equation (2.65)] => WEP, [Equation (2.65)] => EEP

(2.66)

However, the above discussion suggests that WEP alone may guarantee Equation (2.65) and thereby EEP. We can demonstrate this directly by carrying out an explicit calculation of the acceleration of a composite test body within the THsfi framework. The resulting restricted proof of Schiff's conjecture was first formulated by Lightman and Lee (1973a). (b) Proof of Schiff's conjecture We work in the global THsfi coordinate system in which JNG has the form Equation (2.46). Variation of / NG gives a complete set of particle equations of motion and "gravitationally modified" Maxwell (GMM) equations, given by (d/dt)(HW~ \ ) + {W- XV(T - Hvl) = aL(xa), aL(x0) = (ea/mOa){VAo(xa) + V[va A(xfl)] - dA{xa)/dt}, V (EE) = 4np, V x ( ^ » B ) = 4TTJ + d{eE)/dt where W = {T-

(2.67)

Hv2a)112,p = Y.aea<53(x - xfl), J = £ a e a y a 5 3 (x - xa), and

aL is the Lorentz acceleration of particle a. These equations are used to

Einstein Equivalence Principle and Gravitation Theory

51

calculate the acceleration from rest of a bound test body consisting of charged point particles. A number of approximations are necessary to make the computation tractable. First, the functions T, H, e, and fi, considered to be functions of
(2.68)

where To = T(x = 0), T'o = dT/d<S>\x=0. As long as the body is small compared to the scale over which d> varies, we can assume that g0 x « 1, and work to first order in g0 x. Second, we assume that the internal particle velocities and electromagnetic fields are sufficiently small so we can expand the equations of motion and GMM equations in terms of the small quantities v1 ~ e2/mor « 1 where r is a typical interparticle distance. By analogy with the postNewtonian expansion to be described in Chapter 4, we call this a postCoulombian expansion; for the purpose of the present discussion we shall work to first post-Coulombian order. We expect the single particle acceleration to contain terms that are O(g0) (bare gravitational acceleration), O(v2) (Coulomb interparticle acceleration), O(gf0t;2) (post-Coulombian gravitational acceleration), O(v4) (post-Coulombian interparticle acceleration), O(g0v*) (post-post-Coulombian gravitational acceleration), and so on. To O(g0v2), we obtain o + itf'oHo'goi;2 + (T'oTo 1 - H'0H0- l)g0 vavfl + Ty2H» X W

(2-69)

To write the Lorentz acceleration aL(xa) directly in terms of particle coordinates, we must obtain the vector potential A^ in this form to an appropriate order. In a gauge in which £Mo,o - V A = 0

(2.70)

the GMM equations take the form V 2 4 0 - ej^o.oo = 4ns~1p-e~1\e2

V A - e/xA.oo = ~^nJ

(\A0 - A,o), x

+ (sp)- V(sp)V A - ^ V / z x (V x A) (2.71)

These equations can be solved iteratively by writing Ao = A^ + A%\

A = A(0) + A(1)

(2.72)

Theory and Experiment in Gravitational Physics

52

where A(v/A(°y ~ O(g0), and solving for each term to an appropriate order in v2. The result is A o = -4> A = A(0) + O(0O)

(2.73)

where

a

The resulting single-particle acceleration is inserted into a definition of center of mass. It turns out that to post-Coulombian order, it suffices to use the simple center-of-mass definition Y-

-iv

-v

2 J wOaXa>

A = m

a

m

(2-75) m

= ZJ

0a

a

We then compute d2X/dt2, substituting the single-particle equations of motion to the necessary order, and using the fact that, at t = 0, X = 0, dX/dt = 0. The resulting expression is simplified by the use of virial theorems that relate internal structure-dependent quantities to each other via total time derivatives of other internal quantities. As long as we restrict attention to bodies in equilibrium, these time derivatives can be assumed to vanish when averaged over intervals of time long compared to internal timescales. Errors generated by our choice of center-of-mass definitions similarly vanish. To post-Coulombian order, the required virial relation is =o (2.76) where angular brackets denote a time average, and where (2.77) ab

a

where xab = xa xb, rab = |xa(,|, and the double sum over a and b excludes the case a = b. The final result is d2Xl/dt2 = g{-

^[TE1'%1rojea/m)

+ 0 J '[ro 1/2 eo 1 (l-T o Ho 1 eo/Xo)] x (JEjf/m + <5yE E »

(2.78)

where Vo

(2-79)

Einstein Equivalence Principle and Gravitation Theory

53

where F o is given by Equation (2.54), and where

Ef? = ,

EES = ( £

Q

")

(2-8°)

The first term gl is the universal acceleration of a neutral test body (governed by Io alone); the other two terms depend on the body's electromagnetic self-energy and self-energy tensor. These terms vanish for all bodies (i.e., WEP is satisfied) if and only if (2.81) at any event 0>, which is equivalent to Equation (2.63). Hence WEP => EEP and Schiff's conjecture is verified, at least within the confines of the THEH formalism.

It is useful to define the gravitational potential U whose gradient yields the test-body acceleration g; modulo a constant U(x)= ~^T'0Ho

x

g0-x

(2.82)

If the functions T, H, e, and fi are now considered as functions of U instead of
(2.83)

(c) Energy conservation and anomalous mass tensors Because the THefi formalism is based on an action principle, it possesses conservation laws, in particular a conservation law for energy, and so is amenable to analysis using the conserved-energy framework described in Section 2.5. The main products of that framework are the anomalous inertial mass tensor Sm[J and passive gravitational mass tensor Sm'j obtained from the conserved energy. These two quantities then yield expressions for violations of WEP, Local Lorentz Invariance, and Local Position Invariance. As a concrete example (Haugan, 1979), we consider a classical bound system of two charged particles. As in the above "proof" of Schiff's conjecture we work to post-Coulombian order and to first order in g 0 x. We first formulate the equations of motion in terms of a truncated action ?NG = h + An,

(2-84)

Theory and Experiment in Gravitational Physics

54

where 7im is rewritten entirely in terms of particle coordinates by substituting the post-Coulombian solutions for A^, Equation (2.73), into I int . Variation of 7NG with respect to particle coordinates then yields the complete particle equations of motion. We identify a Lagrangian L using the definition 7NG = $Ldt

(2.85)

We next make a change of variables in L from xu x2, vl5 and v2, to the center of mass and relative variables X = ( m ^ + m2x2)/m, x = xt x2, V = dX/dt, v = dx/dt (2.86) where m = mx + m2, n = mlm2/m. A Hamiltonian H is constructed from L using the standard technique Pj s dL/dV{ pi = BL/dv3, H s PJVj + pV - L

(2.87)

The result is H = Tj'2m(l + iTiTo »g0 X) + n'2HE lP2/2m

eie2/r)go X - Tj/2Ho 2[(p P)2 o \e,e2lr)\P2 + (n P)2]/2m2} + hT0li0Ho \m2 - m1)(e1e2/r)(p P + ii pfi + O(p4) + O(P4) (2.88) 4 4 where n = x/r. The post-Coulombian terms O(p ) and O(P) neglected in Equation (2.88) do not couple the internal motion and the center-of-mass motion and thus do not lead to violations of EEP. We now average H over several timescales for the internal motions of the bound two-body system, assumed short compared to the timescale for the center-of-mass motion. The average is simplified using virial theorems obtained from Hamilton's equations for the internal variables derived from H. The relevant expressions are + post-Coulombian terms), H

j

n (n p)]>

(2.89) (2.90)

Einstein Equivalence Principle and Gravitation Theory

55

Notice that although the post-Coulombian corrections in Equation (2.89) may depend on the center of mass variables P or X, this dependence does not affect the form of if; it is only the explicit dependence on P and X in Equation (2.88) that generates the center-of-mass motion. The resulting average Hamiltonian is then rewritten in terms of V using VJ = d/dPJ. The conserved energy function Ec used in Section 2.5 is then defined to be Ec = Tll2Ho\H}, so that at lowest order, Ec = mc\ = m(ToHo *). The result is Ec = M(T0Ho ') + i x (£«%

)]

|[

^^o}

xT'oHo'go-X

(2.91)

where M = m +
(2-92)

By defining the "binding" energy and energy tensor by Ef = -c o 2 {i/ o -y/2/i + To 1 ES

E

&

1/2

eo'e^/r)

+ [post-Coulombian corrections],

l

Ef?

(2.93)

and using Equation (2.82), we cast Equation (2.91) into the form £ c = MRC2, - MRt/(X) + \MKV2

(2.94)

MRC2. = mcl - Ef + tymiWVi - dm\!Ui}

(2.95)

dm? = 2(1 - Toffo'eoMoX^w + £^)/cS. j/ i s i)<5 ij '

(2.96)

where

with

Substitution of these formulae into Equation (2.35) for the center-of-mass acceleration of the system yields precisely Equation (2.78). One advantage of the Hamiltonian approach is that it can also be applied to quantum systems (Will, 1974c). This is especially useful in discussing gravitational red-shift experiments since it is transitions between quantized energy levels that produce the photons whose red shifts are measured. For the idealized gravitational red shift experiments discussed in Section 2.5, only the anomalous passive mass tensor 5m^ is needed. The simplest quantum system of interest is that of a charged

Theory and Experiment in Gravitational Physics

56

particle (electron) moving in a given external electromagnetic potential of a charged particle (proton) at rest in the SSS field, i.e., a hydrogen atom. For such a system the truncated Lagrangian [Equation (2.85)] has the form L = - me(T0 - Hov2)112 - eA^if

(2.97)

where m0 = me and e + \e\ for the electron. We shall ignore the spatial variation of T, H, e, and \i across the atom, hence we evaluate each at x = 0. The Hamiltonian obtained from L is given by H = n / 2 [m e 2 + Ho > + eA| 2 ] 1/2 + eA0

(2.98)

where Pj = dL/dvK Introducing the Dirac matrices

where / is the two-dimensional unit matrix and
H = Ti^lmJ

(2.100)

The gravitationally modified Dirac (GMD) equation is then H\\l>y = ih(8/dt)\il/y

(2.101)

For most applications it is more convenient to use the semirelativistic approximation to if obtained by means of a Foldy-Wouthuysen transformation, yielding H = Ty\me

+ Ho J |p + eA\2/2me - Ho 2p*/Sm2 + HQ leho B/2m,]

+ eA0-HQ

l

{eh/4m2)a - ( E x p - % i h \ xE) (2.102)

where we have made the usual identification p -» ih\ and have ignored the effects of spatial variations in T0,H0,s0, or fi0 on the atomic structure. For a charged particle with magnetic moment M p at rest at the origin, the vector potential as obtained from the GMM equations is given (to the necessary accuracy) by Ao = -e/sor,

A = iu 0 M p x x/r 3

(2.103)

The Hamiltonian then takes the form H = Hr + Hs + H( + HM + O(p6)

(2.104)

Einstein Equivalence Principle and Gravitation Theory

57

where Hr = Tl'2me, H(=-Tlo'2Ho2p4/^m2 - Hvh^{e2hl4m2er3)o Hu = T^ 2 [Ho l(ehl2me)o B]

L, (2.105)

where L = r x p is the angular momentum of the electron. The four pieces of H are the usual rest mass (Hr), Schrodinger (Hs), fine-structure (H{) and hyperfine-structure (Hhf) contributions. We have ignored the Darwin term (oc V E). The magnetic field produced by the proton is given by B = V x A = - i ^ o { [ M p - 3n(fi M p )]/r 3 - (87t/3)Mp<53(x)}

(2.106)

We must first identify the proton magnetic moment. From the hyperfine term Hhl, it is clear that the magnetic moment of the electron is given by M e = T£ /2 #o H - eh/2me)o

(2.107)

It is then reasonable to assume that the magnetic moment of the proton has the same dependence on T o and Ho, Mp = T^Ho

l

(gpeh/2mjap

(2.108)

where gp is the gyromagnetic ratio of the proton and mp is its mass. Then Hbf =

-yoToHo2(gpe2h2/4memp) x ae {|>p - 3fi(il ffp)]/r3 - (8rt/3)«Tp^3(x)}

(2.109)

Solving for the eigenstates of the Hamiltonian using perturbation theory yields £ = Ty\me

+ £p(HoTolEo2) + *AH0TZ h^ 2 ) 2 l

(2.H0)

where ip, Su and SM are the usual expressions for the principal, finestructure, and hyperfine-structure energy levels in terms of atomic constants me, e, mp, gp, h, and quantum numbers. In order to calculate the anomalous mass tensors <5mj/, we must determine the manner in which E varies as the location of the atom is changed. Expanding E to first order in g X, substituting Equation (2.82), and converting to the conserved energy function Ee - E(Tk'2/H0), we obtain Equation (2.30) (with V = 0), with E% = Ef + El-¥EW

(2.111)

Theory and Experiment in Gravitational Physics

58

where £ B e 0 <5p, F

£ B ttoio

3

£g =-Wo 'hf

e0 0(,

(2-112)

and H*

2r o (£| s /cg)^ ii

(2.113)

= 4ro{El/cl)Sij

(2.114) 2

= (3r 0 - A0)(E^/c )8^

(2.115)

Compare Equation (2.113) with Equation (2.96). A useful fact that emerges from the solution for the energy eigenstates is that the Bohr radius is given by a = (e0Ti'2/H0){h2/mee2)

(2.116)

This will be important in analyzing the gravitational red shift of microwave cavities. (d) Limitations of the THefi formalism The THefi formalism is a very strong - perhaps overly strong idealization of the coupling of electromagnetism to gravity. The question naturally arises, can the formalism be applied to realistic physical situations where there are no SSSfieldsand where strong and weak interactions may be present? We shall discuss each of these points in turn. (i) SSS Fields In practical experimental situations, say in an Earthbound laboratory, there are, strictly speaking, no SSS fields: orbital and rotational motions of the planets cause the gravitational potentials to change with time, and the superposition offieldsfrom the Sun and planets leads to asphericalfields.However, the evolution of the gravitational fields occurs on a much longer timescale than the internal (atomic) timescales of typical laboratory experiments, and so the fields can be treated quasistatically. Furthermore, most experiments of interest single out one static, nearly spherical gravitational field by exploiting a symmetry, by modulation, or by some other technique. (For example, singling out of the solar field by searching for a torque with a 24 h period in the DickeBraginsky versions of the Eotvds experiment.) A potentially more serious criticism of the SSS restriction is the possibility of relativistic, nonisotropic effects due for example to the orbital motion of the planets, or to the motion of the solar system relative to the mean rest frame of the universe. These ef-

Einstein Equivalence Principle and Gravitation Theory

59

fects would produce off-diagonal terms in the action / NG , such as F v in Jo or G (E x B) in / em , where F and G are vector gravitational functions. In the case of the overall motion of the solar system, one can see that the frame in which the solar potential is spherical is in motion relative to the frame in which the cosmic background field is spherical (isotropic), therefore there must be two limiting actions of the THe/i form, one applicable to each situation. These limiting cases can be handled by a single action of the THefi form only if the theory is Lorentz invariant, i.e., only if TH~1e/i = 1. Nevertheless, if either of these off-diagonal effects occurs, they will be smaller than the dominant SSS effects by factors of order |v| ~ [orbital velocity of planets] ~ 10 ~4 or |w| ~ [solar system velocity] ~ 10"3. The simplest way to summarize is as follows: the restriction to SSSfieldsis an approximation that may overlook observable effects, however, the experimental consequences that emerge from the pure SSS version are sufficiently interesting and, we believe, sufficiently generic to a broad class of gravitational theories, that powerful conclusions about the nature of gravity can be made within the standard THs/i framework. With this caveat in mind, for most of the remainder of this chapter we will assume that every experiment discussed takes place in a SSS field. (ii) Weak and strong interactions The coupling of classical electromagnetic fields to gravitation is well understood within metric theories of gravity (see Section 3.2) and has been formulated in many nonmetric theories. By contrast, the laws of weak and strong interactions have only recently been given an adequate mathematical representation even in the absence of gravity, and the problem of their coupling to gravity is made even more complicated by the fact that the theories of these interactions fundamentally involve quantum field theory. Thus, at present, electromagnetism is the only interaction amenable to a detailed analysis of EEP using something like the THe/j. formalism. Nevertheless, a violation of EEP by electrodynamics alone can lead to many observable effects, barring fortuitous cancellations, and to several important experimental tests. Consequently, for the remainder of this discussion we shall simply ignore the strong and weak interactions, or if necessary assume that they obey EEP. (e) Application to tests of EEP We now turn to the experiments that test EEP and study the constraints they place on the coupling of electromagnetism to gravity in SSS gravitational fields.

Theory and Experiment in Gravitational Physics

60

Tests of WEP Equation (2.78) gives the acceleration of a composite body through post-Coulombian order in an external SSS field. However, for the purpose of comparing the predicted acceleration with the results of Eotvos experiments, that expression is not accurate enough. The WEP-violating terms in Equation (2.78) are of order EBS/m ~ 10" 3 for atomic nuclei; therefore, WEP-violating terms of order (EES/m)2 ~ {EES/m)v2 ~ 10 ~6 would also be strongly tested by Eotvos experiments accurate to a part in 1012. To obtain these terms, Haugan and Will (1977) extended the Lightman-Lee computation to post-post-Coulombian order (the Hamiltonian method could also have been used). When specialized to composite bodies that are spherical on average (a good approximation for experimental situations), the resulting acceleration is given by d2X/dt2 = g{l + (Efs/Mc2)[2r0 - f(l - ToH

(2.117) where [cf. Equations (2.77), (2.80), and (2.93)]

Ef = ab

o Vo ( l eaebr;b\va yb + (vfl ab

2 V Vo ( abI W a V [ v a yb- (ya *ab)(yb x j r i ] ) (2.118) \ I Because we shall shortly obtain a very tight upper limit on the coefficient 1 T0HQ 1e0^i0 from the Hughes-Drever experiment, we shall simply set it equal to zero in Equation (2.117). Then the results for the Eotvos ratios defined in Equation (2.2) are

^ES = |2T0|,

r, = |2A0|

(2.119)

The quantities E| S an< l £ B S given by Equation (2.118) were estimated for various substances in Section 2.4 [Equations (2.8) and (2.9)] and provided experimental limits on nES and nm that are equivalent to |r o | < 2 x 10- 10 ,

|A0| < 3 x 10- 6

Recall that if EEP is satisfied, r 0 = Ao = 0.

(2.120)

Einstein Equivalence Principle and Gravitation Theory

61

Tests of LLI The Hughes-Drever experiment can now be analyzed in detail using the TH&n formalism (Haugan, 1978). Equations (2.95) and (2.96) demonstrate the possibility of an inertial mass anisotropy 3m\j that leads to a contribution to the binding energy given by SEB = -$8m\'ViVj

(2.121)

where V is the velocity of the body relative to the THefi coordinate system. This term could lead to energy shifts of states having different values of 5m\j and thus to observable effects in a quantum mechanical transition between these states. In the case of the Hughes-Drever experiment, the system, a 7Li nucleus, can be approximated as a two-body system consisting of a J = 0 core (two protons and four neutrons) of charge + 2, and a valence proton in a ground state with angular momentum of 1. The spin of the proton couples to its angular momentum to yield a total angular momentum J = f. In an applied magnetic field, the four magnetic substates Af, = ±j, + § are split equally in energy, giving a singlet emission line for transitions between the three pairs of states. How does SEB alter the energies of these four states? The isotropic part of dm\J oc EBsd'J simply shifts all four levels equally, since < JMi\e1e2r~ l\ JM3} is independent of Mj. However, the other contribution to 8m\J oc £{[? does shift the levels unequally. We first decompose V into a component V^ parallel to the applied magnetic field and a component V± perpendicular to it. Then 0]

(2.122)

where 0, (j> are polar coordinates appropriate to the orbital wave function Aim, =/('")5inii(0>0)- By combining the orbital wave function and spin states into states of total J, Ms, we then calculate the expectation value of (x'xJ/r3)ViVJ in states of different M,. Inserting these results into the formula, (2.121), for 8EB, and taking the difference in the energy shifts between adjacent Af, states, we find that the singlet line splits into a triplet with relative energies

Mi - -i) = o, M-i--!)=-<5

(2-123)

S = &(£f/cg)(l - ToHo lHH0){Vl - 2VD

(2.124)

where

Theory and Experiment in Gravitational Physics

62

In the notation of Section 2.4, Equation (2.13), we have <5ES = £ ( 1 - ToHo 'eolhKVl - 2F(j)

(2.125)

The limit set by the experiment was |<5ES| < 10~ 22 . If we treat the laboratory as being in motion in the SSS field of the Sun, then V^ ~ VL ~ 10" 4 ; hence, as evaluated at the Earth, |1 - ToHo'Bo/ioU = I1 - (Co/clighl)2U < 10" 1 3

(2.126)

We can also assume the laboratory to be moving in the quasistatic, spherically symmetric background field of the universe, with velocity V^ ~ VL ~ 10" 3 , then for that portion of the THEH fields associated with the asymptotic cosmological model (labeled by the subscript oo), we obtain Il-TVO^^HT 15

(2.127)

Although there may be observable effects due to the possible nonmeshing of these two SSS fields into a single THefi field, they are unlikely to cancel the effects we have derived and negate the limits obtained above. The central conclusion is that to within at least a part in 1013, Local Lorentz Invariance is valid. Tests of Local Position Invariance Consider gravitational red-shift experiments. Suppose, for example, one measures the gravitational red shift of photons emitted from various transitions of hydrogen, such as principal transitions, fine-structure transitions within a principal level, or a hyperfine transition in, say, the ground state (21 cm line, basis for hydrogen maser clocks). Then, substituting Equations (2.113)(2.115) into Equation (2.37), we obtain (Will, 1974c) Zp=[l-2r0]AC//c2, Zf=[l-4r0]Al//c& Z hf = [1 - (3r 0 - Ao)] AU/c2.

(2.128)

Notice that the three shifts are different in general. Thus the gravitational red shift depends on the nature of the clock whose frequency shift is being measured unless F o = Ao = 0, i.e., unless LPI is satisfied [Equation (2.59)]. The red-shift parameters a discussed in Section 2.4(c) can thus be read off from Equations (2.128). The Vessot-Levine Rocket red-shift experiment thus sets the limit |(3r 0 - Ao)| < 2 x KT 4

(2.129)

Einstein Equivalence Principle and Gravitation Theory

63

To analyze the Stanford null gravitational red-shift experiment, we must calculate the energy of a microwave cavity. The energy in question is that of an electromagnetic mode whose wavelength is determined by the length of the cavity. The vector potential for the mode can be written, in second quantized notation, A = N(a t eexp[i(k x - cot)] + h.c.)

(2.130)

+

where a is a creation operator, e a polarization vector, N a normalization constant, and h.c. denotes Hermitian conjugate. We have suppressed the sum over k and e. The GMM equations (2.67) yield the dispersion relation |k| 2 -

E0H0CO2

=0

(2.131)

The energy of the electromagnetic field obtained from the canonical Hamiltonian is E = %(aa< + a^a)hco

(2.132)

However, for a stationary mode, the wave number k must satisfy k L = nn

(2.133)

where |L| is the length of the cavity and n is an integer. But it is clear that |L| is proportional to an integer (number of atoms in a line along the length of the cavity) times the Bohr radius a (which determines the interatomic spacing). But from Equation (2.116) we find L cc(eoTo/2Ho l) x (atomic constants, integers), hence, |k| cc H0TQ 1/2SQ 1. Combining Equations (2.131) and (2.132), we finally obtain E = ^csoHoV/Vo^eo3/2 (2-134) where <^SGSO depends only on atomic constants and integers. Expanding in terms of g0 x, and calculating the conserved energy function Ec, we obtain Equation (2.30) with pSCSO _ _ v

C

B

0

-1/2.-3/2

scsoA t o

fc

o

>

6n#= i(3r 0 + A o )(EF°/co)* y

(2.135)

Thus for a superconducting-cavity stabilized oscillator clock Zscso = [1 - l(3r 0 + Ao)] AL//c2

(2.136)

or, in the comparison between a cavity clock and a hydrogen maser clock [see Equation (2.31)] + f (r 0 - A 0 )t//c 2 ]

(2.137)

Theory and Experiment in Gravitational Physics

64

The experimental limit is thus |r o -A o |
(2.138)

(f) The Belinfante-Swihart nonmetric theory As a specific example of the application of the THz\i formalism to the analysis of gravitational theory and experiment we consider the Belinfante-Swihart (1957a,b,c) theory. This theory treats gravity as a symmetric second rank tensor field B on a Riemann-fiat background metric (prior geometry), t\. We first define a "particle metric" g^ according to

( 2 - 139 )

H ~ &,) = $

where K is an arbitrary constant, and where indices on B^ and A^v are raised and lowered using n^. In a coordinate system in which t\ = diag( 1, 1,1,1), the nongravitational action can be put into the form (Lee and Lightman, 1973) r(

dx11 dxv\1/2

/NO = - 1 «o. J ( - 0,v -£-ft)

c

dt + ^ea JA.ixl) dx" -

(2.140) where, through second order in B, H^ is related to the Maxwell field H, v = FMV(1 + i B + i^ 2 ) + 2FAUBJ,(1 + B) - 2Fx(MBt}Bl - 2Fi.B£,B;, + O(Ffi3)

(2.141)

It turns out that, to first order in B, the electromagnetic part of the action can be put into metric form (see Section 3.2 for discussion of this form), but not to second and higher orders. The particle and interaction parts of / N G are already in metric form. The action for the gravitational field IG = -(167T)-1 §(aB»JB& + fB
(2.142)

where a and / are arbitrary constants. In the weak field, post-Newtonian limit appropriate for application to solar system experiments (see Chapter 5), the theory can be made to agree with all experiments performed to date. Thus, the theory was thought

Einstein Equivalence Principle and Gravitation Theory

65

to be a completely viable alternative to general relativity. However, because of the deviations from metric form in the electromagnetic action, the theory violates EEP. We therefore expect it to violate WEP, although at second order in B^. To demonstrate that this is indeed the case, we first compute B^ for a SSS field, then recast /NG into THefi form. The solution of the gravitational field equations (Section 5.5) yields the form B oo = b0,

Bij & b&j

(2.143)

where b0 and b t are functions of a gravitational potential U. Then, from Equations (2.143) and (2.139), we find to O(b2), 0Oo = -(l-bo-2Kb + fb2 + 2Kbb0 + K2b2), 2Kb + Jfef - 2Kbb1 + K2b2) g..= ^.(l + bt-

(2.144)

where b = b0 + 3b ^ We have assumed for simplicity that far from the gravitating source, b0 and b^ vanish (see Section 5.5 for discussion). Substituting Equations (2.144) and (2.141) into Equation (2.140) puts / NG into THe/j. form to O(b2), with T = 1 - b0 - 2Kb + Ibl + 2Kbb0 + K2b2, + K2b2, / * = [ ! + i(*o + *»i)]

(2-145)

In the weak-field limit, it turns out that the SSS solutions for b0 and fcx have the form (see Section 5.5) bo = 2CoU,

b1 = 2C1U

(2.146)

where U is the Newtonian gravitational potential and Co and Ct are arbitrary constants. Then T = 1 - 21/ + 2t/ 2 [i + C o ] + O(l/ 3 ), H = 1 + 2U[C0 + d - 1] + C/2[(C! + C0)(3C1 + Co) - 4CX - 2C 0 + 1] + O(t/ 3 ), £ = 1 + U(C0 + d ) + U2(C0 + Cx)2 + O(t/ 3 ), H = 1 + U(C0 + C t ) + O(t/ 3 ) where we have chosen the values of C o , d , Co + 2K(3Cl - Co) = 1

an<

i^

(2.147)

sucn tnat

(2.148)

in order to ensure that T = 1 1U + .... This will guarantee that the particle Lagrangian will yield the correct Newtonian limit. Notice that,

Theory and Experiment in Gravitational Physics

66

to first order in U, these functions satisfy the EEP constraint in Equation (2.63), but to second order they do not in general. Now in solar system tests of post-Newtonian effects, where the consequences of electromagnetic violations of EEP are negligible, the coefficients (^ + Co) and (Co + C t 1) in T and H are simply the PPN parameters /? and y (see Chapter 4). Solar system measurements of light deflection, radar-time delay, and the perihelion shift of Mercury (see Chapter 7) constrain these parameters by |2C0 + 4C t - 7| < 0.1, \C0 + d - 2| < 0.002

(2.149)

Equations (2.83) and (2.147) then yield

r0 = -2c o (c o + cx)u + o(t/2), Ao = 2C1(C0 -I- d)U + O(U2)

(2.150)

Using the above constraints on Co and C t along with the value U = [/Q s 10"8, the relevant local potential for the Princeton-Moscow Eotvos experiments, we obtain |r o | ^ 1.7 x 10- 8

(2.151)

which violates the experimental limit, Equation (2.120), by a factor 80. Thus, the Belinfante-Swihart theory is unviable.

Gravitation as a Geometric Phenomenon

The overwhelming empirical evidence supporting the Einstein Equivalence Principle, discussed in the previous chapter, has convinced many theorists that only metric theories of gravity have a hope of being completely viable. Even the most carefully formulated nonmetric theory - the Belinfante-Swihart theory - was found to be in conflict with the Moscow Eotvos experiment. Therefore, here, and for the remainder of this book, we shall turn our attention exclusively to metric theories of gravity. In Section 3.1, we review the concept of universal coupling, first defined in Section 2.5. Armed with EEP and universal coupling, we then develop, in Section 3.2, the mathematical equations that describe the behavior of matter and nongravitational fields in curved spacetime. Every metric theory of gravity possesses these equations. Metric theories of gravity differ from each other in the number and type of additional gravitational fields they introduce and in the field equations that determine their structure and evolution; nevertheless, the onlyfieldthat couples directly to matter is the metric itself. In Section 3.3, we discuss general features of metric theories of gravity, and present an additional principle, the Strong Equivalence Principle that is useful for classifying theories and for analyzing experiments. 3.1

Universal Coupling

The validity of the Einstein Equivalence Principle requires that every nongravitational field or particle should couple to the same symmetric, second rank tensorfieldof signature 2. In Section 2.3, we denoted this field g, and saw that it was the central element in the postulates of metric theories of gravity: (i) there exists a metric g, (ii) test bodies follow geodesies of g, and (iii) in local Lorentz frames, the nongravitational laws of physics are those of special relativity.

Theory and Experiment in Gravitational Physics

68

The property that all nongravitational fields should couple in the same manner to a single gravitational field is sometimes called "universal coupling" (see Section 2.5). Because of it, one can discuss the metric g as a property of spacetime itself rather than as a field over spacetime. This is because its properties may be measured and studied using a variety of different experimental devices, composed of different nongravitational fields and particles, and, because of universal coupling, the results will be independent of the device. Consider, as a simple example, the proper time between two events as measured by two different clocks. To be specific, imagine a Hydrogen maser clock and a SCSO clock at rest in a static spherically symmetric gravitational field. If each clock is governed by a Hamiltonian H, then the proper time (number of clock "ticks") between two events separated by coordinate time dt is given by

where E is the eigenstate energy of the Hamiltonian (or energy difference, for a transition). The results of Section 2.6 show that if, for instance, the THefi formalism is applicable, and if EEP is satisfied, e0 = /x0 = (Ho/To)112 everywhere, thus using Equations (2.110) and (2.134) we obtain for each clock JVH oc dt(H0To oc dt (H0TZ

m

Mo 1/2 6o 3/2 ) = TV2 dt

where the proportionality constants are fixed by calibrating each clock against a standard clock far from gravitating matter. Thus, each clock measures the same quantity T o (in metric theories of gravity, in SSS fields, To gOo) a n d the proper time between two events is a characteristic of spacetime and of the location of the events, not of the clocks used to measure them. Consequently, if EEP is valid, the nongravitational laws of physics may be formulated by taking their special relativistic forms in terms of the Minkowski metric r\ and simply "going over" to new forms in terms of the curved spacetime metric g, using the mathematics of differential geometry. The details of this "going over" are the subject of the next section. 3.2

Nongravitational Physics in Curved Spacetime In local Lorentz frames, the nongravitational laws of physics are those of special relativity. For point, charged, test particles coupled to electromagnetic fields, for example, these laws may be derived from the

Gravitation as a Geometric Phenomenon

69

action

a

-(167T)- 1 U A V^F A v -F^(-f?) 1 / 2 d 4 x

(3.1)

where F .= A

A

fj = det||^,||

(3.2)

Here, n^ is the Minkowski metric, which in Cartesian coordinates has the form In the local Lorentz frame, t}^ is assumed to have this form only up to corrections of order [x s x s (^)] 2 , where x s (^) is the coordinate of a chosen fiducial event in the local frame, in other words, r\^ is described more precisely as -1,1,1,1), According to the discussion in Section 3.1, the general form of these laws in any frame is obtained by a simple coordinate transformation from the freely falling frame to the chosen frame. This transformation is given by (3.4) Then, the vectors and tensors that appear in JNG transform according to n-^ = (dx«/dx»)(dxll/dxi)rlxlh dx* = {dx*/dx*)dx*, where J is the Jacobian of the transformation. Partial derivatives of fields, as for example in the formula for F^, transform according to a

'" + dx* dx* *

(

}

However, in the local frame, n^^ = 0. Thus, d2x*

_dx*_dif_dx>_ n

- ^ * - ^ ? a ? &? *"-

y+

82xp

dx»

a* d

&

ex

1 n< +

*

dx* &

dx* dx dx*n"

{

'

Theory and Experiment in Gravitational Physics

70

Using the fact that (dxydx^idx^/dx*)

= <5f

(3.8)

we obtain n y

^'

dx^dx* d2xs dx* dx° d2xd n ~ ~ fa* ~W dx^dx^ »~fa7~fa7 "dx^dx1* n<*

{

'

If we now define 9*e = 1ae,

(3-10)

M»IUl--gg-r.

(in)

then Equation (3.9) can be written

or, using Equations (3.11), (3.12), and (3.13), the Christoffel symbols F^y (also known as connection coefficients) take the form %

P,y

y,D

- gyfi,s)

(3.14)

Then Equation (3.6) becomes dx" dx0 ^^faJdx1^11'1^^

(115)

We define the covariant derivative ";" by Ax;P = A^ - n,A,

(3.16)

and notice that it transforms as a tensor; it can be shown that Atf^A'f + VnA1

(3.17)

where A* = gafiAp. Taking the determinant of Equation (3.5) yields n

= [det(Sx7dx*)]2g

(3.18)

where g = det g^, then 1 / 2

y i

2

(3.19)

Gravitation as a Geometric Phenomenon

71

Substituting these results into / NG gives JNG

_

r(

dx" dxv\112 .

= ~ L mOa J I -g^ I $

(

3

r,

Ju A + 2, eB Jf iA^dx"

.

2

0

)

where We notice that the transformation to an arbitrary frame has resulted simply in the replacements %v by # "comma" by "semicolon" (-f7)1/2d4x by (-gY'Wx

(3.22)

This is the mathematical manifestation of EEP. We must point out that the specific mathematical forms given above for the Christoffel symbols, transformation laws, and so on are valid only in coordinate bases (see MTW, Chapter 10 for further discussion). However, in this book we shall work exclusively in coordinate bases. Generally speaking, then, the procedure for implementing EEP is: put the local special relativistic laws into a frame-invariant form using Lorentzinvariant scalars, vectors, tensors, etc., then make the above replacements. It is simple to show that the same rules apply to the field equations and equations of motion derived from the Lagrangian. In the local frame they are (3.23) where dx = ( - f,p. dx" dx")112,

if = dx*ld%, rfr (3.24)

However, these are not in frame-invariant form. We must write =

MvU/iip,

(3.25)

Theory and Experiment in Gravitational Physics

72

where we have used the fact that for the four-dimensional delta function (fj)~ll2SA is invariant (since J<S4d4x = 1 or 0 regardless of the frame). Then in the general frame the equations are mOaDuJD% = ej^ul,

(3.26)

Ffvv

(3.27)

= 47r./"

where fa = (-9^ dx" dxv)112, DuJDx = u"umv, /" = I ea(-g)-V25\x

u" = dx"/dt, - xa)dx»/dt

(3.28)

a

However, here there is a potential ambiguity in the application of EEP to electrodynamics if one writes Maxwell's equations, (3.27), in terms of the vector potential A^. In the local Lorentz frame, Maxwell's equations have the special relativistic form Ay~A»-»,v=

-47tJ"

(3.29)

It is always possible to choose a gauge (Lorentz gauge) in which A" = 0, thus, since AV'")V = A"^11, we have = A»'\v = -47tJ*

(3.30)

It is tempting then to apply the rules of EEP to this equation to obtain ngA" s A":v.v =
(3.31)

However, there is another alternative. The curved-spacetime Maxwell equation, Equation (3.27), yields (3.32)

But covariant derivatives of vectors and tensors do not commute in curved spacetime, in fact in general A?* = Ke + *UAV

(3-33)

where R%ap is the Riemann curvature tensor, given by

*U = rf^. - r^

+ r ^ r j , - rj^rj.

(3.34)

Then Ar-».v = A " , * + R^A",

(3.35)

Gravitation as a Geometric Phenomenon

73

where R% is the Ricci tensor given by Ri = g**Ry»

Ryf = R-U

(3.36)

This version of Maxwell's equations in Lorentz gauge becomes n0A»-R$Al>= -4nJ", A], = 0

(3.37)

It is generally agreed that this second version is correct (although there is no experimental evidence one way or the other). To resolve such ambiguities, the following rule of thumb should be applied: the simple replacements (i; -* g, comma -* semicolon) should be used without curvature terms in equations involving physically measurable quantities (F"v is physically measurable, A* is not); and coupling to curvature should occur only with good physical reason (as in tidal coupling). (For a fuller discussion, see MTW, box 16.1.) An uncharged test body follows a trajectory given by Equation (3.26) with e = 0, namely Du^/Dx = 0. This equation can be written using Equations (3.17) and (3.28) in the form d V / d r 2 + r^(dxx/dr)(dxp/dz) = 0

(3.38)

This is the geodesic equation. The mathematics of measurements made by atomic clocks and rigid measuring rods follow the same rules since the structure of such measuring devices is governed by solutions of the nongravitational laws of physics. In special relativity, the proper time between two events separated by an infinitesimal coordinate displacement dx", as measured by any atomic clock moving on a trajectory that connects the events, is given by dT = (_^ v dx"dx v ) 1/2

(3.39) v

if the separation is timelike, i.e., »/ dx" dx < 0. The proper distance between two events as measured by a rigid rod joining them is given by ds = (r,liydx»dxv)112

(3.40)

if the separation is spacelike, i.e., n^ dx* dxv > 0. These results are independent of the coordinates used. Then in curved spacetime we have

[timelike] «> g^dx"dxv < 0, [spacelike] «» Sllv dx" dx" > 0

(3.41)

Theory and Experiment in Gravitational Physics

74

There is a third class of separation dx* between events, those for which rjllvdx"dxv = 0

(3.42)

These are called null or lightlike separations, and pairs of events that satisfy this condition are connectible by light rays. It is a tenet of special relativity that light rays move along straight, null trajectories, i.e., if k" = dx"/da is a tangent vector to a light-ray trajectory, then dW/d
iffc'ifc,= 0

(3.43)

where a is a parameter labeling points along the trajectory. It should not be forgotten, however, that this is at bottom a consequence of Maxwell's equations, valid only in the "geometrical optics" limit, in which the characteristic wavelength X [a^fc 0 )" 1 ] is small compared to the scale £P over which the amplitude of the wave changes. (For example, if might be the radius of curvature of a spherical wavefront.) Since the first of equations (3.43) can be written, in flat spacetime dkf/do = {dx*/da)k*y = kvk% = 0

(3.44)

then EEP yields the equations /cv/c?v = 0,

fe"/cv^v

= 0

(3.45)

i.e., the trajectories of light rays in the geometrical optics limit are null geodesies. It is useful to derive this result directly from the curved-spacetime form of Maxwell's equations, in order to illustrate the role and the limits of validity of the geometrical-optics assumption. In curved spacetime, the geometrical-optics limit requires that X be small compared both to ££ and to ffl, the scale over which the background geometry changes {01 is related to the Riemann curvature tensor), i.e., A/(min{&, Si}) = 1/L « 1

(3.46)

In this limit, the electromagnetic vector potential can be written in terms of a rapidly varying real phase and a slowly varying complex amplitude in the form (see MTW, Section 22.5 for details)

K = (a, + <* + y / £

(3.47)

where 6 is the real phase, a,,, b^,... are complex, and e is a formal expansion parameter that keeps track of the powers oiXjL. Ultimately, one takes only the real part of A^ in any physical calculations. We define the

Gravitation as a Geometric Phenomenon

75

wave vector K = e,v>

k

" = /V0 v

(3-48)

Then Maxwell's equations in Lorentz gauge [Equations (3.37)], yield 0 = A% = [(i/s)kv(av + sbv) + a]v + 0(e)]ei9'E, 0 = DgA" - R$A' = [ - s - 2Jfc,fcV +fib")+ 2(i/e)fc'aJ + (i/e)fef^a" + 0(e °j]emie

(3.49)

Setting the coefficients of each power of e equal to zero, we obtain for the leading terms in each equation fa, = 0,

(3.50)

k% = 0,

(3.51)

in other words, the amplitude is orthogonal to the wave vector, and the wave vector is null. Taking the gradient of Equation (3.51) and noting that &. = kv;il since /cM itself is a gradient, we get fcyc" = 0

(3.52)

which is the geodesic equation for k". The trajectory x^a) of the ray can then be shown to be related to k" by the differential equation dx"(a)/da =

fe"(xv)

(3.53)

where a is an affine parameter along the ray. For further discussion of the higher-order terms in Equation (3.49), see MTW, Section 22.5. Another useful and important form of the equations of motion for matter and nongravitational fields can be derived in the case where the equations are obtained from a covariant action principle. This will essentially always be the case, for the following reason: in special relativity, all modern viable theories of nongravitationalfieldsand their interactions take an action principle as their starting point, leading to an action / NG . The use of EEP does not alter the fact that the equations of motion are derivable from an action. Consequently, one is led in curved spacetime to an action of the general form

=

J

( 3 - 54 ) where qA and qAifl are the nongravitational fields under consideration and their first partial derivatives (e.g., M", A^, A^,...) and g^ and g^^ are

Theory and Experiment in Gravitational Physics

76

the metric and its first derivative. (The extension to second and higher derivatives is straightforward). The action principle <5/NG = 0 is covariant, thus, under a coordinate transformation, i? N G must be unchanged in functional form, modulo a divergence [see Trautman (1962) for discussion]. Consider the infinitesimal coordinate transformation x* -> x" + <5x",

<5x" = £"

(3.55)

Then the metric changes according to [cf. Equation (3.5)], a

= - g^% - gvx^ - g^J"

(3.56)

Assume the matter and nongravitational field variables change according to . - «A.*£V

<5<7A = «

(3-57)

where d^v are functions of x". Under this transformation, JSfNG changes by S 9

^

(3 58)

-

Substituting Equations (3.56) and (3.57), integrating by parts, dropping divergence terms, and demanding that JS?NG be unchanged for arbitrary functions £*, yields the "Bianchi identities"

w

J

l

=0

(3.59)

where S£CNO/3qx is the "variational" derivative of i ? N G defined, for any variable \j/, by

dx" V # / and T*" is the "stress-energy tensor," defined by T^^2(-g)-^8^NG/dg,v

(3.61)

Using the fact that (-ff).V2 = ( - 0 ) 1 / 2 n . we can rewrite Equation (3.59) in the form

(3.62)

Gravitation as a Geometric Phenomenon

11

However, the nongravitational field equations and equations of motion are obtained by setting the variational derivative of J£NG with respect to each field variable qA equal to zero, i.e., .= 0

(3.64)

which by Equation (3.63) is equivalent to Tl,y = 0

(3.65)

Thus, the vanishing of the divergence of the stress-energy tensor T"v is a consequence of the nongravitational equations of motion. This result could also have been derived, first working with i ? N G in flat spacetime, to obtain the equation T)j>v = 0 by the above method, then using EEP to obtain Equation (3.65). Notice that Equation (3.65) is a consequence purely of universal coupling (EEP) and of the invariance of the nongravitational action, and is valid independent of the field equations for the gravitational fields. The stress-energy tensor T*"1 for charged particles and electromagnetic fields may be obtained from the action 7 NG , Equation (3.20), by first rewriting it in the form .

_

^

,,

dx'^W*

-NG

(3.66) Since only g^ (and not its derivatives) appears in 7 NG , we obtain

= £ mOau"u\uorl{-g)-ll28\x

- xa(x)]

a

+ (4*)- \F^F\

- kg"vF^Fa0)

(3.67)

where we have used the fact that (3-68) Throughout most of this book we shall use the perfect fluid as our model for matter. This model is an average of the properties of matter over scales that are large compared to atomic scales, but small compared to the scales over which the bulk properties of the fluid vary. Thus, one can speak of density, pressure, velocity of fluid elements at a point within

Theory and Experiment in Gravitational Physics

78

the fluid. A perfect fluid is one that has negligible viscosity, heat transport, and shear stresses. It is then possible to show that the stress-energy tensor for the fluid has the following property: in a local Lorentz frame, momentarily comoving with a chosen element of the fluid, the stress-energy tensor for that element has the form T"v = diag[p(l + n),p,p,p-]

(3.69)

where p is the rest-mass-energy density of atoms in the fluid element, II is the specific density of internal kinetic and thermal energy in the fluid element, and p is the isotropic pressure. This can also be written in the covariant form + J/"V)

(3.70)

where u" = dx^jdi is the four-velocity of the fluid element (=<58 m the comoving frame). Then in curved spacetime, T"v has the form T"v = (p + PU + p)u"Mv + pgT

(3.71)

This can also be derived from Equation (3.67) using suitable techniques in relativistic kinetic theory (see Ehlers, 1971). To obtain a complete metric theory of gravity one must now specify field equations for the metric and for the other possible gravitational fields in the theory. There are two alternatives. The first is to assume that these equations, like the nongravitational equations can be derived from an invariant action / G which will be a function of the gravitational fields (£A (which could include #): IG = IG(A,<1>AJ

(3.72)

The complete action is thus 1=

IG(4>A,\J

+

/NGOZA^A.^V.^V./S)

(3.73)

Variation with respect to K yields the gravitational field equations

or, using Equation (3.61), dSeol64>K= -U-g)ll2T^dgJdcl>A

(3.75)

Theories of this type are called Lagrangian-based covariant metric theories of gravity. Many important general properties of such theories are described by Lee, Lightman, and Ni (1974). The other alternative is to specify gravitational field equations that are not derivable from an action.

Gravitation as a Geometric Phenomenon

79

These are called non-Lagrangian-based theories. Although many such theories have been devised, they have not met with great success in agreeing with experiment. All the metric theories to be described in Chapter 5 that agree with solar system experiments are Lagrangian based. 3.3

Long-Range Gravitational Fields and the Strong Equivalence Principle

In any metric theory of gravity, matter and nongravitational fields respond only to the spacetime metric g. In principle, however, there could exist other gravitational fields besides the metric, such as scalar fields, vectorfields,and so on. If matter does not couple to thesefieldswhat can their role in gravitation theory be? Their role must be that of mediating the method by which matter and nongravitationalfieldsgenerate gravitational fields and produce the metric. Once determined, however, the metric alone interacts with the matter as prescribed by EEP. What distinguishes one metric theory from another, therefore, is the number and kind of gravitational fields it contains in addition to the metric, and the equations that determine the structure and evolution of these fields. From this viewpoint, one can divide all metric theories of gravity into two fundamental classes: "purely dynamical" and "prior geometric." (This division is independent of whether or not the theory is Lagrangian based.) By "purely dynamical metric theory" we mean any metric theory whose gravitational fields have their structure and evolution determined by coupled partial differential field equations. In other words, the behavior of each field is influenced to some extent by a coupling to at least one of the otherfieldsin the theory. By "prior geometric" theory, we mean any metric theory that contains "absolute elements,"fieldsor equations whose structure and evolution are given a priori and are independent of the structure and evolution of the other fields of the theory. These "absolute elements" could include flat background metrics IJ, cosmic time coordinates T, and algebraic relationships among otherwise dynamical fields, such as

where h^ and k^ may be dynamicalfields.Note that afieldmay be absolute even if it is determined by partial differential equations, as long as the equation does not involve any dynamicalfields.For instance, a flat background metric is specified by the field equation Riemfa) = 0

(3.76)

Theory and Experiment in Gravitational Physics

80

or a cosmic time function is specified by the field equations

v w v v r = 0,

VT VT = - 1

where the gradient and inner product are taken with respect to a nondynamical background metric, such as i\. General relativity is a purely dynamical theory since it contains only one gravitational field, the metric itself, and its structure and evolution is governed by a partial differential equation (Einstein's equations). BransDicke theory is a purely dynamical theory; thefieldequation for the metric involves the scalar field (as well as the matter as source), and that for the scalar field involves the metric. Rosen's bimetric theory is a priorgeometric theory: it has a flat background metric of a type described in Equation (3.76), and thefieldequations for the physical metric g involve t\. In Chapter 5, we will discuss these and other theories in more detail. By discussing metric theories of gravity from this broad, "Dicke" point of view, it is possible to draw some general conclusions about the nature of gravity in different metric theories, conclusions that are reminiscent of the Einstein Equivalence Principle, but that will be given a new name: the Strong Equivalence Principle. Consider a local, freely falling frame in any metric theory of gravity. Let this frame be small enough that inhomogeneities in the external gravitational fields can be neglected throughout its volume. However, let the frame be large enough to encompass a system of gravitating matter and its associated gravitationalfields.The system could be a star, a black hole, the solar system, or a Cavendish experiment. Call this frame a "quasilocal Lorentz frame". To determine the behavior of the system we must calculate the metric. The computation proceeds in two stages. First, we determine the external behavior of the metric and gravitational fields, thereby establishing boundary values for thefieldsgenerated by the local system, at a boundary of the quasilocal frame "far" from the local system. Second, we solve for thefieldsgenerated by the local system. But because the metric is coupled directly or indirectly to the otherfieldsof the theory, its structure and evolution will be influenced by thosefields,particularly by the boundary values taken on by thosefieldsfar from the local system. This will be true even if we work in a coordinate system in which the asymptotic form of g^ in the boundary region between the local system and the external world is that of the Minkowski metric. Thus, the gravitational environment in which the local gravitating system resides can influence the metric generated by the local system via the boundary values of the auxiliary fields. Consequently, the results of local gravitational experiments may depend

Gravitation as a Geometric Phenomenon

81

on the location and velocity of the frame relative to the external environment. Of course, local nongravitational experiments are unaffected since the gravitational fields they generate are assumed to be negligible, and since those experiments couple only to the metric whose form can always be made locally Minkowskian. Local gravitational experiments might include Cavendish experiments, measurements of the acceleration of massive bodies, studies of the structure of stars and planets, and so on. We can now make several statements about different kinds of metric theories (Will and Nordtvedt, 1972). (a) A theory that contains only the metric g yields local gravitational physics that is independent of the location and velocity of the local system. This follows from the fact that the only field coupling the local system to the environment is g, and it is always possible to find a coordinate system in which g takes the Minkowski form at the boundary between the local system and the external environment. Thus, the asymptotic values of g^ are constants independent of location, and are asymptotically Lorentz invariant, thus independent of velocity. General relativity is an example of such a theory. (b) A theory that contains the metric g and dynamical scalar fields A yields local gravitational physics that may depend on the location of the frame but which is independent of the velocity of the frame. This follows from the asymptotic Lorentz invariance of the Minkowski metric and of the scalar fields, except now the asymptotic values of the scalar fields may depend on the location of the frame. An example is Brans-Dicke theory, where the asymptotic scalarfielddetermines the value of the gravitational constant, which can thus vary as <j> varies. (c) A theory that contains the metric g and additional dynamical vector or tensor fields or prior-geometric fields yields local gravitational physics that may have both location- and velocity-dependent effects. This will be true, for example, even if the auxiliary field is a flat background metric IJ. The background solutions for g and t\ will in general be different, and therefore in a frame in which g^ takes the asymptotic form diag ( 1,1,1,1), r\^ will in general have a form that depends on location and that is not Lorentz invariant (although it will still have vanishing curvature). The resulting location and velocity dependence in q will act back on the local gravitational problem. (For a clear example of this, see Rosen's theory in Chapter 5.) Be reminded that these effects are a consequence of the coupling of auxiliary gravitational fields to the metric and to each other, not to the matter and nongravitational fields. For metric theories of gravity, only g^ couples to the latter.

Theory and Experiment in Gravitational Physics

82

These ideas can be summarized in the form of a principle called the Strong Equivalence Principle that states that (i) WEP is valid for selfgravitating bodies as well as for test bodies (GWEP), (ii) the outcome of any local test experiment is independent of the velocity of the (freely falling) apparatus, and (iii) the outcome of any local test experiment is independent of where and when in the universe it is performed. The distinction between SEP and EEP is the inclusion of bodies with self-gravitational interactions (planets, stars) and of experiments involving gravitational forces (Cavendish experiments, gravimeter measurements). Note that SEP contains EEP as the special case in which gravitational forces are ignored. It is tempting to ask whether the parallel between SEP and EEP extends as far as a Schiff-type conjecture; e.g., "any theory that embodies GWEP also embodies SEP." As in Section 2.5, we can give a plausibility argument in support of this, for the special case of metric theories of gravity with a conservation law for total energy (Haugan, 1979). Generally speaking, this means Lagrangian-based theories. Consider a local gravitating system moving slowly in a weak, static, and external gravitational field. We assume that the laws governing its motion can be put into a quasi-Newtonian form, with the conserved energy Ec given by (3.77) where MR = M 0 - E B ( X , V ) , £B(X, V) = £g + 8my UiJ(X) - $Sm[j VlV> + O ( E g t / 2 , . . . ) (3.78) (see Section 2.5 for detailed definitions). Here, we use units in which the speed of light as measured far from the local system is unity. The position and velocity dependence in £ B can manifest itself, for example, as position and velocity dependence in the locally measured gravitational constant. For two bodies in a local Cavendish experiment, the gravitational constant is given by Gcavendish = r2Fr/mim2

= r\dE^dr)lmxm1

(3.79)

and thus the anomalous mass tensors will contribute to GCavendUh (see Section 6.4). However, a cyclic gedanken experiment identical to that presented in Section 2.5 shows that the anomalous mass tensors bml4 and 8m\J also generate violations of GWEP A1 = g' + (Smf/MJU*

- (<5mjJ/AW

(3.80)

Gravitation as a Geometric Phenomenon

83

where g = \U. Hence, GWEP (dm$ = Sm{k = 0) implies no preferredlocation or preferred-frame effects, thence SEP. In Chapters 4, 5, and 6 we shall see specific examples of GWEP and SEP in action in the postNewtonian limits of arbitrary metric theories of gravity, and in Chapter 8, shall study experimental tests of SEP. The above discussion of the coupling of auxiliary fields to local gravitating systems indicates that if SEP is valid, there must be one and only one gravitational field in the universe, the metric g. Those arguments were only suggestive however, and no rigorous proof of this statement is available at present. The assumption that there is only one gravitational field is the foundation of many so-called derivations of general relativity. One class of derivations uses a quantum-field-theoretic approach. One begins with the assumption that, in perturbation theory, the gravitational field is associated with the exchange of a single massless particle of spin 2 (corresponding to a single second-rank tensor dynamicalfield),and by making certain reasonable assumptions that the S-matrix be Lorentz invariant or that the theory be derivable from an action, one can generate the full classical Einstein field equations (Weinberg, 1965; Deser, 1970). Another class of derivations attempts to build the most general field equation for g out of tensors constructed only from g, subject to certain constraints (no higher than second derivatives, for instance). By demanding that the field equations should imply the matter equations of motion Tfvv = 0, one is led (except for the possible cosmological term) to Einstein's equations. For a review of these and other derivations of general relativity the reader is referred to MTW, box 17.2. However, the implicit use of SEP in all these derivations cannot be emphasized enough. Empirically, it has been found that every metric theory other than general relativity introduces auxiliary gravitational fields, either dynamical or prior geometric, and thus predicts violations of SEP at some level. General relativity seems to be the only metric theory that embodies SEP completely. Thus, the wide variety of derivations of general relativity assuming SEP, plus evidence from alternative theories lends some credence to the conjecture SEP => [General Relativity]

(3.81)

In Chapters 8 and 12, we shall discuss experimental evidence for the validity of SEP. This qualitative discussion of alternative metric theories of gravity has neglected two subjects, each of which could generate a monograph of its

Theory and Experiment in Gravitational Physics

84

own. The first is "torsion." In applying EEP to the nongravitational laws of physics we assumed the rule "comma goes to semicolon," where semicolon denoted covariant derivative with respect to the metric g [Equations (3.14), (3.16), and (3.17)]. However, it is possible that the correct covariant derivative is given by A% = A'j, + §y}A>

(3.82)

{?,} = T% + S$y

(3.83)

where

with Sjiy antisymmetric on ft and y, i.e., Sfryi-Si,

(3.84)

In general, S^y is a tensor called the "torsion" tensor, and thus does not vanish in the local Lorentz frame. Torsion has been introduced into gravitation theory either as a means to incorporate quantum mechanical spin in a consistent way, as a byproduct of attempts to construct gauge theories of gravitation, and as a possible route to a unified theory of gravity and electromagnetism. However, in almost all experiments discussed in this book, the observable effects of torsion are negligible [see, however, Ni (1979)]. Instead, torsion has an effect primarily in the realm of elementary particle physics or in the very early universe. Thus, we shall neglect torsion completely for the rest of this book, and shall refer the interested reader to the review by Hehl et al. (1976). The second topic to be neglected falls under the heading "general relativity with R2 terms." Although this is an old subject (Weyl, 1919; Eddington, 1922), it has recently attracted some interest. The standard gravitational action of classical general relativity (Section 5.2) has the form (3.85) where R is the Ricci scalar given by R = g^R,,

(3.86)

However, some attempts to make a renormalizable quantum theory of gravity based on general relativity lead to the introduction of "counter terms" into the action, to eliminate the nonrenormalizable infinities. These counter terms are quadratic and higher in the Riemann tensor, Ricci tensor, and Ricci scalar, leading to a gravitational action of the form

1G = (16TT)-l J(R + aR2 + bR^R*"* + cR^R^i-g)1'2dAx

(3.87)

Gravitation as a Geometric Phenomenon

85

Since the theory has only one gravitational field #, one suspects that it satisfies SEP, and so represents a possible counter example to our conjecture that SEP => general relativity. However, in most theories of this type, the constants a, b, and c [units of (length)2] have sizes ranging from the Planck length, 10~ 33 cm, to nuclear dimensions, 10" 1 3 cm, so the observable effects of these terms will be confined to elementary particle interactions or to the very early universe. Thus the issue of "R2 terms," too, will be ignored throughout this book (see Havas, 1977).

The Parametrized Post-Newtonian Formalism

We have seen that, despite the possible existence of long-range gravitational fields in addition to the metric in various metric theories of gravity, the postulates of metric theories demand that matter and nongravitational fields be completely oblivious to them. The only gravitational field that enters the equations of motion is the metric g. The role of the other fields that a theory may contain can only be that of helping to generate the spacetime curvature associated with the metric. Matter may create these fields, and they, plus the matter, may generate the metric, but they cannot interact directly with the matter. Matter responds only to the metric. Consequently, the metric and the equations of motion for matter become the primary theoretical entities, and all that distinguishes one metric theory from another is the particular way in which matter and possibly other gravitational fields generate the metric. The comparison of metric theories of gravity with each other and with experiment becomes particularly simple when one takes the slow-motion, weak-field limit. This approximation, known as the post-Newtonian limit, is sufficiently accurate to encompass all solar system tests that can be performed in the foreseeable future. The post-Newtonian limit is not adequate, however, to discuss gravitational radiation, where the slowmotion assumption no longer holds, or systems with compact objects such as the binary pulsar, where the weak-field assumption is not valid, or cosmology, where completely different assumptions must be made. These issues will be dealt with in later chapters. In Section 4.1, we discuss the post-Newtonian limit of metric theories of gravity, and devise a general form for the post-Newtonian metric for a system of perfectfluid.This form should be obeyed by most metric theories, with the differences from one theory to the next occurring only in the

The Parametrized Post-Newtonian Formalism

87

numerical coefficients that appear in the metric. When the coordinate system is appropriately specialized (standard gauge), and arbitrary parameters used in place of the numerical coefficients, the result, described in Section 4.2, is known as the Parametrized post-Newtonian (PPN) formalism, and the parameters are called PPN parameters. In Section 4.3, we discuss the effect of Lorentz transformations on the PPN coordinate system, and show that some theories of gravity may predict gravitational effects that depend on the velocity of the gravitating system relative to the rest frame of the universe (perferred-frame effects). In Section 4.4, we analyze the existence of post-Newtonian integral conservation laws for energy, momentum, angular momentum, and center-of-mass motion within the PPN formalism and show that metric theories possess such laws only if their PPN parameters obey certain constraints. This formalism then provides the framework for a discussion of specific alternative metric theories of gravity (Chapter 5) and for the analysis of solar system tests of relativistic gravitational effects (Chapters 7-9). Most of this chapter is an updated version of Chapter 4 of TTEG (Will, 1974a). 4.1

The Post-Newtonian Limit (a) Newtonian gravitation theory and the Newtonian limit In the solar system, gravitation is weak enough for Newton's theory of gravity to adequately explain all but the most minute effects. To an accuracy of about one part in 105, light rays travel on straight lines at constant speed, and test bodies move according to a = \U

(4.1)

where a is the body's acceleration, and U is the Newtonian gravitational potential produced by rest-mass density p according to 1

= -4np,

U(x, t) = [-~Ar

d2x'

(4.2)

|x x | A perfect, nonviscous fluid obeys the usual Eulerian equations of hydrodynamics dp/dt + V (pv) = 0, pdv/dt^

pVU - Vp,

d/dt = d/dt + v V

(4.3)

1 We use "geometrized" units in which the speed of light is unity and in which the gravitational constant as measured far from the solar system is unity.

Theory and Experiment in Gravitational Physics

88

where v is the velocity of an element of the fluid, p is the rest-mass density of matter in the element, p is the total pressure (matter plus radiation) on the element, and d/dt is the time derivative following the fluid. From the standpoint of a metric theory of gravity, Newtonian physics may be viewed as a first-order approximation. Consider a test body momentarily at rest in a static external gravitational field. From the geodesic Equation (3.38), the body's acceleration a* = d2xk/dt2 in a static (t,x) coordinate system is given by

«*=-n<> = ie*W,

(4-4)

Far from the Newtonian system, we know that in an appropriately chosen coordinate system, the metric must reduce to the Minkowski metric (see subsection (c)) ff,,,-»if,, = diag(-1,1,1,1)

(4.5)

In the presence of a very weak gravitational field, Equation (4.4) can yield Newtonian gravitation, Equation (4.1) only if gfi-S*,

goo^-l+2U

(4.6)

It can be straightforwardly shown that with this approximation and a stress-energy tensor for perfect fluids given by T00 = p,

TOj = pvJ,

TJk = pvV + p5Jk

(4.7)

the Eulerian equations of motion, (4.3), are equivalent to

T?; ~ r?; + rgoT 00 = o

(4.8)

where we retain only terms of lowest order in v2 ~ U ~ p/p. But the Newtonian limit no longer suffices when we begin to demand accuracies greater than a part in 105. For example, it cannot account for Mercury's additional perihelion shift o f ~ 5 x 10 ~7 radians per orbit. Thus we need a more accurate approximation to the spacetime metric that goes beyond or "post" Newtonian theory, hence the name postNewtonian limit. (b) Post-Newtonian bookkeeping The key features of the post-Newtonian limit can be better understood if we first develop a "bookkeeping" system for keeping track of "small quantities." In the solar system, the Newtonian gravitational potential U is nowhere larger than 10" 5 (in geometrized units, U is dimensionless). Planetary velocities are related to U by virial relations

The Parametrized Post-Newtonian Formalism

89

which yield v2 Z U

(4.9)

The matter making up the Sun and planets is under pressure p, but this pressure is generally smaller than the matter's gravitational energy density pU; in other words P/P £ U,

(4.10)

5

10

{p/p is ~10~ in the Sun, ~10~ in the Earth). Other forms of energy in the solar system (compressional energy, radiation, thermal energy, etc.) are small: the specific energy density II (ratio of energy density to rest-mass density) is related to U by

nzu

(4.ii)

(II is ~ 10" 5 in the Sun, ~ 10~9 in the Earth). These four small quantities are assigned a bookkeeping label that denotes their "order of smallness": U ~ v2 ~ p/p ~ n ~ O(2).

(4.12)

Then single powers of velocity i; are O(l), U2 is O(4), Uv is O(3), UH is O(4), and so on. Also, since the time evolution of the solar system is governed by the motions of its constituents, we have d/dt ~ v V and thus,

\s/et\ \d/e>

O(l)

(4.13)

We can now analyze the "post-Newtonian" metric using this bookkeeping system. The action, Equation (3.20), from which one can derive the geodesic Equation (3.38) for a single neutral particle, may be rewritten

-i

-

Cf

dx»dx*yi2

- ' o - -m0 Jl ~a^~Jf~^fJ

dt

(- 000 - 2gop' - gjkvV)112 dt

(4.14)

The integrand in Equation (4.14) may thus be viewed as a Lagrangian L for a single particle in a metric gravitational field. From Equation (4.6), we see that the Newtonian limit corresponds to L = (1 - 2(7 - v2)112

(4.15)

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90

as can be verified using the Euler-Lagrange equations. In other words, Newtonian physics is given by an approximation for L correct to O(2). Post-Newtonian physics must therefore involve those terms in L of next highest order, O(4). What happened to odd-order terms, O(l) or O(3)? Odd-order terms must contain an odd number of factors of velocity v or of time derivatives d/dt. Since these factors change sign under time reversal, odd-order terms must represent energy dissipation or absorption by the system. But conservation of rest mass prevents terms of O(l) from appearing in L, and conservation of energy in the Newtonian limit prevents terms of O(3). Beyond O(4), different theories may make different predictions. In general relativity, for example, the conservation of post-Newtonian energy prohibits terms of O(5). However, terms of O(7) can appear; they represent energy lost from the system by gravitational radiation. In order to express L to O(4), we must know the various metric components to an appropriate order: L = {1 - 2V - v2 - 0oo[O(4)] V<} 1/2

2gOJ[p0)y (4.16)

Thus the post-Newtonian limit of any metric theory of gravity requires a knowledge of 0OO to O(4), g0J to O(3), gJk to O(2)

(4.17)

The post-Newtonian propagation of light rays may also be obtained using the above approximations to the metric. Since light moves along null trajectories (dx 0), the Lagrangian L must be formally identical to zero. In the first order Newtonian limit this implies that light must move on straight lines at speed 1, i.e., 0 = L = (1 - v2)112,

v2 = 1

(4.18)

In the next, post-Newtonian order, we must have 0 = L={l-2U

-v2-

gjk[O(2)~]vJvk}112

(4.19)

Thus to obtain post-Newtonian corrections to the propagation of light rays, we need to know goo to O(2), gjk to O(2)

(4.20)

The Parametrized Post-Newtonian Formalism

91

In a similar manner, one can verify that if one takes the perfect-fluid stress-energy tensor T"v = (p + pU + p)u"u" + pg»v

(4.21)

expanded through the following orders of accuracy: T 00 T0J TJk

to pO(2), to pO(3), to pO(4)

(4.22)

and combined with the post-Newtonian metric, then the equation of motion 7?vv = 0 will yield consistent "post-Eulerian" equations of hydrodynamics. (c) Post-Newtonian coordinate system To discuss the post-Newtonian limit properly, we must specify the coordinate system. We imagine a homogeneous isotropic universe in which an isolated post-Newtonian system resides. We choose a coordinate system whose outer regions far from the isolated system are in free fall with respect to the surrounding cosmological model, and are at rest with respect to a frame in which the universe appears isotropic (universe rest frame). In these outer regions, one expects the physical metric to vary according to ds2 = -dt\+

[a(t)/ao]2(l + kr2IAal)'2dijdxidxi

+ h^dx^dx1

(4.23)

where the first two terms comprise the Robertson-Walker line element appropriate to a homogeneous isotropic cosmological model and the third term represents the perturbation due to the local system. Here, r is the distance from the local system to the field point, a = a{t)[aQ = a(toj] is the cosmological scale factor, and k is the curvature parameter (k = 0, + 1). At a given radius r0 and at a particular moment t0, we can transform to a coordinate system t' == t,

xy = x\l - krl/4al)-1

(4.24)

in which ds2 = (ifc, +fcJJdx»' dxv'

(4.25)

This must be done at a value of r0 large enough that we can then regard n^ as the asymptotic form of g^, i.e., that h^ ~ M/r0 « 1, where M is the mass of the isolated system, yet small enough that the deviation of the cosmological metric from n^ for r « r0 is small, in fact smaller than

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the post-Newtonian terms in h^ of order (M/r)2. The value of r0 that optimizes these constraints is given by (M/r0)2 > {ro/ao)2, or M « r0 <, (Mao)1/2. Since a0 ~ 1010 light yr, we have, for the solar system r0 <: 1011 km ^ 103 a.u., with maximum deviations from n^ of order (ro/ao)2 ~ 10"24. These are much smaller than the expected postNewtonian deviations (M/r)2 > 10" 16 that influence solar system experiments. Thus, to a precision of about one part in 1022, we can regard the space time metric of the solar system as being asymptotically Minkowskian in its outer regions, out to 103 a.u., with deviations of order M/r and {M/r)2 in its interior. The above discussion ignores the variation of the cosmological scale factor a(t) with time. However, because this variation takes place over a timescale (1010 yr) long compared to a dynamical timescale (1 yr) for the solar system, we can treat the effects of the variation adiabatically. The coordinate system thus constructed we shall call "local quasiCartesian coordinates." In this coordinate system it is useful to define the following conventions and quantities: (i) Unless otherwise noted, spatial vectors are treated as Cartesian vectors, with x* = xk. (ii) Repeated spatial indices or the symbol |x| denotes a Cartesian inner product, for example xkxk = Xkxk = xkxk = |x|2 s x2 + y2 + z2

(4.26)

3

(iii) The volume element d x = dxdydz. (d) Post-Newtonian potentials We assume throughout that the matter composing the solar system can be idealized as perfect fluid. For the purposes of most solar system experiments in the coming decades, this is an adequate assumption (see, however, Section 9.2). As we shall see in more detail in Chapter 5, the post-Newtonian limit for a system of perfect fluid in any metric theory of gravity is best calculated by solving the field equations formally, expressing the metric as a sequence of post-Newtonian functionals of the matter variables, with possible coefficients that may depend on the matching conditions between the local system and the surrounding cosmological model and on other constants of the theory. The evolution of the matter variables, and thence of the metric functionals, is determined by means of the equations of motion Tfvv = 0 using the matter stress-energy tensor and the post-Newtonian metric all evaluated to an

The Parametrized Post-Newtonian Formalism

93

order consistent with the post-Newtonian approximation. The evolution of the cosmological matching coefficients is determined by a solution of the appropriate cosmological model. Thus, the most general postNewtonian metric can be found by simply writing down metric terms composed of all possible post-Newtonian functionals of matter variables, each multiplied by an arbitrary coefficient that may depend on the cosmological matching conditions and on other constants, and adding these terms to the Minkowski metric to obtain the physical metric. Unfortunately, there is an infinite number of such functionals, so that in order to obtain a formalism that is both useful and manageable, we must impose some restrictions on the possible terms to be considered, guided in part by a subjective notion of "reasonableness" and in part by evidence obtained from known gravitation theories. Some of these restrictions are obvious: (i) The metric terms should be of Newtonian or post-Newtonian order; no post-post-Newtonian or higher terms are included. (ii) The terms should tend to zero as the distance |x x'| between the field point x and a typical point x' inside the matter becomes large. This will guarantee that the metric becomes asymptotically Minkowskian in our quasi-Cartesian coordinate system. (iii) The coordinates are chosen so that the metric is dimensionless. (iv) In our chosen quasi-Cartesian coordinate system, the spatial origin and initial moment of time are completely arbitrary, so the metric should contain no explicit reference to these quantities. This is guaranteed by using functionals in which the field point x always occurs in the combination x x', where x' is a point associated with the matter distribution, and by making all time dependence in the metric terms implicit via the evolution of the matter variables and of the possible cosmological matching parameters. (v) The metric corrections h00, h0J, and hJk should transform under spatial rotations as a scalar, vector, and tensor, respectively, and thus should be constructed out of the appropriate quantities. For variables associated with the matter distribution, examples are: scalar, p, |x x'|, v'2, v' (x x') etc.; vector, v), (x x')f, and tensor, (x x')/* x')k, VjVk, etc. For variables associated with the structure of the field equations of the theory or with the cosmological matching conditions, there are only two available quantities in the rest frame of the universe: scalar cosmological matching parameters or numerical coefficients; and a tensor, Sjk. In the rest frame of an isotropic universe, no vectors or anisotropic

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tensors can be constructed. [If the universe is assumed to be slightly anisotropic, other terms may be possible (Nordtvedt, 1976).] (vi) The metric functionals should be generated by rest mass, energy, pressure, and velocity, not by their gradients. This restriction is purely subjective, and can be relaxed quite easily if there is ever any reason to do so. No reason has yet arisen. A final constraint is extremely subjective: (vii) The functionals should be "simple." With those restrictions in mind, we can now write down possible terms that may appear in the post-Newtonian metric. (1) gJk to O(2): From condition (v), gjk must behave as a threedimensional tensor under rotations, thus the only terms that can appear are gjk[O(2)-]:U3jk,Ujk

(4.27)

where Ujk is given by Ujk s

f PV,t)(x-x')M-x')k ^

(4 2g)

X X

The term Ujk can be expressed more conveniently in terms of the "superpotential" %(x, t), given by 3

X(x,t)=-jp(x',t)\x-x'\d Xjk=-SjkU+UJk,

x', V 2 x=-2t/

(4.29)

Thus, the only terms that we shall consider are gjk[O(2)l. U5jk,x,Jk

(4.30)

(2) gOj to O(3): These metric components must transform as threevectors under rotations, and thus contain only the terms : VJtWj where

y

CPWMp J \x x'\

(4.31)

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The functionals V} and Wj are also related to the superpotential % by X.oj =VJ-WJ

(4.33)

(3) goo t° O(4): This component should be a scalar under rotations. The only terms we shall consider are 0oo[O(4)]: U\
(4.34)

where

Y ,,,

2

2

x-x

f,,,,

J

x-x'

(4.35)

|

|

|x-x'|

dt

Restriction (vii) has been used liberally to eliminate otherwise possible metric functionals, for example VJVJU-\ '\'Y

d3x\...

Should one of these terms ever appear in the post-Newtonian metric of a gravitational theory, the formalism could be modified accordingly. There are a number of simple and useful relationships satisfied by the functionals that we have included in the metric: ! = -4npv2,

V2O2 = -AnpU, V2O4 = -4np,

= rf + ® - <E>X

(4.36)

To derive many of these relationships one makes repeated use of the formula, obtained using the continuity Equation (4.3), 8 r

r n(x'

at J

t)f(x x')d3x' = o(x' tW V'/Yx x'1ii3jc'ri + O<2)1 >

(4 371

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96

The Standard Post-Newtonian Gauge

We can restrict the form of the post-Newtonian metric somewhat by making use of the arbitrariness of coordinates embodied in statement (ii) of the Dicke framework. An infinitesimal coordinate or "gauge" transformation [see Equations (3.5), (3.13), and (3.17)] xu = x" + £"(xv)

(4.38)

changes the metric to

*«» - £*,

U

(439)

We wish to retain the post-Newtonian character of g^ and the quasiCartesian character of the coordinate system, and to remain in the universe rest frame, thus the functions £ must satisfy: (i) £mv + £v;/, are post-Newtonian functions; (ii) £. + £v;A, -* 0, far from the system; and (iii) |^"|/|x"| -> 0, far from the system. The only "simple" functional that has this property is the gradient of the superpotential x,»- Thus, we choose

and obtain, to post-Newtonian order 9~jk

=

9jk ~

2A2Xjk>

9oo = 9oo ~ 2AiX,oo + 2X2HoX,j

(4.41)

To the necessary order, the Christoffel symbol rJ00 is equal to Uj [see Equation (3.14)]. We must also transform the functional integrals over xk' that appear in g^ into integrals over x F . The only place where this changes anything is in g00 ^ 1 + 2U(x,l), where

- f P(X'J Now the quantity p(x',T) is an invariant; it is the rest-mass density as measured in a comoving local Lorentz frame. Furthermore, the quantity (g)ll2u°d3x is an invariant proper volume element, where u° is the fourvelocity of the matter. Thus, d3x' = d3x'[(-g)ll2i/i/(-g)ll2u0']

(4.42)

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97

Using Equation (4.41) plus the relation u° = dt/dt, we get to the required order, p'dV = p'd3x'[l + 212[/(x',l)]

(4.43)

We also have k

|x x'l ~ |x-x'|

2

|x-xf

Thus, l/(x,T) = U{x,7) + 212O2 - k2 J P ^ _ ~ _ ^ , | 3 ? T d3x

(4.45)

Using Equations (4.33), (4.36), and (4.41), we obtain, finally 9jk = 9jk ~

^iXjk,

055 = 0oo - 2A2(C/2 +OW-

$ 2 ) - 2X^

+<%-*>!)

(4.46)

By an appropriate choice of kt and X2 w e c a n eliminate certain terms from the post-Newtonian metric. We will thus adopt a standard postNewtonian gauge - that gauge in which the spatial part of the metric is diagonal and isotropic (i.e., x,jk eliminated) and in which g00 contains no term Si. There is no physical significance in this gauge choice; it is purely a matter of convenience. We now have a very general form for the post-Newtonian perfect-fluid metric in any metric theory of gravity, expressed in a local, quasi-Cartesian coordinate system at rest with respect to the universe rest frame, and in a standard gauge. The only way that the metric of any one theory can differ from that of any other theory is in the coefficients that multiply each term in the metric. By replacing each coefficient by an arbitrary parameter we obtain a "super metric theory of gravity" whose special cases (particular values of the parameters) are the post-Newtonian metrics of particular theories of gravity. This "super metric" is called the parametrized post-Newtonian (PPN) metric, and the parameters are called PPN parameters. This use of parameters to describe the post-Newtonian limit of metric theories of gravity is called the Parametrized Post-Newtonian (PPN) Formalism. A primitive version of such a formalism was devised and studied

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by Eddington (1922), Robertson (1962), and Schiff (1967). This EddingtonRobertson-Schiff formalism treated the solar system metric as that of a spherical nonrotating Sun, and idealized the planets as test bodies moving on geodesies of this metric. The metric in this version of the formalism reads 9oj = 0. 9jk

= (1 + 2yM/r)6jk

(4.47)

where M is the mass of the Sun, and )5 and y are PPN parameters. These two parameters may be given a physical interpretation in this formalism. The parameter y measures the amount of curvature of space produced by a body of mass M at radius r, in the sense that the spatial components of the Riemann curvature tensor are given to post-Newtonian order by [see Equations (3.14) and (3.34)] Riju = (3yM/r3)(«j«^a + n^S^ - nfi^j, - n/i,<5tt - f 8jkSa + ^Sikdjt) where n = x/r independent of the choice of post-Newtonian gauge. The parameter ft is said tb measure the amount of nonlinearity (M/r)2 that a given theory puts into the g00 component of the metric. However, this statement is valid only in the standard post-Newtonian gauge. The coefficient of U2 = (M/r)2 depends upon the choice of gauge, as can be seen from Equation (4.46). In general relativity, for example (/? = y = 1), the (M/r)2 term can be completely eliminated from g00 by a gauge transformation that is the post-Newtonian limit of the exact coordinate transformation from isotropic coordinates to Schwarzschild coordinates for the Schwarzschild geometry. Thus, this identification of fi should be viewed only as a heuristic one. Schiff (1960b) generalized the metric [Equation (4.47)] to incorporate rotation (Lense-Thirring effect, Section 9.1), and Baierlein (1967) developed a primitive perfect-fluid PPN metric. But the pioneering development of the full PPN formalism was initiated by Kenneth Nordtvedt, Jr. (1968b), who studied the post-Newtonian metric of a system of gravitating point masses. Will (1971a) generalized the formalism to incorporate matter described by a perfect fluid. A unified version of the PPN formalism was then presented by Will and Nordtvedt (1972) and summarized in TTEG. The Whitehead term Ow was added by Will (1973). Henceforth, we shall

The Parametrized Post-Newtonian Formalism

99

adopt the Will-Nordtvedt version (as augmented with the Whitehead term), altered to conform with MTW signature and index conventions, and with minor notational modifications (see Table 4.1). As in the EddingtonRobertson-Schiff version of the PPN formalism, we introduce an arbitrary PPN parameter in front of each post-Newtonian term in the metric. Ten parameters are needed; they are denoted y, j8, & alt a2, a3, d, d> (3. and C4. In terms of them, the PPN metric reads 0Oo = - 1 + 217 - 2pU2 - 2&bw + (2y + 2 + a3 + Ci + 2(3y - 2/J + 1 + £2 + £)3 9oj = -i(4y + 3 + ax - a2 + d - 2 $ ^ - | ( 1 + a2 - Ci ^ = (1 + 2yU)5jk

(4.48)

Although we have used linear combinations of PPN parameters in Equation (4.48), it can be seen quite easily that a given set of numerical coefficients for the post-Newtonian terms will yield a unique set of values for the parameters. The linear combinations were chosen in such a way that the parameters a t , a2, a3, £l5 f2, £3, and £4 will have special physical significance. Other versions of the formalism have been developed to deal with point masses with charge (Section 9.2), fluid with anisotropic stresses (MTW Section 39), and isolated systems in an anisotropic universe (Nordtvedt, 1976). 4.3

Lorentz Transformations and the PPN Metric

In Section 4.1, the PPN metric was devised in a coordinate system whose outer regions are at rest with respect to the universe rest frame. For some purposes - for example, the computation of the post-Newtonian metric in a given theory of gravity - this is a useful coordinate system. But for other purposes, such as the computation of observable postNewtonian effects in systems, such as the solar system, that are in motion relative to the universe rest frame, it is not a convenient coordinate system. In such cases, a better coordinate system might be one in which the center of mass of the system under study is approximately at rest. Again, this is a matter of convenience; the results of experiments cannot be affected by our choice of coordinate system. Because many of our computations will be carried out for such moving systems, it is useful to reexpress the PPN metric in a moving coordinate system. This will also yield some insight into the significance of the PPN parameters au oe2, and <x3 (Will, 1971c).

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To do this we make a Lorentz transformation from the original PPN frame to a new frame which moves at velocity w relative to the old frame. In order to preserve the post-Newtonian character of the metric, we assume that |w| is small, i.e., of O(l). This transformation from rest coordinates (t,\) to moving coordinates (T,£) can be expanded in powers of w to the required order: this approximate form of the Lorentz transformation is sometimes called a post-Galilean transformation (Chandrasekhar and Contopoulos, 1967), and has the form x = | + (1+

»

+ ±({ w)w + O(4) x {,

T 4

t = T(1 + W + fw ) + (1 + W)S w + O(5) x T

(4.49)

where wr is assumed to be O(0). We use the standard transformation law,

and express the functional that appear in gaf(x, t) in terms of the new coordinates. Since p, n , and p are all measured in comoving local Lorentz frames, they are unchanged by the transformation: for any given element of fluid, p(x,t) = p(Z,i), p(x,t) = p«,t)

(4.51)

If v(x, t) and v(£, T) are the matter velocities in the two frames, they are related by v = v + w + O(3)

(4.52)

The elements of volume d3x' and d3£' in the two frames are related by the transformation law [Equation (4.42)] - v' w - W + O(4)]

(4.53)

The quantity x(t) x'(t) that appears in the post-Newtonian potentials transforms according to - T') + K«T) - S'(T')] WW + O(4), 0 = (t - T')(1 + W) + [«t) - £'(*')] w + O(3) (4.54)

The Parametrized Post-Newtonian Formalism

101

But in the (§,T) coordinates, the quantity £ §' must be evaluated at the same time x, hence we must use the fact that £'(T') = %{x) + v'(t' - t) + O ( T ' - T)

2

(4.55)

Combining Equations (4.54) and (4.55), we obtain 1

, |X -

1

.,, {1 + i(w fi')2+ (w fi')(v' n') + O(4)} S II S

,, = .« .,, X|

|S

(4.56)

where S'|

(4.57)

We then find, using Equations (4.51)-(4.53), and (4.56), along with the definitions of the metric functions, Equations (4.2), (4.32), and (4.35), that

U(x,t) = (1 - WWlr)

- wtVj&z) +

T) + 2WiVj($, T) + W2 U(Z, T) + O(6), .4)($,t) + O(6),

+ 2 ^ W J « , T ) + w V ^ ( { , t ) + O(6), Vj(x, t) = Vj{l t) + wy!/(«, T) + O(5), )

W5(x, t) = Wj(Z, x) + wkUjk($, T) + O(5)

(4.58)

Applying the transformation Equations (4.49) and (4.50) to the PPN metric Equation (4.48) and making use of Equations (4.58) we obtain, for the metric in the moving (£, T) system, to post-Newtonian order,

- 2)5 + 1 + C2 + )« 2 «,T) + 2(1

2(3y + 3 (a, - a2 (2a3 - a ^ F / t r ) - (1 - a2 3 + «t - a2 ,T) - | ( 1 - a 2 - Ci T)],5;t

J

(4.59)

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102

Because we now have available an additional post-Newtonian variable, w, we have an additional gauge freedom that can be employed without altering the standard PPN gauge, which is valid in the frame in which w = 0 (and which, incidently, was not affected by the post-Galilean transformation). By making the gauge transformation T = T + J(l - <x2 - d + 2Z)w%j,

V =V

(4.60)

we can eliminate the terms - ( 1 - a 2 - Ci + 2£)wJx>Oj from g00, -Ul - « 2 ~ Ci + 2f)w*JU from g0J This then becomes part of the standard PPN gauge in a coordinate system moving at velocity w relative to the universe rest frame: that gauge in which gjk is diagonal and isotropic, and in which the terms 36 and wJx,Oj are absent from g00. It is then possible to show that a further post-Galilean transformation (plus a possible gauge transformation to maintain the standard gauge) does not alter the form of the PPN metric, it merely changes the value of the coordinate system velocity w that appears there. At first glance, one might be disturbed by the presence of metric terms that depend on the coordinate system's velocity w relative to the universe rest frame. These terms do not violate the principles of special relativity since they are purely gravitational terms, while special relativity is valid only when the effects of gravitation can be ignored; but they do suggest that the gravitation generated by matter may be affected by motion relative to the universe (violation of the Strong Equivalence Principle). Nevertheless, the results of physical measurements must not depend on the velocity w (this is a consequence of general covariance). For a system such as the Sun and planets, the only physically measureable velocities are the velocities of elements of matter relative to each other and to the center of mass of the system, and the velocity, w0, of the center of mass relative to the universe rest frame (as measured for example by studying Doppler shifts in the cosmic microwave radiation). Thus, the PPN prediction for any physical effect can depend only on these relative velocities and on w0, never on w. Therefore, the terms in the PPN metric that depend on w must signal the presence of effects that depend on w0. This can be seen most simply by working in a coordinate system in which the system under study is at rest, i.e., where w = w0. Then, if any one of the set of parameters {ai,a 2 ,a 3 } is nonzero, there may be observable effects which depend on w0; if a t = a2 = a3 = 0, there is no reference to w or w0 in the metric in

The Parametrized Post-Newtonian Formalism

103

any coordinate system, and no such effects pan occur. Thus, we see that the parameters au a2, and <x3 measure the extent and manner in which motion relative to the universe rest frame affects the post-Newtonian metric and produces observable effects. These parameters are called "preferred-frame parameters" since they measure the size of post-Newtonian effects produced by motion relative to the "preferred" rest frame of the universe. If all three are zero, no such effects are present, and there is no preferred frame (to post-Newtonian order). Notice that even if one works in the universe rest frame, where w = 0, physical effects will be unchanged, for even though the explicit preferredframe terms are absent, the velocities of elements of matter vJ that appear in the PPN metric and in the equations of motion must be decomposed according to v = w0 + v" where v is the velocity of each element relative to the center of mass, and, unless alt <x2, and <x3 are all zero, the same effects dependent upon w0 will result. At this point the PPN metric has taken on its standard form. Table 4.1 summarizes the basic definitions and formulae that enter the PPN formalism and compares the present version with previous versions. Table 4.1. The parametrized post-Newtonian formalism A. Coordinate system: the framework uses a nearly globally Lorentz coordinate system [Section 4.1(c)] in which the coordinates are (t,xl,x2,x3). Three-dimensional, Euclidean vector notation is used throughout. All coordinate arbitrariness ("gauge freedom") has been removed by specialization of the coordinates to the standard PPN gauge (Section 4.2). B. Matter variables: 1. p = density of rest mass as measured in a local freely falling frame momentarily comoving with the gravitating matter. 2. v' = (dx'/dt) = coordinate velocity of the matter. 3. w' = coordinate velocity of PPN coordinate system relative to the mean rest frame of the universe. 4. p = pressure as measured in a local freely falling frame momentarily comoving with the matter. 5. n = internal energy per unit rest mass. It includes all forms of nonrest mass, nongravitational energy - e.g., energy of compression and thermal energy. C. PPN parameters: v, P, L «i, &2, «3, Ci, £2, C3> C4

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Table 4.1. (continued) D. Metric: goo= -1 + 2U- 2PU1 - 2^w + (2y + 2 + a 3 + Ci - 2{)®t + 2(3y - 20 + 1 + C2 + £)* 2 + 2(1 + C3)O3 + 2(3y + 3 - (£, - 2§st - (a t - « 2 - a 3 )w 2 l/ - a 2 wWl7 y + (2a3 0oi = - i ( 4 y + 3 + a, - <x2 + Ct - 2{W - ftl + a2 - i ( B l - 2*2)wtU > gtj = (1 + 2yU)Sij E. Metric potentials: | - xI ®W =

/ p'p"(X - X') ( X' - X" -j ^ j -j 77s

J

|x - x | 3

J

J

Xn

|x x | \ |x x I '[v'-(x-xQP |x - x'| 3

X

d

X

|x x I / pV^ f J |x - x I

| J |x - x'|

X" \ 7T ) d

J

I

|x - x I

|x x |

F. Stress-energy tensor (perfect fluid) T00 = p(l + U + v2 + 217) T0i = p(l + Tl + v2 + 2U + p/p)v' TiJ = pt>V(l + n + v2 + 217 + p/p) + p5iJ(l - 2yU) G. Equations of motion 1. Stressed Matter, 7?vv = 0 2. Test Bodies d2x"/dX2 + TUdxVd).)(dxx/dX) = 0 3. Maxwell's Equations

H. Differences between this version and the TTEG version 1. Adoption of MTW signature ( 1,1,1,1) and index convention (Greek indices run 0,1,2,3; Latin run 1,2,3) 2. New symbol for Whitehead parameter: ^ instead of Cn- as in Will (1973) 3. Modified conservation-law parameters incorporating effects of Whitehead term (see Lee et al., 1974)

The Parametrized Post-Newtonian Formalism

105

4.4

Conservation Laws in the PPN Formalism Conservation laws in Newtonian gravitation theory are familiar: for isolated gravitating systems, mass is conserved, energy is conserved, linear and angular momenta are conserved, and the center of mass of the system moves uniformly. This does not apply to every metric theory of gravity, however. Some theories violate some of these conservation laws at the post-Newtonian level, and it is the purpose of this section to explore such violations using the PPN formalism. One can distinguish two kinds of conservation laws: local and global. Local conservation laws are laws that are valid in any local Lorentz frame, and are independent of the metric theory of gravity. They depend rather, upon the structure of matter that one assumes. Global conservation laws, however, are statements about gravitating systems in asymptotically flat spacetime. Because they incorporate the structure of both the matter and the gravitational fields, they depend on the metric theory in question. (a) Local conservation laws Conservation of baryon number is one of the most fundamental laws of physics, and should certainly be valid in the presence of gravity. This law can be expressed as a continuity equation for the baryon number density n: in a local Lorentz frame momentarily comoving with the matter, the equation expressing conservation of baryon number 5A 0 = d(SA)/dt = dind V)/dt is equivalent to dn/dt + V (nv) = 0

(4.61) (4.62)

where v is the baryon velocity in the comoving frame (v = 0 but V v = SV~1d(SV)/dt # 0). The Lorentz-invariant version of this continuity equation, valid in any local Lorentz frame is 0 = ^(n«°) + ^ ( n ^ ) = (nu")>,

(4.63)

where w" is the baryon four-velocity given by u" = dx^jdx. Equation (4.63) can then be generalized to any frame in curved spacetime using the standard "comma-goes-to-semicolon" rule 0 = (nu").^

(4.64)

This is the law of baryon conservation in covariant form. If the matter is assumed to have a chemical composition that is homogeneous and

Theory and Experiment in Gravitational Physics

106

static, then there is a direct proportionality between the baryon number density (we assume negligible numbers of antibaryons) and the rest mass density p of the atoms in the element of fluid, namely p = \m

(4.65)

where \i is the mean rest mass per baryon in the element and a constant. Proceeding by a similar argument to the one presented above, one obtains the law of rest-mass conservation, (pu% = 0.

(4.66)

By combining this equation with the equations of motion for stressed matter Tfvv = 0 along with the assumption that matter is a perfect fluid, we obtain a third local law, the law of local energy conservation or the law of isentropic flow. The equation ujt:

= 0

(4.67)

may be evaluated, using Equation (3.71). We work in a local Lorentz frame, momentarily comoving with the element 8V of fluid. From Equation (4.67), (d/dt)(p + pU) + V (p + pll + p)\ = 0

(4.68)

This can be rewritten (d/dt){p + pU) + (p + pU) V v + p V v = 0

(4.69)

or, {d/dt)[(p + pU)8V\ + pd(5V)/dt = 0

(4.70)

So, in a local comoving inertial frame, the change in the total energy (restmass plus internal) of an element of fluid is balanced by the work done [pd(8V)2: this simply expresses Local Conservation of Energy or Isentropic Flow, since from the First Law of Thermodynamics, and from Equation (4.70) d(energy) + pdV = 2f(heat) = TdS = 0

(4.71)

Actually, the absence of heat flow was built into the stress-energy tensor from the start by assuming the perfect-fluid form. Had we permitted heat transport, we would have added an additional piece to T"v, 1

v

heat

ZM

H

where q is a "heat-flux four-vector." For further discussion of nonperfect fluids see MTW, Section 22.3 and Ehlers (1971).

The Parametrized Post-Newtonian Formalism

107

Because of the conservation of rest mass, pSV is constant, and Equation (4.70) can be written in the form pdTl/dt - (p/p)dp/dt = 0

(4.72)

Then in frame-invariant language, Equation (4.72) has the form «*[n + p(l/p)J = 0

(4.73)

We can obtain a useful form of the law of conservation of rest mass (or baryon number) by noticing that for any four-vector field, A11, A^ = (-grll2i{-g)mA"lll

(4.74)

hence (pu% = (-gr1/2l(-g)ll2pu"l,

=0

(4.75)

In a coordinate system (t, x), Equation (4.75) can thus be written 0 = ip(-g)ll2u0l0 + ip(-g)ll2u°v% J

(4.76)

J

since u = u°v . By defining the "conserved density" p* P* = p(-g)ll2u°

(4.77)

we can cast Equation (4.75) in the form of an "Eulerian" continuity equation, valid in our (t,x) coordinate system: dp*/dt + V p*v = 0

(4.78)

This "conserved" density is useful because for any function /(x,t) defined in a volume V whose boundary is outside the matter (d/dt) j y p*f d3x = J K p*(df/dt)d3x

(4.79)

Notice that Equation (4.79) implies dm/dt^Q,

m=[ p * d 3 x

(4.80)

where m is the total rest mass of the particles in the volume V; from Equation (4.77), we get, m \ [pu°(g)ll2~\d3x = Jpd(proper volume) = total rest mass of particles

(4.81)

(b) Global conservation laws The conservation laws discussed above are purely local conservation laws; they depend only on properties of matter as measured in local,

Theory and Experiment in Gravitational Physics

108

comoving Lorentz frames, where relativistic and gravitational effects are negligible (hence they are theory independent). Equation (4.80) represents our first "global" or "integral" conservation law; it is really nothing more than conservation of baryons coupled with our specific model for matter. However, when we attempt to devise more general integral conservation laws, such as for total energy (as opposed to exclusively rest mass), total momentum, or total angular momentum, we run into difficulties. It is well known that integral conservation laws cannot be obtained directly from the equation of motion for stressed matter Tfvv = 0 because of the presence of the Christoffel symbols in the covariant derivative. Rather, one searches for a quantity 0" v which reduces to T"v in flat spacetime and whose ordinary divergence in a coordinate basis vanishes, i.e., 0?vv = 0

(4.82)

Then, provided 0*" is symmetric, one finds that the quantities

P" = £ ©"v p ^

J"v = 2 £ xl»®vU <*%

(4.83)

are conserved, i.e., the integrals in Equation (4.83) vanish when taken over a closed three-dimensional hypersurface E. If one chooses a coordinate system (t, x) in which £ is a constant-time hypersurface that extends infinitely far in all spatial directions, then, provided 0" v vanishes sufficiently rapidly with spatial distance from the matter, P" and J1" are independent of time and are given by

p* = J©*° d3x,

J"v = 2 J x ^ 0V>° d3x

(4.84)

An appropriate choice of 0" v allows one to interpret the components of P" and J* in the usual way: as measured in the asymptotically flat spacetime far from the matter, P° is the total energy, PJ is the total momentum, JiJ is the total angular momentum, and J0J determines the motion of the center of mass of the matter. If 0" v exists but is not symmetric, then P" is conserved but J"v varies according to

dJ"v/dt= -2 J © M d 3 x

(4.85)

The quantity 0" v , normally called the stress-energy complex, has been found for the exact versions of general relativity (Landau and Lifshitz, 1962), Brans-Dicke theory (Nutku, 1969b), and others (Lee et al, 1974). A wide variety of nonsymmetric stress-energy complexes have been devised and discussed within general relativity, but only the symmetric version guarantees conservation of angular momentum.

The Parametrized Post-Newtonian Formalism

109

There is a close connection between integral conservation laws and covariant Lagrangian formulations of metric theories. It has been shown (Lee et al., 1974) that every Lagrangian based, generally covariant metric theory of gravity that either (i) is purely dynamical (possesses no absolute variables), or (ii) contains prior geometry, with a simple constraint on the symmetry group of its absolute variables (a constraint satisfied by all specific metric theories known), possesses conservation laws of the form 0?vv = 0 where 0" v is a function of certain variational derivatives of the Lagrangian of the theory that reduces to T** in the absence of gravity. When there are no absolute variables, the conservation laws are the result of invariance under coordinate transformations, and the stress-energy complexes 0" v are not tensors (or tensor densities); moreover, there may be infinitely many of them. When absolute variables are present, their symmetry group produces the conservation laws and 0" v typically are tensors (or tensor densities). Although &lv is guaranteed to exist for any Lagrangian-based metric theory, there is no guarantee that it will be symmetric, and no general argument is known to determine the conditions under which it will be symmetric. In the post-Newtonian limit, the existence of conservation laws of the form of Equation (4.82) can be translated into a condition on values of some of the PPN parameters. The form of 0" v that we shall attempt to construct is given by 0*v = (1 - aU)(T"v + t"v)

(4.86)

where a is a constant, and t"v is a quantity (which may be interpreted under some circumstances as "gravitational stress energy") which vanishes in flat spacetime, and which is a function of the fields U, UJk, ®w, Vp Wp ..., their derivatives, and w (and may also contain the matter variables p, II, p, and v). We reject terms in 0" v of the form w2T»" since such terms do not vanish in general in regions of negligible gravitational field. By combining Equations (3.65), (4.82), and (4.86), we find that, to postNewtonian order, t"v must satisfy "v

(4.87)

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110

In our attempt to integrate Equation (4.87) we will make use of Table 4.1 and Equations (4.36) along with the following identity, which is valid for any function/: + 17,,V2/

(4.88)

where riJ(f)=UAif,j)-iSijVU-\f

(4.89)

Another useful identity is -2rij(^w

+ 3/4172 - VX Vl/)

^U^ - <5,/l/,0)2] + (2n)-l(d/dt)(U,iU,0) + U,£(4ny*V2^ + pv2 + 2p- (87t)-^VL/j2]

(4.90)

where \j/j is the solution of the equation VV ; = -4npUj

(4.91)

Then, Equation (4.87) can be put into the form 4nt°; = 47i(t°0° + t°>) 2a- 5)|Vt/| 2 ] + a - 3 ) l / j ^ , n + (3y + a - 2)UtiU<0\

(4.92)

5/5t[(4y + 4 + OLXWJVM + i(4y + 2 + a, - 2a2 + 2C1)C/,iC/,0 - (5y + a - l)UV2Vi + ia1wI-[/V2t/ + a2Uti(
2ri7(0») + (2a3 (1 + a 2 - Ci 2(4y + 4 + «I)(T^, {Ay + 4 + aiMl/ 2 + a i - 2a2 + 2tl)5ij(U,0)2 Vl/) 2 - 17.ow Vl/] (5y + a- l)U(pv'vJ + pdij) + ziJ} + 4nQ'

(4.93)

The Parametrized Post-Newtonian Formalism

111

where <£ = i(2y + 2 + « 3 + Ci - 2{)®i + (3y - 2/3 + 1 + C2 + + (1 + £3)<J>3 + (3y + 3C4 - 2 0 * 4 ,

(4-94)

x'J = iaiWit/V 2 ^- + ajWj-t/^-t/.o - OL2WJU,,{W VC7),

Q' = t/j[i(«3 +

CI)P»2

2

+ (8n)-K2\vu\

(4.95)

+ c 3 pn

2

+ 3Up + (SnrKiV ** + «3pv w]

(4.96)

It has been found to be impossible to write Q, as a combination of gradients and time derivatives of gravitational fields and matter variables. Thus, integrability of Equations (4.92) and (4.93) requires that each of the terms in Q( vanish identically, i.e., « 3 = fl = Ci = t 3 = C4 3E 0

(4.97)

These constraints must be satisfied by any metric theory in order that there be conservation laws of the form of Equation (4.82). If these conditions hold, then expressions for the conserved energy and momentum can be obtained using Equations (4.84), (4.86), (4.92), and (4.93). The results are (after integrations by parts): ,

(4.98)

n + p/p] - | ( i + 0L2)Wi - fawjUtj} d3x

(4.99)

where we have used the PPN version of the conserved density [Equation (4.77)] p* = p[i + %V2 + 3yU + O (4)] (4.100) In the expression for P°, the first term is the total conserved rest mass of particles in the fluid. The other terms are the total kinetic, gravitational, and internal energies in the fluid, whose sum is conserved according to Newtonian theory (which can be used in any post-Newtonian terms). Thus, P° is simply the total mass energy of the fluid, accurate to O(2) beyond the rest mass, and is conserved irrespective of the validity of the conditions in Equation (4.97). However, if those conditions were violated, one would expect violations of the conservation of P° at O(4). An alternative derivation of the conserved momentum uses Chandrasekhar's (1965) technique of integrating the hydrodynamic equations of motion T = 0 over all space, and searching for a quantity P' whose time derivative vanishes. This procedure is blocked by a term ^Qtd3x where Qt is given by Equation (4.96). This integral can be written as a total time

Theory and Experiment in Gravitational Physics

112

derivative only if (?, can be written as a combination of time derivatives and spatial divergences (which lead to surface integrals at infinity that vanish). But according to the reasoning given above, this can be true only if the five parameter constraints of Equation (4.97) are satisfied. Then Qi 25 0 and the conserved P' derived by this method agrees with Equation (4.99). We now see the physical significance of the parameters <x3, £u £ 2 , C3, and £4: they measure the extent and manner in which a given metric theory of gravity predicts violations of conservation of total energy and momentum. If all five are zero in any given theory, then energy and momentum are conserved; if some are nonzero, then energy and momentum may not be conserved. According to the theorem of Lee, et al., 1974, every Lagrangianbased metric theory of gravity has all five conservation law parameters zero. Notice that the parameter a 3 plays a dual role in the PPN formalism, both as a conservation-law parameter and as a preferred-frame parameter. In order to guarantee conservation of the angular momentum tensor J^, t"v must be symmetric. Equations (4.92) and (4.93) show that there are nonsymmetric terms, xiJ [Equation (4.95)], in tiJ, and that tOi # t'°. However, in integrating Equations (4.92) and (4.93), we have the freedom to add to the nominal solutions for t"v any quantity S"v that satisfies Sfvv = 0

(4.101) iJ

However, we have been utterly unable to find an S that will eliminate or symmetrize the offending terms tiJ in t'J. As for the toi and ti0 components, the best we can do is to make use of the identity d/dt(UV2U + |VC/|2) + d/dxJ(UW2Vj-

Ui0Uj - 2U_kVlkJ s 0

(4.102)

to eliminate or symmetrize one of the offending terms. A convenient choice is to match the term involving f/V2 Vt in ti0 with an identical term in toi. With this choice, all dependence on the constant a is eliminated from ®"v. The result is 2a2)UtOUti 2

-&lWiUV U 2ti-n

=

- a2l/>;w VU,

(4.103)

2 T WI

= ai U Wli V 2 V n - 2a 2 l/ >o w [| .l/, jl + 2a2w[il/,J.jW Vl/

(4.104)

v

Symmetry of t" requires that each of the terms in Equations (4.103) and (4.104) vanish identically, i.e., a!=«2E0

(4.105)

The Parametrized Post-Newtonian Formalism

113

We apply the name Fully Conservative Theory to any theory of gravity that possesses a full complement of post-Newtonian conservation laws: energy, momentum, angular momentum, and center-of-mass motion, i.e., whose PPN parameters satisfy ttl

= a2 s a 3 = d = £2 = C3 = U = 0

(4.106)

A fully conservative theory cannot be a preferred frame theory to postNewtonian order since a t = a2 = a3 = 0. For such theories, only three PPN parameters, y, /?, and f may vary from theory to theory, and &"v and t"v have the form 0"v = [1 + (5y - 1)[/](T"V + t"v), too= (8w)-1(4y + 3) |Vt/|2, tot = t.o = (47t)-1[(2y + lyUjUjo + 4(y tu = [i _ ( 5 y + 4 | - 8( T

-i(2y + 1)<50([/,0)2, O == i(2y + 2 2^)Oj + (3y 2/? + 1 + (3y - 2{)0>4

(4.107)

and the conserved quantities are

P° = J p *(l + ^ 2 - \\J + Jl)d3x F = Jp*[V(l + iu2 - ^{7 + n + p/p) - i^ £ ] d3x, J" = 2 Jp*x[I>J1[l + *»2 + (2y +1)C/ + n + p/p] JOi = Jp*x'(l + if2 - i l / + n)
(4.108)

1

By defining a center of mass X given by fp*x'(l+ ?v2 - i[7 + U)d3x X' = ^

(4.109)

Theory and Experiment in Gravitational Physics

114

we find from Equations (4.108) and the constancy of J 0 ' that (4.110) i.e., the center of mass moves uniformly with velocity P'/P°. Some theories of gravity may possess only energy and momentum conservation laws, i.e., their parameters may satisfy a3 = d = C2 = C3 = U s 0,

one of { ai ,a 2 } # 0

(4.111)

We call such theories Semiconservative Theories. Their conserved P" may be obtained from Equations (4.98) and (4.99); their nonconserved J"v may be obtained from Equations (4.84), (4.92), and (4.93). A peculiar feature of the semiconservative case is that in a coordinate system at rest with respect to the universe, w = 0, and the spatial components t'J are automatically symmetric, irrespective of the values of at and a2 (since rij = 0 if w = 0). Thus, spatial angular momentum J'j is a conserved quantity in this frame, whereas it is not in a moving frame. The center-ofmass component J°\ however, is not conserved in any frame, since %oi _£ T>o for a n v w jjjj s discrepancy Can be understood by noting that the distinction between JiJ and J0J is not a Lorentz-invariant distinction. Because the PPN metric is post-Galilean invariant, the quantities P" and J"v should transform as a vector and antisymmetric tensor respectively under post-Galilean transformations. This can be verified explicitly by applying the transformation Equation (4.49) to the integrals that comprise P" and J"v, with the result, valid to post-Newtonian order P0' P' fr Ji0'

= P°(l + |u 2 ) - u P, = P - (1 + i«2)uP° + |u(u P), = fJ _ j * W + 2(1 + %u2)JoliuJ\ = J'°(l + lu 2 ) - uJJiJ - yu}JJ0

(4.112)

J

where u = u e, is the velocity of the boost. Thus a boost from the universe rest frame where (d/di)JiJ = 0 to a frame moving with velocity w yields jtr

=

2J°(V1[1 + O(w2)]

(4.113)

thus, the violation of angular momentum conservation is intimately connected with.the violation of uniform center-of-mass motion. This is our reason for stating that semiconservative theories of gravity possess only energy and momentum conservation laws. Equation (4.113) may be verified explicitly using Equations (4.103), (4.104), and the fact that

JiJ = 2 J tmd3x,

joi = 2 J tli0]d3x

(4.114)

Every Lagrangian-based theory of gravity is at least semiconservative.

The Parametrized Post-Newtonian Formalism

115

Nonconservative Theories possess no conservation laws (other than the trivial one for P°); their parameters satisfy

oneoffo.fc.Ca.CW^O

(4.115)

Table 4.2 summarizes these conservation law properties of metric theories of gravity, and Table 4.3 summarizes the significance of the various PPN parameters. Table 4.2. Post-Newtonian integral conservation laws PPN parameter values {C1.C2.C3.C4,1*3}

{«i.«2}

Type of theory

all zero all zero may be nonzero

all zero may be nonzero any values

Fully conservative Semiconservative Nonconservative

Conserved quantities P", J"v P" pOa

" In nonconservative theories, P° is only conserved through lowest Newtonian order, i.e., to O(2) beyond the conserved rest mass. Table 4.3. The PPN parameters and their

Parameter y /} £

What it measures, relative to general relativity" How much space-curvature is produced by unit rest mass? How much "nonlinearity" is there in the superposition law for gravity? Are there preferred-location effects? Are there preferred-frame effects? Is there violation of conservation of total momentum?

significance Value in Value in general semiconservative relativity theories

Value in fully conservative theories

1

y

y

1

/?

j?

Of 0 0 0 0 0 0 0

( at a 02 0 0 0 0

0 0 0 0 0 0 0

" These descriptions are valid only in the standard PPN gauge, and should not be construed as covariant statements. For examples of the misunderstandings that can arise if this caution is not heeded, especially in the case of P, see Deser and Laurent (1973), and Duff (1974).

Post-Newtonian Limits of Alternative Metric Theories of Gravity

We now breathe some life into the PPN formalism by presenting a chapter full of metric theories of gravity and their post-Newtonian limits. This chapter will illustrate an important application of the PPN formalism, that of comparing and classifying theories of gravity. We begin in Section 5.1 with a discussion of the general method of calculating post-Newtonian limits of metric theories of gravity. The theories to be discussed in this chapter are divided into three classes. The first class is that of purely dynamical theories (see Section 3.3). These include general relativity in Section 5.2; scalar-tensor theories, of which the Brans-Dicke theory is a special case in Section 5.3; and vector-tensor theories in Section 5.4. The second class is that of theories with prior geometry. These include bimetric theories in Section 5.5; and "stratified" theories in Section 5.6. The theories described in detail in these five sections are those of which we are aware that have a reasonable chance of agreeing with present solar system experiments, to be described in Chapters 7, 8, and 9. Table 5.1 presents the PPN parameter values for the theories described in these five sections. The third class of theories includes those that, while perhaps thought once to have been viable, are in serious violation of one or more solar system tests. These will be described briefly in Section 5.7. 5.1

Method of Calculation

Despite the large differences in structure between different metric theories of gravity, the calculation of the post-Newtonian limit possesses a number of universal features that are worth summarizing. It is just these common features that cause the post-Newtonian limit to have a nearly universal form, except for the values of the PPN parameters. Thus, the computation of the post-Newtonian limits of various theories tends to

Table 5.1. Metric theories of gravity and their PPN parameter values PPN parameters"

Cosmological Theory and its gravitational fields

Arbitrary functions or constants

ma idling

parameters

y

P

(a) Purely dynamical theories (i) General relativity (g) (ii) Scalar-tensor (g, <j>)

none

none

l

1

0

0

0

0

00

1 +0) 2 + 0)

1+A

0

0

0

0

BWN

«1

a2

(«3,0

Bekenstein's VMT

o)(0),r,

th

1 +0) 2 + 0)

1+A

0

0

0

0

Brans-Dicke

CO

ih

1+0) 2 + o)

1

0

0

0

0

0)

K K K

y

a'2

i

1

0 0 0

0 0 0

i i

1 1

(iii) Vector-tensor (g, K) General Hellings-Nordtvedt Will-Nordtvedt (b) Theories with prior geometry (iv) Bimetric theories Rosen (g, 9) Rastall(g,ir,K) BSLL(g, V) B) (v) Stratified theories

none

none none a,f,k

Wo
K

l

co,cua,b,c,d co,cua,b,c,d

0

0 0

P

0 0 0

bc0

0

%

(co/ci)-l a'2 a'2

aco/cl

«2

0 0 0 0 0

" Prime over a PPN parameter (e.g., / ) denotes a complicated function of arbitrary constants and cosmological matching parameters. See text for explicit formulae.

Theory and Experiment in Gravitational Physics

118

have a repetitive character, the major variable usually being the amount of algebraic complexity involved. In order to streamline the presentation of specific theories in the following sections, and to establish a uniform notation, we present a "cookbook" for calculating post-Newtonian limits of any metric theory of gravity. Step 1: Identify the variables: (a) dynamical gravitational variables such as the metric g^, scalar field , vector field K", tensor field J5^v, and so on; (b) prior-geometrical variables such as a flat background metric n^, cosmic time function t, and so on; and (c) matter and nongravitational field variables. Step 2: Set the cosmological boundary conditions. Assume a homogeneous isotropic cosmology, and at a chosen moment of time and asymptotic coordinate system define the values of the variables far from the post-Newtonian system. With isotropic coordinates in the rest frame of the universe, a convenient choice that is compatible with the symmetry of the situation is, for the dynamical variables,

9^ -> gfv =

diag{-co,ci,cuci),

0 , *->(*:, o,o,o), B^ -* B$ = diag(coo,
(5.1)

and for the prior-geometric variables (these values are valid everywhere, since these variables are independent of the local system),

t = t, with

Vt = (l,0)

(5.2)

The relationships among and the evolution of these asymptotic values will be set by a solution of the cosmological problem. Because these asymptotic values may affect the values of the PPN parameters, a complete determination of the post-Newtonian limit may in fact require a complete cosmological solution. This can be very complicated in some theories. For the present, we shall avoid these complications by simply assuming that the cosmological matching constants are arbitrary constants (or more precisely, arbitrary slowly varying functions of time). In Chapter 13, we shall turn to the cosmological question and discuss the relationship between cosmological models and observations that may fix the asymptotic values of the fields and post-Newtonian gravity. Notice that if a flat background metric q is present, it is almost always most convenient to work in a coordinate system in which it has the Minkowski form, for in

Post-Newtonian Limits

119

many theories the resulting field equations involve flat-spacetime wave equations, which are easy to solve. Then the asymptotic form of g shown is determined by the cosmological solution. If r\ is present it is not generally possible (unless in a special cosmology or at a special cosmological epoch) to make both it and g have the asymptotic Minkowski form simultaneously. Of course, once the post-Newtonian metric g has been determined, one can always choose a local quasi-Cartesian coordinate system [see Section 4.1(c)] in which it takes the asymptotic Minkowski form. The form that IJ now takes is irrelevant since, unlike g, it does not couple to matter. In theories without if, it is usually convenient to choose asymptotically Minkowski coordinates right away. Step 3: Expand in a post-Newtonian series about the asymptotic values: Guv

Gfiv ' 'Vv)

= 4>0 +
K^iK + ko,fcl5fe2,/c3), B,v = &°> + &

(5.3)

Generally, the post-Newtonian orders of these perturbations are given by ~ O(2) Hh 0(4), 9 ~ O(2) H- 0(4), k0 ~ O(2) HH 0(4),

"00

boo~ O(2) H- 0(4),

» /
htj ~ 0(2),

/c,- ~ 0(3), Z»oi ~ 0(3),

fty ~ O(2)

(5.4)

Step 4: Substitute these forms into the field equations, keeping only such terms as are necessary to obtain a final, consistent post-Newtonian solution for h^. Make use of all the bookkeeping tools of the postNewtonian limit (Section 4.1), including the relation (d/dt)/(d/dx) ~ O(l). For the matter sources, substitute the perfect-fluid stress-energy tensor T"v and associated fluid variables. Step 5: Solve for h00 to O(2). Only the lowest post-Newtonian order equations are needed. Assuming that h00 -»0 far from the system, one obtains the form /loo = 2aU

(5.5)

where U is the Newtonian gravitational potential [Equation (4.2)], and where a. may be a complicated function of cosmological matching parameters and of other coupling constants that may appear in the theory's field equations (such as a "gravitational constant"). To Newtonian order,

Theory and Experiment in Gravitational Physics

120

the metric thus has the form 0 o o = -co + 2ccU,

gOJ = 0,

giJ = Sif1

(5.6)

To put the metric into standard Newtonian and post-Newtonian form in local quasi-Cartesian coordinates, we must make the coordinate transformation x5 = (c o ) 1 / 2 x 0 ,

x1 = (Cl)ll2xJ

065 = Co ^ o o ,

06J = ( c o c i ) " il2g0j,

(5.7)

then 9iJ = cf

£7 = c x t7

1

gii, (5.8)

and goo = - 1 + 2(cc/coCl)U,

0sj = 0,

gij = dtj

(5.9)

Because we work in units in which the gravitational constant measured today far from gravitating matter is unity, we must set Gtoday = a/coC! = 1

(5.10)

The constraint provided by this equation often simplifies other calculations, however there is no physical constraint implied; it is merely a definition of units. Step 6: Solve for hu to O(2) and h0J to O(3). These solutions can be obtained from the linearized versions of the field equations. The field equations of some theories have a gauge freedom, and a certain choice of gauge often simplifies solution of the equations. However, the gauge so chosen need not be the standard PPN gauge (Section 4.2), and a gauge (coordinate) transformation into the standard gauge [Equations (4.40) and (4.46)] may be necessary once the complete solution has been obtained. Step 7: Solve for h00 to O(4). This is the messiest step, involving all the nonlinearities in the field equations, and many of the lower-order solutions for the gravitational variables. The stress-energy tensor T"v must also be expanded to post-Newtonian order. Using Equations (3.71), (5.6), and (5.10), we obtain T00 = Co V [ l + n + 2cxU + Co'cy T' J = Co'pv'vJ + c^'pS1' + pO(4)

+ O(4)], (5.11)

Step 8: Convert to local quasi-Cartesian coordinates [Equation (5.7)] and to the standard PPN gauge (Section 4.2).

Post-Newtonian Limits

121

Step 9: By comparing the result for g^ with Equation (4.48), or with Table 4.1 (with w = 0), read off the PPN parameter values. In obtaining these post-Newtonian solutions, the following formulae are useful

u = -iv 2 x , \\U\2 = V2(iU2 - O2)

(5.12)

along with Equations (4.29), (4.33), (4.36), and (4.37). 5.2 General Relativity (a) Principal references: Standard textbooks such as MTW and Weinberg (1972). (b) Gravitationalfieldspresent: the metric g. (c) Arbitrary parameters and functions: None (we shall ignore the cosmological constant, which is too small to be measured in the solar system). (d) Cosmological matching parameters: None. (e) Field equations: The field equations are derivable from an invariant action principle 51 = 0, where +W«A,0U

(5-13)

where R is the Ricci scalar [Equation (3.86)] and JNG is the universally coupled nongravitational action, and G is the gravitational coupling constant. By varying the action with respect to g^, we obtain the field equations (5.14) (f) Post-Newtonian limit: Because g is the only gravitational field present, we can choose it to be asymptotically Minkowskian without affecting any otherfields.Thus we have initially c0 = cl = 1. It is convenient to rewrite the field Equation (5.14) in the equivalent form R^ = 8TTG(T,V - i ^ T ) where T = T^^. the form

(5.15)

To the required order in the perturbation h^, R^v has

j nk0,jk + nkk,Oj ~ fcy - fcoo.« + Kk.ii ~ hki,kj ~ hkJ,k,i)

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(i) h00 to O(2): To the required order, Roo = -iV 2 fc 0 0 ,

TOo = - T * p,

0oo = - 1

(5-17)

thus V2h00 = -8nGp,

h00 = 2GU

(5.18)

We now choose units in which G = 1, hence Ko = 21/

(5.19)

(ii) hy to O(2): If we impose the three gauge conditions (i = 1,2,3) K, - \Ki = o,

K = yf*hH

(5.20)

Equation (5.16) for Rtj becomes V2fcy=-8«piw,

hiJ = 2UStJ

(5.21)

(iii) h0J to O(3): If we impose the further gauge condition ^ - R o =

-ifcoco

(5-22)

Equation (5.15) becomes V2h0j + U.oj = 16npvj

(5.23)

or, using Equations (4.29), (4.32), and (4.33), h0J = -4Vj + ix.o; =-ty-iWj

(5.24)

It is useful to check that the solutions for h00, hOj, and htJ do satisfy the gauge conditions, Equations (5.20) and (5.22), to the necessary order. (iv) h00 to O(4): In the chosen gauge, Roo evaluated correctly to O(4) using the known lower-order solutions for h^v where possible, has the form Roo = -iV 2 (/j 0 0 + 2U2 - 8
(5.25)

To the necessary order, we also have Too - k o o T = M l + 2(v2 -U + ±I1 + fp/p)]

(5.26)

Then the solution to Equation (5.15) is h00 = 21/ - 2U2 + 4 $ t + 4
(5.27)

(v) g^ and the PPN parameters: The final form for the metric is 4, gtJ = (1 + 2l/)5y

(5.28)

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Since the metric is already in the standard PPN gauge, the PPN parameters can be read off immediately y = p = 1, £ = 0, ai

= a 2 = a 3 = Ci = C2 = C3 = C4 = 0

(5.29)

(g) Discussion: Notice that general relativity is a fully conservative theory of gravity (af = £; = 0) and predicts no preferred-frame effects (a* = 0). 5.3

Scalar-Tensor Theories A variety of metric theories of gravity have been devised which postulate in addition to the metric, a dynamical scalar gravitational field . The most general such theory was examined by Bergmann (1968) and Wagoner (1970), and special cases have been studied by Jordan (1955), Thiry (1948), Brans and Dicke (1961), Nordtvedt (1970b), and Bekenstein (1977). We shall examine the Bergmann-Wagoner theory in detail, then shall discuss the various special cases. (a) Principle references: Bergmann (1968), Wagoner (1970). (b) Gravitational fields present: the metric g, a dynamical scalarfield<j>. (c) Arbitrary parameters and functions: Two arbitrary functions of (j), the coupling function a>{4>) and the cosmological function A(#). (d) Cosmological matching parameters: 0. (e) Field equations: The field equations are derived from the action

^

(5.30)

The resulting field equations are

(5.31) (532)

The field equation for (j> can be rewritten by substituting the contraction of Equation (5.31) into Equation (5.32), with the result dco ^ 4 The cosmological function l() causes two effects in this theory. First, in the field equation for g, it plays the same role as the cosmological

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constant in general relativity. Second, in the field equation for <j>, it gives the scalar field a range I related to X, co and their derivatives, in the sense that the solutions for cf> for an isolated system contain Yukawa-like terms exp(r/l). The result in g00 (Wagoner, 1970) is a "Newtonian gravitational potential" U of the form

^x,0 = J ^ j ^ ^ ^ V

(5.34)

where the effective gravitational "constant" is given by G(x - x') = a + bexp(- |x - x'\/l)

(5.35)

Experiments that test the inverse square law for gravitation (see Section 2.2) could thus set limits on the cosmological function X. However, henceforth we shall assume X = 0. (f) Post-Newtonian limit: We choose coordinates (local quasi-Cartesian) in which g is asymptotically Minkowskian; <j) takes the asymptotic value (f>0 (which presumably varies on a Hubble timescale as the universe evolves). We define

co = co(
co =

A ==
(5.36)

Following the method of Section 5.1, we obtain for the post-Newtonian metric

goo= -1 +2U - 2 ( 1 +A)U2 + 4 ( , . - A )<E>2 + 2«D3 + 6 ( ^ - |
In going to geometrized units, we have set 1

<5J8)

Notice that if c/>0 changes as a result of the evolution of the universe, then Gtoday may change from its present value of unity (see Section 8.4).

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The PPN parameters may now be read off:

a t = a 2 = a 3 = d = C2 = £3 = U = 0

(5.39)

For details of the derivation see Nutku (1969a), Nordtvedt (1970b). (g) Other theories and special cases: (i) Nordtvedt's (1970b) scalar-tensor theory is equivalent to the Bergmann-Wagoner theory in the special case of zero cosmological function X = 0. Its PPN parameters are the same as in the Bergmann-Wagoner theory. We shall denote these general versions the BWN scalar-tensor theories. (ii) Brans-Dicke theory is the special case a> = constant, 1 = 0. Its PPN parameters may be obtained from the BWN PPN parameters by setting a>' = 0 s A. In the limit a> -* oo, the Brans-Dicke theory reduces to general relativity. (iii) Bekenstein's (1977) Variable Mass Theory (VMT) is a special case of the BWN theory with a restricted form for the coupling function oi((j)). Beginning with a theory in which the rest masses of elementary particles are allowed to vary in spacetime via a scalar field , the variation being determined by a field equation with two arbitrary parameters r and q, Bekenstein has shown that, when transformed to a metric representation, the theory is a BWN scalar-tensor theory with l][r + (1 - r)qf()y2, 0 = [1 - qf(4>)~\f(rr

(5-40)

Note that for chosen values for r and q, the present values of a> and A are determined by the asymptotic value 0, which in turn is found through a cosmological solution using the theory. For further details, see Bekenstein and Meisels (1978,1980) and Bekenstein (1979). (iv) Barker's Constant G Theory (1978) is the special case in which 0 * 0 = (4 - 30)/(20 - 2)

(5.41)

G,oday = 1 = [constant]

(5.42)

A = (1 - o)/2
(5.43)

thus

and

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(h) Discussion: We note that scalar-tensor theories are all fully conservative theories (a, = £, = 0), with no preferred-frame effects (<x; = 0). In the limit a -* oo, they reduce to general relativity, both in the postNewtonian limit and in the exact, strong-field theory, for all except a set of measure zero of pathological coupling functions co(). In particular, this is true for Brans-Dicke theory, Bekenstein's VMT, and Barker's theory. Generally speaking, for large values of a>($0), these theories make predictions at the current epoch for all gravitational situations - postNewtonian limit, neutron stars, black holes, gravitational radiation, cosmology - that differ from general relativity at most by corrections of O(l/co). However, in theories in which co is a function of , there could be significant differences with general relativity in the early universe, even if the present value of co(0o) is large (Chapter 13). In Brans-Dicke theory (constant co) all predictions are within O(l/co) of those of general relativity [see Ni (1972) for an extensive list of references for Brans-Dicke theory]. 5.4

Vector-Tensor Theories Within the class of purely dynamical metric theories of gravity, one simple way to devise a theory that is different from the scalar-tensor theories is to postulate a dynamical four-vector gravitational field K" in addition to the metric, thus obtaining a vector-tensor theory of gravity. A broad class of such theories can be analyzed if we restrict attention to Lagrangian-based theories, and to theories whose differential equations for the vector field are linear and at most of second order. The most general gravitational action for such theories is given by i G = (167CG)-1 §[atR + a2KliK"R + aJPlTR^ 2

+ a4Kp.vK":v (5.44)

(we have ignored the possible term K^ICg^, since it presumably plays the same role as the cosmological function A in scalar-tensor theories). In fact this action is too general; it can be simplified by an integration by parts, dropping divergence terms which do not contribute to the variation of /. Thus the sixth term in JG can be eliminated. Furthermore, the constant a! can be absorbed into G, resulting in a four-parameter set of vector-tensor theories. (a) Principal references: Will and Nordtvedt (1972), Hellings and Nordtvedt (1973).

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(b) Gravitational fields present: The metric g, a dynamical vector field K (assumed timelike). (c) Arbitrary parameters and functions: Four arbitrary parameters co, rj, e, T. (d) Cosmological matching parameters: K. (e) Field equations: The field equations are derived from the action

+ xK^K^J - gf'2 d4x + ING(qA, gj

(5.45)

where F,v = *,. - KK,

(5.46)

The resulting field equations are = SnGT^, eFfvv

v

v

+ \tK% - ^coK'R - iriK R$ = 0

(5.47) (5.48)

where 0£> - K^R

+K ^

- \g^K2R - (K%v

= 2K"K(I1RV)X - faj

+ (K*K(fl;V) K*(flKv) K ( / J K^). a

(5.49)

where K2 = K^K". Throughout, we assume that one of {e, T} is nonzero in order to have a well-defined free dynamical vector field. An important property of these equations is worth examining here. If one takes a co variant divergence of the left-hand side of Equation (5.47), one finds explicitly that it vanishes, in agreement with the law Tfvv = 0, in other words, no additional constraint on the fields is imposed by the vanishing divergence of T"v. This is a result of the fact that the action / is generally covariant and contains no prior-geometric variables [see Lee, Lightman, and Ni (1974) for discussion]. However, a divergence of the left-hand side of Equation (5.48) yields the constraint ll

- (a)K»R + tilPR/i).,, = 0

(5.50)

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This is a result of the fact that the action is not fully gauge invariant, i.e., invariant under the transformation K, -> K^ + A

(5.51)

where A is a scalar function. Only the term involving F p v is gauge invariant. A Lagrangian that admits such a partial gauge group is called "singular," and can be shown to satisfy a "Bianchi identity," which, in the case of the partial gauge group of a vector field, has the form . .,. = 0

(5.52)

This is equivalent to Equation (5.50). This means that, in general, the solution for K^ will be constrained. It is useful to examine the form that this constraint takes in the linearized approximation, in which we write g^ = n^ + Ky

K

? =

K3

° +K

( 5 - 53 )

If we adopt a coordinate system (coordinate "gauge" as opposed to vector gauge) in which >,*; - ±h-° = 0

(5.54)

where indices on h^ and kv are raised and lowered using if, and where h == hi, Equation (5.50), to first order in h^ and fcp, takes the form {3v{?k*v jK(a> + ^n \x)h 0 } = 0

(5.55)

Since this equation must be satisfied for arbitrary sources, then to first order in h^ and k^ we must have + %n- %i)ht0 = 0

(5.56)

In the weak field limit, in the chosen gauge, h^v must have the form h00 = 2(7,

hu = 2yUStJ - (y - l)x,,j

(5.57)

where y is the PPN parameter. Then Equation (5.56) becomes T/cyv - 2K{co + \n - ±r)(2y - 1)17.0 = 0

(5.58)

In the case x =£ 0, this represents a constraint on the gauge of the vector field k^ imposed by the lack of full gauge invariance of the action /. In the case x = 0, no constraint is placed on the vector field; however, in order to obtain consistent solutions of the equations, with a hope of agreeing with experiment, we must have co + \r\ = 0, since K # 0 and t / 0 ^ 0 in general, and since experiments (Chapter 7) place the value of

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y close to unity, so that 2y 1 ^ 0. These constraints will be important in our discussion of the post-Newtonian limit. (f) Post-Newtonian limit: We choose local quasi-Cartesian coordinates in the universe rest frame, with K^ taking the asymptotic form K8®, where K may vary on a Hubble timescale. Following the method of Section 5.1, we compute the post-Newtonian limit, and obtain for the PPN parameters _ 1 + K2[co - 2co{2co + n~ -Q/(2e - T)] 7 ~ 1 - K2[co + 8« 2 /(2e - T )] /*= i(3 + y) + M i

ai

+ y(v - 2)/G],

= 4(1 - y)[l - (2e - T)A] + 4coK2 Aa,

a2 = 3(1 - y)[l - |(28 - T)A] + 2coK2 Aa -

\bK2IG,

<*3 = Ci = Ci = Cs = U - 0

(5.59)

The quantities a, A, a, and b are given by (1 - coK2)(2co -n + 2£) _ (1 - T) - 8o>2K2 A = {(2e - t)[l - K2(co + n - T)] + i ( ^ -

2

T)2K2}-\

a = (2B - r)(3y - 1) - 2(n - T)(2 7 - 1), f(2o> + n - x)[(2y - l)(r + 1) +
%# 0 T= 0

(5.60)

and G is related to the other parameters by our choice of geometrical units, namely Gtoday s G[i(y + 1) + f coK2(y - 1) - i(ij - T)K 2 (1 +
(5.61)

(g) (Mer theories and special cases: (i) The Will-Nordtvedt (1972) theory is the special case a> = n = s = 0, x = 1. Its PPN parameters are given by

y = fi = 1, ox = 0,

f = a 3 = d = C2 = C3 = C4 = 0, a 2 = K 2 /(l + | K 2 )

(5.62)

with 2

)=l

(5.63)

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(ii) The Hellings-Nordtvedt (1973) theory is the special case T = 0, e = 1, r\ = 2ft). Its PPN parameters are given by l + coK2

o

.

.

/ 1 + coy

= « 3 = d = C2 = Cs = C* = 0, 4coK\2(l + co)y + co(y - 1)] 1 + a)X 2 (l + co) 0(2

_2c»K[y + c o ( y l ) ] ~ 1 + coKHl + co)

(5>64)

with G,oday = G[cDK\y + 1)] - 1 = 1

(5.65)

We point out that the original computations of Hellings and Nordtvedt (1973) were in error, since their method failed to take into account the constraint Equation (5.58). (h) Discussion: These vector-tensor theories are semiconservative (a3 = C, = 0) with possible post-Newtonian preferred-frame effects (one of{a x ,a 2 } ¥" 0). In the limit {co, r\, E, T} -» 0, they reduce to general relativity both in the post-Newtonian limit, and in the exact, strong field theory. However, there are other possible limiting cases in which the theories may agree with general relativity (and thus with experiment) in the postNewtonian limit. For instance, in the limit K -»0, the PPN parameters coalesce with those of general relativity. However, the present value of K depends upon a solution of the cosmological problem, and in the early universe K could be sufficiently large to produce significant differences. 5.5

Bimetric Theories with Prior Geometry Theories in this class contain dynamical scalar, vector, or tensor gravitational fields, and a nondynamical metric ij of signature + 2. In typical theories, t\ is chosen to be Riemann flat everywhere in spacetime, that is Rlem(i/) = 0

(5.66)

(in some versions, IJ is chosen to correspond to a spacetime of constant curvature). Because of the above constraint, we can always choose global coordinates in which t]^ = diag( 1,1,1,1); this is usually the most convenient choice for the computation of the post-Newtonian metric.

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Rosen's bimetric theory (a) Principal references: Rosen (1973,1974,1977,1978), Rosen and Rosen (1975), Lee et al. (1976). (b) Gravitationalfieldspresent: the metric g, a flat, nondynamical metric flic) Arbitrary parameters and functions: None. (d) Cosmological matching parameters: co,cy. (e) Field equations: The field equations are derived from the action = (647TG)x (-V)ll2d*x + ING(qA,g,v)

(5.67)

where the vertical line " |" denotes covariant derivative with respect to The field equations may be written in the form Riemfo) = 0 (5.68) 1 where , is the d'Alembertian with respect to q, and T s T^g . (f) Post-Newtonian limit: We choose coordinates in which if has the form diag( 1,1,1,1) everywhere. In the universe rest frame, g then has the asymptotic form diag(c^c^c^c^) [see Equation (5.1)], where c0 and c t may vary on a Hubble timescale. Following the method of Section 5.1, we obtain for the PPN parameters (Lee et al., 1976) y = p = 1, Kl = 0,

£ = « 3 = Ci = C2 = C3 = U = 0, a2 = (c o / Cl ) - 1

(5.69)

with Gtoday = G{coCl)112

= 1

(5.70)

(g) Discussion: The PPN parameters are identical to those of general relativity except for a 2 , which may be nonzero if c0 # cx. Notice that the ratio cjco is equal to the square of the velocity of weak gravitational waves, in units in which the speed of light is unity. This can be seen as follows. In a quasi-Cartesian coordinate system, in which gffl = diag( 1, 1,1,1 ),»; must have the form n^ = diag( - Co \ c^ \ erf *, c r ' ) and the vacuum, linearized field equations for g^v (wave equations for weak gravitational waves) take the form (co/cite^oo -

V

V =0

(5.71) 112

whose solution is a wave propagating with speed vg = (cjco) . Thus, in Rosen's theory, the PPN parameter a2 measures the relative difference

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in speed (as measured by an observer at rest in the universe rest frame) between electromagnetic and gravitational waves. The values of c 0 and Cj are determined by a solution of the cosmological problem. They can also be related to the covariant expressions CQ i + 3cf* = /7JIV0
c 0 + 3cj = n^gtg!,

Rastall's theory (a) Principal references: Rastall (1976, 1977a,b,c, 1979). (b) Gravitational fields present: the metric g, a dynamical timelike vector field K, a nondynamical flat metric r\. (c) Arbitrary parameters and functions: None. (d) Cosmological matching parameters: K. (e) Field equations: The physical metric g is an algebraic function of the fields if and K, given by g = (1 + rfKJS.,)-^2^

+ K ® K)

(5.72)

where \\rfp\\ = H^H" 1 . The field equations are derivable from the action :+ W«A»M

(5-73)

where indices on K^ are raised using g, and where F(N) = - N(2 + N)~ \

N = g^K^

(5.74)

We also have Riem(if) = 0. The resulting field equations are

0)-

1/2

(0" v - k

= 87tG(l + n^KxK^yll2{T"v

- ^VT)KV

(5.75)

where F'(N) = dF/dN,

+ F'{N)KX'IIKO[.I)K'1KV

(5.76)

and & = ©''"^v, T= T" v ^ v . In varying the action /, with respect to Kp, we have taken account of the fact that the dependence on K^ is both explicit and implicit via gMV, thus for example, although the action for matter and nongravitational fields 7NG contains only g^, we have (5.77)

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133

(f) Post-Newtonian limit: We choose coordinates in which n^ = diag( 1,1.1.1), then from Equation (5.72) g^ takes the form, to postNewtonian order 0oo = - c o ( l - Kco2k0 - |CQ 4feg), gOJ =

Kc^kj,

gij = ColSJk(l + Kco2k0)

(5.78)

where c 0 = (1 K2)112, \K\ < 1. Solving the field equations for k^ to the required order, substituting into Equation (5.78), and transforming to local quasi-Cartesian coordinates in the standard PPN gauge yields the PPN parameters y = /? = 1,

Z. = a 3 = d = C2 = C3 = U = 0,

In choosing geometrical units, we set Gtoday = G = 1

(5.80)

(g) Discussion: RastalFs theory is semiconservative (a 3 = Ci 0), with preferred-frame effects (a2 # 0). Its PPN parameters are identical to those of general relativity, except for a2, which maybe nonzero. The value of <x2 depends upon K, whose value is determined by a solution of the cosmological problem. The BSLL bimetric theory This theory is a variant of the Belinfante-Swihart nonmetric theory of gravity, discussed in Section 2.6. Instead of the nongravitational action / N G shown in Equations (2.140) and (2.141), one chooses a universally coupled action, thereby obtaining a metric theory of gravity (Lightman and Lee, 1973b). Otherwise the equations of the theory are the same as those presented in Section 2.6. (a) Principal references: Belinfante and Swihart (1957a,b,c), Lightman and Lee (1973b). (b) Gravitational fields present: the metric g, a dynamical second rank tensor field B, a nondynamical flat metric i\. (c) Arbitrary parameters and functions: three arbitrary parameters a,

f,K. (d) Cosmological matching parameters: a>0, (o^.

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(e) Field equations: The metric is constructed algebraically from t\ and B according to the equations

; - ± 3 D = <5v

(5-81)

where indices on Apv and B^ only are raised and lowered using n^; indices on all other tensors are raised and lowered using g^; B = B^n*". The field equation for i\ is Riem(//) = 0. The field equations for B are derived from the action / = -(167c)" 1 j(aB"^B^x

+ /B^X-i/)1'2**** +

W«A,0,«)

(5-82)

where vertical line denotes a covariant derivative with respect to r\. The resulting field equations are , ,

(5.83)

which may be rewritten in the form D ^

= -(4w/fl)to/f7)1/27^[0|5 - f(a + 4f)-ieil>r,»%d]

(5.84)

where Kl = 8gxl,/dB,v

(5.85)

(f) Post-Newtonian limit: We work in the universe rest frame, choose coordinates in which n^ = diag( 1,1,1,1), and assume that o.co^co^co!). We further assume that |coo| « 1, \a>^ « 1, assumptions that turn out to be consistent with experimental limits. Then to the necessary order, g^ has the form fifoo = -Do + E0b00 - Fob - K2b2 - 2Kbb00 - |fcg 0 . g0J = HbOj, SiJ

= Ddtj + EbtJ + FdiP

(5.86)

where Do = 1 - 2Kco -coo + K2co2 + 2Kcoco0 + |coo + O(co3), Eo = 1 - 2Kco - f a»0 + O(co2), Fo = -2K

+ 2K2co + 2Kco0 + O(co2),

H = 1 - 2Kco - |(co o - co^ + O(a>2), D = 1 - 2Kw + w1+ K2co2 - 2Kcoco1 + |cof + O(co3), E = 1 - 2Kco + Icoj + O(co2), F= -2K

+ 2K2co - 2Kco1 + O(co2),

co = 3(0! coo

(5.87)

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135

Solving the field equations for fcJlv, substituting into Equation (5.86), then transforming to quasi-Cartesian coordinates and to the standard PPN gauge yields the PPN parameters p = i [ l + l a " 1 - ia-^Sa2

- 3a)1/2(a + 4/)- 1 / 2 ] + O(co),

i = a 3 = Ci = £2 = C3 = U = 0, a t = (2a)~1[ojo + » ! - (8X - 2)w] + O(co2), a 2 = (
(5.88)

In using geometrized units, we set _a + 3/-4Xa-16*2a "today

/) j

. jn

T- vj^ti); 1.

(J.O?,)

2a(a + 4/) (g) Discussion: The BSLL Theory is semiconservative (a3 = Ct = 0), with potential preferred-frame effects if a>0 or col are nonzero. However, solar system experiments (Chapter 8) demand that la^ and |a 2 | be small, in keeping with our original assumption that jcoo| « 1, leo^ « 1. Whether a>0 and co1 in fact satisfy this constraint depends upon a solution of the cosmological problem. Notice that if m^ ~ a>0 0, the PPN parameters can be made identical to those of general relativity if

0 = (i -A,*}

(5-90)

5.6

Stratified Theories These theories are characterized by the presence, in addition to a flat background metric t\, of a nondynamical scalar field t whose gradient is covariantly constant and timelike with respect to tj, i.e.,

This scalar field selects out preferred spatial sections or "strata" in the universe that are orthogonal to \t. In a frame in which V / = <5°, the equations of a stratified theory take on some special form. (a) Principal references: Lee, Lightman, and Ni (1974), Ni (1973). (b) Gravitational fields present: the metric g; dynamical scalar, vector, and symmetric tensor fields ), fii^ parameters e, KU K2. (d) Cosmological matching parameters: co,cud,a,b, c.

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(e) Field Equations: The field equations for the prior-geometric variables are Riem(i;) = 0, *l,* = 0,

t/^=-l,

= 0, = 0

(5.91)

The last two equations constrain the vector and tensor fields to have components only in the strata orthogonal to \t. The metric g is constructed algebraically from t/, <j>, t, B, and K according to 9 = / 2 ( # / - E/iW>) - / 2 (0)]dt ® d t + K ® d t + d t ® K + B

(5.92)

The field equations for the dynamical variables are derived from the action - < £ > * - \_Mcj>) + / N G (q A ,^ v )

(5.93)

where all indices on the variables , t, B, and K are raised and lowered using i\. The result is

\_M) (5.94) The constraints on the prior geometric variables allow one to choose a global coordinate system in which t]^ = diag( 1,1,1,1), ttll = d°, Ko = B^o = 0, and in which the field equations simplify to

'1((/») -

Tilf'2(4>)l (5.95)

In this preferred frame (presumably the universe rest frame), g^v has the form 9oj =

Kj,

9u = 8M4)

+ Bij

(5-96)

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137

(f) Post-Newtonian limit: In the preferred frame we expand <j>, Kit and Bij about cosmological boundary values: (j> = 4>0 + ) = c0 - 2c
) = (ci - cOi) + 2ac
2), M4) = d + O(
(5.98)

0oo = - c 0 + 2ccp - 2bczcp2, 9oj = kj,

Qii = ci3u + 2acq>5tj + bu

(5.99)

Solving the field Equation (5.95) for $, kj, and bip and transforming to quasi-Cartesian coordinates and the standard PPN gauge, we obtain the PPN parameters y = aco/ci,

P = bc0 + (K 2 /8K 1 C)(C 0 /C 1 ),

£, = (K 2 /8K 1 C)(C 0 /C 1 ), 12

a t = 2e/(coc1) '

a 3 = Ci = t2 = C3 = U = 0,

- 4a(co/Cl) - 4,

a 2 = - 1 - ( c o / c J t a ^ c + (d + K22/4Kl)(l + K2/4Kl)-x]

(5.100)

In choosing geometrical units, we set Gtoday = c2c\'2co3/2(l

+ K1/4K!)-1 = 1

(5.101)

(g) Other theories and special cases: (i) Ni's (1973) stratified theory is the special case K^ * = K2 = 0 (no tensor field). Its PPN parameters can be obtained from Equation (5.100) by setting K2 = 0, with the result y = acQ/cu

P = bc0,

i = a 3 = Ci = C2 = C3 = U = 0, ax = 2e/(coci)1'2 - 4a(co/Cl) - 4, a 2 =-l-d(c 0 / Cl )

(5.102) l

(ii) Ni's (1972) stratified theory is the special case e = K^ = K2 = 0 (no vector or tensor field). However, as we shall see in the next section, this theory is not viable because its PPN parameter a1 satisfies ai

= -(4y + 4)

which is in serious violation of experiment.

(5.103)

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138

5.7

Nonviable Theories All the metric theories of gravity previously discussed have the property that by making an appropriate choice of values for arbitrary constants and for cosmological matching parameters, one can produce PPN parameter values in agreement with present-day solar system experiments, to be described in Chapters 7,8, and 9. In some theories, a particular choice of these quantities can yield PPN parameters that are identical with general relativity at the current epoch. Therefore, in order to test and possibly rule out some of these competing theories, we will have to explore new arenas for testing relativistic gravity outside the solar system, such as gravitational radiation, the binary pulsar, and cosmology. However, there is a sizable set of metric theories that, while perhaps once thought to have been viable, are now known to be in serious violation of solar system experiments. Some of these theories agree with the "classical" tests: deflection of light, time delay, perihelion shift of Mercury (see Chapter 7 for discussion). But this is not enough. There are now many further solar system tests, discovered through the use of the PPN formalism, that place tight limits on the preferred-frame parameters a1? and a2, on conservation-law parameters such as a3, and on the parameter £,. Many theories violate these limits. The lesson to be learned is that it is no longer sufficient for the inventor of an alternative gravitation theory to compare the predictions of the theory with experiment by simply deriving the static spherically symmetric solution (analogue of the Schwarzschild solution in general relativity), obtaining the PPN parameters ft and y. He or she must determine the full post-Newtonian metric for a dynamical system of bodies or fluid, possibly moving relative to the universe rest frame, including cosmological matching parameters. Only with a complete set of values for the PPN parameters can the theory be compared with the results of solar system experiments. Many of the nonviable theories that we shall describe were discussed in more detail in TTEG. We shall touch upon them here only briefly, referring the interested reader to TTEG and the original references for details. (a) Quasilinear theories Quasilinear theories of gravity are theories whose postNewtonian metric, in a particular post-Newtonian gauge, contains only linear potentials, in particular lacks the potentials U2 and <SW. This is a property of many theories that attempt to describe gravity by means of a linearfieldtheory on a flat spacetime background. If the gauge in which

Post-Newtonian Limits

139

this occurs is not the standard PPN gauge, then a gauge transformation, as in Equations (4.38) and (4.40) yields 06o = 0oo ~ 2X2(U2 + ®w-
(5.104)

Since g00 did not contain U2 or
(5.105)

We shall see that this is in severe violation of Earth-tide measurements (Chapters 8 and 9). The most famous example of a quasilinear theory is Whitehead's (1922) theory. The theory has a nondynamical flat background metric IJ, and a physical metric constructed algebraically from IJ and the matter variables according to = n,,v 2

( / ) - = X" - (*)-, w2

= (/)-(u M )-, da = rifl,dx"dxy

(w-)3iA

»/

r>

(yT(y»)~ = o, u" = dx^/da, (5.106)

where the superscript () indicates quantities to be evaluated along the past i/-light cone of the field point x*. The post-Newtonian metric has y = P = £, = 1, {Ci,C2,C3,C*}^0

oti = <x2 = a3 = 0, (5.107)

Although the theory was thought for a long time to have been viable, the value £ = 1 is now known to be in violation of Earth-tide measurements. Another group of theories in this class is known as Linear FixedGauge (LFG) theories. The standard field theoretic approach to the construction of a tensor gravitation theory on a flat spacetime background is to use the gauge invariant action for a spin-two tensor field h^, combined with the universally coupled nongravitational action to yield

- h*v[ V l J ( - f ) 1 / 2 d 4 x + /NG(<2A,0/1V)

(5.108)

where g^ = */ + h^. However, the Lagrangian is singular: the gravitational part is invariant under the gauge transformation hpv -> h ^ £(!)

(5.109)

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140

while / N G is not. The Bianchi identity associated with this partial gauge invariance is (5.110)

= 0

which is in conflict with the equation of motion that results from the general coordinate invariance of / [Equation (3.63)], T?v=0

(5.111)

LFG theories seek to remedy this by breaking the gauge invariance of the gravitational action through the introduction of auxiliary gravitational fields that couple to h in such a way as to fix the gauge of h. Nevertheless these theories, devised by Deser and Laurent (1968) and Bollini et al. (1970), turn out to be quasilinear in the sense defined above, and predict <* = P in violation of experiment (see Will, 1973). (b) Stratified theories with time-orthogonal space slices These theories are special cases of the stratified theories discussed in Section 5.6, in which there is no vector field K^, i.e., e = 0. Table 5.2. Nonviable metric theories of gravity Theory" (a) Quasilinear theories Whitehead Deser-Laurent Bollini-Giambiagi-Tiomno

Description

For some gauge, U2 Predict galaxy induced perihelion shifts and and Q>w are absent from 0oo; thus £ = fi Earth tides, in violation of observation

(b) Stratified theories with time-orthogonal space slices Einstein (1912) Metric is given by Whitrow-Morduch g = /idt ® it + f2t\; Rosen thus a : = 4(y + 1) Papapetrou Ni (2 versions) Yilmaz Page-Tupper Coleman (c) Conformally flat theories Nordstrom Einstein-Fokker Ni (2 versions) Whitrow-Morduch Littlewood-Bergmann 1

Reasons for nonviability

Predict preferred-frame effects on Earth's rotation rate and on perihelion shifts, in violation of observation

Metric is given by Predict no deflection or g = fii; thus y = 1 time delay of light, in violation of observation

For discussion and references, see TTEG, Ni (1972), and Will (1973).

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141

They therefore have the property that f "0T + W)g^

= - K" = 0

(5.112)

independently of the nature of the source. In the preferred frame, this means goj = 0. However, under a possible coordinate transformation to put the post-Newtonian limit of the theory into the standard PPN gauge, g0J becomes goj = sx,oj = sVj-eWj

(5.113)

By comparing this with the PPN metric [Equation (4.48)], it is possible to obtain in a straightforward manner, independently of £, Bl

= -(4y + 4)

(5.114)

This is a gross violation of geophysical experiments that demand loc^ « 1, while time-delay measurements demand y « 1 (see Chapters 7 and 8). Prior to the placing of the limit on ax, theories of this type were popular alternatives to general relativity, largely because of their mathematical simplicity. Table 5.2 lists nine theories of this type, all nonviable. (c) Conformallyflattheories These theories typically possess a flat background metric IJ and a scalar field . The metric g is constructed from r\ and according to g = /(#/

(5.H5)

where / is some function of . However, in order to obtain the correct Newtonian limit, /($) must have the form (in a suitable coordinate system) / = 1-217 + 0(4)

(5.116)

Thus, flfy = [1 - 21/+ O(4)]5 y

(5.117)

hence y = 1. We shall see in Chapter 7 that this implies zero bending of light and zero time delay, in violation of experiment. This result can also be deduced from the conformal invariance of Maxwell's equations (i.e., invariance under the transformation g^ -» #): propagation of light rays in the metric f(4>)ti is identical to propagation in the flat spacetime metric i/, namely straight-line propagation at constant speed. Table 5.2 lists six conformally flat theories, all nonviable.

Equations of Motion in the PPN Formalism

One of the consequences of the fundamental postulates of metric theories of gravity is that matter and nongravitational fields couple only to the metric, in a manner dictated by EEP. The resulting equations of motion include Tfvv = 0, [stressed matter and nongravitational fields] wvwfv = 0, [neutral test body: geodesies] F?vv v

= 4nJ", [Maxwell's equations]

/c /cfv = 0, [light rays: geodesies]

(6.1) (6.2) (6.3) (6.4)

(see Section 3.2 for discussion). In Chapter 4, we developed the general spacetime metric through post-Newtonian order as a functional of matter variables and as a function of ten PPN parameters. If this metric is substituted into these equations of motion, we obtain coupled sets of equations of motion for matter and nongravitational field variables in terms of other matter and nongravitational field variables. For specific problems, these equations can be solved using standard techniques to obtain predictions for the behavior of matter in terms of the PPN parameters. These predictions can then be compared with experiment. It is the purpose of this chapter to cast the above equations of motion into a form that can be simply applied to specific situations and experiments. That application will be made in Chapters 7, 8, and 9. In Section 6.1, we carry out this procedure for light rays. Section 6.2 deals with massive, selfgravitating bodies and presents appropriate n-body equations of motion. In Section 6.3, we derive the relative acceleration between two bodies, including the effects of nearby gravitating bodies and of motion with respect to the universe rest frame, and put it into a form from which one

Equations of Motion in the PPN Formalism

143

can identify a "locally measured" Newtonian gravitational constant. Section 6.4 specializes to semiconservative theories and presents an n-body action from which the semiconservative n-body equations of motion can be derived. We also develop in Section 6.4 a conserved-energy formalism of the type discussed in Section 2.5, and discuss the Strong Equivalence Principle from this viewpoint. In Section 6.5, we analyze equations of motion for spinning bodies. 6.1

Equations of Motion for Photons We begin with the geodesic equation obtained from Maxwell's equations in the geometrical-optics limit [Equation (6.4)]: fev/cfv = 0 (6.5) 11 where k is the wave vector tangent to the "photon" trajectory, with *"* = 0 (6.6) Substituting k" = dx*/da where a is an "affine" parameter measured along the trajectory, we obtain

We can rewrite Equation (6.7) using PPN coordinate time t = x° rather than a as affine parameter by noticing that

Then the spatial components of Equation (6.7) can be rewritten

~dW + Equation (6.6) can be written ^v^!^L = 0

(6.10)

To post-Newtonian accuracy, Equations (6.9) and (6.10) take the form (see Table 6.1 for expressions for the ChristorTel symbols T*k): 2 =[/,.

dt

1 + y

dx2S dt

0 = 1 - 2C7 - \dx/dt\2(l + 2yU)

(6.11)

The Newtonian, or zeroth order solution of these equations is x£ - n\t - t0),

\n\ = 1

(6.12)

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Table 6.1. Christoffel symbols for the PPN metric

3 + a, - a2 + C, - 2 © ^ + i(l + a2 t/,J-) + a2 + y)U2 3 + B l - a, + Ci - 2 « ^ +1(1 + a2 - d (at - 2a2)w'U + ajvWl/y],

C/>0 - i(4y + 4 + a j ^ ^ - i where 2 + <x3 + {, - 2^a>x + (3y - 2)3 (3y + 3C4

in other words, straight-line propagation at constant speed |dx N /di| = 1. By writing xj = n\t - to) + xJp

(6.13)

and substituting into Equation (6.11) we obtain post-Newtonian equations for the deviation xJp of the photon's path from uniform, straight line motion: ^ £ = (l + y)[yu dx i'-£=-(l+y)U

- 2n(n VC7)],

(6.14) (6.15)

In Chapter 7 we shall use these equations to derive expressions for the deflection and the time delay of photons passing near the Sun. 6.2

Equations of Motion for Massive Bodies One method of obtaining equations of motion for massive bodies is to assume that each body moves on a test-body geodesic in a spacetime whose PPN metric is produced by the other bodies in the system as well as by the body itself (with proper care taken of infinite self-field terms). However, the resulting equations of motion cannot be applied to massive self-gravitating bodies, such as planets, stars, or the Sun (except in general relativity, as it turns out), because such bodies do not necessarily follow geodesies of any PPN metric. Rather, their motion may depend

Equations of Motion in the PPN Formalism

145

upon internal structure (a violation of GWEP). This was first demonstrated by Nordtvedt (1968b). Therefore, one must treat each body realistically, as a finite, selfgravitating "clump" of matter and solve the stressed-matter equations of motion [Equation (6.1)] to obtain equations of motion for a suitably chosen center of mass of each body. For the purposes of solar system experiments, it is adequate to treat the matter composing each body as perfect fluid (see Will, 1971a for discussion). In Newtonian gravitation theory, this program is straightforward. By defining an inertial mass and a center of mass for each body according to pd3x,

ma = I Joth body

xa = m- 1 f pxd3x

(6.16)

Ja

one can show, using the Newtonian equation of continuity [Equation (4.3)] that dmjdt = 0, va = dxjdt = m~1 ja p\d3x, = dyjdt = m;l I p{dv/dt)d3x

K

(6.17)

By using the Newtonian perfect-fluid equations of motion [Equation (4.3)] we obtain the following expression for aa

S \

\ Ql # L

b*a \Jab

r

ab

(^)]

(6-18) J

where mb is the inertial mass of the bth body, Q'J is its quadrupole moment defined by

J

i

| |

2

)

3

(6.19)

and \ab and rah are given by xai, = xa ~ xb,

r^ = IxJ

(6.20)

We now wish to generalize these equations to the post-Newtonian approximation, using the PPN formalism. Because there are many different "mass densities" in the post-Newtonian limit - rest-mass of baryons p, mass-energy density p{\ + U), "conserved" density p*, and so on there is a variety of possible definitions for inertial mass and center of mass. The definition we shall adopt is chosen in order to yield the simplest closed-form result for the equations of motion. It turns out that as long

Theory and Experiment in Gravitational Physics

146

as we average the equations of motion over several internal dynamical timescales of each body (assumed short compared to the orbital dynamical timescale), the final equation of motion is insensitive to the precise form of the definition. We define the inertial mass of the ath body to be ma = f p*(l + iF 2 - \V + n)d3x

(6.21)

Ja

where p* is the conserved density [Equation (4.77)], v = v va(0), where vY = f n*\d3x (6 77\ o(0) I r

and

" " *

\v.£^.)

U = £ p(x',t)\x -x'l'1

d3x'

(6.23)

Note that, roughly speaking, ma is the total mass energy of the body rest mass of particles plus kinetic, gravitational, and internal energies - as measured in a local, comoving, nearly inertial frame surrounding the body. As long as we ignore tidal forces on the ath body, then according to our discussion of conservation laws in the PPN formalism [see Equation (4.108)], ma is conserved to post-Newtonian accuracy, i.e., dmjdt = 0

(6.24)

This can also be shown by explicit calculation using Equations (6.21), (6.22), and (6.23). We now define the center of inertial mass xa s m~1 f p*(l + iv2 - \V + U)xd3x

(6.25)

Ja

By making use of the equation of continuity for p* [Equation (4.78)] and by using Newtonian equations of motion in any post-Newtonian terms, we obtain vfl = dxjdt = m~1 f [p*(l + iu 2 - iU + IT)v + pv - |p*W] d3x

(6.26)

where

The acceleration aa is thus given by ao = d\Jdt = m'1 < Ja p*(l + it; 2 - if/ + U)(d\/dt)d3x + v{ £ pjf d3x + £ |>>ov - (p/p*)Vp] d3x 3

P*Wd

x + \g-a - \«r*a + g>>\

(6.28)

Equations of Motion in the PPN Formalism

147

where &~a, 3~*, and 0>a are determined purely by the internal structure of the ath body. Formulae for these and other "internal" terms are given in Table 6.2. Notice that the acceleration of our chosen center of mass is more than just the weighted average of the accelerations of individual fluid elements, as it is in Newtonian theory. We now evaluate the first integral in Equation (6.28) using the PPN perfect-fluid equations of motion. We substitute the post Newtonian expressions for T"v (Table 4.1) and Y*x (Table 6.1) into the equation of motion, (6.1), and rewrite it in terms of the conserved density p*. The Table 6.2. Integrals for massive bodies in the PPN equations of motion. Vector integrals ' "II '13 X X I [X X I

o*p*'p*"(x' - x") (x - x')(x - x'Y

r

H

y

u,

**v

~>d3Xd3X',

^

|x - x'| 3

- I P*P*'lr " (x ~ x')]2(* ~ x')3

y*i

X - X'

5

Tensor and scalar integrals: P*v'vJd3x

n

« = -ij»prz^p r f

xd x

> "-= -*J.-prr7j-J xd

I'J = £ p*(x - xj'(x - x a )^ 3 x,

/. = ja p*\x - xa\2 d3x

Theory and Experiment in Gravitational Physics

148

result is p*dvJ/dt = p*Utj - |>(1 + 3yU)lj + Pj&2

+ U + pip*)

v\p*Ui0 - p,0) - i(l + <x2 iP*[(4y + 4 + atf + (ax p*(d/dxJ)[® - £«V - i ( d -

- (2jS - 2)U + 3yp/p*]

(6.29)

where O is given in Table 6.1. We now substitute this expression for p*d\/dt into Equation (6.28) and perform the integration, using Newtonian equations where necessary to simplify post-Newtonian terms. Considerable simplification of the equations results if we average over several internal dynamical timescales of each body. Then we can set equal to zero any total time derivatives of internal quantities. This is a reasonable approximation for the solar system, since any secular changes in the structure of the sun or planets that would prevent the vanishing of such averaged time derivatives occurs over timescales much longer than an orbital timescale. This allows us to use several Newtonian virial relations to simplify post-Newtonian expressions. These relations, easily derived using the Newtonian equations of motion have the form for each massive body

H"= - = 0,

ST*> + 33T**J - Q*i - 0>J = (~ \dt J

=(j

[p*WWx) /

Jp*VJ d3x)=0

= 0,

(6.30)

Equations of Motion in the PPN Formalism

149

The final form of the equation of motion is K = (aAelf + (aa)Newt + (aJnbody

(6-31)

where

{ai)seK = -m^iu**

+ CM + Ci(ri - ir**J) vttfHkJ,

(6.32) (6.33)

(27 + 2f}) {(7 f) rr± (y

I ^r

b*a

ab

rr

(.

p

C2) ^

fa

r

ab

ab

+(2j8-l-2«-C 2 ) E ^+(2y + 2/?-2£) ^ c*a* r6c

c*ab rac

.,2 > + 2 + a2+<x3)»i

a 2 )(v 6 Kb)2 + | « 2 ( w fla!,)2 + 3a 2 (w hab)(vb nab)

|(7 +

^+

1

2

+ C1)E

I

*#a rab c*ab

# r

bc

- t Z ^ ( ^ - 3 « 4 ^ ) E "C^ + I 5 xa6 [(2y + 2)va - (2y 1

*w

~ 5z I ^x a 6 -[(47 + 4+a1)vfl b*a "ab

~ * I z

^ r x«fc' [«i v « - ( a i ~ 2«2)T6 + 2a2w]w^'

(6.34)

i>#n "aft

where nab = \ab/rab The first six terms in (a a ) self , Equation (6.32), involving terms such as t{, 3~'a, and so on, depend only on the internal structure of the ath massive body, and thus represent "self-accelerations" of the body's center of mass. Such self-accelerations are associated with breakdowns in conservation of total momentum, since they depend on the PPN conservation-law parameters a 3 , £i> ^2> C3, and £4- In any semiconservative theory of

Theory and Experiment in Gravitational Physics

150

gravity, « 3 EE d = C2 = C3 s C4 = 0

(6.35)

and these self-accelerations are absent. Also note that spherically symmetric bodies suffer no acceleration regardless of the theory of gravity, since for them the terms t{, PJa, ^~**j, Q.{, &{, and &[ are identically zero. The same is true for a composite massive body made up of two bodies in a nearly circular orbit, when the self-acceleration is averaged over an orbital period. Thus, there is little hope of testing the existence of these terms in the solar system. However, in the binary pulsar, for instance, where the orbit eccentricity is large, there may be a potential test. We shall discuss this possibility in Section 9.3. The next term in Equation (6.32), m~1a.3(w + vafHk.j, is a selfacceleration which involves the massive body's motion relative to the universe rest frame. It depends on the conservation-law/preferred-frame parameter <x3, which is zero in any semiconservative theory of gravity. For any static body, v = 0, thus HkJ is zero, but for a body that rotates uniformly with angular velocity o>, v =
(6.36)

and \X X I llm

= e/ co\Sla)

Jm

(6.37)

For a nearly spherical body, the isotropic part of QJm makes the dominant contribution to Equation (6.37), i.e.,

(Qaym * &jmna,

HkJ =s ±e*WQ.

(6.38)

Then the acceleration term in Equation (6.32) becomes -!<x3(Qa/ma)(w + ya) x to

(6.39)

In Chapter 8, we shall see that this term may produce strikingly large observable effects in the solar system, if a 3 is different from zero. The next term, (ao)Newt in Equation (6.31) is the quasi-Newtonian acceleration of the massive body. Here (mP)a* is the "passive gravitational mass tensor" given by (mP)ik=ma{<5*[l + (4/J - y - 3. - 3{ - a, + a 2 - Ci W.M, - 3£nafcnam^i7ma]

C2)fiJ*MJ

(6.40)

Equations of Motion in the PPN Formalism

151

and U(xa) is the quasi-Newtonian potential, given by

U(xa) = £ & ^ ) i r

ab

where [mA(habf\b is the "active gravitational mass" of the bth body, given by (6-42) Note that the active and passive gravitational mass tensors may be functions of direction n^ relative to the other bodies. It is useful to rewrite the quasi-Newtonian acceleration in a form involving inertial, active and passive mass tensors that are independent of position, and a gravitational potential U'm, as follows

W'm= £

frJTK&r*

(6.43)

where

(ax - a2 + C i R M ] + (a2 - d (4/J - y - 3 - 3fl«./<| - ZflT/ (4/? - y - 3 - 3{ - i« 3 - Ki + Ca^M, - (fa3 + Ci - X&M - « - KiJOfM} (6-44) In Newtonian theory, the active gravitational mass, the passive gravitational mass, and the inertial mass are the same, hence each massive body's acceleration is independent of its mass or structure ("Equivalence principle"). However, according to Equation (6.44), passive gravitational mass need not be equal to inertial mass in a given metric theory of gravity (and in fact both may be anisotropic); their difference depends on several PPN parameters, and on the gravitational self energy (Q and Qik) of the body. This is a breakdown in the gravitational Weak Equivalence Principle (GWEP) (see Section 3.3), also called the "Nordtvedt effect" after its discoverer (Nordtvedt, 1968a, b). The possibility of such an effect was first noticed by Dicke [1964b; see also Dicke (1969), Will (1971a)]. The observable consequences of the Nordtvedt effect will be discussed in Chapter 8. Its existence does not violate EEP or the Eotvos experiment (Chapter 2), because the laboratory-sized bodies considered in those situations have negligible self gravity, i.e., (il/m)^^ bodies < 10~39. In Section 6.3, we shall see that there is a close connection between violations of GWEP and the existence of preferred-location and preferred-frame effects in postNewtonian gravitational experiments.

Theory and Experiment in Gravitational Physics

152

According to Equation (6.44), active gravitational mass for massive bodies may also differ from inertial mass and from passive gravitational mass. In Newtonian gravitation theory, the uniform center-of-mass motion of an isolated system is a result of the law "action equals reaction," i.e., of the law "active gravitational mass equals passive gravitational mass." In the PPN formalism, one can still use such Newtonian language to describe the quasi-Newtonian acceleration (ao)Newt. From Section 4.4, we know that uniform center-of-mass motion is a property of fully conservative theories of gravity, whose parameters satisfy a, = a2 = oc3 = Ci = C2 = C3 = U = 0

(6.45)

By substituting these values into Equation (6.44), we find that for fully conservative theories, the inertial mass is equal to ma, and the active and passive mass tensors are indeed equal, and are given by («ptf=(«A)^ = ^ { ^ [ l + ( 4 / » - 7 - 3 - 3 W . / m J - { n f / m . } (6.46) equivalently, (a£)Newt can be written to post-Newtonian order in the form

(ae^/TS*^ \ma mj\ r%b

sxjjj^Xj r^ ))

{6A7)

The term in braces is manifestly antisymmetric under interchange of a and b, hence action equals reaction, and £ , ma(a{)Newt = 0. Note that in general relativity, the mass tensors of Equation (6.44) are isotropic and equal to the inertial mass, i.e., (dropping the Kronecker deltas) fhl = fhp = mA = ma [general relativity]

(6.48)

There is no Nordtvedt effect in general relativity. However, in scalartensor theories, there is in general a Nordtvedt effect, since mP = mA = ma{\ + [(2 + co)"1 + 4A]fta/ma}

(6.49)

For most practical situations, we may assume that the bodies in question are spherically symmetric, then using the equation ClJak m %SJkQa to simplify the mass tensors, we may write

« = I (MAV^ b*

(6.50)

Equations of Motion in the PPN Formalism

153

where (we combine (m{k)~1 and ml into one quantity mP)

K) fl /m a = 1 + (40 - y - 3 - Aft -

Kl

+ | a 2 - f£

K ) » M = 1 + (4/J - y - 3 - ^ - i« 3 - Ki + Cs^/m - (|a 3 + d - SCJPJm, (6.51) The remaining term (aj nbody in Equation (6.31) is called the n-body term. It contains the post-Newtonian corrections to the Newtonian equations of motion which would result from treating each body as a "point mass" moving along a geodesic of the PPN metric produced by all the other bodies, assumed to be point masses, taking account of certain post-Newtonian terms generated by the gravitational field of the body itself. It is the n-body acceleration which produces the "classical" perihelion shift of the planets, as well as a host of other effects, to be examined in Chapters 7 and 8. For the case of general relativity, the n-body terms in Equation (6.34) are in agreement with the equations obtained by de Sitter (1916) [once a crucial error in de Sitter's work has been corrected], Einstein, Infeld, and Hoffmann (1938), Levi-Civita (1964), and Fock (1964). 6.3

The Locally Measured Gravitational Constant

Here, we derive an equation which is not really an equation of motion, but is nevertheless a fundamental result in the PPN formalism. In the previous section, we found that some metric theories of gravity could predict a violation of GWEP (Nordtvedt effect). Such effects would represent violations of the Strong Equivalence Principle (SEP). As discussed in Section 3.3, the existence of preferred-frame and preferredlocation effects in local gravitational experiments would also represent violations of SEP. One such local gravitational experiment is the Cavendish experiment. In an idealized version of such a Cavendish experiment one measures the relative acceleration of two bodies as a function of their masses and of the distance between them. Distances and times are measured by means of physical rods and atomic clocks at rest in the laboratory. The gravitational constant G is then identified as that number with dimensions cm3 g" 1 s" 2 which appears in Newton's law of gravitation for the two bodies. This quantity is called the locally measured gravitational constant GL. The analysis of this experiment proceeds as follows: a body of mass mt ("source") falls freely through spacetime. A test body with negligible mass moves through spacetime, maintained at a constant proper distance rp from the source by a four-acceleration A. The line joining the pair of masses is nonrotating relative to asymptotically flat inertial space. An invariant "radial" unit vector Er points from the test mass toward the source.

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154

Then according to Newton's law of gravitation the radial component of the four acceleration of the test mass is given by /KEr=-GlmJr2p (6.52) for rp small compared to the scale of inhomogeneities in the external gravitational fields. Since the quantity A Er is invariant, we can calculate it in a suitably chosen PPN coordinate system, then use Equation (6.52) to read off the locally measured GL. Before carrying out the computation, however, it is instructive to ask what might be expected for the form of A Er to post-Newtonian order. We imagine that the source and the test body are moving with respect to the universe with velocity w1 and are in the presence of some external sources, idealized as point masses of mass ma at location xa. It is simplest to do the calculation in a PPN coordinate system in which the source is momentarily at rest. Then we would expect A Er to contain postNewtonian corrections to the equation A E, = m1frl rl of the form m1ml A E

mt ma

m^ ma

r -2'

72>

-rzr>

mYY 22 72-K)

V

P ia

'P l"

P

;

Y

r

P

r

r

r

m

(6-53)

where rla = |xx xa|. In obtaining this form, we have neglected the variation of the external gravitational potentials across the separation rp. This variation will produce the standard Newtonian tidal gravitational force, which is of the form (AE) - m a r r

la

and post-Newtonian corrections to this force. The latter we shall neglect throughout. The first term in Equation (6.53) represents post-Newtonian modifications in the two-body motion of the test body about the source, which can be understood and analyzed separately from a discussion of GL. The third term represents effects due to the gradients of the external fields; however, if we fit A Er to an r~2 curve in order to determine GL, these terms will have no effect [in most practical situations, they are negligibly small anyway (Will, 1971d)]. Both of these types of terms will be dropped throughout the analysis. Thus, we retain only terms of the form (m,/ rl)(mjrla) or (m1/^)(wj). The form of the PPN metric that we shall use is given by the expression in Table 4.1, where now the velocity w is the source's velocity relative to the mean rest frame of the universe, denoted w t . We label the test body by a = 0, the source by a 1, and the remaining bodies by a = 2, 3 , . . . Initially, both the source and test body are at rest, i.e., Vl (t

= 0) = vo(t = 0) = 0

(6.54)

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155

We separate the Newtonian gravitational potential Ux due to the source from that due to the other bodies in the system: l/(x) = Ufa) + £ mjra

(6.55)

where rx = |x xx\, ra = |x xo|, and Ut is assumed for simplicity to be spherically symmetric. The proper distance between the test body and the source is given by [see Equation (3.41)] rp = £ [1 + yU(x(X)) + O(4)]|dx/dA|dk

(6.56)

where to sufficient accuracy we may choose a straight coordinate line to join the two points: x(l) = x o (l - X) + \tX,

0< X < 1

(6.57)

Then

Neglecting the variation of the external gravitational potential across the separation r 01 leads to

rP = rj\ + y E - ) + ? \T U^da

(6.59)

The proper distance rp is to be kept constant by the four-acceleration A, thus drp/dt s d\/dt2 = 0

(6.60)

with the result, at t 0, ) where we have used the fact that Vj = v0 = 0 at t = 0, and have neglected time derivatives of the external potential. For the rest of this discussion, it is sufficient to drop the final term in Equation (6.59) (it leads only to terms that we previously decided to ignore) and to treat the coefficient of r01 as a constant. Thus, (

)

(6-62)

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We now assume that the source follows a geodesic of spacetime, but that the four-acceleration of the test body is A. Thus, ^source^source;v

*A

V

«, eS,«rest;v = A", uJU,^ = 0 In PPN coordinates, Equation (6.53) may be written, at t = 0,

uuv o /

dt '

(6-63)

\dt

,4° = 0

(6.64)

where, for the test body, j )

= 1 - 2l/ 1 (x 0 ) - 2 £ mjrla

+ O(4)

(6.65)

where we have again ignored the variation of the external potential in evaluating it at xt instead of at x 0 . We make use of the PPN Christoffel symbols (Table 6.1) evaluated for the external point masses [substitute p = p*{l jv2 3yU), \ap* d3x = ma] and use the Newtonian equations of motion to simplify any post-Newtonian terms. We retain only the terms discussed above; for illustration we also keep the Newtonian tidal force. Substituting Equations (6.64) into (6.61) yields, finally, A - x 1 0 __ '10

^ a*l

marloeJek(3n{ n\. -3a

~ $Jk)

"la

'10

a* I r l a

L

+ Vt/f'(x0) |"1 a 2 wX - { X ^"'""'"l r Z 10

1_

#]

?"la

(6-66)

J

where p*(x',t)d3x' 9

- XI

-x)(xo-x)dx

(6^7)

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157

For a spherically symmetric source, it is possible to show straightforwardly that O(6), VjUf (x0) = (mJr\o){Moe e - 2x%5l)i) l - 2x%dl» - 4o<5*') + O(4) l

k l

(6.68)

where mt and It are the rest mass and spherical moment of inertia of the source, given by mx = I p*d3x,

/ t = I p*r2d3x

(6.69)

We must now compute the invariant radial unit four-vector Er. Its components at x 0 are simply those of the tangent vector to the curve x(A) joining the two bodies, E} = adx\k)ldk = - oxJ01,

£r° = 0

(6.70)

ErvE; == 1 = a 2 |x 01 | 2 1 + 2V X ~ I

(6-71)

The normalization a is obtained from

\

0*1

~\aj

where we have retained only the necessary terms. Thus (6-72) Then the invariant radial component of the four-acceleration A is (6.73) The final result is (Will, 1971d, 1973; Nordtvedt and Will, 1972) A E, = X m a r 10 [3(n lo e) 2 - l>r. 3

T(WI a 2 ot3)w1 5 a 2 ( w i ' e ) + t z^ fl#l

( n io ' e ' '"la

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The first term in Equation (6.74) is simply the Newtonian tidal acceleration. From the second term we may read off the locally measured gravitational constant,

(675)

H ^ y i ^ ) ^ where U& = X man{an\Jrla,

UeU = [/£,

(6.76)

Here, we see a direct example of the possibility of violations of the Strong Equivalence Principle, via preferred-frame or preferred-location effects in local Cavendish experiments. The preferred-frame effects depend upon the velocity Wj of the source relative to the universe rest frame, and are present unless the PPN preferred-frame parameters a1; a2, and <x3 all vanish. The preferred-location effects depend upon the gravitational potentials Unt and (/£*, of nearby bodies, and are present in general unless the PPN parameters satisfy £, = (4)3 y 3 £2) = 0- ^n next section we shall develop a conserved energy formalism for the special case of semiconservative theories of gravity that will reveal a direct connection between violations of local Lorentz and position invariance in Cavendish experiments, and the violations of GWEP described in Section 6.2. We note here that general relativity predicts GL = 1 6.4

(6.77)

N-Body Lagrangians, Energy Conservation, and the Strong Equivalence Principle

In the previous two sections we showed that some metric theories of gravity may predict violations of GWEP and of LLI and LPI for gravitating bodies and for gravitational experiments. In the special case of theories of gravity that possess conservation laws for energy and

tne

Equations of Motion in the PPN Formalism

159

momentum, namely semiconservative theories, it is possible to derive a direct relationship between these violations. The method is the same as that developed in Section 2.5: derive a conserved energy expression for a composite system in a quasi-Newtonian form, from which one can read off the anomalous inertial and passive gravitational mass tensors Sm[J and 5m'J, respectively. The use of cyclic gedanken experiments, parallel to those used in Section 2.5, then reveals that violations of GWEP as well as of LLI and LPI depend upon these anomalous mass tensors. The derivation of these results proceeds as follows (Haugan, 1979): We first restrict attention to semiconservative theories of gravity, thus <x3 = d = £2 = £3 = £4 == 0, and to systems in which the basic particles are point masses. We then build composite bodies out of point masses moving in their mutual gravitational fields. We work in a PPN coordinate frame at rest with respect to the universe rest frame. The equations of motion for the particles then consist of the standard Newtonian acceleration plus the post-Newtonian n-body acceleration anbody, Equation (6.34) with w = 0 and with semiconservative PPN parameters,

b*a

'ab

L

'ab

'ab

c*ab rbc

c*ab rac

a2)

r

c*ab

ac

- i(4? + 4 + ax)va \b + i(2y + 2 + a 2 ) ^ - f (1 + a2)(>

'ab c±ab 'be

-11b*a 5ab 0* - Wto I * fir - T r r

\'

'a

r

b

- (2y 'ab

4 + a i )v o - (4y + 2 + ax - 2a 2 )v fc ]^

(6.78)

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It is then possible to show straightforwardly that these equations of motion can be derived from the Euler-Lagrange equations obtained by varying the trajectory xq(t), vq(t) of the qth particle in the action ^

(6.79)

where

3+

ai

- a2)va vfc - i(l + a2)(vo nab)(v6 fij

Consider a system consisting of a body of mass mQ and a composite body made up of bodies of mass ma. We assume that m0 » Xom«> anc* that the massive body is situated at rest at the origin a distance |X| from the composite body, where |X| is large compared to the size of the composite body. Because it is more massive, the distant body may be assumed to remain at rest, thereby providing an external potential in which the composite body resides and moves. (We ignore coupling of the body to inhomogeneities in the external potential.) We now make a change of variables in L from xa to center-of-mass and relative variables X and x a , respectively, where

X = m~1 £ maxa,

m = Ym<"

a

a

xa=xa-X

(6.81)

We also have \a = dxjdt,

V = dX/dt

(6.82)

A Hamiltonian H is then constructed from L using the standard technique PJ = 8L/dVJ,

pJa = dL/dvJa,

H = PJVJ + X rfpl - L

(6.83)

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161

The result is P2

H = m + 2m

v-i Pa

mm0

ab

*** **

1

> 2ma R + ~

2 % rab K

r

ab

'ab

ab

ab

r

ab

ab

m

'ab

+ Jd + «2) - I ? (».,' P)(«.. P.) + O(p') + O(P*) m

(6.84)

aft 'ab

where /? = |X|, n = X/R, and nofc = xab/fab. We have neglected postNewtonian terms O(p 4 ) and O(P*) in H that do not couple the internal motion and the center-of-mass motion of the composite system. We now average H over several timescales of the internal motions of the composite system, and make use of virial theorems for the internal variables, + mO(4)\

(6.85)

As in Section 2.6, we argue that although the post-Newtonian terms in Equation (6.85) may depend on P or X, this dependence does not affect the form of H. The resulting average Hamiltonian is then rewritten in terms of V using the equation V = 5/5P. The result for the conserved energy function is £ c = M + \{M5ij + [(a t - a2)Q<5y + a 2 Q 0 ]} VW - {M8'J + [(4j8 - y - 3 - 3£)Q<5ij' - ^QiJ2}mon'nj/R

(6.86)

where p

«

2m

a

1

V m-m>\

l

ab

T

ab I

(6.87) ab

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162

By comparing Equation (6.86) with Equations (3.77) and (3.78) we may read off the anomalous mass tensors dm\J = (a! - tx2)Q8iJ + <x2QiJ,

3my = (4/J - y - 3 - 3{)IM<> - &J

(6.88)

Substituting these results into Equation (3.80) yields 3ak = M-l\jAP - y - 3 - 3£)Q<5U - £&J~\(d/dXk)(mon'nJ/R) + A T '[(o^ - a 2 )iW w + oc2Qk}]m0xJ/R3 (6.89) This is in complete agreement with the GWEP-violating terms in ajSjewt. Equations (6.40), (6.43), and (6.44) if we substitute the semiconservative values of the PPN parameters, and take into account that the potential U'm is that due to a single distant point mass, i.e., U im = monlnm/R

(6.90)

To determine the influence of the internal structure of the composite body on its center-of-mass motion, we fixed its structure and focussed on the explicit P and X (or V and X) dependence of H. Now, to study the effect of a body's motion on its internal structure, and thereby obtain an expression for GL, we must fix the center-of-mass motion (P,X), and focus on the explicit p and x dependence of H. Using the Newtonian virial theorem [Equation (6.85)] to simplify the post-Newtonian terms in H, we obtain the conserved energy function Ec = M + i{M8iJ - [(<*! - oc2)Qdij + a 2 Q y ] }PiPj/M2 - {M5ij + [(4)3 - y - 3 - 3£)Q<5>V - ^Q iJ ]}m o n i « J 7^

(6-91)

where M and Q'v are given by Equation (6.87). Notice that the quantities in square brackets are precisely 5m[J and 5m^, of Equation (6.88), but that the sign in front of dm[J is opposite to that in Equation (6.86) (a result of expressing Ec in terms of P rather than V). Let us suppose for simplicity that the composite body is composed of two point masses in a local Cavendish experiment. Then with Ec written in the above form, it is possible to show straightforwardly that the effective force between the two particles is given by F=-(V£c)p,XfUed (6.92) Then the effective local gravitational constant is

- [(4/3 - y - 3 - 3£)8iJ - ^e^monW/R

(6.93)

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163

where e = x 12 /r 12 , and Q'J = mlm2e'eJ/r12. This is precisely Equation (6.75), with P/M = w1; monlnJ/R == U'Jxt, and with lx = 0. Again, we see the explicit connection between violations of GWEP and violations of LLI and LPI, for the case of semiconservative theories of gravity. 6.5

Equations of Motion for Spinning Bodies

The motion of spinning bodies (gyroscopes, planets, elementary particles) in curved spacetime has been a subject of considerable research for many years. This research has been aimed at discovering (i) how a body's intrinsic angular momentum (spin) alters its trajectory (deviations from geodesic motion), and (ii) how a body's motion in curved spacetime alters its spin. No really satisfactory solution is available for the first problem, outside of approximate solutions, or solutions in special spacetimes, because of the difficulties in defining rigorously a center of mass of a spinning body in curved spacetime. The most successful attempts at a solution have been made by Mathisson (1937), Papapetrou (1951), Corinaldesi and Papapetrou (1951), Tulczyjew and Tulczyjew (1962) and Dixon (1979). The central conclusion of these calculations has been that the intrinsic spin S1" (i.e., J"v evaluated in the body's "center-of-mass" frame) of a body should produce deviations from geodesic motion of the form mSa* ~ Sv V K ? a

(6.94)

where W is the body's four velocity, and R*lX is the Riemann curvature tensor. However, these calculations differ greatly in details and interpretation. For a spinning body moving with velocity v in a Newtonian gravitational potential U ~ M/r, these deviations are, in order of magnitude: 5a ~ (|S*|/m)|v|(M/r3) ~ (i'A/rHM/r) 1 ' 2 ^

(6.95)

where b is the radius of the body, and k its rotational angular velocity. For a planet rotating near break-up velocity (X2 ~ m/b3), we have Sa £ (m/b)1'2(M/r)1'2(b/r)aIhwl % 10 ~J 2aNewt

(6.96)

and for a 4 cm-radius gyroscope orbiting the Earth (frequency 200 rps), 5a £ 10~20 aNewt

(6.97)

Thus, for the most part, spin-induced deviations from geodesic motion can be ignored in the solar system. In our derivation of massive-body

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164

equations of motion (Section 6.2), we ignored the effects of tidal gravitational forces (Riemann curvature tensor); and thus our equation of motion, (6.31), does not include the effects of spin. Even for a rapidly rotating neutron star such as the binary pulsar (b ~ 10 km, A ~ 102 Hz, m ~ lmG, r ~ 106 km), (5a;glO-10aNewt

(6.98)

and can be ignored (see Chapter 12). It is problem (ii), the effects of a body's motion on its spin, which is better understood. All calculations to date have shown that, as long as the direct effects of tidal gravitational forces (Riemann curvature tensor) on the spinning body can be neglected, the spin S is Fermi-Walker transported along the body's world line. Here the four-vector S has the components S"s^Vi,> u"S,, = 0

(6.99)

The equation of Fermi-Walker transport is then uvS?v = ul'id'S,,)

(6.100)

where a" is the body's four-acceleration, given by a" = uvufv

(6.101)

The reader is referred to MTW, Section 40.7 for further discussion of Fermi-Walker Transport. The following derivation is patterned after that section. It is convenient to analyze Equation (6.100) in a local Lorentz frame which is momentarily comoving with the body. The basis vectors of this frame are related to those of the PPN coordinate system by a Lorentz transformation plus a normalization, and are given by e% = W, e°j =vj + O(3), 4 = (1 - yU)8) + %Vjvk + O(4)

(6.102)

where all quantities in Equation (6.102) are assumed to be evaluated along the world line of the body. Thus, because of Equation (6.99), the spin is a purely spatial vector in this frame, i.e., S6 s egS, = ! « = 0

(6.103)

Equations of Motion in the PPN Formalism

165

We now calculate the precession of the spatial components of the spin Sj. Since efu^ = 0, we have, from Equation (6.100), 0 = efifS^ = i?Sj.v - SMuv4v

(6-104)

and since Sj is a scalar (scalar product of two vectors), we have uvS/;v = wvS;>v = dS}/dT

(6.105)

The second term in Equation (6.104) is most easily evaluated in the PPN coordinate frame. Using Equation (6.102), we first obtain relations between SM and Sf S0=-VJSJ+O(3)SJ,

Sj=Sj+O(2)S}

(6.106)

Then after some simplification, we get, to post-Newtonian order, dSj/dx = SlVuak] + g0lKSi - (2y + l)vuU,k{\

(6.107)

This can be written in three-dimensional vector notation dS/dx = ft x S, ft = -%\ x a - ^V x g + (y + i)v x Vt/, g= In Equation (6.108) it does not matter whether the vectors entering into ft are evaluated in the PPN coordinate frame or in the comoving frame, since their spatial basis vectors differ only by terms of O(2). We have calculated the precession of the spin relative to a comoving frame which is rotationally tied to the PPN coordinate frame, and whose axes are fixed relative to the distant galaxies. Thus, we have calculated the spin's precession angular velocity ft relative to a frame fixed with respect to the distant galaxies. We shall discuss the observable consequences of this precession in Chapter 9.

The Classical Tests

With the PPN formalism and its associated equations of motion in hand, we are now ready to confront the gravitation theories discussed in Chapter 5 with the results of solar system experiments. In this chapter, we focus on the three "classical" tests of relativistic gravity, consisting of (i) the deflection of light, (ii) the time delay of light, and (iii) the perihelion shift of Mercury. This usage of the term "classical" tests is a break with tradition. Traditionally, the term "classical tests" has referred to the gravitational redshift experiment, the deflection of light, and the perihelion shift of Mercury. The reason is largely historical. These were among the first observable effects of general relativity to be computed by Einstein. However, in Chapter 2 we saw that the gravitational red-shift experiment is really not a test of general relativity, rather it is a test of the Einstein Equivalence Principle, upon which general relativity and every other metric theory of gravity are founded. Put differently, every metric theory of gravity automatically predicts the same red-shift. For this reason, we have dropped the red-shift experiment as a "classical" test (that is not to deny its importance, of course, as our discussion in Chapter 2 points out). However, we can immediately replace it with an experiment that is as important as the other two, the time delay of light. This effect is closely related to the deflection of light, as one might expect, since any physical mechanism in Maxwell's equations (refraction, dispersion, gravity) that bends light can also be expected to delay it. In fact, it is a mystery why Einstein did not discover this effect. It was not discovered until 1964, by Irwin I. Shapiro. The simplest explanation seems to be that Shapiro had the benefit of knowing that the space technology of the 1960s and 1970s would make feasible a measurement of a delay of the expected size (200 us for a round

Classical Tests

167

trip signal to Mars). No such technology was known to Einstein. He was aware only of the known problem of Mercury's excess perihelion shift of 43 arcseconds per century, and of the potential ability to measure the deflection of starlight. But the lack of available technology may not be the whole story. After all, Einstein derived the gravitational red-shift at a time when the hopes of measuring it were marginal at best (a reliable measurement was not performed until 1960), and other workers such as Lense and Thirring, and de Sitter derived effects of general relativity, with little or no hope of seeing them measured using the technology of the day. Why then, did no one at the time take the step from deflection to time delay, if only as a matter of principle? Nevertheless, despite its late arrival, the time delay deserves a place in the triumvirate of "classical" tests, not the least because it has given one of the most precise tests of general relativity to date! We begin this chapter with the deflection of light (Section 7.1), turn to the time delay (Section 7.2), and finally to the perihelion shift of Mercury (Section 7.3). 7.1

The Deflection of Light

An expression for the deflection of light can be obtained in a straightforward way using the PPN photon equations of motion, (6.14) and (6.15). Consider a light signal emitted at PPN coordinate time te at a point xe in an initial direction described by the unit vector ft, where n n = l. Including the post-Newtonian correction xp, the resulting trajectory of the photon has the form x°(t) = t, x(t) = x£ + %t - O + xp(t)

(7.1)

where we have imposed the boundary condition xp(te) = 0. We decompose xp into components parallel and perpendicular to the unperturbed trajectory: Xp(f)|| = ft Xp(t), x p « i = xp(t) - n[n xp(t)]

(7.2)

Equations (6.14) and (6.15) then yield ^

(7.3)

= (1 + y){Uj - n^n Vt/)]

(7.4)

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168

For simplicity, we assume that the Newtonian gravitational potential U is produced by a static spherical body of mass m at the origin (Sun), i.e., (7.5)

Along the unperturbed path of the photon, U then has the form m fi(t - te)\

r(t)

(7.6)

To post-Newtonian order, then, Equation (7.4) can be integrated along the unperturbed photon path using Equation (7.6) with the result d r ( dt' M

}

mA lx(t) a i

xe ft

d> \ r(t)

(7.7)

where d =n

X

(X,

xft)

(7.8)

Note that d is the vector joining the center of the body and the point of closest approach of the unperturbed ray (see Figure 7.1). Equation (7.7) represents a change in the direction of the photon's trajectory, toward the sun (in the direction -d). We then have

Consider an observer at rest on the Earth (©) who receives the photon from the source and a photon from a reference source located at a different Figure 7.1. Geometry of light-deflection measurements. Reference Source

Source

Earth

Classical Tests

169

position on the sky, x r . The angle 9 between the directions of the two incoming photons is a physically measurable quantity, and can be given an invariant mathematical expression. The tangent four-vectorsfeM= dx"/dt andfefr)= dx$r)/dt of the paths x"(t) and x("r)(t) of the two incoming photons are projected onto the hypersurface orthogonal to the observer's four-velocity t/ using the projection operator

( 71 °)

PI = K + "X

The inner product between the resulting vectors is related to the cosine offlby

If we ignore the velocity of the Earth, which only produces aberration, then Equation (7.11) simplifies to coSe=l-(g00)-1gllvk%)

(7.12)

By substituting Equations (7.1) and (7.9) into Equation (7.12) we get, to post-Newtonian accuracy, '*

where

M M

(x, x 8,)

(7.14)

It is useful to note that, to sufficient post-Newtonian accuracy in Equation (7.13), d = n x (x e x fl), dr = nr x (x e x

fir)

(7.15)

We now define the angle 60 to be the angle between the unperturbed paths of the photons from the source and from the reference source, i.e., cos0 o sfl-ii r

(7.16)

and we define the "deflection" of the measured angle from the unperturbed angle to be d6 = 6-d0

(7.17)

There are two interesting cases to consider. This first is an idealized situation that leads to a simple formula. We suppose that the Sun itself is the

Theory and Experiment in Gravitational Physics

170

reference source, then, dr = 0, the second term inside the braces in Equation (7.13) vanishes, (7.18) and d \ re

re

For a photon emitted from a distant star or galaxy, re»r®,

xe-n/re^-l

(7.20)

Also, to sufficient accuracy, x e fi/r®~ nr n = cos 90

(7.21)

thus,

(! + ,) * ( . + - * ) For general relativity (y = 1) this is in agreement with results obtained by Shapiro (1967) and Ward (1970). It is interesting to note that the classic derivations of the deflection of light that used only the principle of equivalence or the corpuscular theory of light (Einstein, 1911, Soldner, 1801) yield only the "1/2" part of the coefficient in front of (Am/d)(\ + cos0o)/2 in Equation (7.22). That does not invalidate these calculations however; they are correct as far as they go. But the result of these calculations is the deflection of light relative to local straight lines, as denned for example by rigid rods; however, because of space curvature around the Sun, determined by the PPN parameter y, local straight lines are bent relative to asymptotic straight lines far from the Sun by just enough to yield the remaining factor "y/2". The first factor "1/2" holds in any metric theory, the second "y/2" varies from theory to theory. Thus, calculations that purport to derive the full deflection using the equivalence principle alone are incorrect (see Schiff, 1960a, and the critique by Rindler, 1968). The deflection is a maximum for a ray which just grazes the Sun, i.e., for 60~0,d^Ro^ 6.96 x 105 km, m = mQ = 1.476 km. In this case, <50max = I d + 7)1"75

(7.23)

The second case to consider is more closely related to the actual method of measuring the light deflection using the techniques of radio interfero-

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metry. There one chooses a reference source near the observed source and monitors changes 80 in their angular separation. If we define $ and <S>r to be the angular separation between the Earth-Sun direction and the unperturbed direction of photons from the two sources, as in Figure 7.1, then cos O = x e ft/r9, cos <J>r = x e nr/rm (7.24) Assuming again that the two sources are very distant, we obtain =

/ I + y\r4m /cos*, cos<J>cos0o\ / I + cos$\ )\ T \ ) ) sin*sin9 0 Am /cos \ / I + cos<J>A~]

~T \

sin*rsin0o

)V

2

) \

K1J5)

If the observed source direction passes very near the Sun, while the reference source remains a decent angular distance away, we can approximate $ « cos x + O(* 2 /^r)

(7.26)

where / is the angle between the Sun-source and Sun-reference directions projected on the plane of the sky (Figure 7.1). The resulting deflection is

This result shows quite clearly how the relative angular separation between two distant sources may vary as the lines of sight of one of them passes near the Sun (d ~ RQ, dr » d, % varying). The prediction of the bending of light by the Sun was one of the great successes of Einstein's general relativity. Eddington's confirmation of the bending of optical starlight observed during a total solar eclipse in the first days following World War I helped make Einstein famous. However, the experiments of Eddington and his co-workers had only 30% accuracy, and succeeding experiments weren't much better: the results were scattered between one half and twice the Einstein value, and the accuracies were low (for reviews, see Richard, 1975; Merat et al., 1974; Bertotti et al., 1962). The most recent optical measurement, during the solar eclipse of 30 June 1973 illustrates the difficulty of these experiments. It yielded a value |(1 + y) = 0.95 ± 0.11

[lo- error]

(7.28)

(Texas Mauritanian Eclipse Team, 1976 and Jones, 1976). The accuracy was limited by poor seeing (caused by a dust storm just prior to the

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172

eclipse, and by clouds and rain during the follow-up expedition in November, 1973) that drastically reduced the number of measurable star images. There were also variable scale changes between eclipse- and comparison-field exposures. Recent advances in photoelectric and astrometric techniques may make possible optical deflection measurements without the need for solar eclipses (Hill, H. 1971). The development of long-baseline radio interferometry has altered this situation. Long-baseline and very-long-baseline (VLBI) interferometric techniques have the capability in principle of measuring angular separations and changes in angles as small as 3 x 10 ~4 seconds of arc. Coupled with this technological advance is a series of heavenly coincidences: each year, groups of strong quasistellar radio sources pass very close to the Sun (as seen from the Earth), including the group 3C273, 3C279, and 3C48, and the group 0111 + 02, 0119 + 11 and 0116 + 08. The angular position of each quasar determines a phase in the radio signal at the output of the radio interferometer that depends on the wavelength of the radiation and on the baseline between the radio telescopes. The angular

Figure 7.2.

Results of radio-wave deflection measurements 1969-75. Value of i (1 + 7) 0.88 i

i

0.92 I

i

0.96 i

i

1.00 r

Radio Deflection Experiments 1969

1.04 i

1.08 r

I

Muhleman et al. (1970) Seielstad et al. (1970)

|

Hill (1971)

I

1970 Shapiro (quoted in Weinberg, 1972)

I

Sramek (1971)

|

a, 1971 Sramek (1974) x w Riley(1973) 1972 Weileretal. (1974)

I

t-

Counselman et al. (1974) 1973 Weileretal. (1974)

1974 Fomalont and Sramek (1975) 1975 Fomalont and Sramek (1976) 5

10 2040=o

Value of ScalarTensor GO

i

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separation between a pair of quasars is determined by a difference in phases. As the Earth moves in orbit, changing the lines of sight of the quasars relative to the Sun, the angular separation 89 varies [Equation (7.25)], resulting in a variation in the phase difference. The time variation in the quantities d, dr, d>, and 3>r in Equation (7.25) is determined using an accurate ephemeris for the Earth and initial directions for the quasars, and the resulting prediction for the phase difference as a function of time is used as a basis for a least-squares fit of the measured phase differences, with one of the fitted parameters being the coefficient ^(1 + y). A number of measurements of this kind over the past decade have yielded an accurate determination of ^(1 + y), which has the value unity in general relativity. Those results are shown in Figure 7.2. One of the major sources of error in these experiments is the solar corona which bends radio waves much more strongly than it bent the visible light rays that Eddington observed. Advancements in dual frequency techniques have improved accuracies by allowing the coronal bending, which depends on the frequency of the wave, to be measured separately from the gravitational bending, which does not. Fomalont and Sramek (1977) provide a thorough review of these experiments, and discuss the prospects for improvement. 7.2

The Time Delay of Light

Because of the presence of the gravitational field of a massive body, a light signal will take a longer time to traverse a given distance than it would if Newtonian theory were valid. An expression for this "time delay" can be obtained simply from Equation (7.3). Integrating the equation using Equation (7.6), we obtain

]

(729)

Then from Equation (7.1), the coordinate time taken to propagate from the point of emission to x is given by

^ l l ^ ]

(7.30)

For a signal emitted from the Earth, reflected off a planet or spacecraft at xp, and received back at Earth, the roundtrip travel time At is given by

At - 2|xe - xp| + 2(1 + y>»to[(r« + * ' - y ' - X ' - * ) ]

(7.31)

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174

where ft is the direction of the photon on its return flight. Here we have ignored the motion of the Earth and planets during the round trip of the signal. To be completely correct, the round trip travel time should be expressed in terms of the proper time elapsed during the round trip, as measured by an atomic clock on Earth; but this introduces no new effects, so we will not do so here. The additional "time delay" 8t produced by the second term in Equation (7.31) is a maximum when the planet is on the far side of the Sun from the Earth (superior conjunction), i.e., when xffi n ~ r$,

x p n ~ rp,

d =* solar radius

(7.32)

then 5t = 2(1 + y)mln(4r9rp/d2)

= i(l + y) [240 ps - 20 JIS In ( ^ - Y (f\\

(7.33)

where R o is the radius of the Sun, and a is an astronomical unit. For further discussion of the time delay see Shapiro (1964,1966a,b), Muhleman and Reichley (1964), and Ross and Schiff (1966). In the decade and a half since Shapiro's discovery of this effect, a number of measurements of it have been made using radar ranging to targets passing through superior conjunction. Since one does not have access to a "Newtonian" signal against which to compare the round trip travel time of the observed signal, it is necessary to do a differential measurement of the variations in round trip travel times as the target passes through superior conjunction, and to look for the logarithmic behavior. To achieve this accurately however, one must take into account the variations in round trip travel time due to the orbital motion of the target relative to the Earth [variations in |x e x p | in Equation (7.31)]. This is done by using radar-ranging (and possibly other) data on the target taken when it is far from superior conjunction (i.e., when the timedelay term is negligible) to determine an accurate ephemeris for the target, using the ephemeris to predict the PPN coordinate trajectory xp(t) near superior conjunction, then combining that trajectory with the trajectory of the Earth xffi to determine the quantity |x$ x p | and the logarithmic term in Equation (7.31). The resulting predicted round trip travel times in terms of the unknown coefficient 5(1 + y) are then fit to the measured travel times using the method of least squares, and an estimate obtained for |(1 + y). [This is an oversimplification, of course. The reader is referred to Anderson (1974) for further discussion.]

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Three types of targets have been used. The first type is a planet, such as Mercury or Venus, used as a passive reflector of the radar signals ("passive radar"). One of the major difficulties with this method is that the largely unknown planetary topography can introduce errors in round trip travel times as much as 5 /zs (i.e., the subradar point could be a mountaintop or a valley), which introduce errors in both the planetary ephemeris and, more importantly, in the round trip travel times at superior conjunction. Several sophisticated attempts have been made to overcome this problem. The second type of target is an artificial satellite, such as Mariners 6 and 7, used as active retransmitters of the radar signals ("active radar"). Here topography is not an issue, and the on-board transponders permit accurate determination of the true range to the spacecraft. Unfortunately, spacecraft can suffer random perturbing accelerations from a variety of sources, including random fluctuations in the solar wind and solar radiation pressure, and random forces from on-board attitude-control devices. These random accelerations c^in cause the trajectory of the spacecraft near superior conjunction to differ by as much as 50 m or 0.1 us from the predicted trajectory in an essentially unknown way. Special methods of analyzing the ranging data ("sequentialfiltering")have been devised to alleviate this problem (Anderson, 1974). The third target is the result of an attempt to combine the transponding capabilities of spacecraft with the imperturbable motions of planets by anchoring satellites to planets. Examples are the Mariner 9 Mars orbiter and the Viking Mars landers and orbiters. In all of these cases, as in the radio-wave deflection measurements, the solar corona causes uncertainties because of its slowing down of the radar signal. Again, dual frequency ranging helps reduce these errors, in fact, it is the corona problem that provides the limiting accuracy for the most recent time-delay measurements. The results for the coefficient |(1 + y) of all radar time-delay measurements performed to date are shown in Figure 7.3. Recent analyses of Viking data have resulted in a 0.1% measurement (Reasenberg et al. 1979). From the results of light-deflection and time-delay experiments, we can conclude that the coefficient ^(1 + y) must be within at most 0.2% of unity. Most of the theories shown in Table 5.1 can select their adjustable parameters or cosmological boundary conditions with sufficient freedom to meet this constraint. Scalar-tensor theories must have co > 500 to be within 0.1% or w > 250 to be within 0.2% of unity.

Theory and Experiment in Gravitational Physics

176

Value of i (1+7) 0.88 I

I

0.92 I

T

0.96 T

I

1.00

1.04

1.08

i

TimeDelay Measurements Passive Radar to Mercury and Venus

Shapiro (1968) Shapiro etal. (1971)

-

1

Active Radar Mariner 6 and 7 Anderson et al. (1975) Anchored Spacecraft Mariner 9 Anderson et al. (1978), Reasenberg and Shapiro (1977) Viking Shapiro et al. (1977) Cain etal. (1978) Reasenberg etal. (1979)

(±0.001)

i

5 10 2040°° Value of Scalar-Tensor co Figure 7.3. Results of radar time-delay measurements 1968-79.

7.3

The Perihelion Shift of Mercury

The explanation of the anomalous perihelion shift of Mercury's orbit was another of the triumphs of general relativity. However, between 1967 and 1974, there was considerable controversy over whether the perihelion shift was a confirmation or a refutation of general relativity because of the apparent existence of a solar quadrupole moment that could contribute a portion of the observed perihelion shift. Although this controversy has abated somewhat, the question of the size of the solar quadrupole moment has yet to be conclusively answered. The PPN prediction for the perihelion shift can be obtained from the PPN equation of motion [Equation (6.31)]. We consider a system of two bodies of inertial masses n^ and m2, and self-gravitational energies Q^ and Q2 The first body has a small quadrupole moment Q'{. We assume that the entire system is at rest with respect to the universe rest frame (w = 0) and that there are no other gravitating bodies near the system. In Chapter 8, we shall return to the effects of motion and of distant bodies (preferred-frame and preferred-location effects) on the perihelion

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shift. For the moment we ignore them. We work in a PPN coordinate system in which the center of mass of the system is at rest at the origin. Making use of the fact that each body is nearly spherical, Qf m we obtain from Equation (6.31) the acceleration of each body

! - - ^ F(2y + 20 ^

a, =

4 + ai )v! v2 - i(2y + 2 + a2 + a3)t>l

+ f(1 + «2)(v2 n) 2 l - ^ £(2y + 2*, - (2y + l)v 2 j v, +

Y7^'

(4y + 4 + ai)Vl

- (4r + 2 + a i -

2a

>2 k ,

a2 = {l<-*2;x-^ - x }

(7.34)

where x = x 21 , n = x/r. Including the Newtonian contribution of the quadrupole moment in the quasi-Newtonian potential produced by body 1, we have xJ (UJi = (mA)2 p-, (Uj)2 = -(mA)x ^-\^r(^nknlW

- 25H1)

(7.35)

where (mA)j and (mA)2 are the active gravitational masses, given by Equation (6.51). For a body which is axially symmetric about an axis with direction e, Qf can be shown to have the form Qf = mxR\J2W^k

- 3^2*)

(7.36)

where J2 is a dimensionless measure of the quadrupole moment, given by J2={C-

A)/mR2

(7.37)

where C = [moment of inertia about symmetry axis], A = [moment of inertia about equatorial axis], R = [radius]

(7.38)

Theory and Experiment in Gravitational Physics

178

(The subscript 2 on J2 denotes that it is associated with the quadrupole, or / = 2 moment of the body.) Since the center of mass of the system is at rest, we may, to sufficient accuracy in the post-Newtonian terms in Equation (7.34), replace vx and v2 by Vi = -(m2/m)v,

v2 = (mi/m)v

(7.39)

where v = v2-v1,

m = m1+m2

(7.40)

We also define the reduced mass fi = mlm2/m

(7.41)

Then the relative acceleration a 2 - a , 5 a takes the form a

=

-

^

+ * "1*^2(1)

ocl+a2+

^ [

[ 1 5 ( g

oc3)^v2 m

^

.

ft)2fi

_

6 ( e

.

m

+ | ( 1 + a2)-^-(v n) m

]

(7.42)

where m* = (mP/w)2(mA)1 + (mp/m = m(l + [self-energy terms for bodies 1 and 2]).

(7.43)

The self-energy terms from Equation (6.51) that appear in the above expression are at most ~ 1 0 ~ 5 for the Sun, and are constant. Thus the difference between m* and m is unmeasurable, so we simply drop the (*) in Equation (7.42). We consider a planetary orbit with the following instantaneous orbit elements (see Smart, 1953, for detailed discussion of the definitions): inclination i relative to a chosen reference plane, the angle Q from a chosen reference direction in the reference plane to the ascending node, the angle co of perihelion from the ascending node measured in the orbital plane, the eccentricity e and semi-major axis a. The sixth element T, the time of periastron passage, is an initial'condition and is irrelevant for our purposes.

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For the solar system, the reference plane is chosen to be the plane of the Earth's orbit (ecliptic) and the reference direction is the Earth-Sun direction at spring equinox. Following the standard procedure for computing perturbations of orbital elements [Smart (1953), Robertson and Noonan (1968)], we resolve the acceleration a [Equation (7.42)] into a radial qomponent M, a component "W, normal to the orbital plane, and a component £f normal to Si and iV, and calculate the rates of change of the orbital elements using the formulae [in the notation of Robertson and Noonan (1968)]: da, -r-=

at

p® ^ip + r) . , iTr . / - -r cos 4> + -, sin 4> cot i sin(<w +
he

he

h

di Wr ~ = cos(co + ),

(7.44)

(7.47)

r sm{w + ) i ; : ~ (7.48) dt h sm i where h is the angular momentum per unit mass of the orbit, <j> is the angle of the planet measured from perihelion, and p is the semi-latus rectum given by p = a(l - e2) (7.49) The variables r and 0 are related to the instantaneous orbit elements by the definitions r == p(l + ecos 4>)~l, r2 d4>/dt = h = (mp)1'2

(7.50)

Now, because observations of the planets are made with reference to geocentric coordinates, the perihelion measured is the perihelion relative to the equinox,
(7.51)

Then the rate of change "of c5 is given by deb pM Sr(p + r) = -y cos 4> H T sm $ dt he he

(7.52)

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180

where we have used the fact that, for all the planets, i is small, so that sin i« 1. For the perturbing acceleration in Equation (7.42) (we drop the subscript " 1 " on m, R and J 2 )

-11 (2y + 20) - - yv2 + (2y + 2)(v ii)2 (2 + «! - 2f2) £ - i(6 + «! + a 2 + a3)

y = - 3(m/?2J2/r4)(e n)(e 2)

+ j j (v n)(v X)|~(2y + 2) - £ (2 - a t + «2) 1

<7-53)

where 2 is a unit vector in the plane of the orbit in the direction of the orbital motion, normal to ii. For Mercury's orbit, the solar symmetry axis is essentially normal to the orbital plane, hence e n ^ 0. Then substituting Equations (7.53) into (7.52) and integrating over one orbit using Equation (7.50) yields AS = (67tm/p)[i(2 + 2y - 0) - a 2 + a 3 + 2£2)fi/m + J2(R2/2mp)-]

(7.54)

This is the only secular perturbation of an orbital element produced by the post-Newtonian terms in Equation (7.42); however the quadrupole terms can be shown to produce secular changes in i and Q proportional to sin 0 and sin 0/sin i respectively, where 6 is the tilt of the Sun's symmetry or rotation axis relative to the ecliptic (9 « 7°). The elements a and e suffer no secular changes under either of these perturbations. The first term in Equation (7.54) is the classical perihelion shift, which depends upon the PPN parameters y and p. The second term depends upon the ratio of the masses of the two bodies (Will, 1975); it is zero in any fully conservative theory of gravity (al = a2 = a3 = £2 = 0); it is also negligible for Mercury, since /x/m ~ m^/mo ~ 2 x 10" 7 . We shall drop this term henceforth. The third term depends upon the solar quadrupole moment J2. For a Sun that rotates uniformly with its observed surface angular velocity, so that the quadrupole moment is produced by centri-

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fugal flattening, one may estimate J2 to be ~1 x 10~7. Normalizing J2 by this value and substituting standard orbital elements and physical constants for Mercury and the Sun (Allen, 1976), we obtain the rate of perihelion shift c5, in seconds of arc per century, 5 = 42795 Ape"1 Xv = [i(2 + 2y - P) + 3 x IO-V2/IO- 7 )]

(7.55)

The measured perihelion shift is accurately known: after the effects of the general precession of the equinoxes (~5000" c""1) and the perturbing effects of the other planets (280" c~1 from Venus, 150" c'1 from Jupiter, 100" c~1 from the rest) have been accounted for, the remaining perihelion shift is known (a) to a precision of about one percent from optical observations of Mercury during the past three centuries (Morrison and Ward, 1975), and (b) to about 0.5% from radar observations during the past decade (Shapiro et al., 1976). Unfortunately, measurements of the orbit of Mercury alone are incapable at present of separating the effects of relativistic gravity and of solar quadrupole moment in the determination of Xp. Thus, in two recent analyses of radar distance measurements to Mercury, J2 was assumed to have a value corresponding to uniform rotation (effect on Xp negligible), and the PPN parameter combination was estimated. The results were 1 s( + 7

fl.005 ± 0.020(1966-1971 data: Shapiro et al., 1972) P) ~ | 1 0 0 3 ± 0.005(1966-1976 data: Shapiro et al, 1976) (7.56)

where the quoted errors are 1CT estimates of the realistic error (taking into account possible systematic errors). The origin of the uncertainty that has clouded the interpretation of perihelion-shift measurements is a series of experiments performed in 1966 by Dicke and Goldenberg (see Dicke and Goldenberg, 1974, for a detailed review). Those experiments measured the visual oblateness or flattening of the Sun's disk and found a difference in the apparent polar and equatorial angular radii of AR = (43'.'3 ± 3'.'3) x 10"3. By taking into account the oblateness of the surface layers of the Sun caused by centrifugal flattening, this oblateness signal can be related to J2 by (Dicke, 1974) J2 = §(A*/KG) - 5.3 x 10" 6

(7.57)

which gives (i?G = 959") J2 = (2.47 ± 0.23) x 10"5 (Dicke and Goldenberg, 1974) (7.58)

Theory and Experiment in Gravitational Physics

182

A value of J2 this large would have contributed about 4" c~1 to Mercury's perihelion shift, and thus would have put general relativity in serious disagreement with the observations, while on the other hand supporting Brans-Dicke theory with a value co ^ 5, whose post-Newtonian contribution to the perihelion shift would thus have been 39" per century. These results generated considerable controversy within the relativity and solar physics communities, and a mammoth number of papers was produced, both supporting and opposing solar oblateness. One recurring line of argument in opposition to the Dicke-Goldenberg result was that their method of measuring the difference in brightness between the solar pole and the solar equator of an annulus of the solar limb produced around an occulting disk placed in front of the Sun, could equally well be interpreted by assuming a standard solar model (with a small J2 ~ 10 ~7 produced by centrifugal flattening) with a temperature difference on the solar surface between the equator and the pole, leading to a brightness difference indistinguishable from that due to a geometrical oblateness. Such a brightness difference, it was suggested, could also be produced by an equatorial excess in the number of solar faculae. Refutations of these arguments by Dicke and his supporters, and counter-refutations abounded in the literature. The controversy abated somewhat in 1973, when Hill and his collaborators performed a similar visual oblateness measurement that yielded AR = (9'.'2 ± 6'.'2) x 1(T3 or J2 = 0.10 ± 0.43 x 10" 5

(Hill et al., 1974)

(7.59)

an upper limit five times smaller than Dicke's value. (See also Hill and Stebbins, 1975). The disagreement between these two observational results remains unresolved. One of the major difficulties in relating visual solar oblateness results to J2 is that a considerable amount of complex solar physics theory must be employed. There is, however, a way of determining J2 unambiguously, namely by probing the solar gravity field at different distances from the Sun, thereby separating the effects of J2 from those of relativistic gravitation through their different radial dependences [see Equation (7.42)]. One method would compare the perihelion shifts of different planets. But the perihelion shifts of Venus, Earth, and Mars are not known to sufficient accuracy, although Shapiro et al. (1972) pointed out that several more years of radar observations of the inner planets may permit such a comparison. Another method would take advantage of Mercury's orbital eccentricity (e ~ 0.2) and search for the different periodic orbital pertur-

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bations induced by J2 and by relativistic gravity. The accuracy required for such measurements would necessitate tracking of a spacecraft in orbit around Mercury, but preliminary studies have shown that J2 could be determined to within a few parts in 107 (Anderson et al., 1977, Wahr and Bender, 1976). Finally, and most promisingly, a mission currently under study by NASA for the 1980s known as the Solar Probe, a spacecraft in a high-eccentricity solar orbit with perihelion distance of four solar radii ("Arrow to the Sun"), could yield a measurement of J2 with a precision of a part in 108 (Nordtvedt, 1977, Anderson et al., 1977). Such missions would also lead to improved determinations of y and /?. The possibility of determining y and j8 from measurements of the precessions of the pericenters of the inner satellites of the gas giant planets has recently been considered by Hiscock and Lindblom (1979).

8 Tests of the Strong Equivalence Principle

The next class of solar system experiments that test relativistic gravitational effects may be called tests of the Strong Equivalence Principle (SEP). That principle states that (i) WEP is valid for self-gravitating bodies as well as for test bodies (GWEP), (ii) the outcome of any local test experiment, gravitational or nongravitational, is independent of the velocity of the freely falling apparatus, and (iii) the outcome of any local test experiment is independent of where and when in the universe it is performed. In Section 3.3, we pointed out that many metric theories of gravity (perhaps all except general relativity) can be expected to violate one or more aspects of SEP. In Chapter 6, working within the PPN framework, we saw explicit evidence of some of these violations: violations of GWEP in the equations of motion for massive self-gravitating bodies [Equations (6.33) and (6.40)]; preferred-frame and preferred-location effects in the locally measured gravitational constant GL [Equation (6.75)]; and nonzero values for the anomalous inertial and passive gravitational mass tensors in the semiconservative case [Equation (6.88)]. This chapter is devoted to the study of some of the observable consequences of such violations of SEP, and to the experiments that test for them. In Section 8.1, we consider violations of GWEP (the Nordtvedt effect), and its primary experimental test, the Lunar Laser-Ranging"E6tvos" experiment. Section 8.2 focuses on the preferred-frame and preferredlocation effects in GL. The most precise tests of these effects are obtained from geophysical measurements. In Section 8.3, we consider preferredframe and preferred-location effects in the orbital motions of planets. Perihelion-shift measurements are important tests of such effects. Another violation of SEP would be the variation with time of the gravitational constant as a result of cosmic evolution. Tests of such variation are de-

Tests of the Strong Equivalence Principle

185

scribed in Section 8.4. In Section 8.5, we summarize the limits on the values of the PPN parameters y, /?, £, al5 a2, and <x3 that are set by the classical tests and by tests of SEP, and discuss the consequences for the metric theories of gravity described in Chapter 5. 8.1

The Nordtvedt Effect and the Lunar Eorvos Experiment

The breakdown in the Weak Equivalence Principle for massive, self-gravitating bodies (GWEP), which many metric theories predict, has a variety of observable consequences. In Chapter 6, we saw that this violation could be expressed in quasi-Newtonian language by attributing to each massive body inertial and passive gravitational mass tensors m{k and m£* which may differ from each other. The quasi-Newtonian part of the body's acceleration may be written [see Equation (6.43)] (mi)i*Kftew« = (mP)lrUlJ

(8.1)

lra

where U is a quasi-Newtonian gravitational potential, and {fh^f and (fhp)'" are given by («i)? = ma{o-}k[l + (ax - a2 + C ^ / m J + (a2 - Ci + C2)OfM,}, (mP)'am = ma{5"»[l + (4/8 - y - 3 - 3£)Qa/ma] - {Qf/m.}

(8.2)

where Qa and Q£* are the body's internal gravitational energy and gravitational energy tensor (see Table 6.2), and ma is the total mass energy of the body. Now, most bodies in the solar system are very nearly spherically symmetric, so we may approximate a* * &a5Jk

(8.3) J

Any "Nordtvedt" effects that arise from the anisotropies in Q * in Equation (8.2) are expected to be too small to be measurable in the foreseeable future (see Will, 1971b, for an example). With the above approximation we write the quasi-Newtonian Equation (8.1) in the form Wkw. = K M . U , .

(8.4)

where (mp/ifi). = 1 + (40 - y - 3 - 4ft - ax + §<x2 - Ki - K ^ M . , U = Z (mAV\,V

(8-5)

The most important consequence of the Nordtvedt effect is a polarization of the Moon's orbit about the Earth [Nordtvedt (1968c)]. Because

Theory and Experiment in Gravitational Physics

186

the Moon's self-gravitational energy is smaller than the Earth's, the Nordtvedt effect causes the Earth and Moon to fall toward the Sun with slightly different accelerations. Including their mutual attraction, we have [from Equations (8.4) and (8.5), and neglecting quadrupole moments],

(8.6) where X and Xo are vectors from the Sun to the Earth and Moon, respectively, and x is a vector from the Earth to the Moon (Figure 8.1). The relative Earth-Moon acceleration a, denned by ®

(8.7)

Figure 8.1. (a) Geometry of the Earth-Moon-Sun system. (b) The Nordtvedt effect - a polarization of the Moon's orbit with the apogee always directed along the Earth-Sun line. Sun

Moon Earth (a)

(b)

Tests of the Strong Equivalence Principle

187

is then given by a = ~m*x/r3 + i/[(Q//n)© - (Q/m)J/n o X/K 3 + {m^m^m^X/R3 - XJR3)

(8.8)

where

G

s (mA)©,

^+1^-1^-^2

(8-9)

The first term in Equation (8.8) is the Newtonian acceleration between the Earth and Moon and the second term is the difference between the Earth's and Moon's acceleration toward the Sun (Nordtvedt effect). The third term is the classical tidal perturbation on the Moon's orbit; since it is a purely nonrelativistic perturbation, we will not consider it for the moment. Hence, the equation of motion of the Moon relative to the Earth, including the perturbation arising from the Nordtvedt effect, is a = -m*x/r3

+ ff[(O/m)e - (n/m\~}mQX/R3

(8.10)

We assume that the Moon's unperturbed orbit is circular with angular velocity co0 and in the x-y plane, and also that the orbit of the Earth around the Sun is circular with angular velocity a>s in the same plane. We work in an inertial PPN coordinate system centered at the Sun. Then the acceleration a and the angular momentum per unit mass of the EarthMoon orbit are given by a = d2x/dt2,

h = x x (dx/dt)

(8.11)

and the following relations hold d2r/dt2 = x a/r + h2/r\ dh/dt= (x x a)

(8.12)

where r = \x\. Thus, by making use of Equation (8.10) and by defining da s ij[(|n|/m)e - (p\/m\-]mQ/R2

(8.13)

we obtain d2r/dt2 = -m*/r2

+ h2fr3 + SacosAt,

dh/dt= -rSa sin At

(8.14)

where cosAt s n x/r,

sin At = (n x x/r) z ,

A = co0 ws

(8.15)

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where n = X/R. Note that At is the angle between the Earth-Sun and Earth-Moon directions. We next linearize about a circular orbit: r = ro + 3r,

h = ho + 5h

(8.16)

2

and use m*jr% = hl/r% = co ,. Integration of the resulting equations yields 5h = (r0/A)5acosAt,

(8.17) (8.18)

Equation (8.18) represents a polarization of the Earth-Moon system by the external field of the Sun. This polarization of the orbit is always directed toward the Sun if r\ > 0 (away from the Sun if r\ < 0) as it rotates around the Earth (see Figure 8.1). Using Equations (8.13) and (8.18) and the values mQ/R2 st 5.9 x 10" 6 km s~2, w° ^ 13.4OJS * 2.7 x 1(T 6 s" 1 , (Q/m)e * - 4.6 x 1(T 10 , and (Q/m)a =* - 0 . 2 x 10~ 10 (Allen, 1976), we obtain dr =s 8.0»/ cos(
(8.19)

Actually, a more accurate calculation would take into account the effect of the Nordtvedt perturbation on the tidal acceleration term in Equation (8.8) and that of the tidal perturbation on the Nordtvedt term; this modifies the coefficient of 8r by a factor of approximately 1 + 2cos/co0 ^ 1.15, giving 8r ^ 9.2/7 cos(a>0 - ojs)t m

(8.20)

Since August, 1969, when the first laser signal was reflected from the Apollo 11 retroreflector on the Moon, the Lunar Laser-Ranging Experiment (LURE) has made regular measurements of the round trip travel times of laser pulses between McDonald Observatory in Texas and the lunar retroreflectors, with accuracies of 1 ns (30 cm) (see Bender et al, 1973, Mulholland, 1977). These measurements were fit using the method of least squares to a theoretical model for the lunar motion that took into account perturbations due to the other planets, tidal interactions, and post-Newtonian gravitational effects. The predicted round trip travel times between retroreflector and telescope also took into account the librations of the Moon, the orientation of the Earth, the location of the observatory, and atmospheric effects on the signal propagation. The "Nordtvedt" parameter, //, along with several other important parameters of the model were then estimated in the least-squares method.

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An important issue in this analysis is whether other perturbations of the Earth-Moon orbit could mask the Nordtvedt effect. Most perturbations produce effects in 5r, which, when decomposed into sinusoidal components, occur at frequencies different from that of the Nordtvedt term (e.g., at angular frequencies co0, 2A), and thus can be separated cleanly from it using a multi-year span of data. However, there is one perturbation, due to the tidal term that we neglected in Equation (8.8), that does have a component at the frequency A. To see this, we expand X and Xo about Xc, the center of mass of the Earth-Moon system, using Xo = Xc + (roe/m*)x,

X = Xc - (mjm*)x

(8.21)

where we now ignore all post-Newtonian self-energy corrections to masses. Then the tidal acceleration in Equation (8.8) becomes

a I l X f e i y e ] I - 2mc/m*) ~ [nc - 5(nc e)% + 2(nc S)e]

(8.22)

Kc

where fic = Xc/Rc, e = x/r. It is the second term, of order {r^/Rf), in the above expression that leads to a perturbation in Sr of frequency A. Applying Equations (8.12), (8.15), (8.16), and integrating, it is possible to show straightforwardly that 2mA

8 [col-A + [terms proportional to cos2At, cos 3At...]

(8.23)

Using the expressions col = rn*/r30,

cof = mQ/R?

(8.24)

we may rewrite Equation (8.23) in the useful form

where Q = toj(o0 x 0.075. Again, a more accurate computation, taking into account the mutual effect of the two terms in Equation (8.22), modifies Equation (8.25) by corrections that depend upon Q, with the result (Brown, 1960)

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where F(Q) = 1 + (81/15)Q + ^ 1.64

(8.27)

Substituting numerical values (Allen, 1976) yields <5>"tidai ^ 110 cos At km

(8.28)

Although this term is ten thousand times larger than the nominal amplitude of the Nordtvedt effect, it turns out, fortunately, that the parameters that appear in Equation (8.26) are known with sufficient accuracy that the tidal term can be accounted for to a precision of about 2 cm. The values of Rc, mQ/m*, and Q. are known to sufficient precision from other data, while the values of mjm* and r0 are estimated using the laserranging data via their effects on the lunar orbit at frequencies other than A. Two independent analyses of the data taken between 1969 and 1975 were carried out, both finding no evidence, within experimental uncertainty, for the Nordtvedt effect. Their results for n were n

_ fO.OO ± 0.03 [Williams et al. (1976)], ~ (0.001 + 0.015 [Shapiro et al. (1976)]

(8.29)

where the quoted errors are la, obtained by estimating the sensitivity of n to possible systematic errors in the data or in the theoretical model. The formal statistical errors that emerged from the data analysis were typically much smaller, of order <x(>7)fOrmai ~ + 0.004. This represents a limit on a possible violation of GWEP for massive bodies of 7 parts in 1012 (compare Table 2.2). For Brans-Dicke theory, these results force a lower limit on the coupling constant a> of 29 (2a, Shapiro result). Improvements in the measurement accuracy and in the theoretical analysis of the lunar motion may tighten this limit by an order of magnitude (Williams et al., 1976), while a comparable test of the Nordtvedt effect may be possible using the Sun-Mars-Jupiter system (Shapiro et al., 1976). Other potentially observable consequences of the Nordtvedt effect include shifts in the stable Lagrange points of Jupiter (measurable by ranging to the Trojan asteroids), and modification of Kepler's third law (Nordtvedt, 1968a, 1970a, 1971a,b). 8.2

Preferred-Frame and Preferred-Location Effects: Geophysical Tests In Section 6.3, we found that some metric theories of gravity predict preferred-frame and preferred-location effects in the locally measured gravitational constant GL, measured by means of Cavendish experiments.

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These effects represent violations of SEP. Unfortunately, present-day Cavendish experiments are only accurate to about one part in 105 in absolute measurements of GL (Rose et al, 1969), and so cannot discern the post-Newtonian corrections to GL in Equation (6.75). However, there is a "Cavendish" experiment that can detect corrections in GL, one in which the source is the Earth and the test body is a gravimeter on the surface of the Earth. A gravimeter is a device that measures the force required to keep a small "proof" mass stationary with respect to the center of the Earth. This is exactly the physical situation assumed in our derivation of GL in Section 6.3. Because of uncertainties in our knowledge of the internal structure and composition of the Earth, it is impossible to determine the absolute value of GL by this method with sufficient precision to detect post-Newtonian effects. Instead, gravimeters are powerful tools for measuring variations in the gravitational force. In Newtonian geophysics, these variations are known as solid-Earth tides; in post-Newtonian geophysics, measurements of these variations can test for variations in GL, with high precision. We therefore shall apply Equation (6.75) for GL to a gravimeter "Cavendish" experiment, and shall focus on the post-Newtonian terms that vary with time. A detailed justification of the application of Equation (6.75) to this situation is given by Will (1971d). Recall that GL = 1 - [4)8 - y - 3 - C2 - «3 + //mr2)] Uext - Aw.n ^/ m r 22Vw. z\22 + «1 - 3//mr 2 )t/f xt e^ - 3//mr )(we e) 2 U

(8.30)

where /, m, and r are the spherical moment of inertia, mass, and radius of the Earth, e is a unit vector directed from the gravimeter to the center of the Earth, and U{kn = Z manJ9an%JrBa,

Uext = UHt

(8.31)

Consider the first post-Newtonian term in Equation (8.30). Because of the Earth's eccentric orbital motion, the external potential produced by the Sun varies yearly on Earth by only a part in 1010, too small to be detected with confidence by Earth-bound gravimeters or Cavendish experiments. The time-varying effects of other bodies (planets, the galaxy) are even smaller. Next, consider the preferred-frame terms. The Earth's velocity w e is made up of two parts, a uniform velocity w of the solar system relative to

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the preferred frame, and the Earth's orbital velocity v around the Sun, thus w | = w2 + 2w v + v2, (w e 6)2 = (w e)2 + 2(w e)(v e) + (v e)2

(8.32)

So because of the Earth's rotation (changing e) and orbital motion (changing v), there will be variations in the gravimeter measurements of GL, given by (we retain only terms which vary with amplitude larger than a 3 - ax)w v + i<x2[(w g)2 + 2(w e)(v e) + (v e) 2 ]

(8.33)

where we have used the fact that, for the Earth, I C* mr2/2

(8.34)

Finally, we consider the preferred-location term. According to our discussion of the PPN formalism (see Section 4.1) the potential I/£, must include all local gravitating matter that is not part of the cosmological background used to establish the asymptotically Lorentz PPN coordinate system. Therefore it must include the Sun, planets, stars, the galaxy, and possibly the local cluster of galaxies. In this case, [/£, is dominated by our galaxy (Ua ~ 5 x 10~7), followed by the Sun (UQ ~ 1 x 10~8), thus, AGJGh = - K t / G ( e eG)2 - ^Uo(e

e0)2

(8.35)

In order to compare this variation in G with gravimeter data, we must perform a harmonic analysis of the terms in Equations (8.33) and (8.35). The frequencies involved will be the sidereal rotation rate of the Earth Q, due to the changing direction of e relative to the fixed direction of w and e G , and its orbital sidereal frequency co due to the changing direction of v relative to w, along with harmonics and linear combinations of these frequencies. We work in geocentric ecliptic coordinates, and assume a circular Earth orbit, with the Earth at vernal equinox at t = 0. Then, e 0 = cos cotex + sin a>tey, v = i;(sin cotex cos atey), w s w[cos /^(cos Xwex + sin lwey) + sin /fwez], eG = cos /SG(cos AGex + sin lGey) + sin /?Ge2

(8.36)

The latter two equations define the ecliptic coordinates (lw,/?w) and For a (^G,PG)gravimeter stationed at Earth latitude L, e = cosLcos(Qt e)ex + [cosLsin(lQt e)cos0 + sinLsin0]e y - [cos L sin(Qt e) sin 6 sin L cos 0]ez

(8.37)

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where £ is related to the longitude of the gravimeter on the Earth, and 0 is the "tilt" (23^°) of the Earth relative to the Earth's orbit (ecliptic). Equations (8.36) and (8.37) give w v = wv cos fiw sin(a>t Xw),

(8.38)

(w e)2 = w 2 [i + | ( i - sin2 <5J(i - sin2 L) + \ sin 2<5W sin 2L cos(fit £ ocw) + icos 2 <5 H ,cos 2 Lcos2(Qf - £ - <xj],

(8.39)

(w e)(v e) = wi;{yCOS)S)1,sin(a»t - AJ + (i sin 2 L)[^cos jSw sin(cot X^ + § sin <5W sin 0 cos cot] + j sin dw(l cos 6) sin 2L sin[(Q + co)t e] 5Cos<5 w sin0sin2Lcos[(Q + (o)t & a w ] \ sin ^ w (l + cos 6) sin 2L sin[(Q a>)t s] jcos 6W sin 9 sin 2Lcos[(Q a>)t £ a w ] \ - cos^)cos 2 Lsin[(2fi + a»)f - 2E - a w ] l + cos0)cos 2 Lsin[(2Q - co)t - 2E - a w ]}, (8.40) 2

2

2

2

(v e) = i> {± + | ( i - sin L)(i - ^sin 0) | ( i sin 2 L)sin 2 0cos2cof + |sin20sin2Lsin(Qf - e) i sin 0(1 - cos 9) sin 2L sin[(fi + 2co)t - e] + jsin 9(1 + cos 0) sin 2L sin[(Q - 2co)t - s] + \ sin2 9 cos 2 L cos 2(Qf - s) i ( l - cos0)2 cos2 Lcos[2(Q + co)t - 2e] | ( 1 + cos 9)2 cos 2 Lcos[2(Q - co)t - 2E]}, 2

2

(8.41)

2

(SG e) = i + | ( i - sin «5G)(i - sin L) + -j sin 2<5G sin 2Lcos(Q( s aG) + \ cos 2 ^ G cos 2 L cos 2(Q{ - £ - aG ),

(8.42)

(e o e)2 = H | ( i - sin2 L)(i - ^sin2 9) + i(j - sin2L)sin29cos2cot + i sin 29 sin 2L sin(Qt - E) + | sin 0(1 - cos 0) sin 2L sin[(Q + 2co)t - e] 5 sin 0(1 + cos 0) sin 2L sin[(O - 2co)t e\ + i sin2 0 cos 2 L cos 2(Q( - E) + | ( 1 - cos0) 2 cos 2 Lcos[2(Q + co)t - 2e] + i ( l + cos0) 2 cos 2 Lcos[2OQ - co)t - 2e]

(8.43)

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where we have used both the ecliptic coordinates (Xw, /?w), (XG, fic) and the equatorial coordinates (OLW,5W), (<XG,<SG) (Smart, 1960) corresponding to the directions of w and eG in order to simplify the various expressions. These coordinate systems are related by sin 8 = sin /? cos 9 + cos /? sin 9 sin X, cos 8 cos a = cosficos X, cos 8 sin a = sin f$ sin 9 + cos /? cos 9 sin A

(8.44)

Equations (8.38)-(8.43) reveal four different types of variations in GL. (i) Semidiurnal variations: These are the terms that vary with frequency around 2Q: 2fi, 2Q, + co,2Q- a>, 2(Q. + co), 2(Q - co); i.e., that have periods around twelve hours (co « Q) and vary with latitude according to cos 2 L. These variations are completely analogous to the twelve hour solid-Earth tides produced by the Sun and Moon, called "semidiurnal sectorial waves" [Melchior (1966)]. The true gravimeter measurements for these tides are affected not only by the variation in G, but also by the displacement of the Earth's surface relative to the center of the Earth, and by the redistribution of mass inside the Earth. This variation in gravimeter readings is related to the variation in G by (AfifMemidiurnal = 1.16(AG/G)semidiurnal

(8.45)

where the factor 1.16 is a combination of "Love numbers," which depend on the detailed structure of the Earth (Melchior, 1966). A more accurate calculation of Ag/g would take into account the fact that in the Earth's interior the perturbing force generated by the variations in GL is proportional to pV U, whereas the tidal perturbing force is proportional to the distance from the center of the Earth. If the Earth's density were uniform, then pWU would be proportional to r and the Love numbers would be the same as in the Newtonian tidal case. However, in Newtonian tidal theory, the Love number for gravimeter measurements, (1.16), is not very sensitive (+ 5%) to variations in. the model for the Earth, thus we do not expect it to be sensitive to a different disturbing force law. (ii) Diurnal variations: These are the terms that vary with a frequency around Q: Q, SI + co, fi co, Q + 2co, Q 2co; i.e., have periods around 24 hours, and vary with latitude according to sin 2L. These variations are analogous to the 24 hour "diurnal tesseral waves" of the solid Earth, and give gravimeter readings related to the variation in G by the same factor: (A<7M,iurnal = 1.16(AG/G)diurnaI

(8.46)

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(iii) Long-period zonal variations: These are the variations with frequencies co and 2co, and with latitude dependence (5 sin2 L), that are analogous to the long-period tides produced by the Sun and Moon, called "long-period zonal waves." These long-period zonal waves produce variations in the Earth's moment of inertia, which in turn cause variations in the rotation rate of the Earth. These rotation-rate variations are related to the amplitude of the zonal variations by (Mintz and Munk, 1953; Melchior, 1966) (AQ/QL^, = 0.41^zonal

(8.47)

where Azonai is related to the zonal variations in G in Equations (8.40), (8.41), and (8.43) by (AG/G)zonal = AnJk

- sin2 L)

(8.48)

(iv) Long-period spherical variations: These are the variations [Equations (8.38) and (8.40)] which have frequency <x>, but no latitude dependence; they represent a yearly variation in the strength of G, and have no counterpart in Newtonian tidal theory. These variations produce a purely spherical deformation of the Earth, as opposed to the sectorial, tesseral, and zonal waves which produce purely quadrupole deformations. This yearly spherical "breathing" of the Earth as G varies causes a variation in the Earth's moment of inertia, which in turn causes a variation in the rotation frequency, given by (AQ/Q)spherioal= -(AJ//) spherical

(8.49)

However, because this effect has no counterpart in Newtonian tidal theory, there is no Love number factor to relate A/// to AG/G. Instead we must do an explicit calculation to determine the factor. We assume the Earth is spherically symmetric and momentarily at rest with respect to the PPN coordinate frame. Since we are focusing on long-period variations of GL (1 yr), we can assume that the Earth is in hydrostatic equilibrium at each moment of time, and changes only quasistatically. Then, from Equations (6.52) and (6.75), or from the PPN perfect-fluid equation of motion, Equation (6.29), keeping only the terms leading to significant long-period spherical perturbations, we find that the equation of hydrostatic equilibrium may be written -T- = P -jr; [1 + i(<*2 + «3 - «i)w©] - j<x2w}@w%p ^

(8.50)

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For a spherically symmetric body, it is straightforward to show that dU__

m(r)

~3r~~ ~~P~'

^

(851)

where m(r) = 4TT P pr 2 dr,

I(r) = 4n f' pr4 dr

(8.52)

%J0

JO

Substituting Equations (8.32), (8.38), (8.40), and (8.51) into Equation (8.50), and keeping only the spherical terms yields

GL(t) = 1 + (a 3 + | a 2 - a t)wt; cos j8w sin(co£ - ^ J

(8.54)

Using m(r) instead of r as independent variable, we may integrate Equation (8.53), to obtain

f =?

(8.55)

where m e is the mass of the Earth. By definition, p must vanish at the surface of the Earth, i.e., pirn®) = 0. As GL(t) changes, the pressure distribution changes, causing a change £ = ^e in the position of each element of matter. For a given shell of matter, the mass inside that shell is constant, by conservation of mass. Then if GL changes by AGL, we get from Equations (8.52) and (8.55), Am = 0, Ap = p(AGL/GL) + O(f)

(8.56)

But the volume of each element of matter changes, and this change can be related to the pressure change using the bulk modulus K (we ignore temperature changes) Ap = - K(AV/V) = - KV S, = - (K/r2)(r20,r

(8.57)

Integrating Equation (8.57) and combining with Equation (8.56), we obtain

£ ('/'V2

d'

O(£2)

(8.58)

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The spherical moment of inertia is given by (8.59) and the change caused by the displacement of each shell of matter is A / = 2 [M r^dm

(8.60)

Combining Equations (8.58) and (8.60) yields A7 = - 8TT ^

J * prdr £ (p'/Ky2 dr'

(8.61)

Numerical integration of this expression for a reasonable Earth model yields A / / / = -0.17AG L /G L

(8.62)

(see Lyttleton and Fitch, 1978; Nordtvedt and Will, 1972). We now substitute numerical values for the quantities that appear in Equations (8.35)-(8.43). For the galaxy, Ua * 5 x 10- 7 , <xG = 265°,

AG = 266°, fio 5G = -29°

= -6°, (8.63)

For the velocity w of the solar system relative to the preferred frame, we use the results of the most recent measurements of the anisotropy of the 3 K microwave background. Our motion through this radiation causes the measured effective temperature to be Doppler shifted differently in the front and back directions. From measurements taken using a 33 GHz Dicke radiometer flown on a U-2 aircraft (to get above a substantial amount of the Earth's atmosphere), Smoot et al. (1977) obtained a value w = 390 ± 60 km s~1 in the direction <xw = 165° ± 9°, 5W = 6° ± 10°. We shall adopt the values <xw^165°,

K-&,

^164°,

j6w^0°

(8.64)

We also have i>^30kms-\

0 = 23.5°

(8.65)

Using these values, we first compute the amplitudes of the dominant components of the Earth tides, as listed in Table 8.1 (unconnected for Love numbers). For comparison, Table 8.1 also gives the amplitudes of the tidal potential for the dominant Newtonian tides in the frequency bands of interest.

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Table 8.1. Amplitudes of earth tides Angular frequency0

Doodson label

PPN tidal amplitude (108 Ag/g)"

(a) Semidiurnal tides (latitude dependence cos2 L) 2fi-3co 0 T2 _ 2fl-2co s 2 2fi - c o 2.9 a2 f 17 a2 K 2fi (9.6 { 2Q + m 2Q + 2a>

-

(b) Diurnal tides (latitude dependence sin 2L) Q-2co Pi 0.7 a2 ii-w s, ("3.5 a 2

a

n+w Q + 2co

0.6 a 2

Predicted Newtonian amplitude (108 Ag/g)*

0.14 2.4 0.02 0.67 0 0 1.3 0.03 4.1 0.03 0.06

" The angular frequencies of the Earth's rotation and the Earth's orbit are Q, and a>, respectively. b Amplitudes are uncorrected for Love numbers. An entry of zero denotes precise absence of a tide at that frequency, while an entry of a dash denotes that the nominal amplitude is smaller than 10 ~9 g.

Recent advances in superconducting techniques in the design and construction of gravimeters have resulted in highly stable devices capable of measuring periodic changes in the local gravitational acceleration g as small as 10" n g. Using such superconducting gravimeters, Warburton and Goodkind (1976) have analyzed an 18 month record of gravimeter data taken at Pinon Flat, California (33°59 N, 116?46W) in search of anomalous PPN tidal amplitudes. From a harmonic analysis of the record, they obtained amplitudes and phases of the tides at the frequencies shown in Table 8.1. They then subtracted (vectorially) from these measured tides the predicted Newtonian tides (corrected by an accurately known Love number factor of 1.160). The remaining amplitudes and phases, known as "load vectors," are thought to be due primarily to the complex effect of ocean tides, which can influence gravimeter readings even at the centers of continents. To take this "ocean loading" into account, they assumed that the anomalous load vectors at the diurnal Pt harmonic and at the semidiurnal T2 (or S2) harmonic, where the PPN effect is negligible or absent, were entirely due to ocean loading. Since the effect of ocean loading is not believed to be strongly frequency dependent over

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199

the narrow (few cycles per year) frequency bands under consideration, the P x and T2 load vectors were simply subtracted from the Kt and from the R2 and K2 load vectors, respectively. Small corrections for barometric effects were also made. The remaining load vectors had amplitudes smaller than 3 x 10" 1 0 g for Ku 1 x 1(T 10 g for K2, and 1 x 1 0 " u g for R2. (Compare with the PPN amplitudes in Table 8.1.) Furthermore, the phases of the remaining load vectors did not agree with the relationships among the phases predicted by Equations (8.39)-(8.43). The result was upper limits on the PPN parameters <x2 and t, given by |a2|<4xl(T4,

|£|
(8.66)

The other important post-Newtonian geophysical effect is the possibility of periodic (co, 2co) variations in the Earth's rotation rate produced by the zonal and spherical variations in GL. The zonal variations have amplitudes [see Equations (8.40), (8.41), and (8.43)] {AGJGh)mnil ~ 3 x 10~8a2[frequency co], ~ 3 x 10" 10 a 2 [frequency 2a>], ~ 3 x 10-10£[frequency 2co]

(8.67)

However, because of the tight limits on a2 and £ set by gravimeter data, we shall ignore these variations. The spherical variations [Equation (8.54)] have amplitude (AGL/GL)spherical * 1.2 x 10" 7 [a 3 + | a 2 - a t ] [frequency co]

(8.68)

resulting in annual variations of the Earth's moment of inertia [Equation (8.62)] with amplitude |A///| =* 2.0 x 10- 8 [a 3 + | a 2 - a,]

(8.69)

Now, the observed annual variations in the Earth's rotation rate, of amplitude |Afl/fi| 2^4 xl0~ 9 can be accounted for as an effect of seasonal variations in the angular momentum J wind of atmospheric winds, to a level of 4 parts in 1010 (Rochester and Smylie, 1974). Then, from conservation of angular momentum, we have A7

T

Q

i/Q

<4xlO~10

(8.70)

Thus, comparing Equations (8.69) and (8.70) we obtain (8.71)

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200

Preferred-Frame and Preferred-Location Effects: Orbital Tests

There are a number of observable effects of a preferred-frame and preferred-location type in the orbital motions of bodies governed by the H-body equation of motion, (6.31). The most important of these effects are perihelion shifts of planets in addition to the "classical" shift discussed in Section 7. To determine these effects, we consider a two-body system whose barycenter moves relative to the universe rest frame with velocity w, and that resides in the gravitational potential UG of a distant body (the galaxy is the dominant such body). In the n-body equations of motion, (6.31), we shall ignore all the self-acceleration terms except the term (6.39) that depends on a3 and w. We shall also ignore the Newtonian acceleration, the Nordtvedt terms, and all the post-Newtonian terms that were included in the classical perihelion-shift calculation. Thus, from Equations (6.32), (6.33), and (6.39) we have the additional accelerations

+I(a 1 -a 2 -a 3 )w 2 +ia 1 w v 1 +|(a 1 -2a 2 -2a 3 )w v2 +fa 2 (wn) 2 .^

r

(w n)(v2 n)

J!° [2(fiG x )n G - 3x(nG n) 2 ] + a 2 - | (x w)v2 rG r

i ^ x3 . r aL i V _(( aai,-2a,)i lVl 2 2 r

"

>

(8.72) where x = x 21 , n = x/r, ra = |x1G|, nG = x 1G /r G . In obtaining Equation (8.72) we have ignored terms of order mGr/rQ, mGr2/rG, and so on. The first two terms inside the braces in Equation (8.72) are constant, therefore they can simply be absorbed into the Newtonian acceleration by redefining the gravitational constant [they are related to the constant corrections to GL in Equation (6.75)]. Since our two-body system will consist of the Sun and a planet, we can ignore Q/m for the planet. If body 1 is chosen to be the Sun, then the relative acceleration <5a = 5a2 d*i

Tests of the Strong Equivalence Principle

201

is given by <5a = \^a.l(dm/m)yi v + |a 2 (w ft)2] + ZULU* [2(flG x)nG - 3x(nG n) 2 ] r r o j - [^a^m/mjv + a2w]w + %tx3(Q/m)QY/ x
(8.73)

where we have made use of Equations (7.39) and (7.40), and where Sm = my m 2 .

Following the method described in Section 7.3, we calculate the secular change in the perihelion position. We assume that m2«m1, that e«l, and that co is perpendicular to the orbital plane, then to zeroth order in e, we obtain for the secular change in a> over one orbit, A<3= - 2

4

rG

w

2

\ m JQ\ me

(8.74)

where vvP, wQ, «P, and nQ are the respective components of w and flG m the direction of the planet's perihelion (wP, nP) and in the direction at right angles to this (wQ,«Q) in the plane of the orbit. The perturbations in Equation (8.73) can also be shown to produce secular changes in e, i, and Q. We now evaluate this additional perihelion shift for Mercury and Earth, using standard values for the orbital elements (Allen, 1976), numerical values for the Sun's gravitational energy and rotational angular velocity ^ 4 x 10~6,

|t»|Q =* 3 x 10~6 s" 1

(8.75)

the direction of the galactic center, and our adopted value for w (see Section 8.2.). Including the "classical" contributions (Section 7.3), the result, in seconds of arc per century, is = 43.0[i(2y + 2 - 123a! + 92a2 + 1.4 x 105a3 = 3.8[i(2y + 2 - 0)] - 198at + 12a2 + 2.4 x 106a3 + 14£ c" l

(8.76)

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Note that the effect of J2 on the Earth's perihelion shift is below the experimental uncertainty. The measured perihelion shifts are (^®)meas^3'.'8 + 0'.'4C-1

(8.77)

By combining Equations (8.76) and (8.77), eliminating the term involving y and /?, and treating J 2 as an experimental uncertainty with maximum value given by Hill's observations, \J2\ < 5 x 10~6 (Section 7.3), we obtain the following limit on the parameters a t , a2, a3, and £ |49at - a2 - 6.3 x 105a3 - 2.2£| < 0.1

(8.78)

It is clear that <x3 must be extremely small, |a3|<2xKT7 (8.79) otherwise there would be major violations of perihelion-shift data. Nonzero values of OLU a2, a3, or t, can also lead to periodic perturbations in orbits, most notably in the lunar orbit, with nominal amplitudes ranging from 70 km, for terms dependent upon oc3, to several meters, for terms dependent upon a1; a2, or £. For a partial catalogue of these effects, see Nordtvedt and Will (1972) and Nordtvedt (1973). In Section 9.3, we shall obtain an even tighter limit on a3 than that shown in Equation (8.79) by considering the effect of the acceleration term equation, (6.39), on the motion of pulsars. 8.4

Constancy of the Newtonian Gravitational Constant

Most theories of gravity that violate SEP predict that the locally measured Newtonian gravitational constant may vary with time as the universe evolves. For the theories listed in Table 5.1, the predictions for G/G can be written in terms of time derivatives of the asymptotic dynamical fields or of the asymptotic matching parameters. Other, more heuristic proposals for a changing gravitational constant, such as those due to Dirac cannot be written this way. Dyson (1972) gives a detailed discussion of these proposals. Where G does change with cosmic evolution, its rate of variation should be of the order of the expansion rate of the universe, i.e., G/G = oH0 (8.80) where Ho is the Hubble expansion parameter whose value is Ho cz 55 km s" 1 Mpc" 1 s ( 2 x 1010 yr)~\ and a is a dimensionless parameter whose value depends upon the theory of gravity under study and upon the detailed cosmological model.

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For very few theories has a systematic study of values of a been carried out. For general relativity, of course, G is precisely constant {a = 0). For Brans-Dicke theory a ranges from a 2= 3qo(a> + 2)" 1 for q0 « 1 to a =s -(co + 2)" 1 for q0 = i (flat Friedman cosmology) to a ^ 3.34<jJ/2(ct) + 2)" 1 for <j0 » 1, where q0 is the deceleration parameter of the cosmology [see Section 16.4 of Weinberg (1972) for review and references]. In Bekenstein's variable-mass theory, generic cosmological models with chosen values of r and q (see Section 5.3) evolve to states at the current epoch in which a < 5 x 10""3 (Bekenstein and Meisels, 1980). But for most other theories, detailed computations of this sort have not been performed (see Chapter 13). However, several observational constraints can be placed on G/G, using methods that include studies of the evolution of the Sun, observations of lunar occultations (including analyses of ancient eclipse data), planetary radar-ranging measurements, lunar laser-ranging measurements, and yet-to-be-performed laboratory experiments. The present status of these experiments is summarized in Table 8.2 [for a review of some of these methods see Halpern (1978)]. Some authors, chiefly Van Flandern (1975,1978), have claimed that the nonzero results for o shown in Table 8.2 are significant and support the hypothesis of a varying gravitational constant, while others, notably Reasenberg and Shapiro (1978) have argued that unavoidable errors in the models used in the numerical estimation Table 8.2. Tests of the constancy of the gravitational constant Method

a = (G/G) x (2 x 1O10 yr)

Reference

Solar evolution Lunar occultations and eclipses

H<2

Planetary and spacecraft radar

H<8 W<3

Viking radar Lunar laser ranging Laboratory experiments

\a\ < 0.6

Chin and Stothers (1976) Morrison (1973) Van Flandern (1975, 1976, 1978) Muller(1978) Newton (1979) Shapiro et al. (1971) Reasenberg and Shapiro (1976, 1978), Anderson et al. (1978) Anderson (1979) Williams et al. (1978) Braginsky and Ginzberg (.1974), Braginsky et al. (1977), Ritterand Beams (1978)

|CT|

< 0.8

a = -(0.6 ±0.3)
' Experiments yet to be performed.

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of parameters such as G/G may seriously degrade such estimates. The laser-ranging and radar-ranging results are regarded as being consistent with G/G = 0. Reasenberg and Shapiro (1976) have pointed out that, because the errors in the radar observations of G/G decrease as T~5/2 where T is the time span of the observations, one can expect from that method an accuracy of A|G/G| < 10" l l yr" 1 by 1985. Anderson et al. (1978) and Wahr and Bender (1976) have shown that radar observations of Viking or of a Mercury orbiter over two-year missions could yield 8.5

Experimental Limits on the PPN Parameters

We now summarize the results of the solar system experiments described in Chapters 7 and 8, in the form of a set of limits on the PPN parameters. For the purposes of this summary, we shall consider only semiconservative theories of gravity, i.e., theories for which <x3 = £i = £2 = (3 = £4 = 0. Our reasons are the following: (i) we wish to keep things simple; (ii) all currently interesting metric theories of gravity are Lagrangian based, and are thus automatically semiconservative; (iii) we have already seen that |a3| < 2 x 10~7; and (iv) decent experimental limits on the parameters Ci, (i> Cs, and £4 are hard to obtain, the only known exceptions being a limit |£3| < 0.06 from the Kreuzer experiment, and a possible limit on |£2| from the binary pulsar (see Chapter 9 for discussion of these tests). We thus have the la experimental limits y = 1.000 ± 0.002

[Viking time delay],

\{2y + 2 - j8) = 1.00 ± 0.02

[perihelion shift, Hill's value for J 2 ],

\40 - y - 3 - ^ |£|<10"

- <*! + fa 2 | < 0.015 [lunar laser ranging],

3

|a2| < 4 x 10"

4

|fa2 - ax| < 0.02 |49ax a2 2.2^| < 0.1

(8.81) (8.82) (8.83)

[Earth tides],

(8.84)

[Earth tides]

(8.85)

[Earth rotation rate],

(8.86)

[anomalous perihelion shifts] (8.87) One useful way to represent these results pictorially is to construct "PPN theory space," a five-dimensional space whose axes are the five semiconservative PPN parameters. A given theory, with chosen values for its adjustable constants and matching parameters, occupies a point in

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this space. If we choose as variables y 1, /? 1, £, al9 and a2, then general relativity occupies the origin, scalar-tensor theories with co > 0 occupy the left hand (y - l)-(j8 - 1) plane, Rosen's bimetric theory occupies the a2 axis, and so on (see Figures 8.2 and 8.3). The results of solar system experiments can be viewed as "squeezing" the available theory space into smaller and smaller portions. For example, Figure 8.2 shows the y-fi-% subspace of PPN theory space, and indicates the constraints imposed by time delay, lunar-laser ranging, perihelion shift, and Earth tide measurements. The resulting available theory space is the "pill box" around the origin (general relativity) shown. Figure 8.3 shows the a t - a 2 plane, and indicates the constraints placed by Earth tide, Earth rotation rate, and perihelion-shift measurements.

Figure 8.2. The (y 1)-(P l)-£ space. Brans-Dicke theory occupies the negative (y l)-axis(/? = 1), while the generalized scalar tensor theories of Bergmann, Wagoner, Nordtvedt, and Bekenstein occupy the half-plane (y 1) < 0. The numbers on the negative (y 1) axis are the corresponding values of co. General relativity resides at the origin. Shown are limits on the PPN parameters placed by the Viking time delay (dotted lines), lunarlaser ranging (dashed lines), and perihelion shift (dot-dashed lines) measurements. The remaining available PPN theory space is the box shown, of thickness 2 x 10" 3 in the direction. 0-

/ 0.03; Scalar-Tensor (BWN, Bekenstein) / 0.02: /

25 :; BransDicke

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For specific theories discussed in Chapter 5, these constraints can be translated into constraints on adjustable constants or matching parameters if the theory is to hope to remain viable. From the la constraints listed above and from the formulae given in Chapter 5, we obtain the limits (i) Scalar-tensor theories: co > 500, A < 10" 3 (ii) Will-Nordtvedt theory: K2 < 4 x 10" 4 (iii) Hellings-Nordtvedt theory: |coX2| < 2 x 10~4, co2K2 < 5 x 10~4 (iv) Rosen's bimetric theory: \co/c1 1| < 4 x 10" 4 (v) Rastall's theory: K2 < 3 x 10" 2 (8.88) Because many theories can be made to agree within experimental error with all solar system tests performed to date, we shall ultimately be forced, beginning in Chapter 10, to turn to new arenas for testing relativistic gravitation.

a/

Ii

I j Hellings-Nordtvedt : I 7 - IK0.002

Figure 8.3. The a!-a 2 plane. The Rosen, Rastall, and Will-Nordtvedt theories occupy parts of the a2-axis shown. The Hellings-Nordtvedt theory, constrained by Viking time-delay measurements of y, occupies the shaded region. General relativity and scalar-tensor theories (ST) reside at the origin. Shown are limits placed by Earth tide (dotted lines), perihelion shift (dashed line), and Earth rotation rate (dot-dashed line) measurements.

Other Tests of Post-Newtonian Gravity

There remains a number of tests of post-Newtonian gravitational effects that do not fit into either of the two categories, classical tests or tests of SEP. These include the gyroscope experiment (Section 9.1), laboratory experiments (Section 9.2), and tests of post-Newtonian conservation laws (Section 9.3). Some of these experiments provide limits on PPN parameters, in particular the conservation-law parameters Ci, d> £3* £4. that were not constrained (or that were constrained only indirectly) by the classical tests and by tests of SEP. Such experiments provide new information about the nature of post-Newtonian gravity. Others, however, such as the gyroscope experiment and some laboratory experiments, all yet to be performed, determine values for PPN parameters already constrained by the experiments discussed in Chapters 7 and 8. In some cases, the prior constraints on the parameters are tighter than the best limit these experiments could hope to achieve. Nevertheless, it is important to carry out such experiments, for the following reasons: (i) They provide independent, though potentially weaker, checks of the values of the PPN parameters, and thereby independent tests of gravitation theory. They are independent in the sense that the physical mechanism responsible for the effect being measured may be completely different than the mechanism that led to the prior limit on the PPN parameters. An example is the gyroscope test of the Lense-Thirring effect, the dragging of inertial frames produced purely by the rotation of the Earth. It is not a preferred-frame effect, yet it depends upon the parameter <xx. (ii) The structure of the PPN formalism is an assumption about the nature of gravity, one that, while seemingly compelling, could be incorrect. This viewpoint has been expounded by Irwin Shapiro (1971) and others. They argue that one should not prejudice the design, performance, and

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interpretation of an experiment by viewing it within any single theoretical framework. Thus, the parameters measured by light-deflection and timedelay experiments could in principle be different according to this viewpoint, while according to the PPN formalism they must be identical [i(l + y)]- We agree with this viewpoint because although theoretical frameworks such as the PPN formalism have proved to be very powerful tools for analyzing both theory and experiment, they should not be used in a prejudicial way to reduce the importance of experiments that have independent, compelling justifications for their performance. (iii) Any result in disagreement with general relativity would be of extreme interest. 9.1

The Gyroscope Experiment

Since 1960, when Leonard Schiff proposed it as a new test of general relativity, much effort has been directed toward the gyroscope experiment (Schiff, 1960b,c; Everitt, 1974; Lipa et al, 1974). The object of the experiment is to measure the precession of a gyroscope's spin axis S relative to the distant stars as the gyroscope orbits the Earth. According to the PPN formalism, this precession is given by (see Section 6.5) dS/dt = ft x S, ft = _ i v x a - |V x g + (y + |)v x VC/, g = QofiJ

(9.1)

where a is the spatial part of the gyroscope's four-acceleration, which is zero for a body in free-fall orbit. In a chosen PPN coordinate system, Equation (9.1) along with the expression for gOj in Table 4.1 yields ft = i(4y + 4 + at)V x V - |a t w x\U

+ (y + frr x \U

(9.2)

where w is the velocity of the coordinate system relative to the universe rest frame, and where V = Vjej

(9.3)

For a system of nearly spherical bodies of masses ma, angular momenta J a , and velocities va, we have

V = £ mavjra - | £ x a x Ja/r3a + O(r8~ 3)

(9.4)

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where xa is the vector from the ath body to the gyroscope. Then " = (7 + i) £ (v - va) x \{mjra) - i(y + i + i«i) I [J. - 3fia(fifl Jfl)]/rfl3 - i*i I (w + vj x V(m>a) - ± £ va x V(ma/ra) a

(9.5)

a

where na = xa/ra. The first term in Equation (9.5) is called the geodetic precession, a consequence of the curvature of space near gravitating bodies. For a circular orbit around the Earth, the Earth's potential (a = ©) leads to a secular change in the direction of the gyroscope spin given, over one orbit, by as = -2n{y + i)(me/a)(S x h) (9.6) Figure 9.1. Precession of gyroscopes in a polar Earth orbit. The gyroscope with its axis in the plane of the orbit undergoes a geodetic precession, while the gyroscope with its axis normal to the orbital plane suffers a precession due to the dragging of inertial frames. 8"/year /

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where a is the orbital radius, and h is a unit vector normal to the orbital plane. For a gyroscope whose initial direction lies in the orbital plane, the angular precession 39 ( = |5S|/|S|) per year is given by «) 5/2 yr~'

(9-7)

There is also a correction of ~0'.'01 yr~' due to the Earth's oblateness. Another secular contribution comes from the Sun's potential (a = Q), given by (^geodetic)© =* O'.'Q2ft(2y + 1)] yr~ x

(9.8)

where we have assumed a circular orbit for the Earth around the Sun. The second term in Equation (9.5) is known as the Lense-Thirring precession or the "dragging of inertial frames" (for further discussion of this effect, see MTW Sections 19.2 and 33.4). For a circular orbit around the Earth, it leads to a secular precession per orbit given by <SS = i(y + 1 + i a i ) ( P / a 3 ) [ J e ~ 3h(B J e ) ] x S

(9.9)

where P is the orbital period of the satellite. For a gyroscope in a polar orbit (fi J @ = 0) or an equatorial orbit (fi Jffi = |J®|), the precession is given by <5SPOL = i(y + 1 + ia 1 )(P/a 3 )J® x S, SSm = - i ( y + 1 + £ a i )(P/a 3 )J e x S

(9.10)

with angular precessions, in arcseconds per year 50poL * 0'.'05[±(y + 1 + i a i )](/le/fl) 3 sin0 yr" 1 , d0EQ ~ O'.'ll[i(y + 1 + W K / V a ) 3 sin «£ yr" 1

(9.11)

where <j> is the angle between the spin vectors of the Earth and gyroscope. The third term in Equation (9.5) is a preferred-frame effect, dependent upon the velocity of the ath body relative to the universe rest frame. For an Earth-orbiting satellite, the dominant effect comes from the solar term (a = O), leading to periodic precession of the form (5S = - i a ^ W o x v e ) x S

(9.12)

where vffi is the Earth's orbital velocity around the Sun and wG = w + v 0 . This leads to a periodic angular precession with a one year period, with amplitude <50p.F. £ 5 x l O " 3 ' ^

(9.13)

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Since the ultimate goal of the experiment is to measure precessions to 10" 3 arcseconds per year, this latter effect is probably too small to be of interest. The last term in Equation (9.5) would appear to be anomalous, since it depends upon the velocity of each body va with respect to our arbitrarily chosen PPN coordinate frame. However, this is simply a result of the fact that the spin precession dSj/dx that we have calculated is not a truly measurable quantity, since the basis vectors e s were not tied to physical rods and clocks. A correct physical choice, and one that is closely related to the actual experimental method, is to use the directions of distant stars as basis directions (Wilkins, 1970). From Equations (7.1) and (7.9), the tangent vector to the trajectory of an incoming photon in the PPN coordinate frame is given by - (1+ y)l/] + (1 + y)93

(9.14)

where |n|2 = 1 is a unit spatial vector in the direction of the unperturbed trajectory from the chosen star, and where 2> is equal to the right-hand side of Equation (7.7), summed appropriately over all the gravitating bodies in the system, and gives the gravitational deflection of the incoming signals. We now project A onto the inertial basis of Equation (6.102), and normalize the spatial components, so that X/(^j)2 = 1, to obtain Aje;= n - n x (v x n)(l + v n) - ^v x (n x v) + (1 + y)@

(9.15)

We now wish to show that the precession of the components of J on this basis is independent of the velocity of the PPN coordinate frame. In Equation (9.5), only the final term has this dependence, so we write it in the form - i I (v. - vB) x V(mjra) - hs * di/dt

(9.16)

a

where vB is the velocity of the solar system barycenter relative to the PPN coordinate frame, and where we have used the fact that, for a freely falling gyroscope, dx/dt = £ V mjra

(9.17)

a

The first term is now independent of the coordinate frame and so may be dropped. The second term may be integrated immediately to obtain <5SB = -i(v B x «5v) x S o

(9.18)

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where the subscript B denotes that we retain only the terms that depend on vB. Then the change in the components of S with respect to A is given by 5(S;A;)B

= <5SB A + S <5AB = - [i(v B x <5v) x S o ] A + i S 0 [vB(<5v A) - c5vvB A] = 0

(9.19)

Thus, as expected, there is no physically measurable dependence on the coordinate-system velocity. In any case, the final term in Equation (9.5) produces only periodic precessions of negligible amplitude. A variety of technical problems has caused the gyroscope experiment to be almost a quarter of a century in the making, from its inception in 1960 to projected launch, in the middle 1980s. Among the more difficult technological hurdles that have had to be overcome in order to produce a spaceworthy experiment that can measure gyroscope precessions accurate to 10" 3 arcseconds per year, or equivalently to 10" 1 6 rad/s, include: (i) Fabrication of a gyroscope that is spherical and homogeneous to a part in a million. For this purpose, a 2 cm radius quartz sphere is used. This constraint is necessary to reduce torques on the gyroscope. Even if this constraint is satisfied, there must be no residual gravitational forces on the gyroscope larger than 10~ 9 g. This necessitates a drag-free satellite. (ii) Readout of the direction of the spin axis. Conventional methods of determining the spin direction of the gyroscope require violations of its sphericity and homogeneity, and thus introduce unacceptable torques. Thus a "London moment" readout method has been adopted. The gyroscope is coated uniformly with a superconducting film. When spinning, the superconductor develops a magnetic dipole moment M parallel to its spin axis. Any change in the direction of M can be determined by measuring the current induced in a superconducting loop surrounding the gyroscope. For this method to be viable, however, it was necessary to develop a magnetic shield that could reduce the ambient magnetic field below 10 ~ 7 G, otherwise the gyroscope could contain trapped magnetic flux of sufficient size to produce anomalous readout signals. By comparison, the ambient magnetic field of the Earth is about 0.5 G. (iii) Determination of basis directions. The precession of the gyroscope's spin axis is measured relative to the direction of a chosen reference star, as observed by a telescope mounted on the gyroscope housing. This direction must be monitored to better than 10" 3 arcseconds per year, so the design of a suitable optical system has been a major problem.

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Further details of the experimental problems and progress are found in Lipa and Everitt (1978) and Cabrera and Van Kann (1978). A variant of the gyroscope experiment has recently been proposed by Van Patten and Everitt (1976) in which the "gyroscope" is itself the orbit of a satellite around the Earth. The dragging of inertial frames causes the plane of the orbit to rotate about an axis parallel to the Earth's rotation axis. Assume the Earth is at rest, and rotates with angular momentum J. The substitution of Equation (9.4) for F, into the equations of motion (Section 4.2) yields the additional acceleration on a body near the Earth da = -i(4y + 4 + a,) \ [2v x J - 3(v n)(ii x J) + 3nv (n x J)]

(9.20)

where v is the body's velocity, and fi = x/r. For an orbit characterized by inclination i relative to the plane normal to J, angle of the ascending node Q and orbit elements p, e, and co, the use of the orbit perturbation Equations (7.47) and (7.48) yields, over one orbit 5i = 0, 8Q = 2n{y + 1 + i^pl/imp3)1'2

(9.21)

Thus the "spin" vector S orthogonal to the orbital plane precesses about the direction of J according to dS/dt = ft x S

(9.22)

ft = (y + 1 + iut)Ja- 3(1 - e2)- 3/2

(9.23)

where For a body in a nearly circular orbit, this yields an annual angular precession 5Q = O'.'22rj(y + 1 + U^jRJa)3

yr" l

(9.24)

In order to eliminate the effects of other sources of precession (such as the quadrupole moment of the Earth) two satellites counterrotating in nearly identical orbits are necessary. With the use of drag-free satellites and with two to three years of orbit data, an experiment with results within 3% accuracy may be possible. 9.2

Laboratory Tests of Post-Newtonian Gravity Because the gravitational force is so weak, most tests of postNewtonian effects in the solar system require the use of the Sun and planets as sources of gravitation. One disadvantage of such experiments

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is that the experimenter has no control over the sources, and so is unable to manipulate the experimental configuration to test or improve the sensitivity of the apparatus, or at the very least, to repeat the experiment. Despite this disadvantage of solar system-sized experiments, the weakness of post-Newtonian gravity has effectively prohibited laboratory experiments, with one exception. That exception is the Kreuzer experiment (Kreuzer, 1968) that compared the active and passive gravitational masses of fluorine and bromine. Kreuzer's experiment used a Cavendish balance to compare the Newtonian gravitational force generated by a cylinder of Teflon (76% fluorine by weight) with the force generated by that amount of a liquid mixture of trichloroethylene and dibromomethane (74% bromine by weight) that had the same passive gravitational mass as the cylinder, namely the amount of liquid displaced by the cylinder at neutral buoyancy. In the actual experiment, the Teflon cylinder was moved back and forth in a container of the liquid, with the Cavendish balance placed near the container. Had the active masses of Teflon and displaced liquid differed at neutral buoyancy, a periodic torque would have been experienced by the balance. The absence of such a torque led to the conclusion that the ratios of active to passive mass for fluorine and bromine are the same to 5 parts in 105, that is (mA/mP)F{ - (mA/wP)Br < 5 x 10-5 (mA/mP)Br

(9.25)

[For further discussion of Kreuzer's experiment, see Gilvarry and Muller (1972) and Morrison and Hill (1973)]. If the active mass were to differ from the passive mass for these substances, the major contribution to the difference would come from the nuclear electrostatic energy (as it does, say in the Eotvos experiment). Since Ee/m ~ 10" 3 , one could regard such effects as post-Newtonian corrections. However, the perfect-fluid P P N formalism of Chapter 4 is poorly suited to a discussion of nuclear matter. A better approximation is one in which the P P N metric is generated by charged point masses, with gravitational potentials generated by masses, microscopic velocities, charges, and so on. Using this metric, one can calculate the active to passive mass ratio of a bound system (nucleus) of point charges, with the result, for a spherically symmetric body (Will, 1976a), mjmv

= 1 + T£(£e/mP)

(9.26)

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where Ee is the electrostatic energy of the system of charges and £ is a combination of PPN parameters derived from the charged-point-mass metric. However, it can be shown that if the perfect-fluid PPN metric of Table 4.1 is simply a macroscopic average of the point-mass metric (as one would expect in most reasonable theories of gravity), then the combination of charged-point-mass parameters that makes up e is precisely the same as the fluid PPN parameter £3. Thus, in any such theory of gravity, mJrn? = 1 + K a ^ / H O

(9.27)

(For further details, see Will, 1976a). The semiempirical mass formula (see Equation 2.8) yields mjmp = 1 + 3.8 x 10~4£3Z(Z - l ) ^ " 4 ' 3

(9.28)

For fluorine Z = 9, A = 19, and bromine Z = 35, A = 80, Equations (9.25) and (9.28) yield |C3| < 6 x 1(T2

(9.29)

This generalizes and corrects a previous result of Thorne et al. (1971). Advancing technology may make several laboratory post-Newtonian experiments possible in the coming decades (Braginsky et al., 1977). The progress that makes such experiments feasible is the development of sensing systems with very low levels of dissipation, such as torque-balance systems made from fused quartz or sapphire fibers at temperatures <;0.1 K, massive dielectric monocrystals cooled to millidegree temperatures, and microwave cavities with superconducting walls. Among some of the experimental possibilities are a measurement of the gravitational spin-spin coupling of two rotating bodies; searches for time variations of the gravitational constant, preferred-frame, and preferred-location effects; and a measurement of the dragging of inertial frames by a rotating body. The reader is referred to Braginsky et al. (1977) for detailed discussion and references. 9.3

Tests of Post-Newtonian Conservation Laws

Of the alternative metric theories of gravity discussed in detail in Chapter 5, all are Lagrangian based, that is, all possess integral conservation laws for energy and momentum. In the post-Newtonian limit, their PPN parameters satisfy the semiconservative constraints a3 = Ci = £2 = £3 s U = 0 What is the experimental evidence for these constraints?

(9.30)

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In Chapter 8, we obtained the upper limit |a3|<2xl(T7

(9.31)

from perihelion-shift data. The effect there was a combined preferredframe effect and self-acceleration of a massive body, in particular of the Sun. However, this limit can probably be tightened considerably, although with somewhat less rigor, by applying the self-acceleration term, Equation (6.39) to pulsars. For these bodies, assumed to be rotating neutron stars, |Q/m| ~ 0.1, and 2 s" 1 < \co\ < 200 s" 1 , thus their self-acceleration has the form Keifl <* 6 x 103|a where v is the pulsar frequency, and 9 is the angle between the pulsar spin axis and its velocity relative to the universe rest frame. Although strictly speaking, the post-Newtonian limit does not apply to pulsars, we feel this is a reasonable estimate of the size of the effect in any theory with a3 # 0. This acceleration will cause a change in the pulse period P p given by = a self -n

(9.33)

where n is a unit vector along the line of sight to the pulsar. Thus, -2 x

independently of Pp, where O is the angle between aseIf and the line of sight ii. For the 90 pulsars reported by Manchester and Taylor (1977) whose values of dPJdt have been measured, those values range between 4 x 10" 13 (Crab Pulsar) and 1 x 10" 18 (PSR 1952 + 29), with half of them lying between 10" 14 and 10" 15 . In all cases, dPJdt > 0, i.e., all pulsars are slowing down. Now for the 40 or so pulsars with 10" 14 > dPp/dt > 10" 15 , it is extremely unlikely that either sin6 = 0 or cos® = 0 for all of them, furthermore if a 3 # 0, we would expect as many pulsars with dPp/dt < 0 as with dPp/dt > 0, assuming their spin directions were oriented randomly. Thus, a conservative limit on <x3 can be obtained by setting sin 0 = cos = 5 in Equation (9.34), and imposing the 10" 14 upper limit on an anomalous dPp/dt, giving |a3l < 2 x 10" 10

(9.35)

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There may be one promising way to set a limit on the parameter £2 involving an effect first pointed out, incorrectly, by Levi-Civita (1937). The effect is the secular acceleration of the center of mass of a binary system. Levi-Civita pointed out that general relativity predicted a secular acceleration in the direction of the periastron of the orbit, and found a binary system candidate in which he felt the effect might one day be observable. Eddington and Clark (1938) repeated the calculation using de Sitter's (1916) n-body equations of motion. After first finding a secular acceleration of opposite sign to that of Levi-Civita, they then discovered an error in de Sitter's equations of motion, and concluded finally that the secular acceleration was zero. Robertson (1938) independently reached the same conclusion using the Einstein-Infeld-Hoffmann equations of motion, and Levi-Civita later verified that result. In fact, the secular acceleration does exist, but only in nonconservative theories of gravity; that is, it depends on the PPN parameters a 3 and £2 (Will, 1976b). The simplest way to derive this result is to treat the two-body system as a single composite "body" in otherwise empty space, and to focus on the self acceleration in the equation of motion, (6.32). For two point masses, Equation (6.32) and the formulae in Table 6.2 give i(a 3 + Ci){mim2x/r3)(vl - v2) + C1(w1rn2/r3)[v2(v2 x) - v ^ x) - fx(v 2 ft)2+ f x ^ ft)2] i ~ tn2)x/r4' + a3m1m2(w + V) vx/r3 (9.36) where x = x 2 - \ u r = |x|,ft= x/r, v = v2 - v ls V is the center-of-mass velocity with respect to the PPN coordinate system, va = va V, and

ms £ mJil+ffi-fa/r)

[b # a]

(9.37)

a=l, 2

Substituting vx s -(m 2 /m)v, v2 s {rnjnifs along with the expressions appropriate for a Keplerian orbit x = p(l + e cos 4>)~l{ex cos + ej,sin<£), v = (m/p)1/2[ e x sin0 + ey(e + cos<£)], r2 dWdt = (mp)112 and averaging (a)self over one orbit, we obtain

(9.38)

(9.39)

<(«)self > = («3 + C2)

- i a 3 ( l - e2y\Q/m)(Y/ + V) x 0

(9.40)

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where eP = ex = [unit vector in the direction of the periastron of m x ], (o = (2n/P)ez = (mean angular velocity vector of orbit], Q = < m1m2/r) = m1m2/a

(9.41)

The second term in Equation (9.40) is the same as the term in Equation (6.39) except for the numerical factor (3 compared to j ( l e 2 )" 1 ], which arises from the difference in averaging for a stationary, nearly spherical body, and for a binary system. However, because of the limit we have already obtained for a 3 , its effects on the self acceleration of a binary system will be negligible. Thus, we shall set a3 = 0, leaving <(a)self > = C 2

In the solar system, this has effects that are utterly unmeasurable. For example, the self acceleration of the Earth-Moon binary system produces a perihelion shift for the Earth of the order dcom ~ 10~ 5 per century. A more promising testing ground for this effect would be a close binary system, such as iBoo, with m t = 1.35mo, m2 = 0.68mQ, P = 0.268 day. The resulting change in the periods (inverse frequencies), say, of the spectral lines of the stars in iBoo would be P~l(dPJdt) = 8.8 x 10- 7 £ 2 e(l - e 2 )" 3 / 2 sin co sin i yr" 1

(9.43)

where i is the inclination of the orbit relative to the plane of the sky, and w is the angle of periastron. Unfortunately, because of Doppler broadening, the frequencies of spectral lines are not known to sufficient accuracy to make such a change observable. However, the discovery of the binary pulsar (Chapter 12) has changed the situation. The characteristics of the orbit are very similar to that of iBoo, however the pulsar provides a much more precise and stable time standard than do spectral lines. This enables one not only to measure changes in the pulse period with high accuracy, but also to determine the parameters of the orbit and thereby the change of the oribit period Pb with high accuracy. The results are (Table 12.1) P^dPJdt l

= (4.617 ± 0.005) x 10~ 9 yr" 1

P b - dPJdt = - (2.4 ± 0.4) x 10 - 9 yr - J

(9.44)

However, because the binary pulsar is a "single-line spectroscopic binary," the individual masses are not known from the velocity curve data (we

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shall see that they can be determined if one assumes a particular metric theory of gravity), rather, the known quantities are (see also Table 12.1) e =* 0.62,

P b =* 27907 s 3

ft = (m2sini) /m2 c^ 0.13mo co si 179° + 4.23°(t - t o )/(l yr)

(9.45)

where / t is known as the mass function, and where t0 [September, 1974]. Then the predicted period change for both the pulsar and the orbit is given by

where X = mjm2 = w pu , sar /m companion , and where we have used the fact that, from Equation (9.45), sin co st - 7 x 10" 2 (t - to)/(l yr),

t - t0 < 10 yr

(9.47)

Note too, that the second derivatives of the periods are given by p ; 1 d2Pp/dt2 = p ^ 1 d2Pb/dt2

Now, from data covering a time span of several years, the error on Pp 1 dPJdt was found to be 10" x 1 y r " l . In other words, P;1 dPJdt did not change by more than 10" 1X yr" 1 in a year, that is, \p-^d2Pvjdt2\ < lO-^yr" 2

(9.49)

Assuming that the secular acceleration is responsible for no more than this amount, in other words, that there is no fortuitous cancellation between this effect and other sources of period change (Section 12.1), we obtain from Equations (9.48) and (9.49) the limit |C2| < 2 x 10" 4 (m o /m) 2/3 |(l + X) 2 /4X(1 - X)\

(9.50)

Now, without assuming a particular metric theory of gravity, we do not know the values of m and X, so the limit on £2 is uncertain (if the masses are equal, for example, X 1, and there is no secular acceleration, by virtue of symmetry). If, for example, we assume that general relativity is valid except for the sole possibility of a violation of momentum conservation manifested

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by £2 # 0, then we can use the values of m and X obtained from periastronshift data and from the gravitational red shift-second-order Doppler shift data (see Chapter 12 for details), m ^ 2.85mo,

X ~ 1.007 ± 0.1

(9.51)

Although the data are not yet sufficiently accurate to exclude X = 1, it is of interest to substitute the nominal value of X into Equation (9.50) to obtain |£2| < 10"2. As long as \X - 1| > 10" 3, we will still have |£2| < 0.1. Of the remaining three conservation-law parameters, only £3 has been tested experimentally, as we saw in the previous section where we obtained the limit |£3| < 0.06 from the Kreuzer experiment. No feasible experiment or observation has ever been proposed that would set direct limits on the parameters £1 or £4. Note, however, that these parameters do appear in combination with other PPN parameters in observable effects, for example in the Nordtvedt effect (see Section 8.1).

10 Gravitational Radiation as a Tool for Testing Relativistic Gravity

Our discussion of experimental tests of post-Newtonian gravity in Chapters 7, 8, and 9 led to the conclusion that, within margins of error ranging from 1% to parts in 10" 7 (and in one case even smaller), the post-Newtonian limit of any metric theory of gravity must agree with that of general relativity. However, in Chapter 5, we also saw that most currently viable theories of gravity could accommodate these constraints by appropriate adjustments of arbitrary parameters and functions and of cosmological matching parameters. General relativity, of course, agrees with all solar system experiments without such adjustments. Nevertheless, in spite of their great success in ruling out many metric theories of gravity (see Sections 5.7, 8.5), it is obvious that tests of post-Newtonian gravity, whether in the solar system or elsewhere, cannot provide the final answer. Such tests probe only a limited portion, the weak-field slow-motion, or post-Newtonian limit, of the whole space of predictions of gravitational theories. This is underscored by the fact that the theories listed in Chapter 5 whose post-Newtonian limits can be close to, or even coincident with, that of general relativity, are completely different in their formulations, One exception is the Brans-Dicke theory, which for large co, differs from general relativity only by modifications of O(l/a>) both in the postNewtonian limit and in the full, exact theory. The problem of testing such theories thus forces us to turn from the post-Newtonian approximation toward new areas of "prediction space," new possible testing grounds where the differences among competing theories may appear in observable ways. The remaining four chapters will be devoted to these new arenas for testing relativistic gravity. One new testing ground is gravitational radiation. Almost from the outset, general relativity was known to admit wavelike solutions analogous

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to those of electromagnetic theory (Einstein, 1916). However, unlike the case with electromagnetic waves, there was considerable doubt as to the physical reality of such waves. Eddington (1922) suggested that they might represent merely ripples of the coordinates of spacetime and as such would not be observable. This lingering doubt was dispelled conclusively in the late 1950s by the work of Hermann Bondi and his collaborators, who demonstrated in invariant, coordinate-free terms that gravitational radiation was physically observable, that it carried energy and momentum away from systems, and that the mass of systems that radiate gravitational waves must decrease (Bondi et al., 1962). The pioneering work of Joseph Weber initiated the experimental search for gravitational radiation. Although no conclusive evidence for the direct detection of gravitational waves exists at present [see Douglass and Braginsky (1979) for a review], gravitational-wave astronomy may ultimately open a new window on the universe. Virtually any metric theory of gravity that embodies Lorentz in variance, on at least some crude level, in its gravitational field equations, predicts gravitational radiation. Thus, the existence of gravitational radiation does not represent a particularly strong test of gravitation theory. It is the detailed properties of such radiation that will concern us here. While the post-Newtonian approximation may be described as the weak-field, slow motion "near-zone" limit, our discussion of gravitational radiation will center on the weak-field, slow motion, "far-zone" limit. In this limit, one finds that metric theories of gravity may differ from each other and from general relativity in at least three important ways: (i) they may predict a difference between the speed of weak gravitational waves and the speed of light (see Section 10.1); (ii) they may predict different polarization states for generic gravitational waves (see Section 10.2); and (iii) they may predict different multipolarities (monopole, dipole, quadrupole, etc.), of gravitational radiation emitted by given sources (see Section 10.3). The use of gravitational-wave speed and polarization as tests of gravitation theory requires the regular detection of gravitational radiation, a prospect that may be far off (see Douglass and Braginsky, 1979). However, the multipolarity of gravitational waves can be studied by analyzing the back influence of the emission of radiation on the source (radiation reaction) for different multipoles. One example is the change in the period of a two-body orbit caused by the change in the energy of the system as a result of the emission of gravitational radiation. Such a test is now possible in the binary pulsar (Chapter 12).

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10.1

Speed of Gravitational Waves The Einstein Equivalence Principle demands that in every local, freely falling frame, the speed of light must be the same - unity, if one works in geometrized units. The speed of propagation of all zero rest-mass nongravitational fields (neutrinos, for example) must also be the same as that of light. However, EEP demands nothing about the speed of gravitational waves. That speed is determined by the detailed structure of the field equations of each metric theory of gravity. Some theories of gravity predict that weak, short-wavelength gravitational waves propagate with exactly the same speed as light. By weak, we mean that the dimensionless amplitude /zMV that characterizes the waves is in some sense small compared to the metric of the background spacetime through which the wave propagates, i.e.,

IIU/IICII«i and by short wavelength, we mean that the wavelength X is small compared to the typical radius of curvature 0t of the background spacetime, i.e., |A/£| « 1 This is equivalent to the geometrical optics limit, discussed in Chapter 3 for electromagnetic radiation. In the case of general relativity, for example, one can show (see MTW, Exercise 35.15) that the gravitational wave vector /" is tangent to a null geodesic with respect to the "background" spacetime, i.e.,

i*r$» = o,

i% = o

where "slash" denotes covariant derivative with respect to the background metric. In a local, freely falling frame, where gfj = rj^, the speed of the radiation is thus the same as that of light. Gravitational radiation propagates along the "light cones" of electromagnetic radiation. General relativity A simple method to derive this result in general relativity, which can then be applied to other metric theories, is to solve the vacuum field equations, linearized (weak fields) about a background metric chosen locally to be the Minkowski metric. Physically, this is tantamount to solving the propagation equations for the radiation in a local Lorentz frame. As long as the wavelength is short compared to the radius of curvature of the background spacetime, this method will yield the same

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results as a full geometrical-optics computation. We thus write G^ = Vw + V

(10.1)

Then the linearized vacuum field equations (5.15) take the form

( 10 - 2 )

Av + K* - <** - K, = °

where indices are raised and lowered using r\. We choose a gauge (Lorentz gauge) in which

Then

D A* = ° whose plane-wave solutions are

V = O'"**'

'"'X* = 0

(10.5)

Thus, the electromagnetic and gravitational light cones coincide, i.e., the gravitational waves are null. Scalar-tensor theories The linearized vacuum field equations are (see Section 5.1 for discussion of notation)

DV + h

-K

= 0, t

(io.6)

>0 V,^v

Choosing a gauge in which 1 4 , - ^ , - ^ =0

(10.7)

we obtain U,9 = D A , = 0

(10.8) |ix

whose plane-wave solutions are proportional to e" " where /"'V = 0

(10.9)

So in scalar-tensor theories, gravitational waves are null. Vector-tensor theories In this case the linearizedfieldequations are much more complex than in the scalar-tensor theories, with the propagation of linearized metric disturbances (h^) being strongly influenced by the background

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cosmological value K of the vector field. In general there are ten different solutions, each with its own characteristic speed and polarization. For one of these solutions, for example (for derivation see Section 10.2) the speed is v2g = (1 - «K 2 )/[1 - (o> - i, - t)K 2 ]

(10.10)

Rosen's bimetric theory We have already discussed weak gravitational waves in this theory, in Section 5.5(g). The resulting speed was given by v\ = Cx/c0 where c1 and c 0 are cosmological matching parameters (see Section 5.5 for discussion). If we take into account not only the cosmological boundary conditions but also a gravitational potential t/ ext due to an external gravitating body (galaxy, sun), with the wavelength of the radiation being short compared to the scale over which l/ext varies, then c 0 and ct may be replaced by c o (l 2l/ cxt ) and c t (l +2£/ ext ), where c 0 and cx denote the purely cosmological values, and thus v2g = (c!/c o )(l + 4[/ ext )

(10.11)

Therefore, the velocity of gravitational radiation may depend both upon cosmological parameters and on the local distribution of matter. Notice that solar system limits on a 2 constrain v2 to be within ~ 4 x 10~ 4 ofunity. RastalFs theory The (extremely complicated) linearized vacuum field equations for the vector field K^ in the rest frame of the universe, where K, = K8° + k^

(10.12)

yield three independent polarizations for k^, one having a different velocity than the other two. However, to first order in the cosmological matching parameter K, which is constrained to be small by Earth-tide measurements (see Sections 5.5, 8.5), the velocities are the same, Vg

= 1 + %K2

and the polarizations for a wave traveling in the z-direction are given by k(1)oce0-ez,

k(2)azex,

k (3) oce y

(10.14)

(These results are valid only in the universe rest frame.) Table 10.1 summarizes the velocities of gravitational waves in these and other theories of grayity discussed in Chapter 5. Generally speaking, there are two ways in which the speed of gravitational waves may differ

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Table 10.1. Properties of gravitational radiation in alternative metric theories of gravity. Gravitational wave speed

Theory General relativity Scalar-tensor theory Vector-tensor theory Rosen's bimetric theory Rastall's theory BSLL theory Stratified theories

1 1 various (cx/c0)1/:2

1 + iK2 + O(K3) 1 + K^o + °>i) + O(co2) a

E(2) class N2 N3

n' ni 5 n6 n«

" Speed is a complicated function of parameters.

from that of light. The first is through the cosmological matching parameters, i.e., vg =* vgc

(10.15)

where vgc denotes the cosmologically determined speed. The second is through the local distribution of matter. If we take into account a nearly constant, but noncosmological gravitational potential t/ ext («1), the matching parameters may be modified by terms of O(l/ ext ), resulting in a speed vg =* 1^(1 + a[/ ext ) (10.16) Solar system experiments limit some of the parameters that appear in the expressions for vgc, but only to accuracies of order 10~ 3 . A crucial test of such theories would be provided by high-precision measurements of the relative speed of gravitational and electromagnetic waves (Eardley et al., 1973). By comparing the arrival times for gravitational waves and for light that come from a discrete event such as a supernova, one could set a limit on the relative speeds that, for a source in the Virgo cluster (11 Mpc from Earth) for example, would yield precision in measuring . . . . time lag, in weeks

(10.17)

Another possible way to test whether vg = 1 has been described by Caves (1980) within the context of Rosen's bimetric theory. If vg < 1, then high-energy particles are prevented from being accelerated to speeds greater than vg by gravitational-radiation damping forces that accompany the nearly divergent gravitational radiation flux emitted by a particle at velocities near vg. The indirect observation of cosmic rays with energies

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exceeding 1019 eV places a very tight upper limit, if this analysis is correct, on 1 - vg in Rosen's theory. Similar conclusions would be expected to follow in any theory in which vg < 1. 10.2

Polarization of Gravitational Waves (a) The E{2) classification scheme General relativity predicts that weak gravitational radiation has two independent states of polarization, the " + " and " x " modes, to use the language of MTW, (Section 35.6), or the + 2 and 2 helicity states, to use the language of quantum field theory. However, general relativity is probably unique in that prediction; every other known, viable metric theory of gravity predicts more than two polarizations for the generic gravitational wave. In fact, the most general weak gravitational wave that a theory may predict is composed of six modes of polarization, expressible in terms of the six "electric" components of the Riemann tensor ROiOj that govern the driving forces in a detector (Eardley et al., 1973; Eardley, Lee, and Lightman 1973). Consider an observer in a local freely falling frame. In the neighborhood of a chosen fiducial world line ^(t), construct a locally Lorentz orthonormal coordinate system {t,xj) with t as proper time along the world line and^(f) as spatial origin ("Riemann normal coordinates"). The metric has the form (MTW, Section 13.6) 9 m = "m + hjn

(10.18)

where + O(|x|3), * + O(|x|3), % = - i / l a y ^ x * + O(|x|3) *

(10.19)

where R^a, are components of the Riemann tensor. For a test particle with spatial coordinates x\ momentarily at rest in the frame, the acceleration relative to the origin is at = 1*66,? = ~ Kof6j* ; are tne

(10.20)

where Roio} "electric" components of Riem due to gravitational waves or other external gravitational influences. Note that despite the possible presence of auxiliary gravitational fields in a given metric theory of gravity, the acceleration is sensitive only to Riem. [This is not necessarily true if the body has self-gravitational energy, as has been emphasized by Lee (1974).]

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Thus, a gravitational wave may be completely described in terms of the Riemann tensor it produces. We define a weak, plane, nearly null gravitational wave in any metric theory [Eardley, Lee, and Lightman (1973)] to be a weak, propagating vacuum gravitational field characterized, in some local Lorentz frame, by a linearized Riem with components that depend only on a retarded time u, i.e., R*yi = KpyM

(10.21)

(henceforth we shall drop the caret on indices) where the "wave vector" ! which is normal to surfaces of constant u, defined by

is almost null with respect to the local Lorentz metric, i.e.,

rfvlX = e,

|e| « 1

(10.23)

where e is related to the difference in speed, as measured in a local Lorentz frame at rest in the universe rest frame, between light and the propagating gravitational wave, i.e., e = (c/vg)2 - 1

(10.24)

We now wish to analyze the general properties of Riem for a weak, plane, nearly null gravitational wave. To do this, it is useful to introduce, instead of the locally Lorentz orthonormal basis (t, xJ), a locally null basis. Consider a null plane wave (light, for instance) propagating in the + z direction in the local Lorentz frame. The wave is described by functions of retarded time u, where u =t - z

(10.25)

(we use units in which the locally measured speed of light is unity). A similar wave propagating in the z direction would be described by functions of advanced time v, where v= t +z

(10.26)

We now define the vector fields I and n to be I = Fe^, n = n^e,,, where

These vectors are tangent to the propagation directions of the two null plane waves. In the (t, xJ) basis they have the form /" = (1,0,0,1),

n" = i ( l , 0,0,-1)

(10.28)

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229

and are null with respect to 17, i.e., I'l'V

=

rt\(

= 0

(10.29)

We also introduce the complex null vectors m and m, where the bar denotes complex conjugation, denned by m = m"e,,, where m" = (2)- ^(O,1, i,0),

m" = (2)" 1/2 (0,1, - i,0)

(10.30)

and where m"mvjjpv = m"mvf/^v = 0

(10.31)

These null vectors obey the orthogonality relations >f v = -2J ( "n v) + 2m("mv)

(10.32)

In a Cartesian basis, they are constant. For the remainder of this section, we shall use roman subscripts (excluding i, j , k) to denote components of tensors with respect to the null tetrad basis I, n, m, in, i.e., Zarb_ = Za$1..
(10.33)

where a, b, c,... run over I, n, m, and m, while p, q, r,... run over only I, m, and m. Because the null tetrad I, n, m, and m is a complete set of basis vectors, we may expand the gravitational wave vector Tin terms of them; however, since the gravitational wave is not exactly null, this expansion will depend in general upon the velocity of the observer's local frame relative to the universe rest frame. Choose a "preferred" observer, whose frame is at rest in the universe, and let /" in this frame have the form I" = f ( l + e,) + enn" + emm" + ejn"

(10.34)

where {£,,£, £,,£} ~ O(E). However, this observer is free (i) to orient his spatial basis so that the gravitational wave and his null wave are parallel, i.e., so that V oc V,

(10.35)

and (ii) to choose the frequency of his positively propagating null wave to be equal to that of the gravitational wave, i.e., 7° = 1°

(10.36)

Hence, em = em = 0, £, = -|fi n , and ? = / " - sj^l" - n")

(10.37)

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Now, because the Riemann tensor is a function of retarded time u alone,

Thus, using the orthogonality relations among the null tetrad vectors,

( 10 - 39 )

Rw = ° The linearized Bianchi identities Rab[cdie] = 0 then yield RatPq,n = O(BnR)

(10.40)

which, except for a trivial nonwavelike constant, implies Rabpq = ^ W = O(£nR)

(10.41)

Thus the only components of Riem that are not O(en) are of the form Rnpnq. There are only six such components and all other components of Riem can be expressed in terms of them. They can be related to particular tetrad components of the irreducible parts of Riem; the Weyl tensor, the traceless Ricci tensor, and the Ricci scalar (see MTW Section 13.5 for definitions). These components are called Newman-Penrose quantities, denoted T, <J>, and A, respectively, (Newman and Penrose, 1962). For our nearly null plane wave in the preferred tetrad, they have the form (i) Weyl tensor: ¥ 0 ~ O(s2nR), V2=~i;RnM

V1 ~ O(enR), + O(£nR),

^ 3 = -iRnlnm +

O(EnR),

*4 = -Rnmnm

(10.42)

(ii) traceless Ricci tensor: $ 0 0 ~ O(en2R),

®i2 = *2i = ¥ 3 + O(enR)

(10.43)

(iii) Ricci scalar: A = - i « F 2 + O(enR) (10.44) To describe the six independent components of Riem we shall choose the set W2, *P3, *P4, and
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231

y

y

o

o (/

/) (a)

(b)

Im * 4

31

o

J) (c)

(d)

i \

(e)

*

\

;

\

/

)

(0

Figure 10.1. The six polarization modes of a weak, plane gravitational wave permitted in any metric theory of gravity. Shown is the displacement that each mode induces on a sphere of test particles. The wave propagates in the +z direction and has time dependence cos cot. The solid lirie is a snapshot at cot = 0, the broken line one at cot = n. There is no displacement perpendicular to the plane of the figure. In (a), (b), and (c) the wave propagates out of the plane; in (d), (e), and (f), the wave propagates in the plane.

are valid for a gravitational wave as detected by the preferred observer. Now in order to discuss the polarization properties of the waves, we must consider the behavior of these components as observed in local Lorentz frames related to the preferred frame by boosts and rotations. However, we must restrict attention to observers who agree with the preferred observer on the frequency of the gravitational wave and on its direction; such "standard" observers can then most readily analyze the intrinsic

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polarization properties of the waves. The Lorentz frames of these standard observers are related by a subgroup of the group of Lorentz transformations that leave \ unchanged. The most general such transformation of the null tetrad that leaves T [cf. Equation (10.37)] fixed is given by I' = n' = m' = m' =

(1 - aa£n)\ - en(am + am) + O(£2), (1 aa£B)(n + aal + am + am) + O(e2), (1 - aaeje'^m + al) - e^e'^n + am) + O(E 2 ), (1 - aaejg-'^m + al) - £nae"'>(n + oan) + O(e2)

(10.45)

where a is a complex number that produces null rotations (combinations of boosts and rotations) asd cp is an arbitrary real phase (0 < q> < 2n) that produces a rotation about ez. The parameter a is arbitrary except for the restriction aa«e n " 1

(10.46)

This expresses the fact that our results are valid as long as the velocity of the frame, w, is not too close either to the speed of light or of the gravitational wave, whichever is less; note that for nearly null waves e~ * » 1 and almost any velocity that is not infinitesimally close to unity is permitted, since aa ~ w2/(l - w2)

(10.47)

For exactly null waves en = 0, and arbitrary velocities w < 1 are permitted. Under the above set of transformations, the amplitudes of the gravitational wave change according to W2 = V2 + O(snR), + 6a2*F2) + O{snR), 2af 3 + 6aa»P2 + O(snR)

(10.48)

Consider a set of observers related to each other by pure rotations about the direction of propagation of the wave (a = 0). A quantity that transforms under rotations by a multiplicative factor e's
O 22 :s = 0,

¥ 3 :s=-l, ¥ 4 :s=-2,

¥ 3 :s=+l, ?4:5=+2

(10.49)

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However, these amplitudes are not observer-independent quantities, as can be seen from Equation (10.48). For example, if in one frame *F2 # 0, *P4 ^ 0, then there exists a frame in which ¥4 = 0. Thus, the presence or absence of the components of various helicities depends upon the frame. Nevertheless, certain frame-invariant statements can be made about the amplitudes, within the small corrections of O(snR). These statements comprise a set of quasi-Lorentz invariant classes of gravitational waves. Each class is labeled by the Petrov type of its nonvanishing Weyl tensor and the maximum number of nonvanishing amplitudes as seen by any observer. These labels are independent of observer. For exactly null waves, the classes are: Class II6; *F2 # 0. All standard observers measure the same value for *F2, but disagree on the presence or absence of all other modes. Class III5: *F2 = 0, ¥3 ^ 0. All standard observers agree on the absence of *F2 and on the presence of ¥3, but disagree on the presence or absence of *P4 and 22. Class N3: f 2 s ^ 5 0> * 4 # 0. $22 # 0. Presence or absence of all modes is observer-independent. Class N2: *¥2 = ¥3 = 4>22 = 0, ¥ 4 =£ 0. Independent of observer. Class O1: x¥2 = x¥3 = *¥4 = 0, 22 = 0. Independent of observer: No wave. For nearly null waves, simply replace the vanishing of modes (=0) with the nearly vanishing of modes [~O(enR)]. This scheme, developed by Eardley et al. (1973), is known as the E(2) classification for gravitational waves, since in the case of exactly null plane waves (en = 0), the transformation equations, (10.45), are the "little group" E(2) of transformations, a subgroup of the Lorentz group. The E(2) class of a particular metric theory is defined to be the class of its most general wave. Although we have confined our attention to plane gravitational waves, one can show straightforwardly (Eardley, Lee, and Lightman, 1973) that these results also apply to spherical waves far from an isolated source provided one considers the dominant l/R part of the outgoing waves, where R is the distance from the source. (b) E(2) classes of metric theories of gravity

To determine the E(2) class of a particular theory, it is sufficient to examine the linearized vacuum field equations of the theory in the limit of plane waves (observer far from source of waves). The resulting

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classes for the theories discussed in Chapter 5 are shown in Table 10.1. Here, we present some examples. Some useful identities that can be obtained from Equations (10.32) and (10.41) are Rnl = RnM + O(snR),

Rm = 2Rnmnih + O(enR),

Rnm = Kinm + O(enR),

R = - 2Rnl + O(snR)

(10.50)

If Riem is computed from a linearized metric perturbation h^{u), then

and W2 = A

+ O(snR),

^ 4 = ifc*» + O{anR),

W3 = O22

O(snR), ^

+ O(enR)

(10.52)

General relativity The vacuum field equations are (10.53)

R,v = 0 The waves are null (en = 0). Thus, KM

= Rnn,nm = Rnlnn, = 0

(10.54)

or V2 = «P3 = O 2 2 = 0

(10.55)

The only unconstrained mode is *F4 ^ 0, so general relativity is of E(2) class N 2 . Scalar-tensor theories In a local freely falling frame, the linearized vacuum field equations are ,9 = o, R = O(
(10.56)

(see Section 5.3 for details), where 0 is the cosmological boundary value of the scalar field (f>. The solution to the first of Equation (10.56) for a plane wave is cp =

(10.57)

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where tj^lT 0. Then, from Equation (10.56),

«(1,= -^oV/'-V»

(10.58)

Thus, Rnn * 0, Rnl s Rnm = 0, thus, V2 = V3 = 0,

¥4#0

(10.59)

and scalar-tensor theories are of class N 3 . Vector-tensor theories In a local freely falling frame in the universe rest frame, the linearized vacuum field equations take the form - coK2h00iltv

- 2coKk0tllv

- (co + \r\

-

- i(co - (if -

?) - ifa ^

» + (i, + TJK/C^O = 0,

/

(10.60)

(6 - |T)D A - efcf,, - i « X ^ ( n ^ - ^ ) + i(»? - t ) X D ^ , - ftg,^) = 0 (10.61) By substituting plane wave forms h^ = h^u) and fcM = k^(u), we can turn the field equations into a set of algebraic equations for the amplitudes hMV and k^. We now project these equations onto the null tetrad I, n, m, and m, and obtain ten homogeneous algebraic equations for the ten unknowns hab, ka, with coefficients that depend upon the parameters a>, n, x, e, and K, and upon en [see Equation (10.37)]. These equations are of the form [(1 - coK2)en(l - K ) - ±fo - t)K2]hmm = 0, «AihM + pBiLm + Pnhu +

PB3K

XAIKM

+ *A3km = 0,

+ PB4.h'nn + pBt'k, + pB6kn = 0

(10.62) (10.63) (10.64)

where A = 1,2,3, and B = 1 , . . . , 6. One mode is given by hmm * 0, en(l - i O = kin ~ r)K\l

- coK2yl

(10.65)

Since -2e n (l - hn) = £ = (vgy2 - 1

(10.66)

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the speed of this mode is given by

In this case it can be shown that (except for special values of the parameters) the remaining amplitudes satisfy Km = fej = K = km = 0,

eJL> - (1 - Ktfi. s 0, (1 - hn)hnl + (1 - a ^ = 0 = (1 - !£)£ + 2£n/I"nn

(10.68)

whose consequence is *P2 = *F3 = $22 = 0. Hence, this mode is N2. There are in principle nine other modes with timm = 0, each with its own speed and characteristic polarization, some as general as II 6 . Rosen's bimetric theory The linearized field equations are of the form D,^MV = 0 with no restrictions on the h^, hence all modes are nonzero in general, hence the theory is of class II 6 . Rastall's theory Since the linearized physical metric g in the universe rest frame has the form [Equation (5.78)]

g0J =

KCQ

l

kp

l

gik = Co 8jk + Kco*k05ik

(10.69)

where k^ = k^u), then we have, after transforming to local Lorentz coordinates, h'tt <x k0 + kz, ^

0,

tilih

oc kg,,

hmih<xk0

(10.70)

However, from the solution of the linearized field equations discussed in the previous section [Equation (10.14)], it is clear that for all solutions, n 'u = O(snR), hence, ¥ 2 = O(enR)

(10.71)

Notice that for thek ( 1 ) mode, only <J>22 °c timjh # 0, so this mode is O t ; for thek (2) andk (3) modes, *P3 oc hm # 0, so these modes are III 5 . The vanishing of *P4 a hmih is valid only in the universe rest frame, a result of the

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special form of g^ there. The most general wave therefore is III 5 , hence Rastall's theory is of class III 5 . It is possible to show that the other theories discussed in Chapter 5 are of class II6 (see Table 10.1). (c) Experimental determination of the E(2) class Consider an idealized gravitational-wave polarization experiment. An observer uses an array of gravitational-wave detectors to determine via Equation (10.20) the six electric components ROiOj of Riem for an incident wave (for discussion of possible devices and arrays see Eardley, Lee, and Lightman, 1973; Paik, 1977; and Wagoner and Paik, 1977). Let us suppose that the waves come from a single localized source with spatial wave vector k (which the observer may or may not know a priori). If the observer expresses his data as a 3 x 3 symmetric "driving force matrix" StJ(t) = Roioj

(10.72)

then, for a wave with k = e2, Equations (10.28), (10.30), (10.42), and (10.43) give the following form for S,7 in terms of the wave amplitudes -2/2Rex¥3 (10.73) 2/2, where the standard xyz orientation of the matrix elements is assumed. Now, if the observer knows the direction k a priori, either by associating the wave with an independently observed event such as a supernova, or by correlating signals detected at two widely spaced antennas (gravitational-wave interferometry), then by choosing a z-axis parallel tok, one can determine uniquely the amplitudes as given in Equation (10.73), and thereby the class of the incident wave. Because a specific source need not emit the most general wave possible, the E(2) class determined by this method would be the least general class permitted by any metric theory of gravity. However, if the observer does not know the direction a priori, it is not possible to determine the E(2) class uniquely, since there are eight unknowns (six amplitudes and two direction angles) and only six observables (Sy). In particular, any observed StJ can be fit by an appropriate wave of class II6 and an appropriate direction. However, for certain observed Sjj, the E(2) class may be limited in such a way as to provide a test of gravitational theory. For example, if the driving forces remain in a fixed plane

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and are pure quadrupole, i.e., if there is a fixed coordinate system in which

n

-X \0

o\

n 0

(10.74)

0 0/

then the wave may be either II 6 (unknown direction), or N 2 (direction parallel to z axis of new coordinate system). If this condition is not fulfilled, the class cannot be N 2 . Such a result would exemplify evidence against general relativity. Eardley, Lee, and Lightman (1973) provide a detailed enumeration of other possible outcomes of such polarization measurements. 10.3

Multipole Generation of Gravitational Waves and Gravitational Radiation Damping It is common knowledge that general relativity predicts the lowest multipole emitted in gravitational radiation is quadrupole, in the sense that, if a multipole analysis of the gravitational field in the radiation zone far from an isolated system is performed in terms of tensor spherical harmonics, then only the harmonics with / ^ 2 are present (see Thorne, 1980 for a thorough discussion of multipole-moment formalisms). For material sources, this statement can be reworded in terms of appropriately denned multipole moments of the matter and gravitational-field distribution within the near-zone surrounding the source: the lowest source multipole that generates radiation is quadrupole. For slow-motion, weak-field sources, such as binary star systems, quadrupole radiation is in fact the dominant multipole. (Some have argued that this is true for any slowmotion source, whether weak field or not. One exponent of this viewpoint is Thorne, 1980.) The result is a gravitational waveform in the radiation zone given by hmm = (2/R)Imm

(10.75)

where R is the distance from the source, 7 y is the moment of inertia of the source, and dots denote derivatives with respect to retarded time. The waveform h^m is related to the measured electric components of Riem by Equation (10.52), «-*=-*4=-i«**

(10.76)

The flux of energy at infinity that results from this waveform is given by

dE/dt = - J < W

(10.77)

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where J y is the trace-free moment of inertia tensor of the system, given to lowest order in a post-Newtonian expansion by / = J p ( x , t){xtXj - !<50x2) d3x

(10.78)

and where angular brackets denote an average over several periods of oscillation of the source (for a recent discussion, see Walker and Will, 1980a). These comments apply to the asymptotic properties of the outgoing radiation field. However, we are interested not in the properties of the outgoing radiation field (those were relevant for Sections 10.1 and 10.2), but in the back reaction of the source to the emission of the radiation. A variety of computations have led to the conclusion that the energy flux at infinity given by Equation (10.77) is balanced by an equal loss of mechanical or orbital energy by the system, and that this energy loss can be derived from a local radiation-reaction force (MTW Section 36.11) F(react)=

_ (2/5)mi\?Xj

(10.79)

where the superscript (5) denotes five time derivatives (Walker and Will, 1980b). However, one school of thought maintains that these conclusions have not been satisfactorily derived from a fully self-consistent, approximate solution of Einstein's equations (Ehlers et al., 1976). It is not the purpose of this section to enter into this controversy. Instead, we shall simply make the assumption that in any semiconservative metric theory of gravity, there is an energy balance between the flux of gravitational-wave energy at infinity and the loss of mechanical energy of the source, provided one averages over several periods of oscillation, and that the energy flux can be determined using a slow-motion, weak-field approximation scheme of a kind suggested by Epstein and Wagoner (1975). If we now focus on binary systems with total mass m = mx + m2, reduced mass ii = m^m^m, orbital separation r, and relative velocity v, quadrupole radiation within general relativity [Equation (10.77)] leads to a loss of orbital energy at a rate (Peters and Mathews, 1963) ^=_/^A(12^-ll^)\ (10.80) dt \ r4 / where r = dr/dt, and where angular brackets denote an average over an orbit. This loss of energy results in a decrease in the orbital period P given by Kepler's third law, (10.81)

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Quadrupole radiation also leads to a decrease in the angular momentum of the system, and to a corresponding decrease in the eccentricity of the orbit (see Wagoner, 1975, for references and a summary of the formulae). Faulkner (1971) has pointed out that these effects of quadrupole gravitational radiation may play an important role in the evolution of ultrashortperiod binary systems (see also Ritter, 1979). But probably the most promising test of the existence of quadrupole radiation has been provided by observations of period changes P in the binary pulsar (Chapter 12). Unlike general relativity, however, nearly every alternative metric theory of gravity predicts the presence in gravitational radiation of all multipoles-monopole and dipole, as well as quadrupole and higher multipoles (Eardley, 1975; Will and Eardley, 1977; and Will, 1977). For binary star systems, the presence of these additional multipole contributions has two effects on the energy-loss-rate formula, (10.80): (a) modification of the numerical coefficients in (10.80) and (b) generation of an additional term (produced by dipole moments) that depends on the selfgravitational binding energy of the stars. The resulting formula for dE/dt may be written in a form that contains dimensionless parameters whose values depend upon the theory under study. Two parameters, KV and K2, are denoted "PM parameters" because they refer to that part of dE/dt that corresponds to the Peters-Mathews (1963) result for general relativity. A parameter KD refers to the dipole self-gravitational contribution, where, at least schematically, we may write {dE/dt)dipole

= - i K f l < D D>

(10.82)

where D is the dipole moment of the self-gravitational binding energy Gla of the bodies D = £ Qaxo

(10.83)

a

(Within each specific theory of gravity the details are more complicated than this, however.) For a binary system, the result is

f = - (~£- iUw2 - K2f2) + i*fl®2])

(10-84)

where <3 is the difference in the self-gravitational binding energy per unit mass between the two bodies. In this section, we shall derive these results using a post-Newtonian gravitational radiation formalism developed by Epstein and Wagoner (1975) and Wagoner and Will (1976). However, because of the complexity of many alternative theories of gravitation beyond the post-Newtonian

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approximation, it has proven impossible to devise a general formalism analogous to the PPN framework, beyond writing Equation (10.84) with arbitrary parameters. But, we can provide a general description of the method used to arrive at Equation (10.84) within a chosen theory of gravity, emphasizing those features that are common to many currently viable theories. Later, we shall describe the specific computations within selected theories. The method proceeds as follows: Step 1: Select a theory. Restrict the adjustable constants and cosmological matching parameters to give close agreement with solar system tests (Chapters 7, 8, and 9). Step 2: Derive the "reduced field equations." Working in the universe rest frame, expand the gravitational fields about their asymptotic values, and, using any gauge freedom available, express thefieldequations in the "reduced" form (v~2d2/dt2 + V2) [terms linear in perturbations of fields] = - ten [source] (10.85) where vg, the gravitational-wave speed, is a function of adjustible constants and matching parameters, and where the "source" consists of matter and nongravitational field stress energies, and of "gravitational" stress energies consisting of terms quadratic and higher in gravitational-field perturbations. If we denote the linear term by xjt (it can be a tensor of any rank) and the source by x, then the solution of Equation (10.85) that has outward propagating disturbances at infinity is ij/(x,t) = 4 J\(r - v;x\x - x'|,x')|x - x'l" 1 d3x'

(10.86)

For field points far from the source (R = |x| » r = "size" of source, \i//\ « 1), we have \j/(x,t)=4R~l jxit-v^R

+ v^t x',x')d3x' + O(r/R)2 (10.87)

where n = x/R. If we assume that the motions of the source are sufficiently slow (source within wave zone, r < k/2n = wavelength/27t « R), then Equation (10.87) may be expanded in the form \x(t-v-lR,x'){n-x')md2x'

(10.88)

m=O

For further use, we note that f; = - V9 %^,o + O(r/R2)

(10.89)

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Step 3: Determine the energy loss rate in terms of \jt. Let us restrict attention to Lagrangian-based theories of gravity (such as the currently viable theories described in Chapter 5). Such theories possess conservation laws of the form (see Section 4.4) ©?vv = 0

(10.90)

where ©*" reduces in flat spacetime to the stress-energy tensor for matter T"v. Hence, we can define quantities P" that are conserved for a localized source, except for a possible flux 0 " j of energy-momentum far from the source: when integration is performed over a constant-time hypersurface, we have

P" = J0«° d3x,

dP"/dt = J0f o ° d3x = - J s 0 W ' d2Sj (10.91)

where S is a closed two-dimensional surface surrounding the region of integration. For each theory, it turns out that 0'"' may be written 0" v = f(il/)T"v + t"v

(10.92)

where /(if/) -> 1 as \j/ -* 0, and t"v is an expression at least quadratic in the first-order perturbations (i/0 of the gravitational fields. If we choose for S a sphere of radius R in the wave zone far from the source, we have for dP°/dt = -R2j>

tOinj dQ

(10.93)

Substituting Equation (10.88) into the expression for t0J provided by the equations of the theory yields an expression for dP°/dt in terms of time derivatives i^>0 of the gravitational fields, evaluated in the far zone. Step 4: Make a post-Newtonian expansion of the "source" x (see Chapter 4 for discussion). For this purpose, use the near-zone, postNewtonian forms for matter variables and gravitational fields obtained in Chapter 5 in the solution for the post-Newtonian metric, appropriately transformed to the gauge adopted in Step 2. Depending on the nature of ^i, the sources x are of even ("electric") order in the post-Newtonian sense [O(0),O(2),...] or odd ("magnetic") order [O(l),O(3),.. .] For electric sources, x typically contains terms of the form l

electric

~ P,pn,pv2,pU,p

(10.94)

modulo total divergences whose moments [monopole, dipole, etc.; see Equation (10.88)] can be shown to be negligible upon integration by

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243

parts [see Epstein and Wagoner (1975) for discussion]. For magnetic sources, x typically has the form (modulo divergences) ^magnetic ~ M P&U P^U, pvh2, PVJ, PV\ PWj

(10.95)

Step 5: Simplify i// using integral conservation laws. Because \jj, Equation (10.88), involves time derivatives of integrated moments of the source T, and since time derivatives of ij/ will ultimately be used, it is convenient to employ the integral conservation laws obtained from Equation (10.90) to extract from the integrated moments terms that are constant in time, linear in time, etc. Some of these terms reflect the imprints of the mass, momentum, angular momentum, and center of mass of the source on the far-zone field, and do not contribute to gravitational radiation. Since these integral conservation laws are to be applied only to near-zone integrals, we neglect surface integrals such as the one in Equation (10.91) (retaining them would only yield higher-order corrections to the energy loss-rate formula). These integral conservation laws give the following useful results (valid in the near zone) (d/dt)

(d2/dt2) §®°°xJxkd3x=2 (d/dt) J0' o (fi x)d3x = j®Jknkd3x

(10.96)

Notice that, because we are dealing with semiconservative theories of gravity, 0" v is not necessarily symmetric, so we have retained the contributions of the antisymmetric parts of 0" v where necessary. However, as we saw in Section 4.4, these terms depend upon the PPN parameters <*! and a2 and so they will be small if we impose the experimental constraints on aj and a2 discussed in Chapter 8, or will be zero if we adopt a version of the theory with a t = a2 = 0 (i.e., a fully conservative version). Step 6: Apply to binary systems. We consider a system made up of two bodies that are small compared to their separations (d « r); that is, we ignore all tidal interactions between them. We may thus treat each body's structure as static and spherical in its own rest frame. We then follow the procedure of Section 6.2: for a given element of matter in body a, we write v = vfl [static structure],

x = Xfl + x

(10.97)

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where

Xa = m-1 £ p*(l + II - if/)xd 3 x, ma = P°a= f p*(l +

ya = dXJdt,

n-^O)d3x,

0 = Jo p*(x', t)|x - x'|"' d3x'

(10.98)

We note that ma is conserved to post-Newtonian order, as long as tidal forces are neglected. The full Newtonian potential U for spherical bodies is given by U(x, t)= Ua+ £ mb\x - Xb\ ~J

[x inside body a]

b*a

= X mb\x Xj.1"1

[x outside body a]

(10.99)

b

Then the total energy of the system P° [cf. Equation (4.108)] is given by (KU00)

where rafc = |Xa Xb|. For a binary system we may evaluate the orbital terms in Equation (10.100) to the required order using Keplerian equations (see Section 7.3 for definitions of orbital elements); r = rab = a(l - e 2 )(l + e cos <^)"r

(10.101)

The result is °

(10.102)

where m = ma + mb, n = mamb/m, and where the semi-major axis a is related to the orbital period P to the necessary order by (P/2n)2 a3/m. In the emission of gravitational radiation whose source is the orbital motion, the quantities ma and mb will be unchanged because of our neglect of tidal forces and internal motions. Invoking energy balance, we thus have (dE/dt)Tadiation

= dP°/dt

(10.103)

We now use the above procedure to split the moments oft that determine \ji into orbital parts (v2 ~ m/r) and "self" parts associated with each body (£7 ~ II ~ p/p ~ m/d » m/r). In terms of the quantities m/r and m/d, we find that electric ij/ fields have the schematic form [Equations (10.88),

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245

(10.94), (10.96)] ^eiectnc ~ 4(m/R) | [constant]

[ml r / m V l [mm]r , J J mono ~ + I ~ I + ~~ 3 L P°' e anc* quadrupole]

[dipole]

-r) r b i v J

(10.104) and magnetic \jt fields have the form [Equations (10.88), (10.95), (10.96)] ^magnetic ~ 4(m/R) \ [constant]

+ j

(10.105)

Because the energy flux tOi [Equation (10.93)] typically depends on (i/'.o)2* the constant terms in Equations (10.104) and (10.105) do not contribute to the radiation. In Equations (10.104) and (10.105) it is the (m/r) term that yields the PM contribution, since t/^0 ~ (m/R)2(m/r2)2v2. The terms of O(m/r)2 and O(m/r)3'2 in Equations (10.104) and (10.105) are postNewtonian corrections of a kind discussed by Wagoner and Will (1976) for general relativity. The terms of O[(w/r)(m/d)] effectively renormalize the masses that appear in the PM result by corrections of O{m/d). The terms of O[(m/d)(m/r) 1/2 ] produce the dipole radiation of interest: (fo) 2 ~ (m/R)2(m/d)2{mll2/r3l2)2v2 ~ (m/R)2(m/r2)2{rn/d)2. Cross terms produce effects that are down from these by powers of (d/r) or that vanish on integration over solid angle [Equation (10.93)]. Hence, we retain only terms in \jj of order (m/r) and (m/d)(m/r)1/2. In evaluating the "self" terms, we employ the standard virial theorem for static spherically symmetric bodies: 3 japd3x

+ na = 0

(10.106)

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where to the necessary order Qfl= -^ap0d3x=

-^ap{x)p{x')\x-x'\-1d3xd3x'

(10.107)

Step 7: Calculate the average energy loss over one orbit, using Newtonian equations of motion to simplify the Newtonian P M contribution and the post-Newtonian dipole contribution to the radiation. To illustrate this method, we shall now focus on three metric theories: general relativity, scalar-tensor theories, and Rosen's bimetric theory. For other theories, such as the BSLL theory and Ni's theory, see Will (1977). General relativity By defining 0"v = /i"v - \rf"h

(10.108)

and choosing a gauge ("Lorentz" gauge) in which 0?vv = 0

(10.109)

where indices are raised and lowered using the Minkowski metric, one can show that Einstein's equations are equivalent to the reduced field equations I|0"v = -

16TCT"V

(10.110)

where T"V = T"v + t"v

(10.111)

v

v

with t" a function of quadratic and higher order in 0^ and its derivatives. Because of the gauge condition, Equation (10.109), T"V satisfies T^VV = 0

(10.112)

Then

e»» = 4R-1

f; (l/ml)(d/dt)m {^(t-R, m=0

x')(n x')md3x'

(10.113)

J

Because of the gauge condition, Equation (10.109), and the retarded nature of 0"v, we need to determine only the 01J components, since

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Now, because the source T"V for 6"v satisfies its own conservation law tfvv = 0, and is symmetric, we may make use of Equations (10.96), with TMV in place of 0*v to show that 6iJ = 2R-\d2ldt2)\

xoo(t - R, x)xixid3x

+ [quadrupole moments of xOj, TJ*] + !

(10.115)

Notice that the monopole and dipole moments of zij have been reexpressed as second time derivatives of quadrupole moments. Since each time derivative (8/dt)x ~ v ~ (m/r)112, there can be no "dipole" contribution to 0** of the form (m/d)(m/r)112. Thus, the only contribution to the field to the required order comes from the lowest order, "Newtonian" part of T 00 , namely t 0 0 = p [ i + O(2)]

(10.116)

For a binary system, Equation (10.115) becomes 9iJ = 2R ~ \d2ldt2) X max\xi + O(m/r)3'2

(10.117)

a

The conservation laws for T"V also imply that the center of mass of the system is unaccelerated, so, decomposing xa into center-of-mass and relative coordinates to Newtonian order, using X = m~i(mtx1 + m2x2),

x = x2 x t

(10.118)

we obtain, modulo a constant, 6iJ = (2n/R)(d2/dt2){xixj) + O(m/r)312

(10.119)

Now, to determine the energy-loss rate, it is most convenient to use for 0'"' the conserved quantity = ( - g)(T»v + til),

0fL,v = 0

(10.120)

where ££L is the Landau-Lifshitz pseudotensor, given for example by MTW, Equation (20.22). (Actually, we could equally well have used T*"1 for this purpose, since one can show that both quantities yield identical equations of motion for matter and identical integral conservation-law results as, for example, in Equation (10.91). The Landau-Lifshitz version is simpler because t£L contains only first derivatives of O1"1.) Evaluating t££ for use in Equation (10.93), using Equation (10.114), and defining the

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248

transverse traceless (TT) part of 6lJ by %PiJpklekl,

p j = <s< - n%

(10.121)

we obtain the energy-loss rate dP°/dt = -(R2/32n) (Jjfl¥r,o0Tr,oda

(10.122)

Substituting Equation (10.119) into Equation (10.122), performing the angular integrations, and averaging over several oscillations of the source yields dp°/dt = -*<*;/«>.

hj = I*(X,XJ - i v 2 )

( 10 - 123 )

Using the Newtonian equations of motion, d\/dt = mx/r3, to evaluate the time derivatives in Equation (10.123) to the required accuracy yields the Peters-Mathews formula, Equation (10.80). Thus, for general relativity Kt = 12, K2 = 11, KD = 0. Scalar-tensor theories By defining f

(10.124)

(see Section 5.1) and choosing a gauge in which 0?vv = 0

(10.125)

we can write the field equations for scalar-tensor theories in the form , 0 " v = - 16TE t"v,

(10.126)

where S = -(6 + 4w)- 1 r[l - $0 - W^o - 2c»'(3 1 ["v + ^ " V y - a»'(3 + 2co)->?"] W,
(10.127)

where co = a>(0), cu' = dco/d^l^, T = g^T"*, and indices on 0"v and . Now, because of the conservation law satisfied by z'"', it is clear that 6Jk can be reexpressed as second time derivatives of quadrupole moments of x 00 , as in general relativity, and thus will not contribute any dipole terms. However, the source, S, does

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lead to dipole terms, as follows. We first evaluate the post-Newtonian forms of 0"v and q> in the near zone. From the post-Newtonian limit as calculated in Section 5.3, for instance, we obtain 600 = 2(1 + y)U + O(4),

60J = 2(1 + y)VJ + O(5),

0° = O(4),

q> = (1 - y)4>0U + O(4)

(10.128)

where y is the PPN parameter, given by y = (1 + o>)/(2 + co),

(10.129)

and where we have used Equation (5.38) to convert to geometrized units. Equations (10.127) and (10.128) then yield, to the necessary order

(10.130) To simplify the source, S, and its moments, we use the post-Newtonian forms for conserved quantities in the near zone as given in Equation (4.107). Then, for a system containing compact objects, the general procedure described above yields, for the far zone to the required accuracy, 6iJ = (1 + y)R-\d2ldt2)

£ rnXxi + O(m/r)3'2, n P + 2[1 + 2ca'(3 -

[1 + 4co'(3

2o,'(3 + 2a,) -

2

+ O(m/r)312

(10.131)

For a binary orbit we obtain (modulo constants), diJ = 2(1 + y)(fj,/R)(v'vj cp=-(l-

mx'xj/r3)

y)4>0{nlR){v2 - (n v)2 + [1 + 4a.'(3 + 2©)"2~\m/r

+ m(a x) 2 /r 3 + 2[1 + 2co'(3 + 2a>)"2]G(n v)}

(10.132)

where S is given by S = Q i M - Q 2 /m 2

(10.133)

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The most useful conserved quantity appropriate for determining the energy flux is given by 0"* = ( - g # 0o \T>" + t£D

(10.134)

where tfx is the scalar-tensor theory analogue of the Landau-Lifshitz pseudotensor, as given by Nutku (1969b) [for alternative conserved quantities, see Lee (1974)]. Evaluating t°{ a n d using Equations (10.114) and (10.121), we obtain dP°/dt = -(R2/32n)(t>o <j> [^'T.O^TT.O + (4© + 6#o V.oP.o] <«

(10.135)

Substituting Equation (10.132) into Equation (10.135) and integrating over solid angle yields Equation (10.84) with = 12 - 5/(2 + co), K2 = 11 - 45(1 + fa + ia 2 )/(8 + 4co), KD = 2(1 + a) 2 /(2 + co) (10.136) Kl

where 2

co)

(10.137)

Rosen's bimetric theory For simplicity, we choose the version of the bimetric theory whose post-Newtonian limit is identical to that of general relativity, that is, we choose c0 = c t (see Section 5.5). This is equivalent to assuming that, far from the local system, both g and i; have the asymptotic form diag( 1,1,1,1). Our final results will then be valid up to corrections of O(l c o /ci), which, according to Earth-tide measurements (limits on a 2 ), must be small. We then define (10.138) and write the field equations, (5.68), in the form

(10.139) v

The post-Newtonian forms for 6" in the near zone are (see Section 5.5) 00° = 41/ + Q(4),

6Oi = 4VJ,

0'-> = O(4)

(10.140)

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To the necessary order, Equation (10.139) then yields T 0 0 = p(l + n + v2 + 2U),

T'J' = pvlv> + p5iJ,

tOj = p{v\\ + n + v2 + AU + pip) - 2V]~]

(10.141)

The conserved quantity associated with the bimetric theory is

The near-zone conserved quantities 0 0 0 and 0 O j can be determined from Equations (10.140) and (10.142) or taken directly from Equation (4.107), since we are using the fully conservative version of the theory. For a system of compact objects, we then obtain

000 = 4R-1 \P° + n P + X a + X mJ(nva)2

O(m/r)3'2,

+ E m.»i»i - | X Qa(n v j a

3 a

312

+ O{m/r)

(10.143)

For a binary orbit, we obtain (ignoring constant terms) 600 = 4(n/R){[ih v)2 - m/r - m(n x) 2 /r 3 ] - <Sn v}, 601 = 4(|i/J?){[uJ'(fi v) - mxJ(ii x)/r 3 ] - | S u J } , 0iJ' = 4(/i/i?)[i;^-i + i®(B v)5 y ]

(10.144)

We now evaluate the energy flux tOj in the far zone using Equations (10.89), (10.138), and (10.142) and obtain dp°/dt = -

(R 2 /32TI)

<j) {efte^o - i e oeiO)

rfn

(10.145)

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Substituting Equation (10.144) into (10.145) and integrating over solid angle yields Equation (10.84), with *i = - ¥ ,

*2= - ¥ ,

*D=- ¥

(10.146)

Other theories Calculation of the PM and dipole parameters within this formalism has been carried through for the BSLL theory and for Ni's stratified theory (Chapter 5), restricting attention to those versions whose post-Newtonian limits are identical to that of general relativity (see Will, 1977, for details). The results are shown in Table 10.2. We note the surprising result that, for all the theories listed in Table 10.2, except scalar-tensor theories and general relativity, the dipole radiation carries negative energy, i.e., increases the energy of the system (KD < 0), and that the PM radiation may carry either positive or negative energy, depending on the theory and on the nature of the orbit. It could be argued (and presumably will be argued by some) that this prediction alone should be sufficient grounds to judge each such theory unviable. However, this is a theorist's constraint that has little experimental foundation in the case of gravitation, and so we will restrict attention to observational evidence for or against such an effect. Such evidence will be provided by the binary pulsar (Chapter 12). The only theory shown in Table 10.2 that automatically predicts no dipole radiation is general relativity. Scalar-tensor theories can also avoid dipole radiation for particular choices of the function co(). For example, if «(<£) = (4 - 3^)/(2<£ - 2) (Barker's constant G Theory), then 1 + 2o>'(3 + 2a>)~2 = 0 = KD. In this case, to post-Newtonian order, the theory satisfies the strong equivalence principle (Section 3.3); the locally measured gravitational constant GL is truly constant, and the theory predicts no Nordtvedt effect (4/? 7 3 = 0). The other theories in Table 10.2 violate SEP. This suggests the general conjecture that a theory of gravity predicts no dipole gravitational radiation if and only if it satisfies SEP to the appropriate order of approximation. In Section 11.3, we shall see more directly how the violation of SEP can manifest itself in dipole gravitational radiation. It is also interesting to note the strong correlation between the sign of the energy carried by gravitational radiation and the E(2) class of the theory, as summarized in Table 10.1. General relativity and scalar-tensor theories predict waves of the least general E(2) classes (N2 and N3), of definite helicity (±2; ±2, 0), and of positive energy. The other theories

Table 10.2. Multipole gravitational radiation parameters in metric theories of gravity PM parameters

Dipole parameter

Theory General relativity Scalar-tensor:

12

BWN, Bekenstein

12

Brans-Dicke

12

5 2 + co

11 -

45

No

Yes

No

Yes

No

No

-125/3

No No No

No No No

-400/3

No

No

2 2 + co a

Vector tensor

Definite helicity? Yes

45

11

Sign of energy

Yes

11

2 + ca

Satisfies SEP?

6

Bimetric : Rosen Rastall BSLL Stratified": Lee-Lightman-Ni

-21/2

-23/2 a

-21/2 18

73/8 19

-20/3 a

" Calculations have not been performed to determine these values. 6 We adopt that version of each theory whose PPN parameters are identical to those of general relativity.

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in Table 10.2 predict waves of more general classes, of indefinite helicity, and of negative or positive energy. It is perhaps not surprising that such theories predict indefinite sign for the emitted energy, since - according to quantum field theory - definite helicity, quantizibility, and positive definiteness of energy go hand in hand. Whether or not a general conjecture along these lines can be proved is an open question. One of the drawbacks of the post-Newtonian method for deriving formulae for energy loss is that it assumes that gravitational fields are weak everywhere. This assumption is no longer valid in systems containing compact objects (neutron stars or black holes), such as the binary pulsar. In the next chapter we shall describe a formalism that retains the essential post-Newtonian features of the orbital motion of such systems but that permits one to take into account the highly relativistic nature of any compact objects in the system. Nevertheless, the basic conclusions summarized in Table 10.2 will be unchanged.

11 Structure and Motion of Compact Objects in Alternative Theories of Gravity

Within general relativity, the structure and motion of relativistic, condensed objects-neutron stars and black holes-are subjects that have attracted enormous interest in the past two decades. The discovery of pulsars in 1967, and of the x-ray source Cygnus XI in 1971, have turned these "theoretical fantasies" into potentially viable denizens of the astrophysical zoo. However, relatively little attention has been paid to the study of these objects within alternative metric theories of gravity. There are two reasons for this. First, as potential testing grounds for theories of gravitation, the observations of neutron stars and black holes are generally thought to be weak, because of the large uncertainties in the nongravitational physics that is inextricably intertwined with the gravitational effects in the structure and interactions of such bodies. Examples are uncertainties in the equation of state for matter at neutronstar densities, and uncertainties in the detailed mechanisms for x-ray emission from the neighborhood of black holes. Second, compared with the simplicity of the post-Newtonian limits of alternative theories and the consequent availability of a PPN formalism, the equations for neutronstar structure and black hole structure are so complex in many theories, and so different from theory to theory, that no systematic study has been possible. Neutron stars were first suggested as theoretical possibilities within general relativity in the 1930s (Baade and Zwicky, 1934). They are highly condensed stars where gravitational forces are sufficiently strong to crush atomic electrons together with the nuclear protons to form neutrons, raise the density of matter above nuclear density (p ~ 3 x 1014 g cm"3), and cause the neutrons to be quantum-mechanically degenerate. A typical neutron-star model has m ^ lm 0 , R =* 10 km.

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However, they remained just theoretical possibilities until the discovery of pulsars in 1967 and their subsequent interpretation as rotating neutron stars. Since that time much effort has been directed toward calculating detailed neutron-star models within general relativity, with particular interest in masses, moments of inertia, and internal structure. These quantities are important in understanding both steady changes and discontinuous jumps ("glitches") in the observed periods of pulses from pulsars. The principle uncertainty in these computations is the equation of state of matter above nuclear density (for a review see Baym and Pethick, 1979). In a certain sense, black hole theory has a longer history than neutronstar theory, as it dates back to a 1798 suggestion by Laplace that such objects might exist in Newtonian gravitation theory (see Hawking and Ellis, 1973, Appendix A). Within general relativity, two key events in the history of black holes were the discovery of the Schwarzschild metric (Schwarzschild, 1916) and the analysis of gravitational collapse across the Schwarzschild horizon (Oppenheimer and Snyder, 1939). However, theoretical black hole physics really came into its own with the discovery in 1963 of the Kerr metric (Kerr, 1963), now known to be the unique solution for a stationary, vacuum, and rotating black hole (with the Schwarzschild metric being the special case corresponding to no rotation). It was the discovery in 1971 of the rapid variations of the x-ray source Cygnus XI by telescopes aboard the UHURU satellite that took black holes out of the realm of pure theory. The source of x-rays was observed to be in a binary system with the companion star HDE 226868; analysis of the nature of the companion and of its orbit around the x-ray source, and detailed study of the x-rays, led to the conclusion that the unseen body was a compact object (white dwarf, neutron star, or black hole) with a mass exceeding 9m© (Bahcall, 1978). Since the maximum masses of white dwarfs and neutron stars are believed to be approximately 1.4m© and 4m©, respectively, the simplest conclusion was that the object was a black hole. The source of the x rays was believed to be the hot, inner regions of an accretion disk around the black hole, formed by gas stripped from the atmosphere of the companion star. Since 1971, other potential black hole candidates in x-ray binary systems have been found, and studies of the central regions of some galaxies and globular clusters have indicated the possible existence of supermassive black holes [see Blandford and Thorne (1979) for a review]. However, a crucial link in the chain of argument that leads to the black hole conclusion for Cygnus XI is that the maximum mass of a neutron star is less than 4m©. There are three

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257

possible sources of uncertainty in this limit (the maximum mass of white dwarfs is much more certain). The first is the equation of state. However, it has been possible to obtain bounds on the maximum mass of between 3 and 5mQ using arguments that are independent of the details of the high-density equation of state (Hartle, 1978). The second is rotation. However, most analyses indicate that rotation cannot increase the maximum mass beyond about 20%. The third is the theory of gravitation. Although alternative theories of gravitation may have post-Newtonian limits close to that of general relativity, their predictions for the highly nonlinear, strong-field regime of neutron-star structure may differ markedly from those of general relativity. Indeed, some theories predict no maximum mass for neutron stars. Since the only present evidence for black holes crucially depends upon the maximum-mass argument, these results within alternative theories are used by many authors as reasons for caution in making the black hole interpretation, rather than as tests of competing theories. As we shall see, some alternative theories do not even predict black holes. However, the discovery of the binary pulsar (Chapter 12) has made the study of neutron-star structure and motion an important tool for testing gravitation theory. The precise orbital data obtained for that system permits for the first time the direct measurement of the mass of a neutron star and the study of relativistic orbital effects (such as periastron shifts) in systems containing condensed objects. In alternative theories of gravity, the nonlinear gravitational effects involved in the neutron star can make significant differences in many relativistic effects, even though in the postNewtonian limit, these effects would have been the same as in general relativity. Crucial tests of competing theories may then be possible. Discussion of these tests will be presented in Chapter 12; this chapter sets the framework for that discussion. In Section 11.1, we analyze the equations of neutron-star structure and, in Section 11.2, the equations of black hole structure in alternative theories of gravitation. In Section 11.3 we present a framework for discussing the motion of compact objects, such as neutron stars, in competing theories. As we noted above, very little systematic study of these issues has ever been carried out, so we shall merely present a few relevant examples. 11.1

Structure of Neutron Stars

In Newtonian gravitation theory, the equations of stellar structure for a static, spherically symmetric star composed of matter at zero temperature (T 0 is an adequate approximation for neutron-star matter)

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258

are given by dp/dr = p dU/dr [Hydrostatic equilibrium], P = P(P) [Equation of state], 2 2 (d/dr)(r dU/dr)= -4nr p [Field equation]

(11.1)

where p(r) is the pressure, p{r) the density, and U(r) the Newtonian gravitational potential. In any metric theory of gravity, it is simple to write down the equations corresponding to the first two of these three equations, because they follow from the Einstein Equivalence Principle (Chapter 2), which states that in local freely falling frames the nongravitational laws of physics are those of special relativity, Tfvv = 0, and p = p(p). Thus, we have in any basis, Tfvv = 0,

p = p(p)

(11.2)

For a perfect fluid, T"v = {p + p)«"uv + pg""

(11.3)

where we have lumped the internal energy pTl into p [compare Equation (3.71)]. It is useful to rewrite these equations in a form that parallels the first two parts of Equation (11.1). For a static, spherically symmetric spacetime, there exists a coordinate system in which the metric has the form ds2 = -e20ir)dt2 - TV{r)drdt + e2Mr)dr2 + e2mr\dQ2 + sin2 0 d(f>2)

(11.4)

For theories of gravity with a preferred frame, this coordinate system must be at rest in that frame. There still exists the freedom to change the t coordinate by the transformation t = t'-f(r)

(11.5)

where f(r) can be chosen to eliminate the off-diagonal term in the metric, namely

f(r) = J ' m(r)e ~ 2o
(11.6)

There is the further freedom to change the coordinate r by r = 9(r')

(11.7)

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If the radial coordinate is chosen so that fi(r) = 0, the coordinates are called "curvature coordinates;" in such coordinates, 2nr measures the proper circumference of circles of constant r. In general relativity, they are known as Schwarzschild coordinates. If the radial coordinate is chosen so that n(r) = A(r), the coordinates are called "isotropic coordinates." However, in two-tensor theories of gravity, such as those with a background flat metric q, these transformations are usually best carried out after the solution to the field equations has been obtained. The reason is that the above transformations will in general make the second tensor a complicated nondiagonal function of r, which may result in worse complications in the field equations than those introduced by starting with the general nondiagonal physical metric, Equation (11.4). For example, if the second tensor field is t\, the field equations may take their simplest form in coordinates in which ij = diag(-l,l,r 2 ,r 2 sin 2 0)

(11.8)

In such a coordinate system there is no freedom to alter <J>, A, T, or fx a priori. Now, for hydrostatic equilibrium, the equations of motion Tfvv = 0 may be written in the form

where j runs over r, 6, (p. For spherical symmetry only the j = r component is nontrivial, and, using the fact that u = (e~0(r), 0,0,0), we obtain dp/dr = (p + p)d
(11.10)

Notice that in the Newtonian limit, p « p, $ =s U and we recover the first two parts of Equation (11.1). Equation (11.10) is valid independently of the theory of gravity. The field equations for <J>, A, *F, and fi will depend upon the theory. In constructing a stellar model, boundary conditions must be imposed. These are dp/dr\r=0 = 0,

p{R) = 0, e2A(r)

R = [stellar surface], e2M

,.

Ci

(1LU)

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The first of these conditions is a continuity condition for the matter, the second defines the stellar surface and its radius R. The remaining four are asymptotic boundary conditions on the metric functions [see Equation (5.6)]. They guarantee that in asymptotically Lorentz coordinates, and in geometrized units (Gtoda}, = 1), 0oo -» - 1 + 2m/r,

gtJ - r\i}

(11.12)

and thus that the Kepler-measured mass of the star will be m. Unfortunately, this exhausts the common features of the equations of relativistic stellar structure, so we must now turn to specific theories. General relativity In curvature coordinates [*F(r) = /x(r) = 0], the field Equation (5.14) takes the form d/dr[r(l - e~ 2A )] = 8nr2p

(11-13)

with the solution e 2A = ( l - 2 r n ( r ) / r r 1

(11.14)

where , . . rr 2 m(r) = 4JI t par, Jo

or

dm = 4nr2 p dr

...... (11.15)

and dO dr

m + 4nr3p r(r 2m)

(11.16)

This equation together with Equations (11.10), (11.14), and (11.15), and the boundary conditions, Equation (11.11) (with c 0 = c1 = 1), are sufficient to calculate neutron-star models, given an equation ofstate. These equations are called the TOV (Tolman, Oppenheimer, and Volkoff) equations for hydrostatic equilibrium. They form the foundation for the study of relativistic stellar structure within general relativity. For reviews of neutron-star structure, see Baym and Pethick (1979), Arnett and Bowers (1977), and Hartle (1978). Scalar-tensor theories Using curvature coordinates [*P(r) = ft(r) = 0] and defining e 2A = [1 - 2 m ( r ) / r ] - 1

(11.17)

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261

we can put the field equations for scalar-tensor theories, Equations (5.31) and (5.33), into the form

2m

drr(r-

2m)(l

1.18) [Note that the equations quoted in Rees, Ruffini, and Wheeler (1975, p. 13) are in error.] The present value of G is related to the asymptotic value of <j>, by [see Equation (5.38)] l

(11.19)

For the special cases of Brans-Dicke theory (co = constant) and the Variable-Mass Theory [co() satisfies Equation (5.40)], it has been shown that for values of co consistent with solar system experiments (i.e., a> ;> 500), all features of neutron-star structure differ from those predicted by general relativity only by relative corrections of O(l/co) (see Salmona, 1967; Hillebrandt and Heintzmann, 1974; Bekenstein and Meisels, 1978). Rosen's bimetric theory In coordinates in which the flat background metric has the form t, = diag( - Co \ el \ cl V 2 , ci V 2 sin2 9)

(11.20)

the field equations take the form V 2 $ + |D~1e""2"2A|VxP - 2»PV|2 1/2 +A+2

= 47tG(c0c1) e 2

l

2

2A y

V A + \D- e- *- \\ ¥ 1/2

\

1/2

''D (p +

- 24*VA|2 *+A

2 1 / 2

= -4nG(c0c1)ii2e">+A+2>lD1i2(p

- p + 2(p - p),

Y - 2*PVA) De 2 A ~ 2 " = 0

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where D = 1 + *p2 e - 2 < I - 2 A , V = tr(d/dr), V2 = r ~ 2{d/dr)r2(d/dr)

(11.22)

Here, we see an example of the loss of freedom to vary the metric functions *P and pi a priori. Thus, for example, the substitutions *F = 0, ju s 0 (Schwarzschild form of the metric) do not lead to a solution of the equations for general matter distributions. However, the metric function *P alone is freely specifiable, for the following reason. If, instead of Equation (11.20) for IJ, we had chosen the equally valid flat metric whose line element is

dsLt = - c o ' [ A + f(r)drf + c^[dr2 + r2(dd2 + sin 2 0# 2 )] where /(r) is an arbitrary function of r, then had changed coordinates to put this metric into the form of Equation (11.20), the result would be to change the function ¥ in g^ by an arbitrary amount. Thus, for example, we are free to choose *P = 0. This is consistent with the fact that *P = 0 is a solution of the fourth field equation, (11.21). The free choice of *F is part of the absolute, prior-geometric character of i\, and represents the freedom to "tip" the null cones of i; relative to those of g. Different choices of *F lead to physically different spacetimes (this point has been overlooked by most authors). The simple choice T s O leads to the field equations V20> = 47tG(coc1)1/2e*+3A(p + 3p), V2A = -47tG(c o c 1 ) 1/2 e* +3A (p - p), H=A

(11.23)

Henceforth, we shall adopt the choice T s O . The boundary conditions on and A are given by $

0,

A

0

(11.24)

Notice that the matching of the tensors i\ and g to the external world influences the structure of the star (violation of SEP) via the effective gravitational constant G^QCJ) 1 ' 2 . We now recast the field equations into the form dfb/dr = Gom*(r)/r2, dA/dr=-GomA(r)/r2

(11.25)

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263

where G o = (c0Ci)1/2G = 1 [geometrized units], = An £ e*+3A(p - p)r2 dr

(11.26)

Outside the star, r > R,
A = MJr

(11.27)

where M^ = m^R), MA = mA(R). In quasi-Cartesian coordinates, the exterior metric then has the form ds2 = - exp( - 2MJr) dt2 + exp(2MJr)(dx2 + dy2 + dz2)

(11.28)

A variety of numerical integrations of the field equations, (11.25), and the hydrostatic equilibrium equations, (11.10), have been carried out using various equations of state (see Rosen and Rosen, 1975; Caporaso and Brecher, 1977; and Will and Eardley, 1977). Generally, neutron stars with Kepler-measured masses M& much larger than those permitted by general relativity are possible, with maximum masses ranging from ~8m o for soft equations of state, to ~80m o for equations of state of the form p = p - p0 for p > p0 ~ 1014 g cm" 3 . NVs stratified theory In coordinates in which i/ = diag(-l,l,r 2 ,r 2 sin 2 0),

(11.29)

we have [see Section 5.6(g)-(i)] e 2 * = /i(*X

V=-Kr,

e2A = e2" = f2(),

X9 = K 0 = O

(11.30)

The field equation for Kr is V2Kr - r-2Kr = -47t e (/ 2 // 1 D) 1 / 2 X r p

(11.31)

where D = 1 + K2(f1f2)~1. One immediate solution of this equation is Kr = 0 (no tipping of null cones). Then, thefieldequation for (f> is given by V20 = 27t(/ 1 / 3 ) 1 ' 2 [p(/' 1 // 1 ) - 3p(/'2//2)]

(11.32)

where f\ = dfjd, f2 = df2/d<j>. The boundary condition on
Tzz^t 0. Outside the star,

(11.33)

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where

M, = 2n J* (fJiy'WJA)

- 3p(/'2//2)]r2 dr

(11.34)

Asymptotically, the functions fx and / 2 are assumed to have the forms M4)

= c 0 - 2c<\> + O(4>2),

/ 2 (0) = c, + O(0)

(11.35)

In coordinates in which g^ is asymptotically Minkowskian, g00 then has the form goo=-l+2GoMJr

(11.36)

where Go = C2C\I2CQ 3/2 = 1

[geometrized units],

Mikkelson (1977) has numerically integrated these equations using reasonable equations of state, after first assuming a specific form for the functions fi(<j>) and f2(4>), designed to yield agreement with general relativity in the post-Newtonian limit. These forms were

with a being an adjustable parameter (note co = ct = c = 1). For a = 1, the maximum Kepler-measured mass was ~1.4m o ; for a = 64, it was ~840m o ; and in the limit a-* oo, the maximum mass was unbounded. The stiffer the equation of state used, the larger the maximum masses. Thus, neutron-star models in alternative theories of gravity can be very different from their counterparts in general relativity, the known exceptions to this rule being scalar-tensor theories. In particular, the maximum mass of a neutron star may be orders of magnitude larger than that in general relativity. 11.2

The Structure and Existence of Black Holes

General relativity predicts the existence of black holes. Black holes are the end products of catastrophic gravitational collapse in which the collapsing matter crosses an event horizon, a surface whose radius depends upon the mass of matter that has fallen across it, and which is a one-way membrane for timelike or null world lines. Such world lines can

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cross the horizon moving inward but not outward. The interior of the black hole is causally disconnected from the exterior spacetime. There is now considerable evidence to support the claim that any gravitational collapse situation, whether spherically symmetric or not, with zero net charge and zero net angular momentum, results in a black hole, whose metric (at late times after the black hole has become stationary) is the Schwarzschild metric, given in Schwarzschild coordinates by ds2 = - ( 1 - 2M/r)dt2 + (1 - 2M/r)-ldr2 + r2(d62 + sin 2 0# 2 ) (11.39) (If the collapsing body has net rotation, the black hole is described by the Kerr metric.) Much is now known about the theoretical properties of black holes within general relativity, and there are strong candidates for observed black holes in Cygnus XI and elsewhere. For reviews of this subject see Giacconi and Ruffini (1978) and Hawking and Israel (1979). However, the existence of black holes is not an automatic byproduct of curved spacetime. To be sure, curved spacetime is essential to the existence of horizons as one-way membranes for the physical interactions, but whether or not a horizon occurs depends crucially on the field equations that determine the curvature of spacetime. In the following examples, we shall illustrate this point. Throughout this section, we restrict ourselves to nonrotating, spherically symmetric systems. Scalar-tensor theories As one might have expected, scalar-tensor theories, being in some sense the least violent modification of general relativity, predict black holes. However, what is unexpected is that they predict black holes whose geometry is identical to the Schwarzschild geometry. The reason is that the scalar field <>/ is a constant throughout the exterior of the horizon, given by its asymptotic cosmological value (f>0. Thus, the vacuum field equation, (5.31), for the metric is Einstein's vacuum field equation, and the solution is the Schwarzschild solution. The scalar field has no effect other than to determine the value of the gravitational constant. (This result also holds for rotating and charged black holes.) In Brans-Dicke theory, for instance, the most direct way to verify this is to use the vacuum field equation for cp = cj) (j)0, \3g

J
(11.40)

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Now the surface integrals over the spacelike hypersurfaces cancel because the situation is stationary; that over the hypersurface at infinity vanishes because along symmetry directions must vanish. Thus, = 0

(11.41)

But cptX is spacelike, since cp is stationary, so (pAqy* > 0 everywhere, and Equation (11.41) thus implies (px = 0. Further details and other arguments can be found in Thorne and Dykla (1971), Hawking (1972), Bekenstein (1972), and Bekenstein and Meisels (1978). Rosen's bimetric theory For the case *P = 0, the static spherically symmetric vacuum field equations are [see Equation (11.23)] fi = A,

V20> = V2A = 0

(11.42)

with solutions
n = A = MJr

(11.43)

and ds2 = - exp( - 2MJr) dt2 + exp(2MJr)(dx2 + dy2 + dz2)

(11.44)

There is no horizon in this spacetime, only a naked singularity at r = 0. Thus, at least within the subset of vacuum spacetimes specified by *F = 0, there are no black holes in Rosen's theory. 11.3

The Motion of Compact Objects: A Modified EIH Formalism In Chapter 6, we derived the n-body equations of motion for massive, self-gravitating bodies within the parametrized post-Newtonian (PPN) framework [see Equations (6.31)-(6.34)]. A key assumption that went into that analysis was that the weak-field, slow-motion limit of gravitational theory applied everywhere, in the interiors of the bodies as well as between them. This assumption restricted the applicability of the equations of motion to systems such as the solar system. However, when dealing with a system such as the binary pulsar in which there is a neutron star with a highly relativistic interior, one can no longer

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apply the assumptions of the post-Newtonian limit everywhere, except possibly in the interbody region between the relativistic bodies. Instead, one must employ a method for deriving equations of motion for compact objects that, within a chosen theory of gravity, involves (a) solving the full, relativistic equations for the regions inside and near each body, (b) solving the post-Newtonian equations in the interbody region, and (c) matching these solutions in an appropriate way in a "matching region" surrounding each body. This matching presumably leads to constraints on the motions of the bodies (as characterized by suitably denned centers of mass); these constraints would be the sought after equations of motion. Such a procedure would constitute a generalization of the Einstein-Infeld-Hoffmann (EIH) approach (see Einstein, Infeld, and Hoffmann, 1938). Let us first ask what would be expected from such an approach within general relativity. In the full post-Newtonian limit, we found that the motion of post-Newtonian bodies is independent of their internal structure, i.e., there is no Nordtvedt effect. Each body moves on a geodesic of the post-Newtonian interbody metric generated by the other bodies, with proper allowance for post-Newtonian terms contributed by its own interbody field. This is the EIH result. It turns out however, that this conclusion is valid even when the bodies are highly compact (neutron stars or black holes). The only restriction is that they be quasistatic, nearly spherical, and sufficiently small compared to their separations that tidal interactions may be neglected. The effects of rotation (Lense-Thirring effects) are also neglected. This would be a bad approximation for a neutron star about to spiral into a black hole, for example, but is a good approximation for the binary pulsar (rpulsar/rorbit ^ 10"5). Although this conclusion has not been proven rigorously, a strong argument for its plausibility can be presented by considering in more detail the matching procedure discussed above. We first note that the solution for the relativistic structure and gravitational field of each body is independent of the interbody gravitational field, since we can always choose a coordinate system for each body that is freely falling and approximately Minkowskian in the matching region and in which the body is at rest. Thus, there is no way for the external fields to influence the body or its field, provided we can neglect tidal effects due to inhomogeneities of the interbody field across the interior of the matching region. Only the velocity and acceleration of the body are affected. Now, provided the body is static and spherically symmetric to sufficient accuracy, its external gravitational field is characterized only by its Kepler-measured mass m, and is independent of its internal structure. Thus, the matching procedure

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described above must yield the same result, whether the body is a black hole of mass m or a post-Newtonian body of mass m. In the latter case, the result is the EIH equations of motion (see Section 6.2), so it must be valid in all cases. A slightly different way to see this is to note that because the local field of the body in the freely falling frame is spherically symmetric, depends only on the constant mass m, and is unaffected by the external geometry, the acceleration of the body in the freely falling frame must vanish, so its trajectory must be a geodesic of some metric. The metric to be used is a post-Newtonian interbody metric that includes post-Newtonian terms contributed by the body itself, but that excludes self-fields. This conclusion has been verified for nonrotating black holes by D'Eath (1975), and for the Newtonian acceleration of post-postNewtonian bodies by Rudolph and Borner (1978). D'Eath (1975) gives a detailed presentation of the matching procedure described above. A key element of this derivation is the validity of the Strong Equivalence Principle within general relativity (see Chapter 3 for discussion), which guarantees that the structure of each body is independent of the surrounding gravitational environment. By contrast, most alternative theories of gravity possess additional gravitational fields, whose values in the matching region can influence the structure of each body, and thereby affect its motion. Consider as a simple example a theory with an additional scalar field (scalar-tensor theory). In the local freely falling coordinates, although the interbody metric is Minkowskian up to tidal terms, the scalar field has a value 0{i) will influence the resulting mass according to m = m((j)0). Thus, the asymptotic metric of the body in the matching region may depend upon its internal structure via the dependence of m on <j)0 (essentially, the matching conditions will depend upon m, dm/d(j),...). Furthermore, the acceleration of the body in the freely falling frame need not be zero, as we saw in Sections 2.5 and 3.3. If the energy of a body varies as a result of a variation in an external parameter, we found, using cyclic gedanken experiments that assumed only conservation of energy, that [see Equation (3.80)] a - a8eodesic ~ \EB(X, V) ~ (dm/dWt

(11.45)

Thus, the bodies need not follow geodesies of any metric, rather their motion may depend strongly on their internal structure. In practice, the matching procedure described above is very cumbersome (D'Eath, 1975). A simpler method, within general relativity, for

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obtaining the EIH equations of motion, is to treat each body as a "point" mass of inertial mass ma and to solve Einstein's equations using a pointmass Lagrangian or stress-energy tensor, with proper care taken to neglect "infinite" selffields.In the action for general relativity we thus write

JGR = (lenG)-1 JR(-g)l/2d*x

- ^ m f l jdxa,

(11.46)

where xa is proper time along the trajectory of the ath body. By solving the field equations to post-Newtonian order, it is then possible to derive straightforwardly from the matter action an n-body EIH Lagrangian in the form J EIH =

u

...xn,\u...

yn)dt

(11.47)

written purely in terms of the variables (xfl, vo) of the bodies. The result is Equation (6.80) with the PPN parameters corresponding to general relativity. The n-body EIH equations of motion are then given by

-ft-a dt dv'a

«-l,...,n

(11.48)

dx'a

In alternative theories of gravity, the only difference is the possible dependence of the mass on the boundary values of the auxiliaryfields.In the quasi-Newtonian limit (Sections 2.5 and 3.3) this was sufficient to yield the complete quasi-Newtonian acceleration of composite bodies including modifications (Nordtvedt effect) due to their internal structure. Thus, following the suggestion of Eardley (1975)1 we merely replace the constant inertial mass ma in the matter action with the variable inertial mass ma{\j/A), where \j/K represents the values of the external auxiliary fields, evaluated at the center of the body (we neglect their variation across the interior of the matching region), with infinite self-field contributions excluded. The functional dependence of ma upon the variable i//A will depend on the nature and structure of the body. Thus, we write 1 = JG ~ £ Jm-{^A[x.(Tj]} dxa

(11.49)

a

In varying the action with respect to thefieldsg^v and i^A the variation of ma must then be taken into account. In the post-Newtonian limit, where the fields i//A are expanded about asymptotic values i//1^ according to 1 Parts of this section are developed from unpublished notes by Douglas Eardley.

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270

= it is generally sufficient to expand ma(i/jA) in the form

+ \ Z (5 W # k O ) # i ? V l M * B +

(11-50)

A,B

Thus, the final form of the metric and of the n-body Lagrangian will depend on ma and on the parameters dmjdxj/^, and so on. We shall use the term "sensitivity" to describe these parameters, since they measure the sensitivity of the inertial mass to changes in the fields \j/ A. Thus, we shall denote s) 3) s<, >' = -d (lnm a )/#£ #B ["second sensitivity"]

(11.51)

and so on. The final result is a "modified EIH formalism." By analogy with the PPN formalism, a general EIH formalism can be constructed using arbitrary parameters whose values depend on the theory under study, and, in this case, on the nature of the bodies in the system. However, to keep the resulting formalism simple, we shall make some restrictions. First, we restrict attention to fully conservative theories of gravity. Technically, this means any theory whose EIH Lagrangian is post-Galilean invariant. Now, every Lagrangian-based metric theory of gravity will possess an EIH Lagrangian (thus all the theories discussed in Chapter 5 fall into this class), however not every theory is fully conservative. Only general relativity and scalar-tensor theories are automatically fully conservative. Other theories can be fully conservative, in their postNewtonian limits, at least, only for special choices of adjustable constants and cosmological matching parameters that make the PPN parameters <*! and oe2 equal to zero (Rosen's bimetric theory with co = cu for example). It is not known whether these choices are sufficient to guarantee that the EIH Lagrangian also be post-Galilean invariant. Nonetheless, the experimental upper limits on the PPN parameters OL1 and <x2 (Chapter 8) obtained from searches for post-Newtonian preferred-frame effects make it unlikely that the analogous effects in the EIH formalism will be of much interest. Therefore we shall adopt a fully conservative EIH formalism. We shall also restrict attention to theories of gravity that have no Whitehead term in the post-Newtonian limit (i.e., £ = 0). The experimental constraints on £ (Chapter 8), from searches for galaxy-induced effects in the solar system, likewise make the analogous effects in the EIH formalism of little interest.

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Each body is characterized by an inertial mass ma, defined to be the quantity that appears in the conservation laws for energy and momentum that emerge from the EIH Lagrangian. We then write, for the metric, valid in the interbody region and far from the system, 000 = - 1 + 2 £ «fl*ma|x - xa| ~ * + O(4), a

9OJ= 0(3),

9u = ( l + 2 1 7>a\* ~ xa| -*) Su

(11.52)

where a* and y* are functions of the parameters of the theory and of the structure of the ath body. For test-body geodesies in this metric, the quantities x*ma and ^a*wJ a are the Kepler-measured active gravitational masses of the individual bodies and of the system as a whole. In general relativity, a* = y* = 1. To obtain the EIH Lagrangian, we first generalize the post-Newtonian semiconservative n-body Lagrangian [Equation (6.80)] in a natural way, to obtain

= - I ma{l - \^vl

- i^<2><] + \a,b

r

ab

a*b

X

h*J

flj(

flj]

(i i-53)

where nab = xjrab. The quantities s/«\ st, 9^,, ®ah, 9abc, <£ttb, and Sah are functions of the parameters of the theory and of the structure of each body, and satisfy ^ab

*(<.»)>

Wab (abY

«ab ~ *(ab),

In general relativity, all these parameters are unity. In the true postNewtonian limit of semiconservative theories (with t, 0), for structureless masses (no self gravity), the parameters have the values [compare Equation (6.80)]

^ab = 7(4? + 3 + at - a2),

Sab= 1 + a2

(11.55)

In the fully conservative case, including contributions of the self-gravitational binding energies of the bodies, one can show to post-Newtonian

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272

order, that 1,

9A = 1 + (4/? - y - 3)(QJma + Qb/mb),

3),

(11.56)

where Qa is the self-gravitational energy of the ath body. We now impose post-Galilean invariance on the Lagrangian in Equation (11.53). We make a low-velocity Lorentz-transformation from (t,x) to (T, £) coordinates, given by x = { + (1 + |W 2 )WT + | ( £ w)w + O(4)

t = T(1 + W

4

+ fw ) + (1 + W)S

x &

w + O(5) x T

(11.57)

We required that L be invariant, modulo a divergence, i.e., L(l T) = L(x, f) dt/dt + # / d t

(11.58)

for some function ^. From the transformation Equation (11.57), we have v = v + w jw2v v(v w) jw(v w), r*1 = C, 1 [1 + i(w fi^)2 + w fi> n^,]

(11.59)

where v = d£/dz, and n^b = %abl£,ab. Substituting these results into Equations (11.53) and (11.58), and dropping constants and total time derivatives, yields L«,T)= - I ma{\ - \ a

2

+ va2(va w) + (v. w) 2 ]}

+i($ab + ^ab - T#ab)(w + 2va W) }

(11.60)

Thus the action is invariant if and only if ab

=

Furthermore, the scale of L and the constant term £ a ma are irrelevant, thus, we can always scale the values of ma so that .a?*,1' = 1, and choose

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the constant to be £ a ma in terms of the rescaled mass. This merely guarantees that the inertial mass obtained from the Hamiltonian constructed from L agrees with that obtained from the equations of motion. Thus, the final form of the modified EIH Lagrangian is

ad

U)]

yb - &Jya HJfo nj I

(11.62)

Since our ultimate goal is to apply this formalism to binary systems containing compact objects, such as the binary pulsar, let us now restrict attention to two-body systems. Denning & =
r3 [

r

r - 2vt v 2 ) -

^

fl)2

(v2 - v j x [(SF

a2 = { 1 ^ 2 , x - > - x }

(11.63)

where a s dsjdt, x = x 2 x l5 r = |x|, n = x/r. It is possible to show straightforwardly from these equations that if we define ma=ma + \mav2a - \^mjnhjrah,

a# b

X = (mjXj + m2x2)/("'i + "»2)

(11.64)

then the "center of mass" X of the system is unaccelerated, i.e., 2

=0

(11.65)

This agrees with the fully conservative nature of the EIH Lagrangian and justifies our identification of ma as the inertial rest mass of each body. If we now choose the center of mass to be at rest at the origin, X = X 2 0, then, to sufficient post-Newtonian accuracy we may write

x 2 = [nti/m + O(2)]x

(11.66)

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As in Section 7.3, we define y=v2-\u

a = a 2 -a 1;

m = ml + m 2 ,

n = m1m2/fn,

dm = m2 mt

(11.67)

then the equations of motion for the relative orbit take the form

dm r

m

f ( )

^ J

m

(11.68)

r

In the Newtonian limit of the orbital motion, we have a=-m^x/r3

(11.69)

with Keplerian orbit solutions x = p(l + ecos^>)~1(exCOS0 + e,,sin), r2 d4>ldt = hs CSmp)112,

p = a{\ - e2),

v = (&m/p)il2[-exsin + ey(cos

(Pb/2n) = a3/&m (11.70) where a, p, and e are the semi-major axis, semi-latus rectum, and eccentricity, respectively; h is the angular momentum per unit mass; and P b is the orbital period. In this solution we have chosen the x direction to be in the direction of the periastron. The post-Newtonian terms in Equation (11.68) can then be viewed as perturbations of the Keplerian orbit. Using the method of perturbations of osculating orbital elements outlined in Section 7.3, we find that the periastron advance is given by 1

(11.71)

where 0> q)(% .). ^<§2 _ ^(/nj:^ 2 1 1 + w 2 ® 122 )/m

(11.72)

This is the only secular perturbation produced by the post-Newtonian terms in Equation (11.68). In the PPN limit, this result agrees with Equation (7.54), for fully conservative theories (with £ = 0). In obtaining the modified EIH equations of motion, we assumed that the field equations obtained from Equation (11.49) were solved for the interbody gravitational fields through post-Newtonian order. However,

Structure and Motion of Compact Objects

275

those equations can also be solved for the gravitational-radiation fields in the far zone, and for the rate of energy loss via gravitational radiation. The method parallels that presented in Chapter 10, except that now, the self-gravitational corrections in the sources of the fields ip (ifr may include the metric itself) are automatically taken into account to all orders via the sensitivities s [see Equation (11.51)]. The only terms that we need to retain in order to determine the lowest order quadrupole and dipole contributions to the energy loss rate are [compare Equations (10.104) and (10.105)] Aeiectric ~ 4(m/R) < - [1 + (s) + ] [monopole and quadrupole]

}, [dipole] ] [dipole] + [1 + (s) + ] 1 [quadrupole]

(11.73)

The only other differences from the post-Newtonian method described in Section 10.3 are the use of the conservation laws and Newtonian equations of motion obtained from the modified EIH Lagrangian. We shall ultimately be interested primarily in the energy loss due to dipole gravitational radiation, so it is useful to rewrite the dipole portion of Equation (10.84) using terms more suited to the modified EIH formalism, namely

where S is related to the difference in sensitivities between the two bodies. As an illustration of these methods, we shall again focus on specific theories: general relativity, Brans-Dicke theory, and Rosen's bimetric theory. General relativity As we have already seen, the EIH equations of motion for compact objects within general relativity are identical to those of the full postNewtonian limit. In other words, &ab = 0&ab = 3)abc = 1, independently of the nature of the bodies. Furthermore, the gravitational radiation produced by the orbital motion is dominated by quadrupole radiation (no

Theory and Experiment in Gravitational Physics

276

dipole radiation), and the energy loss rate is the same as in the pure postNewtonian case, obeying the Peters-Mathews formula, Equation (10.80) with Kj = 12, K2 = 11, KD = 0. (The same caveats regarding the rigor with which this result has been established apply here as in Section 10.3.) Brans-Dicke theory The modified EIH formalism was first developed by Eardley (1975) for application to Brans-Dicke theory. It makes use of the fact that only the scalar field (f> produces an external influence on the structure of each compact body via its boundary values in the matching region. In fact the boundary value of is related to the local value of the gravitational constant as felt by the compact body by (j)

= G'1(4 + 2o})/(3 + 2co)

(11.75)

Hence, we shall regard the inertial mass ma of each body as being a function of G, or more specifically, of In G. Then, if post-Newtonian, interbody gravitational fields lead to variations in 4> away from its asymptotic (cosmological) value = 4>o + tnen w e m a v write

(1L76)

Defining the sensitivities sa and s'a of the inertial mass of body a to changes in the local value of G, following Equation (11.51), by

s'a=-[d2(\nma)/d(\nG)2]0

(11.77)

and dropping the 0 subscript, we obtain

ma(4) = mil + sa{cpl4>Q) - Wa - st + sa)(cp/ct>0)2 + O(W0o) 3 ]

(11.78)

The action for Brans-Dicke theory is then written

l z

a

(11.79)

where the integrals in the matter action are to be taken along the trajectories of each particle, and where infinite "self-fields" are to be ignored.

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The resulting field equations are (compare Section 5.3)

Dg = Y

^

[T - 24>dT/8(j>-]

(11.80)

where T*v = (-gy112

I m s ( ^ « ' ( u 0 ) - ^ 3 ( x - xfl),

dT/d = - ( - 9 ) - 1 / 2 2 (5ma(
(11.81)

The equations of motion take the form T?vv - (dT/dtfrW = 0

(11.82)

Performing a post-Newtonian expansion following the method outlined in Chapter 5, we obtain to lowest order
0 = (2 + a ) - 1 X m a (l -

lsa)/ra,

a

g00 = - 1 + 2 X (ma/ra)[l - s./(2 + <»)] + O(4), a

9iJ

= dJl + 2y X (ma/ra)[l + s./(l + ai)]J

(11.83)

where rfl = |x xo|, y = (1 + a;)/(2 + co), and we have chosen units in which G s l . Notice that the active gravitational mass as measured by test-body Keplerian orbits far from each body is given by W , = oi*ma = ma[l - sj(2 + <»)]

(11.84)

In the full post-Newtonian limit, where sa =* QJma, this agrees with Equation (6.49). If we define a "scalar mass" (ms)a by K ) a = i ( 2 + o J )- 1 m a (l-2s a )

(11.85)

so that ( )

(11.86)

the metric can be written S'oo = ~ 1 + 2 X a

gtj = dv {l + 2 I [(mA)a - 2(ms)J/r.J

(11.87)

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From the active mass and the scalar mass, it is useful to define a "tensor mass" (mT)o, (mT)a = (mA)a - (ms)a = ( | ^ )

*.

(11.88)

It can then be shown (Lee, 1974) that the tensor mass (mT)a is associated with a conservation law of the form

U-gMT**+ ni* = 0

(11.89)

where V is a symmetric stress-energy "pseudotensor" given by Lee (1974). This result is consistent with the identification of ma as the inertial mass of the ath body. The full post-Newtonian solution for g^ and cp may now be obtained, and the results substituted into the matter action, which, for the ath body takes the form I.=

- jma(cj>)dt(-g00

- 2g0Jvi - g,/^1'2

(11.90)

To obtain an n-body action in the form of Equation (11.47), we first make the gravitational terms in / manifestly symmetric under interchange of all pairs of particles, then take one of each such term generated in la, and sum over a. The resulting n-body Lagrangian then has the form of Equation (11.62) with <§ab = 1 - (2 + coy ^s. + sb-

2Sasb),

1

®ab = 4(2y + 1) + i ( 2 + oi)- (so + sb- 2sasb), 9abc = 1 - (2 + Co)"x(2sfl + sb + se) + (2 + co)- 2 [(l - 4sa)sksc + (5 + 2(o)sB{sb + sc) - (s'a - s2a)(l - 2sb)(l - 2s c )]

(11.91)

The quasi-Newtonian equations of motion obtained from the EIH Lagrangian are

K = ~ I K x X ) [ l - (2 + coy l(sa + sb- 2Sasby]

(11.92)

b*a

In the full post-Newtonian limit, the product term s^ may be neglected, and the acceleration may be written a. a -[(mp)./mj £ {m^Jrl

(11.91)

b*a

where (mA)b is given by Equation (11.84) and where W . = m.[l - sj(2 + co)-]

(11.92)

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279

in agreement with our results of Section 6.2, Equation (6.49). However, if the bodies are sufficiently compact that sa ~ sb ~ 1, then because of the product term s^,,, it is impossible to describe the quasi-Newtonian equations simply in terms of active and passive masses of individual bodies. Roughly speaking, the sensitivity s ~ [self-gravitational binding energy]/[mass], so s e ~ 10~10, s o ~ 1(T6, swhitedwarf ~ 10~3. For neutron stars, whose equation of state is of the form p = p(p), a model is uniquely determined (for a given value of <w) by the local value of G and by the central density pc or the total baryon number JV. Now, the sensitivity s is to be computed holding N fixed; it can then be shown that

fdlnm\

(dlnm\

dlnN\ /SlnnA fdlnN\

For fixed equation of state and fixed central density, a simple scaling argument reveals that m and N scale as G~3/2, so

Note that (d In m/d In N)G is the injection energy per baryon. Then it can be shown that S'NS = ( ! - sNS)(dsNS/d In m) G

(11.95)

Equations (11.94) and (11.95) actually hold in any theory of gravity in which the local structure depends upon a single external parameter whose role is that of a gravitational "constant." For a variety of neutron star models, Eardley (1975) has shown that s ranges from s ^ 0.01 for m = 0.13mo to s ^ 0.39 for m = 1.41mo. For black holes, we have seen (in Section 10.2) that the scalar field is constant in the exterior of the hole, thus from Equation (11.83) sBH = | ; equivalently mBH scales as G~1/2. Note that the quasi-Newtonian equation of motion for a test black hole and a companion (mBH « mc) is thus given, from Equation (11.92), by = - [(3 + 2cu)/(4 + 2o))]mcx/r3

(11.96)

therefore the Kepler-measured mass experienced by a test black hole is the tensor mass mT of the companion (Hawking, 1972). The energy loss rate due to dipole gravitational radiation can be computed simply in this formalism (the PM radiation can also be calculated, but the result is not particularly illuminating). The wave-zone form of cp

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280

is given, from Equations (11.78), (11.80), and (11.81) by

^

^ | E ma(l - 2s a )[l + ii va + O(m/r)]|

(11.97)

For a binary system, modulo constants, we obtain 9

= - [ 4 / ( 3 + 2co)]K~ V<»(n v)

(11.98)

where ©ss 2 - S l

(11.99)

Then, following the method of Section 10.3, we find that the rate of energy loss is given by Equation (11.74) with KD = 2/(2 + co). Rosen's bimetric theory In Rosen's theory, the flat background metric ij can influence the structure of a compact object, in spite of the fact that it is a nondynamical field. In a coordinate system in which the physical metric g is asymptotically Minkowski, and thus in which i\ has the form i; = diag(-Co 1 ,cr 1 ,cr 1 r 2 ,c 1 -V 2 sin 2 0)

(11.100)

we found in Section 11.1 that the equations of structure for static stars depend only on the quantity (CQCJ) 1 ' 2 . This quantity, as we discovered from the post-Newtonian limit of Rosen's theory (Equation 5.70), plays the role of the local gravitational constant G. Let us now adopt the fully conservative version of the theory, i.e., the version in which the cosmological values of the matching parameters are c 0 = ct = 1. Consider a body moving with velocity v through some given postNewtonian interbody field. In asymptotically Minkowski coordinates, the metric has the form ds2 = -e2*'dt2

+ ix'jdxJdt + e2A'(dx2 + dy2 + dz2)

(11.101)

where
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We must also transform to coordinates xf in which the body is at rest. For |v| ~ O(l), we have, using Equation (4.49), ri-oo = - e - 2 * ' [ l - 2v2(& - A') + 2X' v], i? 8 j=-z} + 2i;/<&/-Af), mi= e-^'dtj + 2viVj(^ - A') - Iv^'n

(11.103)

Now the nondiagonal components of nm are of O(3) or O(4), and by the nature of the local field equations, (11.21), for static spherically symmetric bodies, they contribute only quadratically, i.e., at O(6) or higher. Thus, to O(4) we can determine the local values of c 0 and cx by (cokcai = -tooo)' 1 = e2
(11.105)

where we have dropped the superscript /. If we view the inertial mass ma as a function of the interbody values of the metric coefficients (and of the velocity), then we may write ma{gj = m.{\- sfl[(D + A + §i>2(«> - A) - | v Z ] - Wa ~ s2a)(® + A)2 + }

(H-106)

where sa and s'a are given by Equation (11.77). The action for the theory can then be written (see Section 5.5)

.igjdr.

(H-107)

where we again ignore infinite "self-field" contributions to ma. In coordinates in which 17 = diag(-1,1,1,1), the field equations are D ^ v - g'^g^yg^s

= - IGnig/r,)1'2^ - k, v T)

(11.108)

where 1

Tfov,

* (0) + 1 (1). 112

= (-g)'

I M . t e ^ u 0 ) - 1 ^ - xfl), a

Tft = -2{-g)-112

E (dmjdg^iu0)-^^

- xa)

(11.109)

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A post-Newtonian expansion using the methods of Chapter 5 yields g00 = - 1 + 2 £ mjra + O(4),

gu = 6U [l + 2 I m.(l - fsj/r.j

(11.110)

hence a? = l,

7fl* = l - | s f l

(11.111)

We note that, unlike the case in Brans-Dicke theory, the active mass as measured by test-body Keplerian orbits is equal to the inertial mass ma. The full post-Newtonian metric can now be obtained (the function A must also be determined to O(4) for use in ma), and the results for it and for mjig^) substituted into the matter action a(gjdxa

(11.112)

The resulting n-body Lagrangian is of the EIH form, with

Values of sa and s^ for neutron stars range from sa m 0.05, s'a ss 0.07 for ma =i 0.4 m o to sa ^ 0.6, s'a ^ 0.2 for m =: 12 m o (Will and Eardley, 1977). Calculation of the dipole gravitational radiation energy loss rate proceeds as in Section 10.3, but using Equations (11.106), (11.108), and (11.109), with the result given by Equation (11.74), with KD = -20/3 as before, and ® = s2 s1.

12 The Binary Pulsar

The summer of 1974 was an eventful one for Joseph Taylor and Russell Hulse. Using the Arecibo radio telescope in Puerto Rico, they had spent the time engaged in a systematic survey for new pulsars. During that survey, they detected 50 pulsars, of which 40 were not previously known, and made a variety of observations, including measurements of their pulse periods to an accuracy of one microsecond. But one of these pulsars, denoted PSR 1913 + 16, was peculiar: besides having a pulsation period of 59 ms - shorter than that of any known pulsar except the one in the Crab Nebula - it also defied any attempts to measure its period to ± 1 us, by making "apparent period changes of up to 80 fis from day to day, and sometimes by as much as 8 us over 5 minutes" (Hulse and Taylor, 1975). Such behavior is uncharacteristic of pulsars, and Hulse and Taylor rapidly concluded that the observed period changes were the result of Doppler shifts due to orbital motion of the pulsar about a companion. By the end of September, 1974, Hulse and Taylor had obtained an accurate "velocity curve" of this "single line spectroscopic binary." The velocity curve was a plot of apparent period of the pulsar as a function of time. By a detailed fit of this curve to a Keplerian two-body orbit, they obtained the following elements of the orbit of the system: Ku the semiamplitude of the variation of the radial velocity of the pulsar; Pb, the period of the binary orbit; e, the eccentricity of the orbit;
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epoch. The results are shown in the first column of Table 12.1 (Hulse and Taylor, 1975). However, at the end of September 1974, the observers switched to an observation technique that yielded vastly improved accuracy (Taylor et al., 1976). That technique measures the absolute arrival times of pulses (as opposed to the period, or the difference between adjacent pulses) and compares those times to arrival times predicted using the best available pulsar and orbit parameters. The parameters are then improved by means of a least-squares analysis of the arrival-time residuals. With this method, it proved possible to keep track of the precise phase of the pulsar over intervals as long as six months between observations. This was partially responsible for the improvement in accuracy. The results of this analysis using data up to August 1980 are shown in column 2 of Table 12.1 (Taylor, 1980). The discovery of PSR 1913 + 16 caused considerable excitement in the relativity community (to say nothing of the editorial offices of the Astrophysical Journal Letters), because it was realized that the system could provide a new laboratory for studying relativistic gravity. Post-Newtonian orbital effects would have magnitudes of order v2 ~ K\ ~ 5 x 10"7, m/r ~ fxlax sin i ~ 3 x 10~7, a factor often larger than the corresponding quantities for Mercury, and the shortness of the orbital period (~ 8 hours) would amplify any secular effect such as the periastron shift. This expectation was confirmed by the announcement in December, 1974 (Taylor, 1975) that the periastron shift had been measured to be 4.0° ± 1.5° yr" 1 (compare with Mercury!). Moreover, the system appeared to be a "clean" laboratory, unaffected by complex astrophysical processes such as mass transfer. The pulsar radio signal was never eclipsed by the companion, placing limits on the geometrical size of the companion, and the dispersion of the pulsed radio signal showed little change over an orbit, indicating an absence of dense plasma in the system, as would occur if there were mass transfer from the companion onto the pulsar. These data effectively ruled out a main-sequence star as a companion: although such a star could conceivably fit the geometrical constraints placed by the eclipse and dispersion measurements, it would produce an enormous periastron shift (>5000° yr" 1 ) generated by tidal deformations due to the pulsar's gravitational field (Masters and Roberts, 1975 and Webbink, 1975). Another suggested companion was a helium main-sequence star, which could accommodate the geometrical and periastron-shift constraints. Estimates of the distance to the pulsar (5 kpc: Hulse and Taylor, 1975) and extinction along the line of sight (~ 3.3 mag: Davidsen, et al., 1975) indicated that such

Table 12.1. Measured parameters of the binary pulsar"

Parameter

Symbol (units)

Right ascension (1950.0) Declination (1950.0) Pulse period Derivative of period Velocity curve half-amplitude Projected semi-major axis Orbital eccentricity Orbital period Mass function Longitude of periastron (9/74) Periastron advance rate Red-shift-Doppler parameter Sine of inclination angle Derivative of orbital period Reference

a 5 PP(s) P P (ss- x ) K^kms" 1 ) ay sin i (cm) e Pb(s) fi(fnQ) CO

cb (deg y r " l )

nsinS)i

P^ss" 1 )

Value from period data (summer, 1974)

Value from arrival-time data (9/74-8/80)

19h13m13s ± 4s +16°00'24" + 60" 0.059030 + 1
19h13m12'469 + K014 16°O1'O8'.'15 + O'.'2O 0.0590299952695 + 8 (8.636 ± 0.010) x 10" 1 8 *** (7.0208 + 0.0012) x 1010 0.617138 + 8 27906.98157 ± 6 *** 178.867° + 0.002° 4.226 + 0.001 0.0044 + 0.0003 <0.96 (-2.1+0.4) x HT 1 2 Taylor (1980)

" The entry of a dash (-) denotes that a determination of the parameter was beyond the accuracy of the data; the entry (***) denotes that an accurate value of the parameter is not needed.

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a helium star would have an apparent magnitude mR ~ 21. A number of searches have failed to detect any such object within an error circle of radius 0'.'5 around the radio pulsar position derived from the analysis of the arrival-time data (Kristian et al., 1976; Shao and Liller, 1978; Crane et al., 1979; and Elliott et al., 1980). Other possible companions to the pulsar are the condensed stellar objects: white dwarf, neutron star, or black hole. None of these is expected to be observable optically, and there is no evidence for a second (companion) pulsar in the system. Attempts to delineate further the nature of the companion involved constructing scenarios for the formation and evolution of the system. The favored scenario appears to be evolution from an x-ray binary phase whose end product is two neutron stars (Flannery and van den Heuvel 1975, Webbink 1975, and Smarr and Blandford 1976). However alternative scenarios have been constructed that lead to white dwarf companions (Van Horn et al. 1975, Smarr and Blandford 1976), black hole companions (Webbink 1975, Bisnovatyi-Kogan and Komberg 1976, Smarr and Blandford 1976) and helium star companions (Webbink 1975, Smarr and Blandford 1976). As we shall see, the nature of the companion is crucial for discussion of various relativistic and astrophysical effects in the system. One of the most important of these effects is the emission of gravitational radiation by the system, and the consequent damping of the orbit (Wagoner, 1975). The observable effect of this damping is a secular change in the period of the orbit. However, the timescale for this change, according to general relativity, is so long (~ 109 yr) that it was thought that 10 to 15 years of arrival-time data would be needed to detect it. However, with improved data acquisition equipment and continued ability to "keep in phase" with the pulsar, Taylor and his collaborators surpassed all expectations, and announced in December 1978 a measurement of the rate of change of the orbital period in an amount consistent with the prediction of gravitational radiation damping in general relativity (Taylor et al. 1979, Taylor and McCulloch 1980, Taylor 1980). This chapter presents a detailed discussion of the confrontation between gravitation theory and the binary pulsar. In Section 12.1, we develop an arrival-time formalism analogous to that used by the observers to analyze the data from the binary pulsar, and we discuss the important relativistic gravitational effects (and some competing nonrelativistic effects) in the system. We encounter a new and unexpected role for relativistic gravitational theory: that of a practical, quantitative tool for measuring astrophysical parameters (such as the mass of a pulsar). In Section 12.2, we use general relativity to interpret the data from the system. In Section 12.3, we

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interpret the data using alternative theories of gravitation, and discover that one theory, Rosen's bimetric theory, faces a killing test. In fact, we conjecture that for a wide class of metric theories of gravity, the binary pulsar provides the ultimate test of relativistic gravity. 12.1

Arrival-Time Analysis for the Binary Pulsar

Because the pulsar is the only object seen to date in the system, the analysis of its radio signal is equivalent to that of optical stellar systems in which spectral lines from only one of the members are observed. Such systems are known as "single-line spectroscopic binaries," and standard methods exist for analyzing them. However, there are important differences in the binary pulsar, including the possibility of large relativistic effects, and the ability to measure directly the arrival times of individual pulses, instead of the pulse period. For this reason it is worthwhile to develop a "single-line spectroscopic binary" arrival-time analysis tailored to systems like the binary pulsar. Such an analysis was first carried out in detail by Blandford and Teukolsky (1976) and extended by Epstein (1977) (see also Wheeler, 1975). We begin by setting up a suitable coordinate system. We choose quasiCartesian coordinates (t, x) in which the physical metric is of post-Newtonian order everywhere except possibly in the neighborhood of the pulsar and its companion, and is asymptotically flat. The origin of the coordinate system coincides with a suitably chosen "center of mass" of the binary system. The "reference plane" (Figure 12.1) is denned to be a plane perpendicular to the line of sight from the Earth to the pulsar (plane of the sky), passing through the origin. The "reference direction" is the direction in the reference plane from the origin to the north celestial pole. At any instant, the orbit of each member of the binary system is tangent to a Keplerian ellipse ["osculating" orbit; see Smart (1953) for discussion of these concepts and definitions]. This ellipse lies in a plane that intersects the reference plane along a line (line of ascending nodes) at an angle Q (angle of nodes) from the reference direction. The orbital plane is inclined at an angle i from the reference plane. The periastron of the osculating orbit of the pulsar occurs at an angle a> from the line of nodes, measured in the orbital plane. The other elements of the osculating relative orbit are the semi-major axis a, the eccentricity e, and the time of periastron passage T o . Then the instantaneous relative coordinate position x = x2 x t (1 = [pulsar], 2 = [companion]) is given by x = -a[(cos£ - e)eP + (1 - e2)1'2sin£§Q]

(12.1)

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To Observer

Figure 12.1. Geometry and orbit elements for the binary pulsar. where eP is a unit vector in the direction of the periastron of the pulsar, and eQ is a unit vector at right angles to this in the orbital plane (measured in the direction of motion of the pulsar). The quantity E is the eccentric anomaly, related to coordinate time t by E - e sin E - {2n/Pb){t - To)

(12.2)

where Pb is the binary orbit period. (For the purposes of this arrival-time analysis, it is more convenient to use E than the true anomaly 4>.) The relative separation is given by r = |x| = a(\ ecos£)

(12.3)

By solving the quasi-Newtonian limit of the modified EIH equations of motion (See Section 11.3), taking into account the modifications due to the self-gravity of the pulsar and its companion, we find [from Equation (11.70)]

PJ2n =

(12.4)

where m = mt + m2 is the sum of the inertial masses of the bodies. To quasi-Newtonian order, the center of mass [Equation (11.64)] may be

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289

chosen to be at rest at the origin, i.e., X = m~ 1 (m 1 x 1 +m 2 x 2 )s0

(12.5)

then Xj = (m2/m)x,

x2 = (m1/m)x

(12.6)

Now, any perturbation of the orbit, whether relativistic or not, is to be viewed as causing changes in the orbit elements Q, i, a>, a, and e of the osculating Keplerian orbit; given a set of values of these elements at any instant, Equations (12.1), (12.2), and (12.6) define the coordinate locations of the two bodies. The changes in the osculating elements produced by perturbations can be either periodic or secular. We next consider the emission of the radio signals by the pulsar. Let x be proper time as measured by a hypothetical clock in an inertial frame on the surface of the pulsar. The time of emission of the Nth pulse is given in terms of the rotation frequency v of the pulsar by N = No + vr + ivt 2 + £vt3 +

(12.7)

where N o is an arbitrary integer constant, and v = dv/di\z=0, v = d2v/dt2\t=0. We shall ignore the possibility of discontinuous jumps ("glitches") in the frequency of the pulsar. We ultimately wish to determine the arrival time of the Nth pulse on Earth. Outside the pulsar and its companion, the metric in our chosen coordinate system is given by Equation (11.52), 0OO = - 1 + 2 X «;»./|x - x a(0| + 0(4), <J=1,2

goj = O(3),

2 X y*mj\x - xa{t)\ + O(4))

(12.8)

where ma is the inertial mass of the ath body, and a* and y* are factors that take into account the possibility of self-gravitational corrections to the "gravitational" masses if any of the bodies are compact (see Section 11.3 for discussion). Because we are interested in the propagation of the pulsar signal away from the system, we shall ignore the possibility of large, beyond-post-Newtonian corrections to the metric in the close neighborhood of the pulsar and the companion. The main result of such corrections will be either constant additive terms in the arrival-time formula equation, (12.7), that can be absorbed into the arbitrary value of No, or constant multiplicative factors (such as the red shift at the surface of the neutron

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star) that can be absorbed into the unknown intrinsic value of v. Modulo such factors, proper time x at the pulsar's point of emission can be related to coordinate time t by dx = dt[\ - <x%m2jr - \v\ + O(4)]

(12.9)

where we have dropped the constant contribution of the pulsar's gravitational potential a}'m1/|xem x t |, and where we have ignored the difference in the potential and the velocity between the emission point and the center x t of the pulsar. The two correction terms in Equation (12.9) are the gravitational red shift and the second-order Doppler shift. We can rewrite Equation (12.9) using Equations (12.1) and (12.2), which yield v\

=
(12.10)

with the result (modulo constants) dx/dt = 1 - afmjr - ^m\jmr

(12.11)

Using Equations (12.2) and (12.3) we may integrate this equation to obtain T = t - (m2/a)(<x£ + gm2/m)(PJ2n)e sin E

(12.12)

Although the constants that have been dropped in integrating Equation (12.11) may actually undergo secular or periodic variations in time due to orbital perturbations and other effects (such as in a), the correction term in Equation (12.12) is already sufficiently small that such variations will have negligible effect. Now, the pulsar signal travels along a null geodesic. We can therefore use the method in Sections 6.1, 7.1, and 7.2 to calculate the coordinate time taken for the signal to travel from the pulsar to the solar system barycenter x 0 , with the result tarr ~ * = |*o('arr) ~ * l W |

+ (a? + yf)m2 ln{2ro(tm)/[r(t) + x(t) i]}

(12.13)

where r0 = |xo|, n = xo/ro, and where we have used the fact that r0 » r. The second term in Equation (12.13) is the time delay of the pulsar signal in the gravitational field of the companion; the time delay due to the pulsar's field is constant to the required accuracy, and has been dropped. Our ultimate goal is to express the timing formula equation, (12.7), in terms of the arrival time farr. In practice, one must take into account the fact that the measured arrival time is that at the Earth and not at the barycenter of the solar system, and will therefore be affected by the Earth's position in its orbit and by its own gravitational red shift and

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Doppler-shift corrections. In fact, it is the effect of the Earth's orbital position on the arrival times that permits accurate determinations of the pulsar position on the sky. It is also necessary to take into account the effects of interstellar dispersion on the radio signal. These effects can be handled in a standard manner [see Blandford and Teukolsky (1976), for example], and will not be treated here. Now, because r 0 » r, we may write \*o(tm) - XiW| = ro{tm) - x,(t) n + O(r/r 0 )

(12.14)

Combining Equations (12.13) and (12.14), and using the resulting formula to express xx(r) in terms of x 1 (t arr ) to the required post-Newtonian order, we obtain t = *arr ~ ^0 + *l(ttn ~ r0) fi + (*i(t m - r0) fi)(Xl(tarr - r0) ft) + [O(3)t arr ]

(12.15)

where the time-delay term is [O(3)t arr ]. We now choose the constant in Equation (12.2) so that E - e sin E = (2n/Ph){tMt - r0)

(12.16)

then, combining Equations (12.12), (12.15), and (12.16), substituting Equations (12.1) and (12.6), and noting from Figure 12.1 that eP n = sinisintu,

eQ n = sinicosco

(12.17)

we find, modulo constants, T = tatI - s?(cosE - e) - (@ + ^ ) s i n £ - (2n/Pb)(l - e cos E)~ l{d sin E-36 cos £)[j/(cos E - e)+& sin £ ] + [O(3)t arr ]

(12.18)

where s4 = at sin i sin co, 0$ = (1 e 2 ) 1/2 a t sin i cos co, V = (w|/ma 1 )(a? + &m2/m)(PJ2n)e

(12.19)

a, = (m2fm)a

(12.20)

with

The timing formula then takes the form N = N0 + vtarr - va/(cos£ - e) - \{08 + ^sinE -v(2n/Pb){l-ecosE)-i(s/sinE-@cosE)[#?(cosE-e)+@sinE'} - e) + & sin E] + |vfa3rr +

(12.21)

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The quantity N is to be regarded as a function of the time tarr and of the parameters No, v, v, v, alsin/, a>, e, P b , tQ, c£. From an initial guess for the values of these parameters a prediction for the arrival time of a given N is made. The difference between the predicted arrival time and the observed arrival time is used to correct the parameters using the method of least squares. Possible variations with time of the parameters resulting from perturbations of the system can also be determined, for example, by substituting CO-KO +
P b - + P b + $ P b t + ---

(12.22)

and so on, into Equation (12.21). (The factor \ in the formula for P b comes from the formal definition of P b in terms of osculating elements.) We now turn to a discussion of the important measurable parameters and their interpretation. (a) The pulsar period The terms linear, quadratic, and cubic in tarr in the timing formula, Equation (12.21), determine the effective pulse period (at a chosen epoch) and its derivatives. The results of least-squares fits using data up to August 1980 were (Table 12.1) p p = v -» = 0.0590299952695 ± 8 s, P p = -vv~2 = (8.636 ± 0.010) x 1(T 18 s s" 1

(12.23)

where the epoch was September, 1974. No determination of P p has been possible to date except for the crude limit set by the fact that P p has not changed by more than the experimental error over timescales of one year, thus, |Pp|<6 x 10"28ss-2

(12.24)

Despite the fact that the pulsar's period is the second shortest known, its "spin-down rate," P p , is anomalously small, i.e., Pp/Pp = (4.617 ± 0.005) x 10" 9 yr" 1 . The most popular explanation for this is that the pulsar has a weak magnetic field, leading to small braking torques caused by magnetic Lorentz forces, thence a small P p . However, its short period is a remnant of an earlier phase in the evolution of the system, during which accretion of matter onto the pulsar caused it to be "spun up," essentially to its present period [for discussion see Smarr and Blandford (1976), BisnovatyiKogan and Komberg (1976)].

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(b) The Keplerian velocity curve The terms vs/(cosE e) v^tsinE in Equation (12.21) will be referred to as the "Keplerian velocity curve." The time derivative of these terms yields simply the first-order Doppler shift of the pulsar frequency, given by Av/v oc vt ii. This variation in frequency is the quantity usually measured in spectroscopic binaries, and was the quantity measured in the binary pulsar until the method of arrival-time measurements was adopted in late 1974. By fitting the measurements of arrival times to cos£ and sin is curves using Equation (12.21), a determination of the parameters P b , e, stf, and 88 can be made. From these parameters it is conventional to determine (i) the periastron direction at a given epoch: tan w = (1 - e2)- 1/2 J%s/

(12.25)

(ii) the projected semi-major axis of the pulsar: ax sin i = [ y 2 + m\\ - e2yxY'2

(12.26)

(iii) the mass function of the pulsar:

A M * ! sin 0 3 (iV2*r 2

(12.27)

The observed values for these quantities are shown in Table 12.1. Using Equations (12.4) and (12.20), we may also express fx in the form / 1 = Sr(m2 sin i)3/m2

(12.28)

These interpretations assume afixedKeplerian orbit with constant values of the orbit elements. In reality, variations with time of the elements as given for example by Equation (12.22), make it necessary to treat the above values as being valid at a chosen epoch, and to perform further least-squares fits to determine rates of change such astit,Pb, e, and so on. We shall discuss some of these quantities below. (c) The periastron shift By substituting <x>-*a> + cat into the expressions for si and 88 in the Keplerian velocity curve, one can make an accurate determination of cb. The best value to date is d> = 4?226 ± 0!001 yr~ 1 . There are several possible sources of periastron shift in the binary pulsar. The first is relativistic: in Section 11.3, we found that in a binary system with compact objects, the periastron shift rate in a fully conservative theory of gravity in the modified EIH formalism took the form

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[from Equation (11.71)] + m23>122)/m

(12.29)

\M here should not be confused with that defined by Equation (12.19).] Substituting the known values of Pb and e and using Equation (12.4), we obtain d»rel = 2!lO(m/m 0 ) 2/3 ^«r 4/3 yr~* (12.30) In general relativity 9 s ^ == 1. The second possible source is a noninverse square gravitational potential produced by tidal deformation of the companion by the pulsar. The resulting rate is given by d)tidal =* 30nk2Ph-\$m1/m2)(R2/a)5f(e), /( e ) = ( l - e 2 ) - 5 ( l + | e 2 + i e 4 )

(12.31)

where R2 is the radius of the companion (Cowling, 1938). The quantity k2 is a dimensionless factor that depends on the mass distribution of the companion and is of order 10 ~2 for white dwarfs or helium stars. In obtaining Equation (12.31), we have assumed that the companion is not a neutron star or a black hole in order to avoid the additional complications of self-gravitational effects on the tidal effects. Because of the {R2/a)5 dependence, the tidal contribution of such objects would be negligible in any case. Substituting numerical values we obtain

where X = mxfm2. For a white dwarf companion (R2 5> 104km), the tidal effect is also negligible. Only for a "helium-star" companion, for which (fc2/10~2)(i?2/105 km)5 ~ 6(m2/mQf-6, can the tidal periastron advance be significant (Roberts et al., 1976). The third possible source of periastron shift is the noninverse square potential produced by rotational deformation of the companion. For a body that rotates with angular velocity Q about an axis that is inclined by an angle 8 relative to the plane of the orbit, the result is d)rol * 27t/c2Pb-1(3Wm2)(i?2/a)5(l + X-1)(n/n)2g(e)P2(cos9), g(e) 3 (1 - e2r2, n = 2n/Pb

(12.33)

This result is valid provided one assumes that the angular momentum of the companion is small compared to that of the orbit, an assumption

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295

that is valid for most reasonable companions [see Smarr and Blandford (1976) for discussion]. With numerical values, we find ,5

\mo) x

K

^ 'V 1 0 ~ 2 A 1 0 5 km

a O?83a(^m/mo)"2/3P2(cos0) yr" 1

(12.34)

where a = %k2(Cl2Rl/m2)(R2/l03 km)2

(12.35)

For stable, uniformly or differentially rotating white dwarf models, for example, a may range from zero to ~ 15 (Smarr and Blandford, 1976). Notice that the rotationally induced periastron motion can be either an advance [P2(cos 6) > 0], for example, when the spin axis is normal to the orbital plane, or a regression [P2(cos 9) < 0], as when the spin axis lies in the orbital plane. If the companion is a neutron star, a black hole, or a nonrotating white dwarf, then only the relativistic periastron precession is present. The observed advance shown in Table 12.1 then yields via Equation (12.30) a relation between the masses of the bodies: m = 2.85m G ^- 3/2 # 2

(12.36)

where & and ^ are functions of mu m2, and the structure of the pulsar and possibly of the companion. If the companion is a rotating white dwarf, only the relativistic and rotational contributions are significant, thus we may write (b = 2°10(/n/m G ) 2/3 ^«r 4/3 + O°83a(m/mGr2/3«r2/3P2(cos0) yr" 1 (12.37) If the companion is a helium star with rotation axis perpendicular to the orbital plane, all three sources of periastron precession may be present, with / m \2

\mQJ

2/3

mQ

(12.38)

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(d) The gravitational red shift and second-order Doppler shift: the way to weigh the pulsar The term <^sin£ in the timing formula, Equation (12.21), represents the combined effects of the gravitational red shift of the pulsar frequency produced by the gravitational field of the companion and of the second-order Doppler shift produced by the pulsar's motion. In some theories of gravity, there is another effect that contributes to the timing formula at the same order as
(12.39)

If the companion is a white dwarf or a helium star, for example, the parameter n* is simply the combination of PPN parameters n* = 4p y 3 (fully conservative theories with £ = 0), however, if the companion is a neutron star or a black hole, rj* could be more complicated and could depend upon the internal structure of the companion. As GL then varies during the orbital motion, the structure of the pulsar, its moment of inertia, and thence its intrinsic rotation frequency will vary, according to

v

/

GL

_

K

,*^ rr

(12 .40)

where K determines the response of the moment of inertia to the changing G. The contribution of this variation to the timing formula is given by J Av dt = - vKn*(m2/a)(Pb/2n)e sin E

(12.41)

modulo constants. Thus the parameter W is actually given by V = (mi/ffMiHaJ + « W m + Kn*){PJ2n)e

(12.42)

With numerical values it takes the form « a* 2.93 x Kr 3 (m 2 /m)(mAn 0 ) 2/3 «r

1/3

(<x| +
(12.43)

However, were it not for the presence of periastron precession in the system, this parameter would be entirely unmeasurable, since for constant values of sd and 8$, the term %> sin E is degenerate with the two Keplerian velocity curve terms, i.e., it cannot be separated from them in a least-squares fit (Brumberg et al. 1975, Blandford and Teukolsky, 1975). However, the variation of co at 4° per year causes $4 and J 1 themselves to vary with approximately a 90-year period. Thus, over a suf-

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ficiently long time span (though much shorter than 90 years, fortunately), a separate determination of si, 88, and < can be made. Using data through August, 1980, Taylor (1980) in fact reports <«?~4.4±0.3 x H T 3 s

(12.44)

Equations (12.43) and (12.44) yield a further relation between the masses of the bodies. When combined with the mass relations provided by the mass function [Equation (12.28)], and by the periastron shift [Equations (12.36), (12.37), or (12.38)], and with assumptions about the nature of the companion and about the theory of gravitation, they permit a unique (within experimental errors) determination of the masses m^ and m2 and of sin i. This is that unique new role of relativistic gravity alluded to in the introduction to Chapter 12. Not only does a relativistic effect, the periastron shift, yield a constraint on the masses of the bodies, it also enables the determination of a second relativistic effect, the red-shift Doppler coefficient (. Nowhere in astrophysics has relativistic gravity played such a direct, quantitive role in the measurement of astrophysical parameters. (e) Post-Newtonian effects and sin i In Equation (12.15), we dropped the explicit term arising from the time delay, and denoted it [O(3)farr]. There are additional terms in the timing formula that are also of [O(3)£arr], produced by post-Newtonian deviations of the orbital motion from a pure Keplerian ellipse. Within general relativity, these terms have been analyzed in detail by Epstein (1977), and included in the data analysis by Taylor, et al. (1979). They provide an independent means to determine the parameters of the system, especially the inclination angle i. This is a valuable consistency check for any interpretation of the data. In fact the data are just accurate enough to be sensitive to these effects, and the limit on sin i quoted in Table 12.1 was obtained from these terms. Note that this particular result is valid only in general relativity; the corresponding analysis of the [O(3)farr] terms using the modified EIH formalism has not been carried out. (f) Decay of the orbit: a test for the existence of gravitational radiation A variety of effects may cause the orbital period P b of the system to undergo a secular change with time, but the most important is the effect of the emission of gravitational radiation. According to the quadrupole formula of general relativity (see Section 10.3), a binary system

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should lose energy to gravitational radiation at a rate given by Equation (10.80), dE_ _ /n2m2 dt ~

where /i is the reduced mass of the system, and F{e) = (1 + He 2 + Me4)(l - e2)'1'2

(12.46)

The resulting rate of change of P b is given, from Kepler's third law, by Pb-1 dPJdt = -IE"1 dE/dt = -¥{nm2/aA)F{e)

(12.47)

where E = \\an/a. For the known parameters of the binary pulsar we find jn\5'* X_ = -(1.91xlO-9)(^l T T - ^ y r - 1 (12.48) (1 KmQ) As we pointed out in Section 10.3, most theories of gravitation alternative to general relativity predict the existence of dipole gravitational radiation. Since the magnitude of the effect in binary systems depends upon the self-gravitational binding energies of the two bodies, the binary pulsar provides an ideal testing ground. In general relativity, neutronstar binding energies can be as large as half their rest masses, and in other theories even larger, so the dipole effect, if present, could produce more rapid period changes than the general relativistic quadrupole effect. The predicted energy loss rate is given by ah/at = 2KD\ & n m vs> /r }

where KD is a parameter whose value depends upon the theory in question, and S is related to the difference in "sensitivities" (s2 Si) between the two bodies, where sa is a measure of the self-gravitational binding energy per unit mass of the ath body. In general relativity, KD = 0. The rate of change of period is thus given by l + | e 2 )(l - e 2 r 5 / 2

(12.50)

where now E = \ 'S^mja. For the parameters of the binary pulsar, we obtain

- "(3.09 x w - ^ J ^ j ^ y , - '

(,2.M,

Binary Pulsar

299

For a theory of gravity with |/cD| ~ 1, this can be several orders of magnitude larger than the general relativistic quadrupole prediction, unless, for instance, the two bodies are identical, in which case there is no dipole radiation, by virtue of the symmetry of the situation. However, before these effects can be used as a reliable test for the existence of gravitational radiation or as a test of alternative gravitation theories, other possible sources of period change must be accounted for. Since we have previously discussed tests of gravitational theory involving detecting changes in the pulsar period as well as in the orbital period (see Section 9.3), we shall review possible sources of both. (i) Tidal dissipation. Tides raised on the companion by the gravitational field of the pulsar will change both the energy of the orbit and the rotational energy of the companion via viscous heating. The corresponding tides raised on the pulsar are negligible because of its small size and by the same token, if the companion is a neutron star or a black hole, tidal dissipation is negligible. For a companion with rotation axis normal to the plane of the orbit, the rate of change of the orbital period is given by (Alexander, 1973) Ph

672TC

3 W ) - 6 ^ ^i fete2 (12.52) 2 Pb 25 ' \m 2)\a ) \ m2 ) \ ' nj where n = 2n/Pb is the orbital mean anomaly, is an "average" coefficient of viscosity of the companion given by

(1-e

2

v

fir8dr

(12.53)

where n is the local coefficient of viscosity in units of gem" 1 s" 1 , and h(e2,Q/n) is a complicated function of e2 and Q/n of the following general form He2,0/n) = ht(e2) - (i/n)h2(e2)

(12.54)

For circular orbits h1 = h2 = 1, however for the binary pulsar (e m 0.6) they could be an order of magnitude larger (but hx ^ h2). We note that if Q < «(companion counter rotates relative to the orbit), tidal dissipation always decreases the orbit energy and thus the period, whereas if Q/n > hx(e2)/h2(e2) (companion rotates faster than the orbit by some factor of order unity), dissipation increases the orbit energy (at the expense of rotational energy) and causes the period to increase. Notice than even if the companion is in synchronous (tidally locked) rotation, Q = n, there can still be tidal dissipation due to the time-changing deformation of the

Theory and Experiment in Gravitational Physics

300

companion resulting from the eccentric motion of the pulsar. Substituting the observed parameters of the system, we obtain = -2x mo R2 \ 9 T73T i3*(e2,n/»)yr~1 (12.55) 105 km/ where . For standard molecular viscosity, (n~) ~ 1, i.e., <ju>13 ~ 10" 13 , and tidal dissipation is completely negligible. However, if the source of viscosity is tidally driven turbulence (Press et al., 1975; Balbus and Brecher, 1976), 13 could be as large as unity. For a helium star companion, PJPb could then be comparable to the general relativistic quadrupole radiation damping rate. For a white dwarf companion {R2 < 104 km), tidal dissipation is negligible unless the white dwarf is very rapidly rotating (|Q| » n), and a very strong source of viscosity, such as magnetic viscosity «M>i3 ~ 103), is present (Smarr and Blandford, 1976). (ii) Mass loss from the system. The emission of energy of various forms (particles, electromagnetic radiation, etc.) from the pulsar results in a decrease in its rotational kinetic energy, and thus in an increase in its pulse period, given by Erol = -l{2nlPpfPJPp

(12.56)

where / is the moment of inertia of the pulsar. The loss of mass energy from the pulsar leads to a change in the orbital period at a rate PJPb = -\rhjm

(12.57)

Now, if the emission of energy is dominated by relativistic particles (photons, for example) then most of the mass loss will occur at the expense of rotational kinetic energy, i.e., £rot < m,

(12.58)

PJPb ~ 1 x 10" 6 {ml2.MmQ)-HA5(PpIPp)

(12.59)

Thus, where 745 = 7/1045 g cm2. Since the observed value for Pp/Pp is ~ 4 x 10~9 yr~x (Table 12.1), then PJPb due to energy loss must be ~10~ 1 4 yr" 1 . (iii) Acceleration of the binary system. If the center of mass of the binary system suffers an acceleration relative to that of the solar system, then

Binary Pulsar

301

both the orbital and pulsar periods will change at a rate given by PJPb = Pp/Pp = r 0 = a n + r0-J [V - (v n)2]

(12.60)

where v and a are, respectively, the relative velocity and acceleration between the binary system and the solar system. The first term is the projection of the acceleration along the line of sight, while the second represents the effect of variation of the line of sight. Accelerations may also lead to observed second-time derivatives of periods, given by PJPb = Pp/Pp = r0 + 2(P/P) 0 r 0 - Tr%

(12.61)

where (P/P)o is the observed relative rate of change for the corresponding period. One possible source of acceleration was discussed in Section 9.3, namely a violation of conservation of total momentum in some theories of gravity. There, we used the observed limits on Pp/Pp to set a potential limit on the PPN conservation law parameter £2 [ m t n a t c a s e > t n e second and third terms in Equation (12.61) are negligible compared to the first]. Another source is the differential rotation of the galaxy. If we assume that the binary system (b) and the solar system (©) are in circular orbits around the galaxy with angular velocities Qb and Q 0 , distances from the galactic center rb and rQ, and longitudes relative to the galactic center cf>b and Q, then Equation (12.60) takes the form

Pp/Pp = PJPb = («o - 0^2r0Vo

'

x [cos(^ b - 0) - rQrbro2sin2(^b

- <£0)]

(12.62)

Estimates of the location and distance of the binary pulsar (Hulse and Taylor, 1975) yield ro~5kpc,

rQ~10kpc,

rb~8kpc,

<£ b -<£ 0 ~3O°

(12.63)

Using the standard galactic rotation law, Q(r) ~ 250 (km s"1)/?-, we find PJPb = K/Pp ~ 2 x lO" 1 3 yr" 1

(12.64)

This is too small to be of importance (Will 1976b, Shapiro and Terzian 1976). Another possible source of acceleration is a third massive body in the vicinity of the binary system. For a body of mass m3, and for a circular orbit with orbit elements a3, co3, and i3, we have

PJPb = PJPp~ - f e Y ( - ^ ~ ) a 3 s i n i 3 c o s ( c o 3 + 4>) (12.65)

Theory and Experiment in Gravitational Physics

302

where <> / is the orbital true anomaly, and Pb/Pb = Pp/Pp * ( ^ J ( - ^ L _ ) a3 sin i, sin(a>3 +
(12.66)

It is then simple to show that if t] represents the observed upper limit on \PP/Pp\, then the contribution of a third body to period changes is limited by \PJPp\ = |P b /P b | < (7 x lO-^if/meY'^/lO-11 yr- 2 ) 4 ' 7 x |cot(0 + a>3)sm(4> + o)3)3/7| yr" 1 (12.67) where / is the mass function of the binary system relative to the third body, given by / = (m3 sin i3)3(m + m 3 )" 2

(12.68)

Since the observed valued of Pp/Pp has not changed by more than its experimental error in a year (Table 12.1), we may conclude that rj < 10"11 yr~2. An explicit determination of r\ from the timing data that improves this limit would help to determine the likelihood that a third body is responsible for part of the observed orbit period change. (g) Precession of the pulsar's spin axis

If the pulsar is a rapidly rotating neutron star, it should experience the same relativistic precession effects on its spin axis as does a gyroscope in orbit around the Earth (see Section 9.1). The dominant effects are the geodetic precession due to the companion's gravitational field, and a Lense-Thirring-type precession due to the companion's "magnetic" gravitational field generated by its orbital motion [see Equations (9.2) and (9.4)]. The Lense-Thirring precession due to the possible rotation of the companion is negligible. By substituting Equations (9.4) and (9.2) with J = 0 into Equation (9.1), inserting the orbital elements for the binary pulsar, and averaging over an orbit, one finds

fdt = ftxS il = (3n/Ph)[m22/ma(l -

2 e

)][i(2y + l) + f(y+ 1 +ia1)(mi/m2)]fi

(12.69)

where y and a t are PPN parameters and h is a unit vector normal to the orbital plane (Barker and O'Connell, 1975; Hari Dass and Radakrishnan, 1975; and Rudolph, 1979). In obtaining this result we have ignored the possibility of modified-EIH-formalism corrections to effective masses in alternative theories of gravity. The magnitude of ft is about one degree per year (compare with an Earth-orbiting gyroscope in Section 9.1); note,

Binary Pulsar

303

however, that no precession occurs if the pulsar's spin axis is normal to the plane of the orbit. If precession does occur, it could be viewed as a means to test gravitational theory. However, it may be more fruitful to use the relativistic precession as a means to probe the nature of the pulsar's emission mechanism. As the pulsar precesses, the observer's line of sight intersects the surface of the neutron star at different latitudes, thus it may be possible to obtain two-dimensional information on the shape of the emitted beam, as well as to study the variation of spectrum and polarization with latitude. Unfortunately, in most pulsar models, the radio pulses are emitted in a pencil beam, so the pulsar might one day disappear altogether. We now turn to the confrontation between the binary pulsar and gravitation theory. It is here that the philosophy of testing gravitation theory must depart somewhat from that adopted in Chapters 2 through 9. There, we regarded experiments as "clean" tests of gravitational theory. Because the underlying nongravitational physics associated with solar system and laboratory experiments was reasonably well understood, the experimental results could be viewed as limiting the possible alternative theories of gravity, in a theory-independent way. The use of the PPN formalism was a clear example of this approach. The result was to "squeeze theory space" in a manner suggested by Figures 8.2 and 8.3. However, when complex astrophysical systems such as the binary pulsar are used as gravitational testing grounds, one can no longer be so certain about the underlying physics. In such cases, a gravitation-theory-independent approach is not useful. Instead, a more appropriate approach would be to assume, one by one, that individual theories are correct, then use the observations to make statements about the possible compatible physics underlying the system. The viability of a theory would then be called into question if the resulting "available physics space" were squeezed into untenable, unreasonable, or ad hoc positions. Such a method would be most powerful for theories that make qualitatively different predictions in such systems. We shall illustrate this philosophy of "squeezing physics space" (using relativistic gravity to determine astrophysical parameters) with general relativity, Brans-Dicke theory, and Rosen's bimetric theory. 12.2

The Binary Pulsar According to General Relativity

The confrontation between relativistic gravity and binary pulsar data takes its simplest and most natural form within general relativity. In general relativity, there are no EIH self-gravitational mass corrections

Theory and Experiment in Gravitational Physics

304

due to violations of SEP (see Section 11.3), and there is no dipole gravitational radiation. Thus, ^ E g = aj H yf = 1 and KD = n* 0. The relevant measured parameters of the system are then given by the following expressions Mass Function: fi = (m2 sin ifjm2

(12.70)

Orbital Period: PJ2n = ( a » 1 / 2 Periastron Shift:

(12.71)

2?10(m/mG)2/3 yr""1, [black hole, neutron star, nonrotating white dwarf companion]

(12.72)

2?10(m/mo)2/3 + O?83a(m/m o r 2/3 P 2 (cos0)yr-\ [rotating white dwarf companion]

(12.73)

a; = 105km/

"V»©/

[aligned rotating helium star companion]

(12.74)

Red-shift-Doppler Parameter: ^ = 2.93 x 10"31

m\213 m

(12.75)

Gravitational Radiation Reaction (Pure quadrupole): VAJgr.quad = -(1.91 x 10-9)(m/mG)5/3X(l + X)'2 yr" 1

(12.76)

Together with the measured values shown in Table 12.1, these equations determine constraints on the possible masses of the pulsar and companion, and on the inclination i. The most convenient way to display these constraints is to plot m1 vs. m2. The results are shown in Figure 12.2. One constraint is provided by the mass function fx and by the fact that sin i < 1. The periastron shift constrains the system to lie along the straight line BH-NS-WD if the companion is a black hole, neutron star, or nonrotating white dwarf [Equation (12.72)]. This line represents a total mass m = 2.85 mQ. It is useful to remark that the maximum mass of a nonrotating white dwarf is ~ 1.4 solar masses. If the companion is a rapidly rotating white dwarf (with "U" denoting uniform rotation and

305

Binary Pulsar

"D" denoting differential rotation), the system could lie in the regions denoted U and D [Equation (12.73)]. The regions to the left of the BHNS-WD line correspond to white dwarfs with spin axes aligned perpendicular to the orbital plane (6 = 0). In this case, the rotational contribution to d> is positive so the inferred system mass m must be less than 2.85m0. The regions to the right of the BH-NS-WD line correspond to white dwarfs with spin axes in the orbital plane (0 = n/2). Here the rotational periastron shift is retrograde and thus m > 2.85mo. Values of the parameter a that depends upon the structure and rotation rate of the white dwarf were given by Smarr and Blandford (1976). Figure 12.2 also shows the configuration if the companion is a helium star, tidally locked into the orbital rotation rate (Q = n). Values for k2 and R2 for helium stars were given by Roberts, Masters, and Arnett (1976). The red-shift-Doppler parameter then constrains the system to lie between the lines marked #. Finally, if we attribute all the observed orbit-period decay to gravitationalradiation damping, then the system must lie between the lines marked Figure 12.2. The mx-m2 plane in general relativity. The shaded region fits all the formal observational constraints. The point marked "a" is the most likely configuration.

3.0

\ \ \ ^H-NS »',

- N.

©

2.0

i

\ \ \

y ,

i

\ \ \

fo, s

D-WD"

I- D

""

^ - - '

gr.§C

\HE

.- 1.0

-^^^BH-NS-WD sin i> 1

i

i

1.0

i

2.0 Mass of Pulsar m,/m0

I

1

3.0

1

Theory and Experiment in Gravitational Physics

306

P b . This leaves the shaded region available. The most natural physical interpretation therefore seems to be that the companion is a black hole, neutron star, or nonrotating white dwarf (point "a" in Figure 12.2) of mass m2 = 1.42 ± 0.07m©. The mass of the pulsar is then mx = 1.43 ± 0.07m© and the sine of the inclination angle (from the mass function) is sin i = 0.72 + 0.04. This interpretation is also consistent with the constraint sin i < 0.96 obtained by taking into account in the timing formula postNewtonian effects such as the time delay [Equation (12.13)] and periodic perturbations of the Keplerian orbit (Taylor, 1980). Before this interpretation can be accepted with confidence, however, some account must be taken of the possible nonrelativistic sources of orbit period change discussed in the previous section, in particular tidal dissipation and a third body. Thus, barring these remote possibilities, general relativity leads to a natural physical configuration for the system, and the results support the conclusion that the measurement of Ph represents the first observation of the effects of gravitational radiation. They also lend support to the validity of the quadrupole formula (see Section 10.3) for radiation damping, at least as a good approximation, and rule out the possibility that gravitational waves are composed of half-retarded plus half-advanced fields and therefore carry no energy at all (Rosen, 1979). 12.3

The Binary Pulsar in Other Theories of Gravity

(a) Brans-Dicke theory Because solar-system experiments constrain the coupling constant a to be large (co > 500) we expect the predictions of scalar-tensor theories to be within corrections of order (1/co) of their general relativistic counterparts for the binary pulsar. The self-gravitational mass renormalizations merely introduce corrections of order (l/co) (see Section 11.3). Thus, the mt m2 plane in scalar-tensor theories is largely indistinguishable from that in general relativity. Even the added possibility of dipoie gravitational radiation does not seriously constrain either "physics" space or the coupling constant co. Substituting the value KD = 2/(2 + co) into Equation (12.51) we find (A/PbWie = - ( 1 x 10-9)(500/a>)(S/0.1)2(AVm©) yr" l

(12.77)

For neutron star models with masses around 1.4m©, s ~ 0.39 (Eardley, 1975). Thus, (PJPb) dipoie could be significant if the companion is a white dwarf or a neutron star whose mass differs from that of the pulsar by greater than ~ 10%. In such an event, it might be possible to push the

Binary Pulsar

307

coupling constant even higher than 500. However, the data can equally well be fit (for <x> ~ 500) by a system with two nearly equal-mass neutron stars, or by one of the above possibilities with a small contribution to Pb from some nonrelativistic source. This is a case in which the theoretical predictions are sufficiently close to those of general relativity, and the uncertainties in the physics still sufficiently large that the viability of the theory cannot be judged reliably. We would expect roughly the same conclusions to be valid in general scalar-tensor theories such as Bekenstein's VMT (see Section 5.3). (b) Rosen's bimetric theory In the bimetric theory, however, the situation is very different. The EIH self-gravitational mass corrections (of Section 11.3) lead to qualitative differences for two reasons. First, the correction terms in ^, 0*, etc. are ~s, and second, s can be much larger for bimetric neutronstar models than for their general relativistic counterparts. Table 12.2 Table 12.2. EIH sensitivities, s, s', in Rosen's bimetric theory." Inertial mass m(mQ) normal star white dwarf neutron stars 0.097 0.165 0.409 0.635 0.865 1.158 1.868 2.371 3.217 4.553 7.177 10.93 12.34 14.45C

s

s'

~10~6 glO"3

~10~6

0.006 0.018 0.048 0.071 0.096 0.128 0.206 0.258 0.331 0.410 0.494 0.561 0.582 0.628

b

£io- 3 b

0.065 0.099 0.132 0.174 0.265 0.294 0.271 0.223 0.175 0.152 0.222 b

" For equations of state from Canuto (1975). Accuracy + 3 in last place. * Accurate value not computed. c Maximum mass.

308

Theory and Experiment in Gravitational Physics

shows values of s and s' for normal stars and white dwarfs, and for neutronstar models with inertial masses up to 14.5wo (Will and Eardley, 1977). From these values, we compute values for'S and 0, given by [see Equations (11.72) and (11.113)] (12.78) and plot the corresponding m1 m2 plane for the bimetric theory, shown in Figure 12.3. [For simplicity, we have ignored the effect of changes in GL on the parameter <, Equation (12.43). It is only significant if the companion is a neutron star {t\* %s2), and is expected to modify <
Figure 12.3. The mx-m2 plane in Rosen's bimetric theory. Note the scale of masses is almost double that of Figure 12.2. The numbers shown are the predicted values of P b /P b due to gravitational radiation, including dipole gravitational radiation.

6.0 -

5.0 -

N.

4.0 -

/+10-* \ +10"' V-+10- 6

a,

<S 3.0 s

s

^

2.0 -

1.0 -

^

\

D-WD

^

JT>VC

' * ' - ' ' ' + 1( ^ - ' ^ + 1 0 ^ - ^ sin i> 1

/"^U-WD I

1.0

i

I

i

2.0 3.0 4.0 Mass of Pulsar nij/nio

i

5.0

Binary Pulsar

309

In Figure 12.3, we notice that the companion cannot be a nonrotating white dwarf, since such a configuration would violate the condition sini < 1. If the companion is a neutron star, the system must lie along the curve "NS," with total inertial mass ~ 7 m o . When the red-shiftDoppler constraint (curves 'V') is folded in, the theory is left with a major problem. Dipole gravitational radiation causes the system to gain energy and the period to increase at a rate ( i V n W i e =* +(2 x l(T 5 )(6/O.3) 2 (Mn 0 ) yr" 1

(12.79)

with specific values for various companions shown in several locations in Figure 12.3. In order to agree with the observed value of PJPb ^ (2.4 + 0.4) x 10~9 y r ~ \ the theory must produce a mechanism (tidal dissipation, third body) to cancel this predicted increase and account for the observed period decrease. The contrived and ad hoc nature of such mechanisms deals a convincing blow to the viability of this theory. (c) The ultimate test of gravitation theory? This result may, in fact, apply to many other theories, particularly those with "prior geometry." In such theories, SEP is violated, and the differences between the theories and general relativity become larger the stronger the gravitational fields. Thus, one can expect qualitative EIH mass renormalizations similar to those in the bimetric theory. Furthermore, all such theories are expected to predict dipole gravitational radiation of magnitude comparable to that in the bimetric theory. So it is very likely that the binary pulsar data will be able to rule out a broad class of alternative gravitation theories. However, the class of "purely dynamical" theories has the property that the effects of the additional gravitational fields can usually be made as small as one chooses, both in weak-field and in strong-field or gravitational-radiation situations, by choosing sufficiently weak coupling constants (co'1 -> 0 in Brans-Dicke, for instance). Thus, Brans-Dicke theory, with to ^ 500, is consistent with the present binary pulsar data, even though it, too, predicts dipole gravitational radiation. Such theories, that merge smoothly and continuously with general relativity, can never be truly distinguished from it (as long as experiments continue to be consistent with general relativity). Except for such cases, the binary pulsar may provide the "ultimate" test of gravitation theory.

13 Cosmological Tests

Since the discovery by Hubble and Slipher in the 1920s of the recession of distant galaxies and the inferred expansion of the universe, cosmology has been a testing ground for gravitational theory. That discovery was thought at the time to be a great confirmation of general relativity for two reasons. First, general relativity, in its original form, predicted a dynamical universe that necessarily either expands or contracts. Of course, Einstein had later modified the theory by introducing the "cosmological constant" into the field equations in order to obtain static cosmological solutions in accord with the current, pre-Hubble observations. To his great joy, following Hubble's discovery, Einstein was allowed to drop the cosmological constant. Second, was simply the fact that general relativity was capable of dealing with the structure and evolution of the universe as a whole, a capability not shared by Newtonian theory (unless special assumptions are made). However, this capability is more a consequence of the Einstein Equivalence Principle (alternatively of the metric-theory postulates) than a property of general relativity. Because of EEP, spacetime is endowed with a metric g which determines the results of observations made using nongravitational equipment (light rays, telescopes, spectrometers, etc.) and the motion of test bodies (galaxies). Via the field equations provided by each metric theory of gravity, the distribution of matter then determines the metric g, and thereby the entire physical spacetime in which observations are made. By contrast, in Newtonian cosmology, space and time are fixed a priori, and one is faced either with the problem of specifying and interpreting the boundary of afiniteuniverse or with the mathematical problems associated with an infinite universe in Newtonian theory [see Sciama (1975) for a discussion of Newtonian cosmology].

Cosmological Tests

311

Despite the success of general relativity in treating the expansion of the universe, there remained doubts. Chief among these was the "timescale problem." The early values for the Hubble constant Ho, the ratio between recession velocity and distance, implied an age of the universe since the beginning of the expansion ("big bang") that was shorter than the estimated ages of the stars (from stellar evolution theory) and of the radioactive elements on the Earth. However, by the late 1950s, revisions in the extragalactic distance scale (increase by a factor of five) and the consequent reduction of the Hubble constant increased the age of the universe to a value greater than that of our galaxy, thus resolving the timescale problem. But the crucial confirmation of the big bang model came in 1965 with the discovery of the 3K cosmic microwave background radiation (Penzias and Wilson, 1965). This discovery implied that the universe was once much hotter and much denser than it is today [see Weinberg (1977) for a detailed account of the discovery and of its interpretation]. In particular, it made the steady-state theory of Bondi, Gold, and Hoyle untenable. It also made it possible to resolve the discrepancy between the observed cosmic abundance of helium (20-30% by weight) and estimates of the production of helium in stars (a few percent at most). Calculations by Peebles (1966) and by Wagoner, Fowler, and Hoyle (1967) of nucleosynthesis in a hot (109 K) big bang yielded helium abundances precisely within the observed range [for a review, see Schramm and Wagoner (1977)]. The hot big-bang model within general relativity is today the standard working model for cosmology [for reviews of general relativistic cosmology, see Peebles (1971), Weinberg (1972), MTW, Sciama (1975) and Zel'dovich and Novikov (1983)]. However, cosmological models within alternative theories of gravity have not undergone a systematic study with a view toward testing them in a cosmological arena. One reason is that in their exact, strong-field formulations, alternative theories are sufficiently different that it has not been possible to date to devise a general scheme, analogous to the PPN formalism, for classification, comparison, and confrontation with observations. Also, cosmological observations are not "clean" tests of gravitation since much "dirty" astrophysics often goes into their interpretation. But many alternative theories of gravity, even those whose postNewtonian limits are identical to or close to that of general relativity, are different enough in their full formulations that they may predict qualitatively different cosmological histories. These may be sufficiently different that observational data such as the mere existence of the microwave background or the observed abundance of helium, however imprecise,

Theory and Experiment in Gravitational Physics

312

may suffice to rule out some theories in spite of the astrophysical and observational uncertainties. Section 13.1 outlines the general approach to be used in building cosmological models in alternative metric theories of gravity. In Section 13.2, we present a brief and qualitative survey of what little is known at present about cosmology in such theories. 13.1

Cosmological Models in Alternative Theories of Gravity

We begin by making two important assumptions about the nature of the universe that should hold in any metric theory of gravity: Assumption 1: The Einstein Equivalence Principle (EEP) is valid. Assumption 2: The Cosmological Principle is valid. As we saw in Chapter 2, the validity of EEP is equivalent to the adoption of a metric theory of gravity. The cosmological principle states that the universe presents the same aspect to all observers at any fixed epoch of cosmic time, or equivalently, that the universe is homogeneous and isotropic, at least on large scales (~ 100 Mpc). The cosmological principle may be justified by noting the observations of isotropy of the universe, especially of the microwave background, and by assuming that we occupy a typical, not special place in the universe (Copernican principle). Neither of these two assumptions is open to much question (see, however, Ellis et al., 1978) although there has been considerable study of cosmological models within general relativity that, while approximately isotropic today, were highly anisotropic in the past (for a review, see MacCallum, 1979). Because of EEP, spacetime is endowed with a metric g in whose local Lorentz frames the nongravitational laws of physics take their special relativistic forms. The cosmological principle then demands that the line element of g must take the Robertson-Walker form (MTW, Section 27.6) ds2 =^gflvdxlidxv

= -dt2 + a(t)2[(l - kr2rldr2

+ r2(d62 + sin2 0 # 2 ) ]

(13.1)

where r, 9, and 4> are dimensionless coordinates, t is proper time as measured by an atomic clock at rest, a(t) is the expansion factor (units of distance), and k e {+1,0} is a constant. Each element of cosmic matter (galaxy) is assumed to be at rest in these coordinates. If k = 1, the universe is closed (i.e., has closed spatial sections), if k = 1, the universe is open, and if k = 0, the universe is open, with Euclidean spatial sections. The alternative form of the metric that was used in Section 4.1 to establish the asymptotically flat PPN metric can be obtained from this by making

Cosmological Tests

313

the transformation to the new radial coordinate r' given by r = {r'lao){l + kr'2IAaly"

(13.2)

where a0 is the value of a(t) at the present epoch. Although the present value of the scale factor a is difficult to measure, its rate of variation with time is subject to observation. In particular one defines the Hubble constant H o and the deceleration parameter q0 by H o = (d/a)0,

qo=-

H« 2(a/a)0

(13.3)

where a dot denotes d/dt and the subscript "0" denotes present values. These parameters may be measured by a variety of techniques, such as the magnitude-red-shift relation or the angular-size-red-shift relation for distant galaxies. The present "best" values for these parameters are H o ^ 60 ± 2 0 km s" 1 M p c ~ \

-l
(13.4)

The large uncertainty in q0 is a result of the uncertain effects of galactic evolution on the intrinsic luminosities of distant galaxies used as "standard candles" in magnitude-red-shift measurements. The validity of EEP also allows one to determine the behavior of the matter in the universe, independently of the theory of gravity. If we idealize that matter as a homogeneous perfect fluid, then the equations of motion Tfvv = 0 can be shown (MTW, Section 27.7) to yield the following equations for the evolution of the mass-energy density p(t) and the pressure p(t): P(t) = p mO |>oMt)] 3 + Pro[ao/a(t)T, Pit) = yrolao/a(t)Y

(13.5)

where pm0 and pr0 denote the present mass-energy densities of matter and radiation, respectively. These equations will be valid for temperatures less than about 1010 K, when the electrons and positrons annihilated. We now turn to an outline of the recommended method for obtaining cosmological models in any metric theory of gravity. Step 1: Use the cosmological principle to determine the mathematical forms in Robertson-Walker coordinates to be taken by all the dynamical and nondynamical fields of the theory. For the dynamical fields listed in Section 5.1, these forms are Metric:

ds2 = -dt2 + a(t)2da2,

Scalar:

0(r),

Vector:

K^dx* = K{t)dt,

Tensor:

B^dx"dxv

= co0(t)dt2 + (ox{t)do2

(13.6)

Theory and Experiment in Gravitational Physics

314

where da2 = (1 - /cr2)"1 dr2 + r2{d62 + s i n 2 0 # 2 )

(13.7)

For a nondynamical flat background metric r\ governed by the equation Riem(i/) = 0, the general form for its line element dy2 = tj^dx" dx" in Robertson-Walker coordinates is dSf2 = - i(t)2 dt2 + %{i)2 da2 2

dSf = -i(t)

2

2

dt + da

2

[k = - 1 ] , [k = 0]

(13.8)

where r(t) is a function of t, with i = dt/dr (as in Section 11.3, we shall ignore the possibility of "tipping" of the t] cones relative to the g cones). Note that there is no solution for the case k = 1. Thus, it is very unlikely that any theory of gravity with a flat background metric can have a closed (k = 1) cosmological model for the physical metric g. For a nondynamical cosmic time coordinate T, it is sufficient to assume that T = T(t). The matter variables have the form p = p(t),

p = p(t),

u" = (1,0,0,0)

(13.9)

Step 2: Substitute these forms into the field equations of the theory. Step 3: Set boundary conditions on the fields, in particular on their present values <j>0,K0,x0,a0, etc. These values are related in general to such measurable quantities as H0,q0, and k, as well as to the PPN parameters and the present rate of variation of G, or (G/G)o. Use the present experimental values or limits on these parameters to limit the class of cosmological models to be considered. Step 4: Integrate the field equations and the equations of motion backward in time (using numerical methods as a rule), taking into account possible changes in the equation of state for the matter variables as the universe becomes hotter and denser (see MTW, Section 28 for discussion). Step 5: The tests. Although cosmological data is sketchy and imprecise, there are two pieces of evidence about the early universe about which there seems to be little disagreement, the 3K cosmic microwave background radiation and the cosmic abundance of helium. (i) The microwave background: There is now general consensus that the microwave background is the relic of a hotter, denser phase of the universe, where the temperature exceeded 4 x 103 K. No reasonable mechanism has yet been devised to produce the background during later epochs (T < 4 x 103 K) that agrees both with the observed high degree of isotropy of the radiation [after the effects of the Earth's motion (see Section 8.2) have been subtracted] and with the close agreement of the

Cosmological Tests

315

spectrum with that of a black body. Prior to the epoch T = 4 x 103 K, a variety of physical processes are consistent with the observed background, ranging from recombination of electrons and protons to form hydrogen to the quantum evaporation of primordial "mini"-black holes (m < 1015 g). Thus, in order to predict cosmological models with the microwave background, the theory must guarantee that the universe evolved from a state with T>4 x 103 K (p/p0 > 109, a/a0 < 10~3). An example of an unviable cosmological model would be one that contracts from some earlier dispersed state to a maximum density and temperature below the above limits, then bounces and reexpands to the present observed state. Such a model would contain no reasonable explanation for the microwave background. A class of models in Rosen's bimetric theory has this property (see Section 13.2). (ii) The helium abundance: It is also generally believed that stellar nucleosynthesis can account for only a small fraction of the observed 20-30% abundance by weight of helium, and thus, most of the helium was produced in the early universe. Similar claims have been made for the deuterium abundance (observed to be ~2 parts in 105 by weight), but in this case the contributions of galactic production (and destruction) and of chemical fractionation are more uncertain, so we shall focus on helium [see Schramm and Wagoner (1977)]. Primordial nucleosynthesis requires temperatures in excess of 109 K and baryon number densities > 10~6 cm" 3 , and therefore a viable cosmological model must predict a state at least this hot and dense. Furthermore, the fraction of helium produced is sensitive to the rate of expansion of the universe at the epoch of nucleosynthesis. The reason is as follows: when nucleosynthesis occurs, essentially all the neutrons go into helium nuclei, so the abundance of helium depends only on the neutron-proton abundance ratio at the time tN of nucleosynthesis, i.e., X(He4) = 2(n/p)(l + n/p)-%N

(13.10)

where X denotes the mass fraction and n/p is the neutron-proton density ratio. This ratio n/p is determined by two factors. First is the (n/p) ratio at the moment ("freeze out") when weak interactions are no longer fast enough to maintain the neutrons and protons in chemical equilibrium; at freeze out their ratio is thus given by (n/p)F exp[(mn mp)//c7V], where mn and mp are the proton and neutron rest masses and TF is the temperature at freeze out. Second is the interval of time between freeze out and nucleosynthesis, during which the neutrons undergo free decay. The faster the expansion rate at a given temperature, the earlier the

Theory and Experiment in Gravitational Physics

316

weak interactions freeze out, thus TF is higher and (n/p) is closer to unity. In addition, the time between freeze out and nucleosynthesis is shorter and fewer neutrons decay. The result is a higher abundance of helium. The opposite occurs for a lower expansion rate. In some cosmological models, the expansion rate during nucleosynthesis can be expressed phenomenologically as a^da/dt^ZQnp)1'2

(13.11)

where ^ is a parameter whose value is 1 in the standard model of general relativity, and p is the total mass-energy density. The resulting helium abundance is given approximately by X(He4) ~ 0.26 + 0.38 log £

(13.12)

for a present density p o ~ 1 0 ~ 3 O g m c m ~ 3 (see Schramm and Wagoner, 1977, for discussion). Thus, a value of £ greater than about 3 or less than about 5 would do serious violation to observed helium abundances. Since the scale a{t) of the universe was 109 times smaller at this epoch than at present, this is a very restrictive result for a generic theory of gravity. Other possible tests of cosmological models, such as the question of galaxy formation or the problem of the observed ratio of the number of photons to the number of baryons (nY/nb ~ 108) are so poorly understood within general relativity that they are unlikely to be useful tools for testing alternative theories in the foreseeable future. 13.2

Cosmological Tests of Alternative Metric Theories of Gravity For specific theories of gravity, results for the confrontation between theory and cosmology are sparse. No systematic study of cosmological models in alternative theories has been carried out, and of those analyses that have been performed within specific theories, few have addressed such questions as the microwave background and the helium abundance. Thus, we shall confine ourselves to a brief list, without details and largely without comment, of those few results that are known.

General relativity The "standard big bang model" (MTW, Section 28) agrees at least qualitatively with all observations, although there remain problems when one pushes for more precision or more detailed comparison with observation such as galaxy formation, the photon-to-baryon ratio puzzle, the initial singularity, the value of k, the abundances of deuterium and the other light elements, the mean density of the universe, and so on.

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317

Brans-Dicke theory Several computations (Greenstein 1968, Weinberg 1972) have shown that a wide class of cosmological models in Brans-Dicke theory are in qualitative agreement with all observations, including the helium abundance. The models begin from a singular big bang as in general relativity, one difference being the uncertainty in the boundary condition to be placed on the scalar field (f> at t = 0. However, choices can be made for this boundary condition that yield results similar to those of general relativity for similar values of the present uncertain matter density p0. Moreover, the larger the value of a>, the closer the agreement with general relativity. In all cases, the present value of G/G is below the experimental uncertainty (see Chapter 8). Bekenstein's variable-mass theory (VMT) By contrast with Brans-Dicke theory, the VMT can have cosmological models that begin the expansion from a nonsingular "bounce" (which presumably was preceded by a contraction phase). Bekenstein and Meisels (1980) have studied a variety of such models that satisfy the following constraints: at the initial moment of expansion, /($) is small (Equation 5.40), i.e., co{4>) ~ § (required for the model to start from a minimum radius), and a c± 1016-1017 cm (appropriate for initial temperatures of ~ 101 * K). After numerical integration of the field equations for a variety of values of the curvature parameter k and the arbitrary constants r and q (see Chapter 5), they reached the following conclusions: (i) Although the initial value of a> was quite small, its present value in many models exceeded 500, thus yielding close agreement with all experimental tests, and with the predictions of general relativity for neutron stars, black holes, gravitational waves, the binary pulsar, etc. (ii) The gravitational constant G decreased by between 36 and 40 orders of magnitude between the initial moment and the present, thereby accounting for the "large number" puzzle that Gm^/hc^ 10" 38 , where mp is the proton mass. Because of the large variation in G, this ratio was initially near unity, (iii) Despite the large variation in G, the present value of G/G, in most cases, was well below the experimental upper limits. Because the universe in these models began from a hot, dense (though nonsingular) state, it permits origins for the cosmic microwave radiation as naturally as does general relativity. However, the helium abundance remains an open question at this writing. At the time of nucleosynthesis (T ~ 109 K) the expansion rate would have been very different from that of general relativity, since ca was very small then (perhaps of order f). Only a

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318

detailed computation can determine whether there are VMT cosmological models that are consistent with the helium abundance. Rosen's bimetric theory Because the theory has a flat background metric i/, there are no closed (k = 1) cosmological models. The Euclidean (k = 0) models have been studied by Babala (1975) and by Caves (1977). There are only two classes of models that have a physically reasonable expansion phase. One class expands from a singular state at a finite proper time in the past. These models make the definite prediction {G/G)o ;> 0.51H0[l + 3Om0(l + «2o)~']

(13.13)

where Qm0 = 4npmO/3Hl, and a2o is the present value of the PPN parameter a2. Experiments (Chapter 8) place the limit |a20| « 1, and observations indicate Qm0 < 0.1 for Ho ^ 55 km s" 1 Mpc" 1 . This prediction could thus be tested by future measurements or limits on (G/G)o. The other class of models have a bounce at a minimum radius given by (Caves, 1977) amjao £ [1 + (1 + a 2 0 )/3Q m 0 r 2 £
(13.14)

too large to permit a natural origin of the microwave background. The open (k = 1) models have, among other possibilities, expansion from a singular state at finite proper time in the past, and a similar expansion from a singular state at an infinite proper time in the past (Goldman and Rosen, 1976). These models have not been meshed with the present values of the PPN parameters, Ho, or {G/G)o. Rosen (1978) has also studied models in which the background metric t\ is notflat,but rather corresponds to a spacetime of constant curvature. The helium abundance has not been studied in any models in the bimetric theory. Rastall's theory As in Rosen's theory, the presence of a flat background metric rules out closed (k = 1) cosmological models. Rastall (1978) has shown that the k = 0 models predict a contraction phase, a nonsingular bounce, then an expansion phase. However, the bounce occurs at a radius fl min/«o TS> t o ° large to provide an explanation of either the microwave background or the helium abundance. Although the results presented here are very sketchy, they illustrate an important lesson. For some theories of gravitation, cosmology may provide do-or-die tests. This applies particularly to theories whose

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319

predictions for present-day gravitational phenomena (post-Newtonian limit, neutron stars, gravitational waves, and present cosmological observations) are indistinguishable from those of general relativity, viz. the VMT. For such theories, gravitational effects in the early universe may be sufficiently different from those predicted by general relativity that the cosmic microwave background and the abundances of the light elements may help to determine the most viable theory of gravitation.

14 An Update

In this chapter, we present a brief update of the past decade of testing relativity. Earlier updates to which the reader might refer include "The Confrontation between General Relativity and Experiment: An Update " (Will, 1984), "Experimental Gravitation from Newton's Principia to Einstein's General Relativity" (Will, 1987), "General Relativity at 75: How Right Was Einstein?" (Will, 1990a), and "The Confrontation Between General Relativity and Experiment: a 1992 Update" (Will, 1992a). For a popular review of testing general relativity, see "Was Einstein Right?" (Will, 1986). 14.1

The Einstein Equivalence Principle

(a) Tests of EEP Several recent experiments that constitute tests of the Weak Equivalence Principle (WEP) were carried out primarily to search for a "fifth-force" (Section 14.5). In the "free-fall Galileo experiment" performed at the University of Colorado (Niebauer, McHugh and Faller, 1987), the relative free-fall acceleration of two bodies made of uranium and copper was measured using a laser interferometric technique. The "Eot-Wash" experiment (Heckel et al., 1989; Adelberger, Stubbs et al., 1990) carried out at the University of Washington used a sophisticated torsion balance tray to compare the accelerations of beryllium and copper. The resulting upper limits on q [Equation (2.3)] from these and earlier tests of WEP are summarized in Figure 14.1 Dramatically improved " mass isotropy " tests of Local Lorentz Invariance (LLI) (Section 2.4(b)) have been carried out recently using lasercooled trapped atom techniques (Prestage et al., 1985; Lamoreaux et al., 1986; Chupp et al., 1989). By exploiting the narrow resonance lines made

Theory and Experiment in Gravitational Physics 1

1

1

1

1

1

1

1

1

1

:T I -

321 i

Renner

10r-9 "

10r io

-

1free-Fall

-

Boulder Princeton

1

10"

10.-12 1 +«2) i

1900

i

EoMVash

|

LURE

Moscow 1

«i

i

I-

1

1920

1

t 1

1940

l

i

i

1960 1970

1

1

1980

1990

Year of experiment

Figure 14.1. Selected tests of the Weak Equivalence Principle, showing bounds on r/, which measures fractional difference in acceleration of different materials or bodies. Free-fall and Eot-Wash experiments originally performed to search for the fifth force. Hatched and dashed line show current bounds on t] for gravitating bodies (test of the Strong Equivalence Principle) from lunar laser ranging (LURE). possible by the suppression of atomic collisions in the traps, these experiments have all yielded extremely accurate results, quoted as limits on the parameter 3 [Equation (2.13)] in Figure 14.2. In the THeju framework 2 (Section 2.6), S = !], where c0 and ce are re1 = spectively the limiting speed of test particles and the speed of light. Also included for comparison is the corresponding limit on 5 obtained from Michelson-Morley type experiments. Recent advances in atomic spectroscopy and atomic timekeeping have made it possible to test LLI by checking the isotropy of the one-way propagation of light (as opposed to the round-trip speed of light, as tested in the Michelson-Morley experiment). In one experiment, for example (" JPL" in Figure 14.2), the relative phases of two hydrogen maser clocks at two stations of NASA's Deep Space Tracking Network were compared over five rotations of the Earth by propagating a light signal one-way along an ultrastable fiberoptic link connecting them (Krisher, Maleki et al.,

An Update

322 1

1

1

1

1

1

1

i

i

i

i

Michelson-Morley «

JPL Joos

TPAl

T 10"8

X,

-

Brillet-Hall

I

I

10"

Hughes-Drever 10"

10,-20

T t

8=

-

e

l

l

1900

NIST

l

J l

1920

lHarvard_ U. Washington

1

1 1940

1

1

1

1960 1970

1 1980

1 *

1 1990

Year of experiment

Figure 14.2. Selected tests of local Lorentz invariance showing bounds on parameter S, which measures degree of violation of Lorentz invariance in electromagnetism. Michelson-Morley, Joos, and Brillet-Hall experiments test isotropy of the round-trip speed of light, the later experiment using laser technology. Two-photon absorption (TPA) and JPL experiments test isotropy of the one-way speed of light. The remaining four experiments test isotropy of nuclear energy levels. Limits assume the speed of Earth is 300 km/s relative to the mean rest frame of the universe. 1990). In another ("TPA"), the isotropy of the Doppler shift was studied as a function of direction using two-photon absorption in an atomic beam (Riis et al., 1988). Although the bounds from these experiments are not as tight as those from mass-isotropy experiments, they probe directly the fundamental postulates of special relativity, and thereby of LLI. A number of novel tests of the gravitational redshift (Local Position Invariance) were carried out. The varying gravitational redshift of Earthbound clocks relative to the highly stable millisecond pulsar PSR 1937 + 21, caused by the Earth's monthly motion in the solar gravitational field around the Earth-Moon center of mass (amplitude 4000 km), has been measured to about 10 % (Taylor, 1987), and the redshift of stable oscillator clocks on the Voyager spacecraft caused by Saturn's gravitational field yielded a one percent test (Krisher, Anderson and Campbell, 1990). The

Theory and Experiment in Gravitational Physics

Pound-Rebka

10"

IT 10,-2

I

,-3

10"

'

-,- Null

T T Y

1

323

Millisecond Pulsar Solar

I

Redshift t Redshift Y Saturn

*

H-Masser

Y I960

1970 1980 Year of experiment

1990

Figure 14.3. Selected tests of local position invariance via gravitational redshift experiments, showing bounds on a, which measures degree of deviation of redshift from the formula Av/v = AU/c2. solar gravitational redshift has been tested to about 2 % using infrared oxygen triplet lines at the limb of the Sun (LoPresto, Schrader and Pierce, 1991). Figure 14.3 summarizes the bounds on a [Equation (2.21)] that result from these and earlier experiments. It is now routine to take redshift and time-dilation corrections into account in making comparisons between timekeeping installations at different altitudes and latitudes, and in navigation systems, such as the NAVSTAR Global Positioning System, which use Earth-orbiting atomic clocks. (b) The c2 formalism The THefi formalism (Section 2.6) can be applied to tests of local Lorentz invariance, but in this context it can be simplified (Haugan and Will, 1987; Gabriel and Haugan 1990). Since most such tests do not concern themselves with the spatial variation of the functions T, H, e, and fi, but rather with observations made in moving frames, we can treat them as spatial constants. Then by rescaling the time and space coordinates, the charges and the electromagnetic fields, we can put the THefi action in Equation (2.46) into the form

f (1 - vy2dt +£ea

>

-c2B2)d4x,

(14.1)

An Update

324

where c2 = // 0 /r o e 0i u 0 = cl/cl. This amounts to using units in which the limiting speed c0 of massive test particles is unity, and the speed of light is c. If c # 1, LLI is violated; furthermore, the form of the action above must be assumed to be valid only in some preferred universal rest frame. The natural candidate for such a frame is the rest frame of the cosmic microwave background. The electrodynamical equations which follow from Equation (14.1) yield the behavior of rods and clocks, just as in the full THsfi formalism. For example, the length of a rod moving through the rest frame with velocity V in a direction parallel to its length will be observed by a rest observer to be contracted relative to an identical rod perpendicular to the motion by a factor 1 V2/2 + O(VA). Notice that c does not appear in this expression. The energy and momentum of an electromagnetically bound body which moves with velocity V relative to the rest frame are given by E = M R + ^M R F 2 + ^ M f F'F^,

(14.2a)

p> = M R V + 8M\V\ s

(14.2b) s

where M R = MoE% , Mo is the sum of the particle rest masses, E# is the electrostatic binding energy of the system, and SMI is the anomalous inertial mass tensor, given by

SMf = - 2 [ ~ llgfif^+JS 8 *!,

(14.3)

where ^

E

(14.4a) (14.4b)

ab

Note that (c~2 1) here corresponds to the parameter S plotted in Figure 14.2. The electromagnetic field dynamics given by Equation (14.1) can also be quantized, so that we may treat the interaction of photons with atoms via perturbation theory. The energy of a photon is ft times its frequency co, while its momentum is fico/c. Using this approach, one finds that the difference in round-trip travel times of light along the two arms of the interferometer in the Michelson-Morley experiment is given by L0(v2/c)(c~2 1). The experimental null result then leads to the bound on

Theory and Experiment in Gravitational Physics

325

(c~2 1) shown on Figure 14.2. Similarly the anisotropy in energy levels is clearly illustrated by the tensorial term in Equation (14.2a); by evaluating £|Sl> for each nucleus in the various Hughes-Drever-type experiments and comparing with the experimental limits on energy differences, one obtains the extremely tight bounds also shown on Figure 14.2. The behavior of moving atomic clocks can also be analysed in detail (Gabriel and Haugan, 1990), and bounds on (c~2 1) can be placed using results from tests of time dilation and of the propagation of light (Riis et al., 1988; Krisher, Maleki et al., 1990; Will, 1992b). The bound obtained from the " J P L " test of the isotropy of the one-way speed of light (see below) was based on the prediction for the time dilation of hydrogen maser clocks (Gabriel and Haugan, 1990) namely ,

(14.5)

where a = -f(l-c2).

(14.6)

(c) Kinematical frameworks for studying LLI There are a number of frameworks for studying tests of special relativity (or of LLI) that are kinematical in nature, dating back to H. P. Robertson [see Haugan and Will (1987) and Mac Arthur (1986) for recent reviews]. One particularly useful version was developed by Mansouri and Sexl (1977a,b,c) (see also Abolghasem, Khajehpour and Mansouri, 1988, 1989). It assumes that there exists a preferred universal reference frame E:(7", X) in which the speed of light is isotropic (with unit speed in the appropriate units). The transformation between £ and a moving inertial frame S:{t, x) is given by T=a-\t-e-x), l

i

i

(14.7a)

X = d- x-(d~ -b- )yvxY//w

2

+ <wT,

(14.7b)

where w is the velocity of the moving frame, a, b, and dare functions of w2, and E is a vector determined by the procedure adopted for the global synchronization of clocks in S. In special relativity, the functions a, b, and a? have the special forms a'1 = b = y = (1 w2)'1'2, and d = 1, but E can be arbitrary, depending upon the procedure for synchronization; with either Einstein (round-trip light signals) or clock-transport synchronization, £ = w.

In the low-velocity limit, it will be useful to expand the functions a, b, d,

An Update

326

and E in powers of w2 using arbitrary parameters. Adopting a slightly different convention from Mansouri and Sexl, we write

a(w)x, l+(a-4)w 2 + ( a 2 - 4 K + . . . , b(w)x l+0?+i)w 2 + 0?2+f)w4 + ...,

(14.8a) (14.8b)

d(w) xl+Sw2

(14.8c)

+ S2w4 + ..., ..

(14.8d)

In SRT, a, a2, fi /?2, 8 and <S2 all vanish, and with standard synchronization, so do s and e2. The physics that results from experiments should not depend on the synchronization procedure, except measurements which depend on a direct, one-time comparison of separated clocks. Thus a measurement of the absolute value of the speed of light in S by a time-of-flight technique between two points will depend on the synchronization of the two clocks (a particularly perverse choice of synchronization can make the apparent speed between those points infinite, for example). However, a study of the isotropy of the speed between the same two clocks as the orientation of the line connecting them varies relative to £ should not depend on how they were synchronized, as long as they were synchronized by some procedure initially. Similarly, a measurement of the Doppler shift of an atomic spectral line using a single "clock" as receiver of the signal should not depend on synchronization, provided that the velocity of the atom is expressed in terms of observables measured by a single clock. This point has been misunderstood by numerous authors who have argued against the efficacy of tests of the one-way speed of light. An advantage of the Mansouri-Sexl framework is that it allows one to understand explicitly the role of synchronization in a given experiment. A disadvantage of this and similar kinematical frameworks is that they do not allow for the dynamical effects revealed by the c2 framework. Thus, the transformation of Equation (14.7) must be understood as being based on measurements made by a standard rod and a standard atomic clock. Measurements made using different rods or clocks would not yield the same relationships between the two frames. Nevertheless, for some experiments, such as the JPL experiment or the two-photon-absorption (TPA) experiment which involved only a single type of atom or atomic clock and the propagation of light, the Mansouri-Sexl formalism can be put to good use (Will, 1992b). In the JPL experiment, for example, the phases of two hydrogen maser oscillators of frequency v separated by a baseline of L = 21 kilometers were

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327

compared by propagating a laser carrier signal along a fiberoptic link connecting them. The phase comparisons could be performed simultaneously at each end using signals propagated in both directions along the fiber. The phase differences were monitored over afive-dayperiod as the baseline rotated relative to the Earth's velocity w through the cosmic microwave background. The predicted phase differences as a function of direction are, to first order in w A^«2a(wn-wn0),

(14.9)

where $ = 2nvL, and where n and n,, are unit vectors along the direction of propagation of the light, at a given time, and at the initial time, respectively. The initial phase difference has been set arbitrarily to zero; this is tantamount to choosing a convention for synchronization. The observed limit on a diurnal variation in the relative phase resulted in the bound |a| < 1.8 x 1(T4; this gives a limit on (c~ 2 -l) using Equation (14.6). The bound from the TPA experiment was |«| < 1.4 x 10~6. The best bound from such isotropy experiments comes from "Mossbauer-rotor" experiments (Champeney, Isaak and Khan, 1963; Isaak, 1970), which test the isotropy of time dilation between a gamma ray emitter on the rim of a rotating disk and an absorber placed at the center; the result is |a| < 9 x 10~8. (d) Other frameworks for analysing EEP A number of alternative formalisms have been developed to analyse EEP and Schiff's conjecture in detail. Ni (1977) devised an extension of the THe/i formalism in which the action for test particles and electromagneticfieldscouples minimally to a metric gm, but in which there is an additional electromagnetic coupling to a scalar field of the form (167T)-1 J \/(-g)£"v'"'FllvFpacl4x. EEP is satisfied if and only if s 0. On the other hand, electromagnetically bound test bodies satisfy WEP, but experience anomalous torques if
/dt < 0.1 Ho, where Ho is the Hubble parameter was set by showing that this electromagnetic coupling would cause rotations in the plane of polarization of radiation from distant radio sources, which are not observed (Carroll and Field, 1992). Ni (1987) has extended this formalism to incorporate non-abelian gauge fields. Bekenstein (1982) focussed on a particular model for violation of EEP: a coupling of electromagnetism to a dynamical, dimensionless scalar field that manifests itself as a spacetime variation of thefinestructure constant. The dynamics of the scalar field is determined more or less uniquely by a

An Update

328

set of reasonable postulates together with the requirement that the fundamental scale that determines its dynamics be of the order of but no smaller than the Planck scale (Gh/c3)1'2 x 10~33 cm. He found, however, that the spatial variation of the field is so severely constrained by the Eotvos experiment that the length scale must be smaller than 10~3 Planck lengths. This, he argued, rules out any variability of the fine structure constant. Coley (1982, 1983a,b,c) studied an extension of the THe/i formalism to non-metric theories that possess both a metric and an independent affine connection, retaining the restriction to static, spherically symmetric (SSS) fields. The model contains seven independent functions, whose forms can be constrained by various experimental tests of EEP. Horvath et al. (1988) extended the THe/x formalism to include weak interactions in a "gravitationally modified" standard model. Such a formalism could be used to calculate explicitly the possible WEP-violating effects of weak interactions, which were only estimated by Haugan and Will (1976) (see also Fischbach et al., 1985; Lobov, 1990). (e) Is spacetime symmetric ? Our statement of the metric theory postulates included the assumption that the metric is symmetric, corresponding to a standard pseudo-Riemannian spacetime. It turns out that a nonsymmetric metric, even if coupled to matter fields in a universal way, does not satisfy the postulates of EEP (Will, 1989; Mann and Moffat, 1981). Consider a class of theories in which the action for charged test particles and electromagnetic fields coupled to gravity is given by the "minimally coupled" form of Equation (3.20) where now gm =£ gyft and g1" is the inverse of gm such that gM*gm = g^g^ = S"v. (Mann, Palmer and Moffat (1989) and Gabriel et al. (1991a) consider a broader class of electromagnetic actions, but the minimally coupled version illustrates the essential features.) We consider nonsymmetric theories having the property that, in an SSS gravitational field, a Cartesian coordinate system can be found in which the nonsymmetric g^ takes the form 9W = -T(r),

g^H^Sij,

gm = -ga = L(r)n,

(14.10)

where T, H, and L are functions of r = |x|, «,. = xjr. The inverse of gm is given by gm = g~l (14.11)

Theory and Experiment in Gravitational Physics

329

where (-g) = H3T{\~L2/HT). Substituting into Equation (3.20), and identifying Fm = Et and Ftj = eijkBk, we obtain

/ = - ! % f(f-W2«fc + £ea UtfA a

J

a

J

+ ^- [{eE2-n-\B2-co{n-Bf]}dix, ore J

(14.12)

where e = (H/T)ll2(\ -L2/HTyw, co = L2/HT.

ft = (/f/T)1/2(l

-L2/HT)i/2, (14.13)

Apart from the term co(n B)2, this action is that of the THefi formalism. The condition for validity of EEP, s = fx = (H/T)v2 for all r is violated by the action (14.12), if L ^ 0 (nonsymmetric metric). For example, in the THs/i formalism, the acceleration of an electrically neutral, composite body of charged particles with total mass M and internal electrostatic energy £ ES is given by Equation (2.117), dropping the magnetostatic terms. In order to apply this directly to nonsymmetric theories, it suffices to show that the to(n-B)2 term in Equation (14.12) makes no contribution to a, to electrostatic order. This can be shown by direct calculation, extending the Lightman-Lee procedure appropriately; it can be seen heuristically by noting that, to the required order, 0[g(EES/M)], the only part of the vector potential A that results in a contribution to the acceleration of the composite body is that part produced by the acceleration of each charged particle in the external gravitational field. This part of A is therefore parallel to g, and thus to n, and hence the relevant part of n B vanishes; as a consequence, the a>(n B)2 term will have no effect, to the electrostatic order considered. Higher-order magnetostatic effects will result from that term, but, as we saw in Section 2.4(a), these are significantly smaller than electrostatic effects. For systems that move through the SSS field with velocity V, the co(n B)2 terms will also produce effects of order V2EEB/M (Gabriel et al., 1991b). The given forms of s and fi imply that T^H^E^ = 1. Assuming that 7"« 1 + 0(m/r), Hx\+ O(m/r), and L2 <4 TH, where m is the mass of the external source, we obtain from Equations (14.13) and (2.83), To as (r2/2m)dL2/dr.

(14.14)

Thus nonsymmetric theories in this class violate WEP, and consequently, Eotvos experiments can test their validity. The significance of the resulting

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constraints on the nonsymmetric part of the metric will depend on the specific form of L(r). In one nonsymmetric theory, Moffat's NGT (Moffat 1979a,b, 1987, 1989; Moffat and Woolgar, 1988; for a recent review see Moffat, 1991), L(r) = / 2 /> 2 , where / 2 is a parameter (which can be negative) defined by

e1 = f^/(-g)S°d3x,

where S" is a conserved current (W(-g)S")i/1 = 0)

of hitherto unspecified microscopic origin, and the integral is over the gravitating source. Thus, in this theory, with the minimal coupling of Equation (3.20), r o = 2^/mr3. However, because of an additional matter coupling in the Lagrangian of NGT, there is an extra WEP-violating term in the gravitational acceleration of a body that depends on the value of its ^-parameter, namely, <5a = g(2/ 2 /r 3 )(/ b 2 /M), where £ refers to the source and
Thus the constraint placed on NGT will depend on the model adopted for the f2 parameter. For bulk, electrically neutral, stable matter consisting of neutrons, protons, and electrons, it is straightforward to show that the most general form of / 2 is f2 =fB2B+fL2L, where B and L are the total baryon and lepton numbers of the body, and / B 2 and fL2 are arbitrary coupling parameters (which can be negative) having units of (length)2. Thus tests of WEP will constrain the fB2 /L2 plane. Because of the r~3 dependence in Equation (14.15), the most sensitive tests use the Earth as the gravitating source, and for this purpose, the Eot-Wash III experiment (Adelberger, Stubbs et al., 1990) is the most stringent. We determine EJM, B/M and L/M for each of the test masses in this experiment, and we note that, for the Earth, L9 « B^/2.05. The experimental limits from E6t-Wash III then provide the constraints on the coupling parameters shown in Figure 14.4. With the coupling parameters constrained by the rough bound 2 x 10"44 cm2, we obtain the limit \£92\ < (100 m)2. It should be noted that this result applies only to the minimally coupled electromagnetic action. Mann, Palmer and Moffat (1989) have presented an alternative class of couplings of F^ to the nonsymmetric metric, one of which satisfies WEP to electrostatic order, and thus evades the bound given above. In this model, e = ju = (H/T)"2, and so the only EEP-violating effects come from the o>(n-B)2 term in Equation (14.12). In fact, this (n B)2 term is generic to all nonsymmetric theories, and has important observable consequences. It will produce perturbations in the

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331

-

-A -

Figure 14.4. Constraints on j \ and J\ of minimally coupled Moffat NGT from the Eot-Wash III experiment. The hatched region is excluded. energy levels of an atomic system that depend on the orientation of the system's wave function relative to the direction n (anisotropies in inertial mass). Such perturbations can be constrained by energy-isotropy experiments of the type used to test local Lorentz invariance (Gabriel et al., 1991b). The violation of EEP by this term also produces observable effects in the propagation of light, such as polarization dependence in the propagation of light near the Sun (Gabriel et al., 1991c). One consequence of this is a depolarization of the Zeeman components of spectral lines emitted by extended, magnetically active regions near the limb of the Sun; observations of the residual polarization of such lines place the stringent bound Ko2| < (535 km)2, substantially smaller than the values preferred by Moffat (1991). 14.2

The PPN Framework and Alternative Metric Theories of Gravity

The PPN framework of Chapter 4 is the standard tool for studying experiments and gravitational theories in the weak-field slow motion limit appropriate to the solar system. Other versions of the PPN formalism have been developed to deal with bodies with strong internal gravity (Nordtvedt, 1985), and post-post-Newtonian effects (Epstein and Shapiro, 1980; Fischbach and Freeman, 1980; Richter and Matzner, 1982a,b; Nordtvedt,

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1987; Benacquista and Nordtvedt, 1988; Benacquista, 1992). A version of the formalism with potentials substantially more complicated than the canonical version has also been proposed (Ciufolini, 1991). Despite the experimental bound of co > 500 on the coupling constant of Brans-Dicke theory, variants of the theory became popular again during the 1980s, as a result of developments in cosmology and elementaryparticle physics. Inflationary models of cosmology involving Brans-Dickelike scalar fields coupled to gravity have been developed and studied in detail. Scalar fields coupled to gravity or matter are also ubiquitous in particle-physics-inspired models of unification, such as string theory. In many models, the coupling to matter leads to violations of WEP, which can be tested by Eotvos-type experiments. In many models the scalar field is massive; if the Compton wavelength is of macroscopic scale, its effects are those of a "fifth force" (see Section 14.5). Only if the theory can be cast as a metric theory with a scalar field of infinite range or of range long compared to the scale of the system in question (solar system) can the PPN framework be applied. If the mass of the scalarfieldis sufficiently large that its range is microscopic, then, on solar-system scales, the scalar field is suppressed, and the theory is typically equivalent to general relativity. In any event, the bounds from solar system experiments can provide constraints on such speculations. The post-Newtonian limit of a class of massive scalar-tensor theories, including the Yukawa potentials that result from the massive scalar field, was derived by Helbig (1991) and Zaglauer (1990). 14.3

Tests of Post-Newtonian Gravity (a) The classical tests

Improvements in the accuracy of very long baseline interferometry (VLBI) to the level of hundreds of microarcseconds made new tests of the deflection of light possible. For example, a series of transcontinental and intercontinental VLBI quasar and radio galaxy observations made primarily to monitor the Earth's rotation ("VLBI" in Figure 14.5) was sensitive to the deflection of light over almost the entire celestial sphere (at 90° from the Sun, the deflection is still 4 milliarcseconds). The data yielded a value 5(1+7)= 1.000 + 0.001, comparable to the Viking test of the Shapiro time delay (Robertson and Carter, 1984; Robertson, Carter and Dillinger, 1991; Shapiro, 1990). A measurement of the deflection of light by Jupiter using VLBI was recently reported (Truehaft and Lowe, 1991); the predicted deflection of about 300 microarcseconds was seen with about 50% accuracy.

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'ft' 1.2

T

1919 Expedition

333

Deflection of light optical o radio

1.1 VLBI

VLBI

1.0

0.9 Expedition

1.1

PSR 1937 + 21 Viking IVoyager JVo

1.0 Shapiro time delay 4 two-way D one way

0.9

0.8

-

J L _L J_ 1920 1930 1940 1950 1960 1970 1980 Year of experiment

1990

Figure 14.5. Measurements of the coefficient (l + y)/2 from light deflection and time delay measurements. The general relativity value is unity. Arrows denote anomalously large values from 1929 and 1936 expeditions. Shapiro time-delay measurements using Viking spacecraft and VLBI light deflection measurements yielded agreement with general relativity to 0.1 per cent. Recent" opportunistic " measurements of the Shapiro time delay include a measurement of the one-way time delay of signals from the millisecond pulsar PSR 1937 + 21 (Taylor, 1987), and measurements of the two-way delay from the Voyager 2 spacecraft (Krisher, Anderson and Taylor, 1991). The results for the coefficient 5(1 + y) of all light deflection and timedelay measurements performed to date are shown in Figure 14.5. Continued radar ranging to Mercury and the other planets has resulted in further improvements in the measured perihelion shift of Mercury. After the perturbing effects of the other planets have been accounted for, the excess shift is now known to about 0.1 % (Shapiro 1990) with the result that & = 42"98 (1.000 + 0.001)^' [see Equation (7.55)]. [For an amusing history of how the theoretical value of 42"98 has been misquoted in

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numerous books, including the first edition of this book, see Nobili and Will (1986).] In addition, the controversy over the solar quadrupole moment may be approaching a resolution. Beginning around 1980, the observation and classification of modes of oscillation of the Sun have made it possible to obtain information about its internal rotation rate, thereby constraining the possible centrifugal flattening that leads to an oblateness; current results favor a value J2 as 1.7 x 10~7 (Brown et al., 1989), making the correction to Xp [Equation (7.55)] from the solar quadrupole moment smaller than the experimental error. If further studies of solar oscillations continue to support this interpretation, the perihelion shift of Mercury will once again be a triumph for general relativity. (b) Parametrized post-Newtonian ephemerides Improvements in the accuracy of planetary and spacecraft tracking and in the ability of theorists to model their motions has made it useful to adopt a slightly different attitude toward tests such as the time delay and the perihelion shift. As we remarked in Section 7.2, the measurement of the time delay of light involves a multiparameter leastsquares fit of tracking data to a model for the trajectory of the planet or spacecraft and for the propagation of the radar signal. The " time delay " as a distinct phenomenon is never measured directly. Similarly the "perihelion shift" of Mercury is not observed, rather the least-squares method estimates various parameters (ft, y, J2, etc.) that determine part of the shift. Although this point of view takes some of the glamour out of the subject, it is the standard approach in the analysis of relativistic solarsystem dynamics. The goal is to determine the parameters in a model for the relativistic motion of bodies in the solar system. One might call this model a "parametrized post-Newtonian ephemeris". The current model (Hellings, 1984; Reasenberg, 1983) includes such parameters as: (i) the initial positions and velocities of the nine planets and the Moon; (ii) the masses of the planets, and of the three asteroids Ceres, Pallas and Vesta; (iii) the mean densities of 200 of the largest asteroids whose radii are known; (iv) the Earth-Moon mass ratio; (v) the value of the astronomical unit; (vi) PPN parameters, y, fi, a,,...; (vii) J2 of the Sun; (viii) other parameters relevant to specific data sets, such as station locations, rotation and libration of bodies, known systematic errors or corrections, etc. The model also includes PPN equations of motion for the bodies, and PPN equations for the propagation of the tracking signal. In some applications, the model also includes equations that tie the coordinate system associated with the

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ephemerides to a system tied to distant stars via VLBI. The output of the model might, for example, be a predicted " range " (round-trip travel time) from a particular station to a planet or spacecraft at a particular epoch, as a function of the parameters. The parameters are then adjusted in the leastsquares sense to minimize the difference between the predicted and observed ranges. One circumstance that has made it possible to obtain improved determinations of the parameters is the ability to combine different data sets in an unambiguous way. In the orbit of Mercury, the effects of /?, y and J2 are large, but not separable using Mercury radar data alone. In the orbit of Mars, their effects are much smaller (and that of J2 smaller still than that of fi and y), but the accuracy of Viking lander ranges is so much better that the effects can be seen more clearly than with Mercury data. Lunar laserranging data has also been incorporated into the data set. In the coming years, analysis of PPN ephemerides will further improve our knowledge of PPN parameters, J2, and the dynamics of the solar system (for reviews, see Kovalevsky and Brumberg, 1986; and Soffel, 1989). (c) Tests of the strong equivalence principle

Recent analyses of lunar laser-ranging data continue to find no evidence, within experimental uncertainty, for the Nordtvedt effect (Section 8.1). Their results for n [Equation (8.9)] are n = 0.003+0.004, (Dickey et al., 1989) n = 0.000 ± 0.005, (Shapiro, 1990) n = 0.0001 ±0.0015, (Muller et al., 1991)

(14.16)

where the quoted errors are \a, obtained by estimating the sensitivity of n to possible systematic errors in the data or in the theoretical model. The third of these results represents a limit on a possible violation of WEP for massive bodies of 7 parts in 1013 (compare Figure 14.1). For Brans-Dicke theory, these results force a lower limit on the coupling constant co of 600. Nordtvedt (1988a) has pointed out that, at this level of precision, one cannot regard the results of lunar laser ranging as a clean test of SEP because the precision exceeds that of laboratory tests of WEP. Because the chemical compositions of the Earth and Moon differ in the relative fractions of iron and silicates, an extrapolation from laboratory Eotvos-type experiments to the Earth-Moon system using various nonmetric couplings to matter (Adelberger, Heckel et al., 1990) yields bounds on violations of WEP only of the order of 2 x 10"!2. Thus if lunar laser

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Table 14.1. Constancy of the gravitational constant Method Lunar Laser Ranging Viking Radar Binary Pulsar" Pulsar PSR 0655 + 64"

G/G (10-12 yr"1) 0+11 2+4

-2±10 11±11 <55

Reference Miiller et al. (1991) Hellings et al. (1983) Shapiro (1990) Damour and Taylor (1991) Goldman (1990)

" Bounds dependent upon theory of gravity in strong-field regime and on neutron star equation of state.

ranging is to test SEP at higher accuracy, tests of WEP must keep pace; to this end, a proposed satellite test of the equivalence principle (Section 14.4) will be an important advance. In general relativity, the Nordtvedt effect vanishes; at the level of several centimeters and below, a number of non-null general relativistic effects should be present (Mashhoon and Theiss, 1991; Gill et al., 1989; Nordtvedt, 1991). An improved limit on the "preferred frame" PPN parameter a, of 4x 10~4 was reported by Hellings (1984), from analyses of Mercury and Mars ranging data. Nordtvedt (1987) has placed an improved bound on the parameter <x2 of 4 x 10~7 by showing that the failure of conservation of angular momentum in a frame moving relative to the universe when a2 / 0 [Equations (4.104) and (4.114)] would lead to anomalous torques on the Sun that would cause the angle between its spin axis and the ecliptic to be arbitrarily far from its observed value. Improved observational constraints have recently been placed on G/G, using ranging measurements to Viking (Hellings et al., 1983; Shapiro, 1990), lunar laser-ranging measurements (Miiller et al., 1991), and pulsar timing data (Damour, Gibbons and Taylor, 1988; Goldman, 1990; Damour and Taylor, 1991). Recent results are shown in Table 14.1. The best limits on G/G come from ranging measurements to Viking. The combination of three factors: (i) extremely accurate range measurements made possible by anchoring of the landers and orbiters, (ii) the unexpectedly long lifetime of the spacecraft (Lander 2 survived for 6 years), and (iii) the ability to combine Viking data consistently with other data sets such as Mercury and Venus passive radar, Mariner 9 radar and lunar laser-ranging data, made it possible to look for G/G at levels below 10~u yr""1. The major factors limiting the accuracy of these estimates (and responsible in part for

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the difference between the two Viking estimates in Table 14.1, despite being based upon similar data sets) are the uncertainty in the masses and distributions of the asteroids, and the level of correlations among the many parameters to be estimated in the model. It has been suggested that radar observations of a Mercury orbiter over a two-year mission (30 cm accuracy in range) could yield A(G/G) ~ l O ^ y r 1 (Bender et al., 1989). Although bounds on G/G using solar-system measurements can be obtained in a phenomenological manner through the simple expedient of replacing G by G0 + G0(t10) in Newton's equations of motion, the same does not hold true for pulsar and binary pulsar timing measurements (Nordtvedt 1990). The reason is that, in theories of gravity that violate SEP, the "mass" and moment of inertia of a gravitationally bound body may vary with variation in G. Because neutron stars are highly relativistic, the fractional variation in the mass can be comparable to AG/G, the precise variation depending both on the equation of state of neutron star matter and on the theory of gravity in the strong-field regime. The variation in the moment of inertia affects the spin rate of the pulsar, while the variation in the mass can affect the orbital period in a manner that can add to or subtract from the direct effect of a variation in G, given by PJPb = -jG/G. Thus, the bounds quoted in Table 14.1 for the binary pulsar PSR 1913 + 16 and the pulsar PSR 0655 + 64 are theory dependent and must be treated as merely suggestive. (d) Tests of post-Newtonian conservation laws Of the five "conservation law" PPN parameters £ f2, £3, £4, and <x3, only three, C2> C3 and <x3, have been constrained directly with any precision. The bound |<x3| < 2 x 10~10 was obtained in Section 9.3 using pulsar timing measurements. A remarkable planetary test of Newton's third law was reported by Bartlett and van Buren (1986), leading to an improved constraint on £3 (Section 9.2). They noted that current understanding of the structure of the Moon involves an iron-rich, aluminum-poor mantle whose center of mass is offset about 10 km from the center of mass of an aluminum-rich, ironpoor crust. The direction of offset is toward the Earth, about 14° to the east of the Earth-Moon line. Such a model accounts for the basaltic maria which face the Earth, and the aluminum-rich highlands on the Moon's far side, and for a 2 km offset between the observed center of mass and center offigurefor the Moon. Because of this asymmetry, a violation of Newton's third law for aluminum and iron would result in a momentum nonconserving self-force on the Moon, whose component along the orbital

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direction would contribute to the secular acceleration of the lunar orbit. Improved knowledge of the lunar orbit through lunar laser ranging, and a better understanding of tidal effects in the Earth-Moon system (which also contribute to the secular acceleration) through satellite data, severely limit any anomalous secular acceleration, with the resulting limit K/«p)Fe

<4xlO- 12 .

(14.17)

The resulting limit on £3 is |C3| < 1 x 10~8. Data from the binary pulsar PSR 1913 + 16 have finally permitted a strong test of the post-Newtonian " self-acceleration" effect described in Section 9.3, Equation (9.42). Assuming a theory not too different from general relativity (but with the possibility of C2 # 0) so that we can use the accurate values for the pulsar and companion masses obtained from timing data (Section 14.6(a)), together with the observational bound on variations in the pulsar period \Pp\ < 4 x 10~30 s~' (Taylor and Weisberg, 1989; J. Taylor, private communication), we obtain from Equation (9.48) the bound \C2\ < 4 x 10~5 (Will, 1992c). (e) Other tests of post-Newtonian gravity A gyroscope moving through curved spacetime suffers a geodetic precession of its axis given by dS/dt = Q x S, where £2 = (7 + j)v x V{7, where v is the velocity of the gyroscope and U is the Newtonian gravitational potential of the source [Equation (9.5)]. The Earth-Moon system can be considered as a " gyroscope ", with its axis perpendicular to the orbital plane. The predicted geodetic precession here is about 2 arcseconds per century, an effect first calculated by de Sitter. This effect has now been measured to about 2 % using lunar laser-ranging data (Bertotti, Ciufolini and Bender, 1987; Shapiro et al., 1988; Dickey, Newhall and Williams, 1989; Shapiro, 1990). Current values or bounds for the PPN parameters are summarized in Table 14.2. 14.4

Experimental Gravitation: Is there a Future?

Although the golden era of experimental gravitation may be over, there remains considerable opportunity both for refining our knowledge of gravity, and for exploring new regimes of gravitational phenomena. Nowhere is the intellectual vigor and continuing excitement of this field more apparent than in the ideas that have been developed for experiments and observations to push us to the frontiers of knowledge.

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Table 14.2. Current limits on the PPN parameters Parameter

Experiment

Value or limit

Remarks

7

Time delay Light deflection

1.000 ±0.002 1.000 + 0.002

Viking ranging VLBI

Perihelion shift Nordtvedt effect

1.000 + 0.003 1.000 ±0.001

J2 = 10~7 assumed rj 4/?y 3 assumed

Earth tides Orbital preferred-frame effects

< 10~3

Gravimeter data

<4xlO"

Combined solar system data

a2

Earth tides Solar spin precession

<4xlO-4 <4xlO~ 7

Gravimeter data Assumes alignment of solar equator and ecliptic are not coincidental

a3

Perihelion shift Acceleration of pulsars

< 2 x 10~7 <2xlO~'°

Statistics of dP/dt for pulsars

V

Nordtvedt effect

< 1.5 xlO" 3

Lunar laser ranging

Self-acceleration Newton's 3rd law

<4xlO~ 5 < 10"8

Binary pulsar Lunar acceleration

i a,

r 4i

C3

" Here rj is a combination of other PPN parameters given by }j = 4/?y 3 y<J a,+3a2§£, |C2. In many theories of gravity, £, = a, = £,. = 0.

(a) GP-B and the search for gravitomagnetism According to general relativity, moving or rotating matter should produce a contribution to the gravitational field that is the analogue of the magnetic field of a moving charge or a magnetic dipole (for reviews of the " gravitoelectromagnetic " analogy for weak-field gravity, see Braginsky, Caves and Thorne, 1977; Ciufolini, 1989). Although gravitomagnetism plays a role in a variety of measured relativistic effects, it has not been seen to date, isolated from other post-Newtonian effects [Nordtvedt (1988b) has discussed the extent to which it has been seen indirectly]. The Relativity Gyroscope Experiment (Gravity Probe B or GP-B) at Stanford University, in collaboration with NASA and Lockheed Corporation, has reached the advanced stage of development of a space mission to detect this phenomenon directly, in addition to the geodetic precession discussed in Section 9.1 (Everitt et al., 1988). A set of four superconducting-niobiumcoated, spherical quartz gyroscopes will be flown in a low polar Earth orbit, and the precession of the gyroscopes relative to the distant stars will

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be measured. For a polar orbit at about 650 km altitude, the predicted secular angular precession rate is j(l + y + |a,) 42 x 10"3 arcsec/yr [Equation (9.11)]. The accuracy goal of the experiment is about 0.5 milliarcseconds per year. A full-size flight prototype of the instrument package has been tested as an integrated unit. Current plans call for a test of the final flight hardware on the Space Shuttle followed by a Shuttle-launched experiment a few years later. Another proposal to look for an effect of gravitomagnetism is to measure the relative precession of the line of nodes-of a pair of laser-ranged geodynamics satellites (LAGEOS), with supplementary inclination angles; the inclinations must be supplementary in order to cancel the dominant relative nodal precession caused by the Earth's Newtonian gravitational multipole moments (Ciufolini, 1989). This is a generalization of the van Patten-Everitt proposal involving pairs of polar-orbiting satellites described in Section 9.1. Current plans involve a joint project of NASA and the Italian Space Agency. A third proposal envisages orbiting an array of three mutually orthogonal, superconducting gravity gradiometers around the Earth, to measure directly the contribution of the gravitomagnetic field to the tidal gravitational force (Braginsky and Polnarev, 1980; Mashhoon and Theiss, 1982; Mashhoon, Paik and Will, 1989). (b) Space tests of the Einstein equivalence principle The concept of an Eotvos experiment in space has been developed, with the potential to test WEP to 10"17 (Worden, 1988). Known as the Satellite Test of the Equivalence Principle (STEP), the project is a joint effort of NASA and the European Space Agency. If approved, it could be launched in the year 2000. The gravitational redshift could be improved to the 10~9 level, and second-order effects and the effects of J2 of the Sun discerned, by placing a hydrogen maser clock on board Solar Probe, a proposed spacecraft which would travel to within four solar radii of the Sun (Vessot, 1989). (c) Improved PPN parameter values A number of advanced space missions have been proposed in which spacecraft orbiters or landers and improved tracking capabilities could lead to significant improvements in values of the PPN parameters (see Table 14.2), of J2 of the Sun, and of G/G. For example, a Mercury orbiter, in a two-year experiment, with 3 cm range capability, could yield improvements in the perihelion shift to a part in 104, in y to 4 x 10~5, in G/G to 10~14 y r 1 , and in J2 to a few parts in 108 (Bender et al., 1989).

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(d) Probing post-post-Newtonian physics in the solar system It may be possible to begin to explore the next level of corrections to Newtonian theory beyond the post-Newtonian limit, into the post-postNewtonian regime. One proposal is to place an optical interferometer with microarcsecond accuracy into Earth orbit. Such a device would improve the deflection of light to the 10~6 level, and could possibly detect the second-order term, which is of order 10 microarcseconds at the limb (Reasenberg et al., 1988). Such a measurement would be sensitive to a new " P P P N " parameter, which has not been measured to date. (e) Gravitational-wave astronomy A significant part of the field of experimental gravitation is devoted to designing and building sensitive devices to detect gravitational radiation and to use gravity waves as a new astronomical tool. This important topic has been reviewed thoroughly elsewhere (Thorne, 1987). 14.5

The Rise and Fall of the Fifth Force A clear example of the role of " opportunism" in experimental gravity since 1980 is the story of the "fifth force". In 1986, as a result of a detailed reanalysis of Eotvos' original data, Fischbach et al. (1986, 1988) suggested the existence of a fifth force of nature, with a strength of about a percent that of gravity, but with a range (as defined by the range A of a Yukawa potential, e~'ix/r) of a few hundred meters. This proposal dovetailed with earlier hints of a deviation from the inverse-square law of Newtonian gravitation derived from measurements of the gravity profile down deep mines in Australia [for a review, see Stacey et al. (1987)], and with ideas from particle physics suggesting the possible presence of very low-mass particles with gravitational-strength couplings [for reviews, see Gibbons and Whiting (1981), Fujii (1991)]. During the next four years numerous experiments looked for evidence of the fifth force by searching for composition-dependent differences in acceleration, with variants of the Eotvos experiment or with free-fall Galileo-type experiments. Although two early experiments reported positive evidence, the others yielded null results. Over the range between one and 104 meters, the null experiments produced upper limits on the strength of a postulated fifth force of between 10~3 and 10~6 the strength of gravity (Table 14.3). Interpreted as tests of WEP (corresponding to the limit of infinite-range forces), the results of the free-fall Galileo experiment, and of the Eot-Wash III experiment are shown in Figure 14.1 (Niebauer, McHugh and Faller, 1987; Adelberger, Stubbs et al., 1990). At the same time, tests of the inverse square law of

Table 14.3. Composition-dependent tests of the fifth force Experiment name or place

Year

Method

Palisades, NY Eot-Wash Boulder, CO Eot-Wash Index, WA Montana Paris Bombay Snake River Eot-Wash II Japan Florence Bombay II Irvine, CA Eot-Wash III Index, WA II Florence II Japan II

1986 1986 1987 1987 1987 1988 1988 1988 1988 1988 1988 1988 1989 1989 1989 1989 1989 1990

Flotation Torsion balance Free fall Torsion balance Torsion balance Torsion balance Beam balance Torsion balance Torsion balance Torsion balance Free fall Flotation Torsion balance Torsion balance Torsion balance Torsion balance Flotation Free fall

Substance compared

Source of force

Fifth force?

Cu/H 2 O Cu/Be Cu/U Be/Al Be/Al Cu/CH 2 Cu/Pb, C/Pb Cu/Pb C/Pb Be/Al Al/Cu, Al/C Plastic/H 2 O Cu/Pb Cu/Pb Cu/Be, Al/Be Cu/CH 2

Cliff Hillside Earth Hillside Cliff Hillside Lead, brass masses Lead masses Water in lock Lead masses Earth Mountain Lead masses Lead masses Hillside Cliff Mountain Earth

Yes No No No Yes No No No No No No No No No No No No No

Plastic/Hp Al/Cu, Al/C, Al/Be

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gravity were carried out by comparing variations in gravity measurements up tall towers or down mines or boreholes with gravity variations predicted using the inverse square law together with Earth models and surface gravity data mathematically "continued" up the tower or down the hole. Early experiments reported significant differences between predicted and observed gravity, but these were subsequently explained as resulting from systematic errors in the upward continuation results caused by insufficiently controlled biases in the distribution of surface gravity measurements, as well as by poorly-accounted-for effects of distant geological structures such as hills and ridges. Independent tower, borehole and seawater measurements now show no evidence of a deviation from the inverse square law (Thomas et al., 1989, Jekeli, Eckhardt and Romaides, 1990; Thomas and Vogel, 1990; Speake et al., 1990; Zumberge et al., 1991). The consensus at present is that there is no credible experimental evidence for a fifth force of nature. For reviews, see Fischbach and Talmadge (1989, 1992), Will (1990b), Adelberger et al. (1991); for a complete bibliography on the fifth force, see Fischbach et al. (1992). 14.6

Stellar-System Tests of Gravitational Theory (a) The binary pulsar and general relativity

The binary pulsar PSR 1913 + 16 has lived up to, indeed exceeded, all expectations that it would be an important new testing ground for relativistic gravity (Chapter 12). Instrumental upgrades at the Arecibo radio telescope where the observations are carried out, and improved data analysis techniques have resulted in accuracies in measuring times of arrival (TOA) of pulses at the 15 /us level. Analysis of this TOA data uses a timing model developed by Damour, Deruelle and Taylor (Damour and Deruelle, 1986; Damour and Taylor, 1992) superceding earlier treatments by Haugan, Blandford, Teukolsky and Epstein that were described in Section 12.1 [see Haugan (1985) and references therein]. The observational parameters of this model that are obtained from a least squares solution of the arrival time data fall into three groups: (i) nonorbital parameters, such as the pulsar period and its rate of change, and the position of the pulsar on the sky; (ii) five "Keplerian" parameters, most closely related to those appropriate for standard Newtonian systems, such as the eccentricity e and the orbital period Ph; and (iii) a set of "postKeplerian " parameters. Thefivemain post-Keplerian parameters are <<»>, the average rate of periastron advance; y, the amplitude of delays in arrival of pulses caused by the varying effects of the gravitational redshift and time dilation as the pulsar moves in its elliptical orbit at varying distances from

An Update

344

the companion and with varying speeds [denoted <$ in Section 12.1(d)]; Pb, the rate of change of orbital period, caused predominantly by gravitational radiation damping; and r and s = sin i, respectively the "range" and "shape" of the Shapiro time delay caused by the companion, where Us the angle of inclination of the orbit relative to the plane of the sky. In general relativity, these post-Keplerian parameters can be related to the masses of the two bodies and to measured Keplerian parameters by the equations (Section 12.2) <«> = 3(27t/Pb)5/3w2/3(l -e2Y\ i3

m

y = e(Pb/27iy rn2m- (\ +m2/m), Pb = -(192 K /5)(27rm/P b ) 5 ' 3 (^/m)(l + g e 2 + ||e 4 )(l

(14.18a) (14.18b) -e2)-1'2, (14.18c)

s^sini,

(14.18d)

r = m2,

(14.18e)

where m, and m2 denote the pulsar and companion masses, respectively, m = m, + m2 is the total mass, and n = mxm2/m is the reduced mass. The formula for > ignores possible non-relativistic contributions to the periastron shift, such as tidally or rotationally induced effects caused by the companion [Section 12. l(c)]. The formula for Pb represents the effect of energy loss through the emission of gravitational radiation, and makes use of the "quadrupole formula" of general relativity. For a recent survey of the quadrupole and other approximations for gravitational radiation, see Damour (1987). It ignores other sources of energy loss, such as tidal dissipation [Section 12.1(f)]. The values for the Keplerian and post-Keplerian parameters shown in Table 14.4 are from data taken through December 1990 (Taylor et al., 1992). Plotting the constraints the three post-Keplerian parameters imply for the two masses w, and m2, via Equations (14.18), we obtain the curves shown on Figure 14.6. It is useful to note that Figure 12.2 corresponds essentially to the inset in Figure 14.6. From > and y we obtain the values m, = 1.4411(7) MQ and w 2 = 1.3873(7) Mo, where the number in parenthesis denotes the error in the last digit. Equation (14.18c) then predicts the value Pb = 2.40243(5) x 10~12. In order to compare the predicted value for Pb with the observed value, it is necessary to take into account the effect of a relative acceleration between the binary pulsar system and the solar system caused by the differential rotation of the galaxy. This effect was previously considered unimportant when Pb was

Table 14.4. Parameters of the binary pulsar PSR 1913 + 16" Parameter (i) 'Physical' parameters Right ascension Declination Pulsar period Derivative of period 2nd derivative of period (ii) 'Keplerian' parameters Projected semimajor axis Eccentricity Orbital period Longitude of periastron Julian ephemeris date of periastron (iii) 'Post-Keplerian' parameters Mean rate of periastron advance Gravitational redshift and time dilation Orbital period derivative 3

Symbol (units)

Value

a <5 Pp (ms)

19h13m12.s46549(15) 16°01'08':i89(3) 59.029997929883(7) 8.62629(8) x 10"18 < 4 x 10^30

ap sin i (light sec) e P*(») «o(°) Ta (MJD)

2.341759(3) 0.6171309(6) 27906.9807807(9) 226.57531(9) 46443.99588321(5)

> C y r 1 ) y(ms) Pb (10-12)

4.226628(18) 4.294(3) -2.425(10)

Numbers in parentheses denote errors in last digit.

An Update

346 i

\

'

\

1

1

' 2

1.41

-

3rovo

1 1

()

1.39

-" "

companion

1.40

1 ^s

1 2

3 -

-

Mass

o 1.38 '-.

1.37 -

' . _ \ .

1.42

1.43 1.44 1.45 Mass of pulsar ( M Q )

-

1.46

Figure 14.6. Constraints on masses of pulsar and companion from data on PSR 1913 + 16, assuming general relativity to be valid. The width of each strip in the plane reflects observational accuracy, shown as a percentage. The inset shows the three constraints on the full mass plane; intersection region (a) has been magnified 400 times for the full figure.

known only to 10% accuracy [Section 12.1(f)(iii)]. Damour and Taylor (1991) carried out a careful estimate of this effect using data on the location and proper motion of the pulsar, combined with the best information available on galactic rotation, and found P ° A L ~ -(1.7 + 0.5) xlO" 14 .

(14.19)

Subtracting this from the observed Pb (Table 14.4) gives the residual P£BS = -(2.408 ± 0.010[OBS]±0.005[GAL]) x lO"12,

(14.20)

which agrees with the prediction, within the errors. In other words, pGR

-5^g= = 1.0023±0.0041(OBS)±0.0021(GAL).

(14.21)

The parameters r and J are not yet separately measurable with interesting

Theory and Experiment in Gravitational Physics

347

accuracy for PSR 1913 + 16 because the 47° inclination of the orbit does not lead to a substantial Shapiro time delay. The internal consistency among the measurements is also displayed in Figure 14.6, in which the regions allowed by the three most precise constraints have a single common overlap. This consistency provides a test of the assumption that the two bodies behave as "point" masses, without complicated tidal effects (conventional wisdom holds that the companion is also a neutron star), obeying the general relativistic equations of motion including gravitational radiation. It is also a test of the Strong Equivalence Principle (SEP), in that the highly relativistic internal structure of the neutron star does not influence its orbital motion or the gravitational radiation emission, as predicted by general relativity. (b) A population of binary pulsars ? In 1990, two new massive binary pulsars similar to PSR 1913 + 16 were discovered, leading to the possibility of new or improved tests of general relativity. PSR 2127+11C. This system appears to be a clone of the HulseTaylor binary pulsar (Anderson et al., 1990; Prince et al., 1991): Pb 28,968.36935 s, e = 0.68141, < cb > = 4.457° yr"1 (see Table 14.5). The inferred total mass of the system is 2.706 + 0.011 MQ. Because the system is in the globular cluster Ml5 (NGC 7078), observed periods Pb and Pp will suffer Doppler shifts resulting from local accelerations, caused either by the mean cluster gravitational field or by nearby stars, that are more difficult to estimate than was the case with the galactic system PSR 1913 + 16. This may limit the accuracy of measurement of the relativistic contribution to Ph to about 2 % . PSR 1534 + 12. This is a binary pulsar system in our galaxy (Wolszczan, 1991). Its pulses are significantly stronger and narrower than those ofPSR1913 + 16,so timing measurements have already reached 3 ^s accuracy. Its parameters are listed in Table 14.5 (Taylor et al., 1992). Because of the short data span, Pb has not been measured to date, but it is expected that in a few years, the accuracy in its determination will exceed that of PSR 1913+16. The orbital plane appears to be almost edge on relative to the line of sight (i « 80°); as a result the Shapiro delay is substantial, and separate values of the parameters r and 5 have already been obtained with interesting accuracy. This system may ultimately provide broader and more stringent tests of the consistency of general relativity than did the original binary pulsar (Taylor et al., 1992).

Table 14.5. Parameters of new binary pulsars" Parameter

PSR 1534+12

PSR 2127+11C

15h34m47.s686(3) 12°05'45"23(3) 37.9044403665(4) 2.43(8) xlO~18

21h27m36.s188(4) 11°57'26!29(7) 30.5292951285(9) 4.99(5) xlO" 18

(ii) ' Keplerian' parameters Projected semimajor axis Eccentricity Orbital period Longitude of periastron Julian ephemeris date of periastron

3.729468(9) 0.2736779(6) 36351.70270(3) 264.9721(16) 48262.8434966(2)

2.520(3) 0.68141(2) 28968.3693(5) 316.40(7) 47632.4672065(20)

(iii) 'Post-Keplerian' parameters Mean rate of periastron advance Gravitational redshift and time dilation Orbital period derivative Range of Shapiro delay r (jis) Shape of Shapiro delay 5 = sin i

1.7560(3) 2.05(11) -0.1(6) 6.2(1.3) 0.986(7)

4.457(12) * * * *

(i) 'Physical' parameters Right ascension Declination Pulsar period Derivative of period

" Numbers in parentheses denote errors in last digit. * Values not yet available from data.

Theory and Experiment in Gravitational Physics

349

(c) Binary pulsars and scalar-tensor theories In Section 12.3, we noted that some theories of gravity, such as the Rosen bimetric theory, are strongly, even fatally, tested by the binary pulsar. Other theories that are in some sense "close" to general relativity in all their predictions, such as the Brans-Dicke theory, are not so strongly tested, because the apparent near equality of the masses of the two neutron stars leads to a suppression of dipole gravitational radiation. Despite this, two circumstances have made it worthwhile to focus in detail on binary pulsar tests of scalar-tensor theories. The first is the remarkable improvement in accuracy of the measurements of the orbital parameters of the binary pulsar since 1980, and the continued consistency of the observations with general relativity, as described above, together with the discovery of new binary pulsars such as PSR 1534+12. The second is the resurrection of scalar-tensor theories in particle physics and cosmology. With this motivation, Will and Zaglauer (1989) carried out a detailed study of the effects of Brans-Dicke theory in the binary pulsar. Making the usual assumption that both members of the system are neutron stars, and using the methods summarized in Chapters 10-12, one obtains formulas for the periastron shift, the gravitational redshift/second-order Doppler shift parameter, and the rate of change of orbital period, analogous to Eqs. (14.18c). These formulas depend on the masses of the two neutron stars, on their internal structure, represented by "sensitivities" s and K* and on the Brans-Dicke coupling constant a>. First, there is a modification of Kepler's third law, given by Pb/2n = (a}/^my'2. Then, the predictions for >, y and Pb are 3

,

i3

1

3

y = e(Pb/2ny m2m- '^-" Pb =

(14.22a)

(a* + #/n 2 /w+ <>/?),

(14.22b)

w

-{\92n/5){2nm/Pby»(ji/m)>$- F{e) (14.22c)

where, to first order in £, = (2+a)~\ assuming cop 1, we have 0 = 1 -i(sl +52-2^2), & = 9[l -#+#(Si a2* = l - £ s 2 ,

(14.23a)

+s2-2slS2)],

(14.23b) (14.23c)

n* = (l-2s2)£,

(14.23d)

2 7/2

' F(e) = ft} -e r [/c,(l +y + P)-K2Qe>

+

p)]t

(14.23e)

An Update

350 (14.23f) 2

-^r )],

(14.23g) (14.23h)

T'=l-sx~s2, 2

G(e) = (1 -e )"

5/2

(14.23i) 2

(l -4e ),

^ = s,-s2.

(14.23J) (14.23k)

The quantities ,?a and K* are defined by d

J^>\ ,* = JdJ±m,

(14.24)

and measure the " sensitivity " of the mass wa and moment of inertia / a of each body to changes in the scalar field (reflected in changes in G) for fixed baryon number N (see Section 11.3). The first term in Pb is the effect of quadrupole and monopole gravitational radiation, while the second term is the effect of dipole radiation (in Section 11.3 we calculated only the dipole contribution). Estimating the sensitivities i a and K* using an equation of state for neutron stars sufficiently stiff to guarantee neutron stars of sufficient mass, and substituting into Equations (14.23), we find that the lower limit on a> required to give consistency among the constraints on , y and Pb as in Figure 14.6 is 105. The combination of > and y gives a constraint on the masses that is relatively weakly dependent on £,, thus the constraint on <J is dominated by Ph and is directly proportional to the measurement error in Pb; in order to achieve a constraint comparable to the solar system value of 2 x 10~3, the error in P^BS would have to be reduced by a factor of five. Damour and Esposito-Farese (1992) have devised a multi-scalar-tensor theory in which two scalar fields are tuned so that their effects in the weakfield slow-motion regime of the solar system are suppressed, with the result that the theory is identical to general relativity in the post-Newtonian approximation. Yet in the regime appropriate to binary pulsars, it predicts strong-field SEP-violating effects and radiative effects that distinguish it from general relativity. It gives formulae for the post-Keplerian parameters of Equations (14.22) as well as for the paramaters r and s that have corrections dependent upon the sensitivities of the relativistic neutron stars. The theory depends upon two arbitrary parameters /?' and /?"; general relativity corresponds to the values fi' = /?" = 0. It turns out (Taylor et al., 1992) that the binary pulsar PSR 1913+16 alone constrains the two parameters to a narrow but long strip in the /?'-/?"-plane that

Theory and Experiment in Gravitational Physics

351

includes the origin (general relativity) but that could include some highly non-general relativistic theories. The sensitivity of PSR 1534+ 12 to r and s provides an orthogonal constraint that cuts the strip. In this class of theories, then, both binary pulsars are needed to provide a strong test. (d) Other stellar-system tests of gravitational theory The suppression of dipole gravitational radiation resulting from the apparent high symmetry of the binary pulsar system suggests that more stringent tests might be found in systems in which the two compact objects are dissimilar, for example, two very unequal mass neutron stars or a neutron star and a white dwarf. Several candidate systems have been suggested. The 11-minute binary 4U1820-30. This system is believed to consist of a neutron star and a low-mass helium dwarf in a nearly circular orbit with a period of 68 5.008 s. It is not the most" clean " system available for testing gravitational theory, because its evolution is affected by mass transfer from the companion low-mass dwarf onto the neutron star, whose X-ray output comprises the data from which the binary nature of the system was established (Stella, Priedhorsky and White, 1987; Morgan, Remillard and Garcia, 1988). In fact the rate of mass transfer is believed to be controlled by gravitational-radiation damping of the orbit. Because of this complication, the analysis of the implications of Brans-Dicke theory for this system is model dependent. Will and Zaglauer (1989) generalized a class of general relativistic mass-transfer models to the Brans-Dicke theory, and showed that, if a limit could be placed on \PJPb\ of 2.7 x 10"7 yr"1, corresponding to an early published limit, then bounds on co as large as 600 could be placed, depending on the assumed mass of the neutron star and on the assumed equation of state. Unfortunately, recent observations of the system using the Ginga X-ray satellite suggest that Ph is opposite in sign to that predicted by a gravitational-radiation-driven mass-transfer model (Tan et al., 1991). Evidently, the binary system is undergoing acceleration either in the mean gravitational field of the globular cluster in which it resides, or in the field of a nearby third body. Whether the effect of such local accelerations on Pb can be sufficiently understood to yield an interesting bound on co remains to be seen at present. PSR 1744-24A. This is an eclipsing binary millisecond pulsar, in the globular cluster Terzan 5 (Lyne et al., 1990), with a very short orbital period of 1.8 hrs, e = 0, and a mass function of 3.215 x 10"4, indicating a low-mass companion of 0.09 MQ. The asymmetry of the system is promising for dipole gravitational radiation, but the observations are

An Update

352

complicated by the possibility of cluster accelerations as well as by the apparent presence of a substantial wind from the companion (the cause of the eclipses), which may complicate the orbital motion. Nevertheless, even if measurements of Pb can only reach 50 % accuracy relative to the general relativistic prediction of Pb/Pbx 1.3 x 10~8yr~', the bound on co could exceed 1000 (Nice and Thorsett, 1992). This discussion illustrates both the promise and the problems inherent in stellar-system tests of gravitational theory. Dipole gravitational radiation and strong violations of SEP resulting from the presence of neutron stars can lead to potentially large observable effects. Offsetting this are the complications of astrophysical effects within the systems, such as mass transfer, and of environmental effects, such as cluster or third-body acceleration^. Under the right conditions, however, a significant test may emerge. 14.7

Conclusions

In 1992 we find that general relativity has continued to hold up under extensive experimental scrutiny. The question then arises, why bother to test it further? One reason is that gravity is a fundamental interaction of nature, and as such requires the most solid empirical underpinning we can provide. Another is that all attempts to quantize gravity and to unify it with the other forces suggest that gravity stands apart from the other interactions in many ways, thus the more deeply we understand gravity and its observational implications, the better we may be able to mesh it with the other forces. Finally, and most importantly, the predictions of general relativity arefixed;the theory contains no adjustable constants so nothing can be changed. Thus every test of the theory is potentially a deadly test. A verified discrepancy between observation and prediction would kill the theory, and another would have to be substituted in its place. Although it is remarkable that this theory, born 77 years ago out of almost pure thought, has managed to survive every test, the possibility of suddenly finding a discrepancy will continue to drive experiments for years to come.

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Index

action principle, 75; n-body, in modified EIH formalism, 273; n-body, in PPN formalism, 158-60; n-body, in THejt formalism, 54 active gravitational mass: comparison with passive mass via Kreuzer experiment, 214; in PPN formalism, 151 advanced time, 228 anomalous mass tensors, 40 Apollo 11 retroreflectors, 188 Bianchi identities, 76, 128, 230 big-bang model, 311 binary pulsar, 11, 257, 283; acceleration of center of mass, 300, 344; arrivaltime analysis, 287-92, 343; in Brans-Dicke theory, 306, 349-50; companion, 284; decay of orbit, 297; detection of gravitational radiation, 306; determination of masses, 297, 344; dipole gravitational radiation, 298; formation and evolution, 286; in general relativity, 303-6, 344-6; gravitational radiation emission, 297; gravitational red shift, 290; mass loss, 300; measured parameters, 285, 345; periastron shift, 284, 293; postNewtonian effects, 297; precession of pulsar spin, 302; pulsar mass in general relativity, 306; in Rosen's theory, 307; second-order Doppler shift, 290, 296; spin-down rate, 292; test of conservation of momentum, 218-20, 338; third body, 301; tidal effects in, 294, 299; as ultimate test for gravitation theory, 309

binary system: single-line spectroscopic, 287; test of conservation of momentum, 217-20, 338 black holes, 256; in general relativity, 264; motion, see modified EIH formalism; motion in Brans-Dicke theory, 279; in Rosen's theory, 266; in scalartensor theories, 265 boundary conditions for post-Newtonian limit, 118 Brans-Dicke theory, 125, 182, 190, 203, 265, 276, 306, 317, 332, 335, 349; see also scalar-tensor theories Cavendish experiment, 82, 153, 191 center of mass, 113, 145, 146, 160 Christoffel symbols, 70; for PPN metric, 144 classical tests, 166; and redshift experiment, 166 "comma goes to semicolon" rule, 71 completeness of gravitation theory, 18 connection coefficient, see Christoffel symbols conservation laws: angular momentum, 108; baryon number, 105; breakdown of, for total momentum, 149; center-ofmass motion, 108; and constraints on PPN parameters, 111; energy-momentum, 108, 111-12; global, 107-8; local, 105-7; rest mass, 106; tests of, for total momentum, 215-20, 337-8 conserved density, 107, 111 constants of nature, constancy: gravitational, 202; nongravitational, 36-8; and Oklo natural reactor 36-8 coordinate systems: curvature coordi-

Index nates, 259; local quasi-Cartesian, 92; preferred, 17; standard PPN gauge, 97 coordinate transformation, 69 Copernican principle, 312 cosmic time function, 80 cosmological principle, 312 cosmology, 7, 310; in Bekenstein's variable-mass theory, 317; in Brans-Dicke theory, 317; in general relativity, 313; helium abundance, 315; microwave background, 311, 314; in Rastall's theory, 318; in Rosen's theory, 318; timescale problem, 311 covariant derivative, 70 Cygnus XI, 256 de Sitter effect, 338 deceleration parameter, 313 deflection of light, 5; derivation in PPN formalism, 167-70; derivation using equivalence principle, 170; eclipse expedition, 5, 171; effect of solar corona, 172; measurement by radio interferometry, 172; optical measurements, 6; radio measurements, 172; VLBI, 332 Dicke, R. H., 10, 16 Dicke framework, 10, 16-18 Doppler shift in binary pulsar, 290, 293, 296 dynamical gravitational fields, 118 E(2) classification, see gravitational radiation Earth-tides, 191 eccentric anomaly, 288 Einstein Equivalence Principle (EEP), 16-22; and cosmology, 312; implementation, 71; and nonsymmetric metric, 328; and speed of gravitational waves, 223; and speed of light, 223; and THt;1 formalism, 46-50 Einstein- Infeld- Hoffmann (EIH) formalism, 267; EIH Lagrangian, 269; see also modified EIH formalism energy conservation: and cyclic gedanken experiments, 39-43; and Einstein Equivalence Principle, 39; in PPN formalism, 158-63; and Scruffs conjecture, 39-43; and Strong Equivalence Principle, 82; in THepi formalism, 53-8 Eotvb's experiment, 24-7; and Belinfante-Swihart nonmetric theory, 66; and fifth force, 341; lunar, 185-90; and Nordtvedt effect. 185-90; Princeton and Moscow versions, 25; in space, 340

376 Eot-Wash experiment, 320 equations of motion: charged test particles, 69; compact objects, see modified EIH formalism; Eulerian hydrodynamics, 87; n-body, 149, 159; Newtonian, for massive bodies, 145; photons, 143; PPN hydrodynamics, 147; self-gravitating bodies, 144-53; spinning bodies, 163-5; in THe/u, formalism, 50 equatorial coordinates, 194 Eulerian equations of hydrodynamics, 87 Fermi-Walker transport, 164 fifth force, 341-3 flat background metric, 79; in Robertson-Walker coordinates, 314 gauge transformation, % general covariance, 17; and preferred coordinate systems, 17; and prior geometry, 17 general relativity, 121-3; black holes, 265; derivations of, 83; EIH formalism, 267; field equations, 121; locally measured gravitational constant in, 158; location in PPN theory space, 205; and modified EIH formalism, 275; motion of compact objects, 267; neutron stars, 260; Nordtvedt effect, 152; polarization of gravitational waves, 234; post-Newtonian limit, 121; PPN parameters, 122; quadrupole generation of gravitational waves, 246-8; with R2 terms, 84-5; speed of gravitational waves, 223; standard cosmology, 316 geocentric ecliptic coordinates, 192 geodesic equation, 73; for compact objects, 267 geometrical-optics limit: for gravitational waves, 223; for Maxwell's equations, 74-5 gravimeter, 191; superconducting, 198 gravitational constant, 120; constancy of, 202-^t, 336; locally measured, 153-8, 191; in scalar-tensor theories, 124 gravitational radiation: in binary pulsar, 297; detection in binary pulsar, 306, 346-7; dipole, 240, 249, 251, 279, 298; dipole parameter, 240, 253; E(2) classification, 226-7; effect on binary system, 239; energy flux in general relativity, 238; energy loss, 90, 238-40; forces in detectors, 237; in general relativity, 223, 234, 246; measurements of polarization, 237; measurement of speed, 226; in

Index modified EIH formalism, 275; negative energy of, 252; PM parameters, 240, 253; polarization, 227-38; post-Newtonian formalism, 240; quadrupole nature in general relativity, 238; in Rastall's theory, 225, 236; reaction force, 239; in Rosen's theory, 225, 236, 250; in scalar-tensor theories, 224, 234, 248, 252, 279; speed, 223-6; speed in Rosen's theory, 131; in vector-tensor theories, 224, 235 gravitational red shift, 5, 32-6, 322; in binary pulsar, 290, 296; and cyclic gedanken experiments, 42-3; derivation, 32-3; null experiment, 36; Pound-Rebka-Snider experiment, 33; in TH^ formalism, 62-4; solar, 322; Vessor-Levine rocket experiment, 35 gravitational stress-energy, 109, 241 gravitational waveform, 238 Gravitational Weak Equivalence Principle (GWEP), 82; breakdown, 151, 185; see also Nordtvedt effect gyroscope precession: derivation in PPN formalism, 208-9; dragging of inertial frames, 210; goedetic effect, 209, 338; and LAGEOS, 340; Lense-Thirring effect, 210; Stanford experiment, 212, helicity of gravitational waves, 227, 232, 252 helium abundance, 315 helium main-sequence star, 284, 294 Hubble constant, 202, 313 Hughes- Drever experiment, 30, 61 hydrogenic atom in THe/* formalism, 55-7 inertial mass, 13, 145; anomalous mass tensor, 40, 55, 162, 323; dependence on gravitational fields, 269; in modified EIH formalism, 273; postNewtonian, 146 isentropic flow, 106 isotropic coordinates, 259 J2, see quadrupole moment Kerr metric, 256 Kreuzer experiment, 214 laboratory experiments as tests of postNewtonian gravity, 213 Lagrangian-based metric theory, 78-9 little group, 233 local Lorenz invariance, 23; and

377 propagation of light, 321; Hughes-Drever experiment, 30; kinematical frameworks, 325; Mansouri-Sexl framework, 325; tests of, 30-1, 320; tests using TH^ formalism, 61-2; in TH formalism, 48, 323; violations of, 40-1 local position invariance, 23; gravitational red-shift experiments, 32-6, 322; tests using TH^ formalism, 62-4; in THW formalism, 49; violations of, 40-1 local quasi-Cartesian coordinates, 92 local test experiment, 22 Lorentz frames, local, 23 Lorentz invariance: local, see local Lorentz invariance; of modified EIH Lagrangian, 272 Lorentz transformations of null tetrad, 232 Lunar Laser Ranging Experiment (LURE), 188 Mansouri-Sexl framework, 325 Mariner 6, 175 Mariner 7, 175 Mariner 9, 175 mass, see active gravitational mass; inertial mass; passive gravitational mass mass function, 283 Maxwell's equations, 72-3; ambiguity in curved spacetime, 72-3; geometricaloptics limit, 74-5; in THe/u formalism, 50 metric, 22, 68; flat background, 79, 118; nonsymmetric, 328 metric theories of gravity, postulates, 22; see also theories of gravitation microwave background, 311, 314; Earth's motion relative to, 197 Minkowski metric, 20, 80, 118 modified EIH formalism: in Brans-Dicke theory, 276; equations of motion for binary systems, 273; in general relativity, 275; gravitational radiation, 275; Keplerian orbits, 274; Lagrangian, 273; Newtonian limit, 274; periastron shift, 274; in Rosen's theory, 280-2; variable inertial mass, 269 moment of inertia of Earth, variation in, 195 momentum conservation: breakdown, in PPN formalism, 149; tests of, 215-20, 337-8 neutron stars, 255; boundary conditions, 259; form of metric, 258; in general relativity, 250; maximum mass, 256; motion, see modified EIH formalism; in

Index Newtonian theory, 257-8; in Ni's theory, 263; in Rosen's theory, 261-3; in scalar-tensor theories, 260 Newman-Penrose quantities, 230 Newtonian gravitational potential, 87, 88, 151 Newtonian limit, 21, 87, 145; conservation laws, 105; empirical evidence, 21; and fifth force, 341-3; inverse square force law, 21, 341-3; in modified EIH formalism, 274 Newton's third law, 152; and Kreuzer experiment, 214; and lunar motion, 337 Nordtvedt, K., Jr., 98 Nordtvedt effect, 151; and lunar motion, 185-90; test of, using lunar laser ranging, 188-90; 335 null separation, 74 null tetrad, 229 oblateness of Sun, 181; Dicke-Goldenberg measurements, 181; Hill measurements, 182; and solar oscillations, 334 Oklo natural reactor, 36-8 orbit elements, Keplerian, 178, 283, 287; perturbation equations for, 179 osculating orbit, 287 parametrized post-Newtonian formalism, see PPN formalism particle physics, 20-1 passive gravitational mass, 13; anomalous mass tensor, 40, 55, 58, 162; comparison with active mass, 214; in PPN formalism, 150 perfect fluid, 77-8 periastron shift: in binary pulsar, 284, 293; for compact objects, 274 perihelion shift: derivation in PPN formalism, 177-80; measured, for Mercury, 181, 333; Mercury, 4, 176-83; preferredframe and preferred-location effects, 200-1 PM parameters, 240 post-Coulombian expansion, 51 post-Galilean transformation, 272 post-Keplerian parameters, 343-4 post-Newtonian limit: for gravitationalwave generation, 240-6; see also PPN formalism post-Newtonian potentials, 93, 104 PPN formalism, 10, 97; active gravitational mass, 151; for charged particles, 214; Christoffel symbols, 144; comparison of different versions, 104; conservation-law parameters, 111; Eddington-Robertson-Schiff version, 98;

378 PPN ephemerides, 334; limits on PPN parameters, 204, 216, 219, 339; metric, 99, 104; n-body action principle, 158-60; n-body equations of motion, 149, 153; passive gravitational mass, 150; PPN parameters, 97; PPN parameter values for metric theories, 117; post-post-Newtonian extensions, 331; preferred-frame parameters, 103; significance of PPN parameters, 115; standard gauge, 97, 102 preferred-frame effects: in Cavendish experiments, 148; geophysical tests, 1909; on gyroscope precession, 210; in locally-measured gravitational constant, 190; orbital tests, 200-2, 336; and solar spin axis, 336; tests from Earth rotation rate, 199; tests using gravimeters, 199 preferred-frame parameters: in PPN formalism, 103; in THe/t formalism, 48 preferred-frame PPN parameters, limits on, 199, 202, 336, 339 preferred-location effects: in Cavendish experiments, 148; geophysical tests, 190-9; in locally-measured gravitational constant, 190; orbital tests, 200-2; tests using gravimeters, 199 prior geometry, 17, 79, 118 projected semi-major axis, 293 proper distance, 73, 155 proper time, 73, 68 PSR 1744-24A, 351 PSR 1534+12,347 PSR 1913 + 15, see binary pulsar PSR 2127+11C, 347 pulsars, 256, 283 quadrupole moment, 145, 177; solar, 180; solar, measurable by Solar Probe, 183; and solar oscillations, 334 quantum systems in THc/x formalism, 55-7 quasi-local Lorentz frame, 80 radar: active, 175; passive, 174; and time delay of light, 174 radio interferometry and deflection of light, 172 reduced field equations, 241 rest frame of universe, 31, 99 rest mass, total, 107 retarded time, 228 Ricci tensor, 73, 230 Riemann curvature tensor, 72; electric components, 227; irreducible parts, 230

Index Riemann normal coordinates, 227 Robertson-Walker metric, 91, 312 Rosen's bimetric theory, 131; absence of black holes, 266; binary pulsar, 307; cosmological models, 317; field equations, 131; gravitational radiation, 225, 236, 250-2; location in PPN theory space, 205; and modified EIH formalism, 280; neutron stars, 261; postNewtonian limit, 131; PPN parameters, 131 rotation rate of Earth, variation in, 195 scalar-tensor theories, 123-6; Barker's constant G theory 125; Bekenstein's variable-mass theory, 125, 317; Bergmann-Wagoner-Nordtvedt, 123; binary pulsar, 306; black holes, 265; Brans- Dicke, see Brans-Dicke theory; cosmological models, 317; field equation, 123; gravitational radiation, 224, 234, 248, 50; limits on m, 175, 335; location in PPN theory space, 205; and modified EIH formalism, 276; neutron stars, 260; Nordtvedt effect, 152; post-Newtonian limit, 124; PPN parameters, 125; and string theory, 332 Schiff, L. I., 38 Schiff s conjecture, 38; proof in THe/u. formalism, 50-3 Schwarzschild coordinates, 259, 265 Schwarzschild metric, 256, 265 self-acceleration, 149; of binary system, 217, 338; of pulsars, 216 self-consistency of gravitation theory, 19 semi-latus rectum, 179 Shapiro, 1.1., 166 Shapiro effect, see time delay of light solar corona, 172, 175 Solar Probe, 183, 340 spacelike separation, 73 special relativity, 20-1; agreement of gravitational theory with, 20-1; and propagation of light, 325-7; tests in particle physics, 20-1 specific energy density, 89 spin, 163; precession, 165; precession in binary pulsar, 302 static spherical space times, form of metric, 258 stress-energy complex, 108 stress-energy tensor, 76; in PPN formalism, 104; vanishing divergence of, 77 Strong Equivalence Principle (SEP), 7983; and dipole gravitational radiation, 252; and motion of compact objects,

379 268; tests of, 184, 335; violations in Cavendish experiments, 153; violations of, 102 superpotential, 94 THe/x formalism, 45-66; limitations, 589 theories of gravitation: Barker's constant G theory, 125; Bekenstein's variablemass-theory, 125, 317; BelinfanteSwihart, 64-6; Bergmann-WagonerNordtvedt, 123; bimetric, 130-5; Brans-Dicke, see Brans-Dicke theory; BSLL bimetric theory, 133; conformally flat, 141; E(2) classes, 233-7; fully-conservative, 113; general relativity, see general relativity; HellingsNordtvedt, 130; Lagrangian-based, 43, 78-9, 109; linear fixed-gauge, 139; Moffat, 330; Ni, 137, 263; nonconservative theories, 115; nonviable, 19, 138-41; postulates of metric theories, 22; PPN parameters for, 117; purely dynamical vs. prior geometric, 79; quasilinear, 138; Rastall, 132, 225, 236, 318; Rosen, see Rosen's bimetric theory; scalar-tensor, see scalar-tensor theories; semiconservative, 114; special relativistic, 7; stratified, 135-7; stratified, with time-orthogonal space slices, 140; vector-tensor, see vector-tensor theories; Whitehead, 139; Will-Nordtvedt, 129; with nonsymmetric metric, 328-30 time delay of light: in binary pulsar, 290; as classical test, 166; derivation in PPN formalism, 173-4; effect of solar corona, 175; measurements of, 174, 333; radar measurements, 176 timelike separation, 73 torsion, 84 transverse-traceless projection, 248 universal coupling, 43, 67-8 vector-tensor theories, 126-30; field equations, 127; gravitational radiation, 224, 235; Hellings-Nordtvedt, 130; post-Newtonian limit, 129; PPN parameters, 129; Will-Nordtvedt, 129 velocity curve, 283 Viking, 175, 336 virial relations, 52, 54, 148, 161, 245 Voyager 2, 333 Weak Equivalence Principle (WEP), 13, 22; and cyclic gedanken experiments,

Index 41-2; and electromagnetic interactions, 28-9; and Ebtvbs experiment, 24-7; and fifth force, 341; and gravitational interactions, 29, 82; of Newton, 13; and nonsymmetric metric, 329; and strong interactions, 28; tests of, 24-9,

380 320; tests using TH£/, formalism, 60; and weak interactions, 29 Weyl tensor, 230 Whitehead PPN parameter, limits on, 199 Whitehead's theory, 139 Whitehead term, 95, 98

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