Ether space-time and cosmology: new insights into a key physical medium, Volume 2

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“Horsehead nebula”

Apeiron

ETHER SPACE-TIME & COSMOLOGY Volume 2 NEW INSIGHTS INTO A KEY PHYSICAL MEDIUM

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Michael C. Duffy and Joseph Levy Editors

A book dealing with experimental and theoretical studies devoted to the exploration of the modern ether concept, evidence of its reality and implications for modern physics.

Apeiron Montreal

Published by C. Roy Keys Inc. 4405, rue St-Dominique Montreal, Quebec H2W 2B2 Canada http://redshift.vif.com Copyright © 2009 by C. Roy Keys Inc. All rights reserved No parts of this book may be reproduced stored in a retrieval system and transmitted in any form or by any means without the written permission of the copyright owner.

First Published 2009 Library and Archives Canada Cataloguing in Publication Ether space-time and cosmology : new insights into a key physical medium / Michael C. Duffy and Joseph Lévy, editors. Includes bibliographical references. ISBN 978-0-9732911-8-6 1. Ether (Space). 2. Cosmology. 3. Relativity (Physics). 4. Space and time. I. Lévy, Joseph, 1936- II. Duffy, Michael Ciaran QC177.E84 2009

530.11

C2009-900610-3

Illustration, front: Horsehead Nebula by Robert Gendler

http://www.robgendlerastropics.com

ETHER SPACE-TIME & COSMOLOGY Volume 2 New insights into a key physical medium EDITORS M. C. Duffy & J. Levy EDITORIAL BOARD M. Arminjon, Laboratoire “Sols, Solides, Structures, Risques” CNRS & Universites de Grenoble France. J. Carroll, Engineering Department, University of Cambridge, Cambridge, United Kingdom G.T. Gillies, Department of aerospace and mechanical engineering, University of Virginia, USA J. G. Gilson, School of Mathematical Sciences, Queen Mary University of London, Mile End Road, London E14NS V. O. Gladyshev, Moscow Bauman State Technical University, Moscow, Russia. A. L. Kholmetskii, Department of Physics, Belarusian State University, Minsk Belarus A. N. Petrov Department of Physics and Astronomy, University of MissouriColumbia, Columbia, MO 65211, USA; Sternberg Astron. Inst.,Universitetskii pr.,13 Moscow, 119992, RUSSIA; P. Rowlands, University of Liverpool, Liverpool, United Kingdom. F. Selleri, Dipartimento di Fisica, Università di Bari, INFN, Sezione di Bari, Bari, Italy. G. Spavieri, Centro de fisica fundamental, University de los Andes, Mérida, 5101, Venezuela T. Suntola, www.sci.fi/~suntola,Finland L. Székely, Institut for philosophical research of the Hungarian Academy of sciences. Web Site of the Program

http://www.physicsfoundations.org/Ether_spacetime/book.htm

Contents of Volume 2 -1-

Introduction Michael C. Duffy & Joseph Levy -3Relativity in Terms of Measurement and Ether Lajos Jánosssy’s Ether-Based Reformulation of Relativity Theory László Székely Institute for Philosophical Research of the Hungarian Academy of Sciences Postal address: HU-1398 Budapest, Post Box: 594 [email protected]

-37AETHER THEORY CLOCK RETARDATION vs. SPECIAL RELATIVITY TIME DILATION

Joseph Levy 4 Square Anatole France, 91250, St Germain lès Corbeil, France E-mail: [email protected]

-53RELATIVITY AND AETHER THEORY A CRUCIAL DISTINCTION

Joseph Levy 4 Square Anatole France, 91250 St Germain lès Corbeil, France E-mail: : [email protected]

-67The Dynamic Universe Zero-energy balance restores absolute time and space Tuomo Suntola Vasamatie 25, 02630 Espoo, Finland E-mail: [email protected]

-135DYNAMICAL 3-SPACE: A REVIEW

Reginald T. Cahill School of Chemistry, Physics and Earth Sciences, Flinders University, Adelaide 5001, Australia

-201Relativistic physics from paradoxes to good sense - 1 F. Selleri Dipartimento di Fisica, Università di Bari INFN, Sezione di Bari

-267Photon-like solutions of Maxwell’s equations John Carroll, Joseph Beals IV, Ruth Thompson Centre for Advanced Photonics and Electronics, Engineering Department, University of Cambridge, 9 JJ Thomson Avenue, Cambridge, CB3 0FA, UK

-323Cosmological Coincidence and Dark Mass Problems in Einstein Universe and Friedman Dust Universe with Einstein’s Lambda James G. Gilson [email protected] School of Mathematical Sciences Queen Mary University of London Mile End Road London E14NS

-355THE SPACE-CURVATURE THEORY OF MATTER AND ETHER 1870-1920

James E. Beichler P.O. Box 624, Belpre, Ohio, 45714, USA [email protected]

-417On microscopic interpretation of the phenomena predicted by the formalism of general relativity Volodymyr Krasnoholovets Indra Scientific bvba, Square du Solbosch 26 B-1050, Brussels, Belgium* E-mail address: [email protected]

-433CAN PHYSICS LAWS BE DERIVED FROM MONOGENIC FUNCTIONS

José B. Almeida Universidade do Minho, Physics Department Braga, Portugal, e-mail:[email protected]

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Abstracts of Volume 1 -481-

Instructions to conributors to future volumes

Introduction The program “Ether space-time & cosmology” to which this book belongs, comprises several volumes designed to inform the physics community of the resurgence of the ether in modern science. The reality of the concept and its importance were evident by the end of the 20th century, and at the beginning of the 21st century. Ether theory plays a creative role even if given different names such as physical vacuum, fundamental plenum or cosmic substratum. The introduction of the ether as a main actor in physical processes, will resolve a number of paradoxes in 20th century physics which arose because of its dismissal. This second volume, like the first, presents articles, written by experienced physicists, dealing with different aspects of the ether concept. The necessity of the ether is not questioned today, even by those who pretend to do so, but who don’t hesitate to attribute qualities to the vacuum, but the nature and the properties of the ether do not find a total consensus and much must be done to this end. This is the reason why the articles presented in these volumes may defend different points of view, allowing the reader to compare the different arguments presented and to make an informed choice. One of the objectives of this series of books is to progressively disclose its properties. The main approaches which are presented in the different articles are based on the Lorentz ether concept, which assumes the existence of a preferred ether frame, and the Einstein ether concept, which assumes the complete equivalence of all inertial frames. For much of the experimental data, and at least before further analysis, it is commonly admitted that these approaches anticipate the same results; it is the reason why they appear difficult to differentiate. And, another objective pursued by many authors is to find criteria capable to discriminate between them. In the subset of the Lorentzian approach, there is a current which aims at demonstrating that the conventional space-time transformations conceal hidden variables which need to be disclosed for a full understanding of physics. “Ether space-time & cosmology” is a development of the Physical Interpretations of Relativity Theory conferences, which began in 1988, in London, and which now take place in London, Moscow, Calcutta and Budapest. Details of these conferences, including names and addresses of contacts and sponsors, are given on the PIRT web site <www.physicsfoundations.org> Drs. Michael C. Duffy & Joseph Levy

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Relativity in Terms of Measurement and Ether Lajos Jánosssy’s Ether-Based Reformulation of Relativity Theory László Székely Institute for Philosophical Research of the Hungarian Academy of Sciences Postal address: HU-1398 Budapest, Post Box: 594 [email protected] Abstract In his monograph Theory of Relativity Based on Physical Reality, Hungarian physicist Lajos Jánossy develops the complete Einsteinian formalism of relativity theory by analysing the process of measurement, the systems of measures created in this process and experimental data expressed in terms of measures. He demonstrates that based on a simple principle (which he calls the Lorentz principle) and its generalization the whole formalism of the original theory may be developed in conformity with the notions of common sense without mathematizing physical reality, so that the new way of development is of the same heuristic power as the original one. His analysis makes it clear that the allegedly revolutionary new notions of space and time follow not from physical experiences but from Einstein’s positivist philosophical commitments. Having established the place and role of a privileged (but not absolute) reference system, at the second level of his theory Jánossy connects this system to the carrier of electromagnetic phenomena which he also assumes to be the carrier of the gravitational and other physical fields. Although he uses the term ‘ether’, he explicitly rejects the old theories of this entity and attributes to it dynamic properties. In the last section of the paper Einstein’s and Jánossy’s ether concepts are compared and it is argued that despite the parallelism between the two concepts, from Jánossy’s point of view Einstein’s ether is too mathematical to cure the inverted relation between mathematics and physics characteristic for Einstein’s relativity. Key words: relativity, ether, propagation of light, privileged reference system, space-time, measurement, ideal solid rod, ideal clock, common sense in physics, mathematics in physics, physical reality, Einstein, Lorentz, Jánossy

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1. Introduction In Physical Relativity, a monograph published by Clarendon Press in 2005, Harvey Brown criticizes the received view of Einstein’s theory and argues for a physical interpretation of relativistic phenomena. [Brown 2005] Both Brown’s book and the regular conferences on the interpretations of relativity theory organized by Michel Duffy [Duffy 1988, 1990 ….2006] clearly indicate that the long tradition of considering the original, Einsteinian-Minkowskian notion of relativity theory too mathematical and claiming that it blurs (or even turns into its opposite) the epistemological relation between mathematics and physics is alive even today, more than 100 years after Einstein’s famous paper. In the introduction to his book Brown mentions the Hungarian physicist Lajos Jánossy as one of his forerunners inspiring his ideas. [Brown 2005, vii.] Jánossy was an important figure in the tradition of alternative interpretations of Einstein’s theory, who (following Lorentz’s ideas) elaborated a comprehensive alternative (“physical”) relativity. He, along with American Herbert Ives (who belonged to a former generation of physicists) and Prokhovnik (a contemporary of Jánossy) may be considered as one of the classics of the field. However, while on the basis of personal communications it seems that his work on relativity theory was well known by those who did research in the topic in the last decades, he (in contrast with Ives and Prokhovnik) is only rarely cited in the literature. (M. Duffy mentions Jánossy’s work in his recent paper [Duffy 2008] and Bell in his famous study How Teach Relativity? also expresses his appreciation for Jánossy’s contribution to the topic [Bell 1976].) The aim of this paper is to give a brief review of Jánossy’s reformulation of relativity theory, which deserves more recognition than it has received until now. 2. Lajos Jánossy’s career Lajos Jánossy was born in Mátyásföld (then a village near Budapest, now part of the Hungarian capital) in 1912. His father Imre Jánossy was an astronomer who died relatively young in 1920. After the death of her husband, his mother, Gertrud Borstrieber (a mathematician belonging to the first generation of Hungarian women with a university degree) married the Hungarian philosopher George Lukács, who was considered by the French philosopher Lucian Goldman the first representative of the existentialist philosophy, but who later gave up his youthful enthusiasm for Kierkegaard and became a famous and highly controversial Marxist philosopher of the 20th century, oscillating permanently between communist movement discipline and sovereign philosophical thought and causing many a disturbance for the party leadership. After the fall of the Hungarian Soviet Republic in 1919 Lajos Jánossy’s family moved to Austria and later to Berlin. Instead of following his stepfather in politics or philosophy, Jánossy became a physicist. He studies physics at the Humbold University in Berlin where he was a student of Edwin Schrödinger whose

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metatheoretical considerations on physics had a determinative influence on him. In the 1930s Jánossy became also a university professor and read physics (and especially relativity theory) at Manchester University (while his stepfather left Hitler’s Berlin for Stalin’s Moscow and lived there with his political and moral compromises). His main research field being cosmic radiations, he became an internationally respected scientist in the field, and his monograph on the topic belongs to the basic literature on the subject [Jánossy 1948, 1950]. After Word War Two George Lukács returned to Hungary and in 1950 Lajos Jánossy (then a professor at the Institute for Advanced Studies in Dublin and a colleague of his former professor, Schrödinger) followed him. While his stepfather was never a “pet” (or with the good German word a “Liebling”) of the communist party, party leaders needed his international respect, as well as Lajos Jánossy’s scientific knowledge. So the latter became head of the Central Institute for Physical Research, a grand new research institute established on a Soviet model. It is generally held that Einstein’s theory of relativity was deemed by the official Soviet ideologists as a bourgeois theory, so Jánossy’s criticism of the Einsteinian notion of relativity may appear in this context as a version of the Soviet criticism of the theory, but it is not the case. On the one hand, although several attempts were made in the Soviet Union to discredit relativity theory as a prototype of false, idealistic physics, and at the turn of the forties to the fifties of the last century a fierce campaign was waged against the theory, the attempts never resulted in its official denunciation. On the contrary, after the death of Stalin, Einstein‘s Soviet followers won the debate and Einstein’s theory came to be glorified as a true dialectical theory, which as such fully corresponds to Marxism-Leninism. [See e.g. Graham 1972, 111-138; Székely 1987] On the other hand, and quite importantly, Lajos Jánossy had never taken part in the antirelativistic campaign. The greater part of his critical considerations on relativity theory was published in a period when official Soviet ideology endorsed Einstein. Hence, beside the criticism his concept received from orthodox Einsteinian physicists, Jánossy’s notion of the relativity theory also became a target of official philosophers of the Soviet block. Although the Hungarian Academy Press undertook the publication of his comprehensive work „Theory of Relativity Based on Physical Reality” [Jánossy 1971], in his last years he was considered by the orthodox Einsteinian physicists who were then dominating the Hungarian physics scene as an anti-relativist dinosaur and (while formally preserving his university position) he was gradually displaced from Hungarian scientific life. He died in 1978. Whereas the ideological, political and sociological contexts of Lajos Jánossy’s scientific work would also offer interesting topics, this contribution will be restricted to reviewing his concept of the theory of relativity only from the point of view of physics and the philosophy of science.

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3. The metatheoretical foundation 3.1. The relation between mathematics and physics and the norm of common sense As indicated in the title of his monograph, Lajos Jánossy characterizes his notion of relativity theory as a theory based on physical reality. This title expresses both a critical and a confirmative aspect. On the one hand, Jánossy argues that the Einsteinian theory is not based on physical reality: while it is an effective mathematical tool for handling the results of measurements and for making predictions, it does not provide an appropriate theory of physical reality. On the other hand, he affirms that the mathematical formulas of Einstein’s theory are correct in the sense that they are in correspondence with observation and empirical data and are able to give correct predictions about the behaviour of physical reality. Of course, Jánossy sees clearly that what Einstein offers us is not only mere mathematics but a definite physical theory. He insists, however, that Einstein turns the relation between mathematics and physics into its opposite: in his view the German physicist projects mathematical formulae into the physical world and in this way constructs physical reality by hypostatisation of mathematical ideas. Consequently in the context of his criticism the so called “geometrization of physics” which is often praised as a great achievement of relativity theory appears as a result of hypostatisation and Jánossy focuses his criticism on this element of the theory: “The theory of relativity in its original formulation is certainly not a mere attempt to describe phenomena by suitable mathematical expressions – the theory is a far reaching attempt to give a theory of space and time. Our criticism of the theory is just connected with this latter feature. We think that the theory reflects correctly certain general physical laws, but these laws – in our opinion – have nothing to do with the “general structure of space and time”. Therefore our attempt is to give a physical interpretation of relativistic formulae, which is different from old one.” [Jánossy 1971, 13] But how do we know that the view Einstein offers of the physical world is inappropriate? An incorrect methodology does not necessarily imply the incorrectness of the theory. Does the theory have any independent, non-methodological features which might make it problematic? In answering this question, Lajos Jánossy represents a view which is typical in the criticism of Einstein’s relativity and which can be characterized as “common sense criticism”.

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“I got acquainted with the theory of relativity at a comparatively early age – I read the famous popular book written by Einstein. Reading the latter I had difficulties with some of Einstein’s concepts: however, having been young and enthusiastic, I convinced myself in the end that I could understand those concepts – to prove this I tried to explain the theory to everybody who was interested. In the course of such attempts I learned the ‘language of relativity’ and I gradually ‘got used’ to the theory. …. Many years later I read several years in succession a course of physics at the university of Manchester. My course contained also the special theory of relativity. As the years went on I developed a technique of presenting the subject so that in the end I could convince my students that they really understood the theory. However, as my technique presenting the theory improved, my own belief in the adequateness of the concepts vanished. In the end I became convinced that from the philosophical point of view the concepts had to be changed. Since about 1950 I have struggled with the problem of the reformulation of the theory and the results of my deliberations are found in this volume.” [Jánossy 1971, 14] As Descartes’s narrative about his schools and education in his Discours de la Methode (Discourse on the Method) expresses a radical criticism of the philosophical views of the epoch and his personal style functions as endorsement and authentication of the criticism, here, in Jánossy’s reminiscence we also encounter a radical philosophical criticism. Jánossy challenges the generally received view that relativity theory requires us to give up our common sense terms. We should not be mislead, he argues, but recognize that there is really something disturbing in Einstein’s theory and the correct attitude is not to suppress this disturbing factor by blaming our common sense for incapacity to grasp physical reality but to face and eliminate it by reformulating the theory. In other writings he is more sanguine and characterizes the received attitude of modern physics to common sense as a cult of irrationality, in the context of which contradiction with common sense becomes a virtue and the scientific character of a theoretical claim is measured by the extent of its absurdity. Rejecting this approach, he insists that “[a] scientific way of thinking cannot be but the refinement, deepening and further development of everyday thought” and that “the whole complex of the theory of relativity can be built up by means of natural methods in conformity with everyday thought”. [Jánossy and Elek 1963, 9, the original is in Hungarian] (Jánossy, influenced by the philosophy of his stepfather, prefers the term “everyday thought” to “common sense” but in his argument the former functionally corresponds to the latter.)

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To summarize, the metatheoretical foundation of the criticism and reformulation of relativity theory by Jánossy consists of two interlaced moments, namely, the priority of physics regarding the mathematical formalism and the conscious acceptance of the terms of common sense as a norm for theory construction. Whereas these moments are common to criticisms of Einstein’s theory, the metatheoretical foundation of the criticism is only seldom formulated so explicitly and definitely as in his case, and this is especially true regarding the role of common sense. The requirement of conformity with the basic notions of common sense as a norm for theory construction emphasized so resolutely by Jánossy may be regarded as Jánossy’s thesis and considered as one of the most important metatheoretical theses concerning modern physics. [Székely 1987; Székely 1988] 3.2. Measures, measurement and relativity theory Metatheoretical norms and principles, however excellent, cannot have any significance if one cannot find the way of their correct application in concrete theories. Jánossy’s main achievement regarding relativity theory is not simply the formulation of the metatheoretical foundation of the criticism but a complete and consistent reformulation of the theory in physical and mathematical terms. In the following parts of our paper Jánossy’s version of relativity theory will be often contrasted with the Einsteinian one. To avoid misunderstandings, it is important to emphasize that in doing so we will always use the terms „Einstein’s theory” or “Einsteinian relativity” in the sense of the version of the theory as it was presented in Einstein’s original (physical) papers and as it is generally taught at universities and presented in textbooks. That is, in our usage the term „Einstein’s theory” will not include any of the metatheoretical and physical reflections made by the German physicist after the publication of the theory. The relation of Jánossy’s notion of relativity theory to Einstein’s subsequent, out-of-theory reflections (which cast a new light on his original formulation of the theory and leave room for a reading which might suggest its reformulation in the direction represented by Jánossy) will be considered at the end of this paper. As a consequence of the heated ideological debates, late in his scientific career Jánossy abandoned philosophical categories regarding relativity theory. Thus in his comprehensive monograph “Theory of Relativity Based on Physical Relativity” published in 1971 we cannot find even such ideologically neutral categories as “common sense” or “everyday thought”. Instead of using philosophical categories he identifies the indicated disturbing aspect of Einstein’s theory in terms of measurement theory. According to him, “ [i]n our approach of physics in general and the theory of relativity in particular we think it very important always to remember that we are dealing with objective physical quantities

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and that we attempt to describe the latter in terms of measures.” [Jánossy 1971, 15] Furthermore, “ [a]n objective physical process develops according to its own laws and it can be described in arbitrary measures.” [Jánossy 1971, 14] Distinguishing measures from things measured, Jánossy definitely commits himself to the traditional concept of physical reality, according to which there exists something „out there” with its own laws and thus he rejects the positivist approach. But emphasizing the arbitrariness of the measures used by physics, he also opposes naive, metaphysical realism which maintains that the investigated objects and the theoretical entities directly correspond to each other (or – in a weaker version – considers the latter the approximations or conceptual pictures of the formers). In his concept physical quantities as characteristics of physical entities are outside of physical theories, while measures (and theoretical construction, so coordinate systems built up of these measures) are the representations of these quantities which physicists can chose arbitrarily. [Jánossy 1971, 72] Consequently, in Jánossy’s interpretation space and time coordinates, as well as their transformations lose the mystical character conferred them by relativity theory: “We may write x=r,t for a four-coordinate of an event. Changing from one system of reference to another we can introduce transformed coordinates x’=f(x) (1) where f(x) is some reversible four-function of its variable x. If the coordinates x are suitable to describe events, then the transformed coordinates are also suitable. Introducing particular measures x or x’ for events we give some kind of names to the events with the help of which we recognize them. … The fact that a transformation type (1) mixes the measures of time and space coordinates does not seem to be of particular importance and it does not imply any properties of space and time.” [Jánossy 1971, 14] This view of physical quantities and their measures is open to contention. However, it is based on acceptable and justified metatheoretical postulates well established in the history of physics which may serve as a foundation for physical theories. Furthermore, it is clear that these postulates contradict the Machianpositivist philosophical background of Einstein’s notion of relativity and thus their definite formulation by Jánossy makes it evident that that notion is not neutral from

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the point of view of physics: it does not follow from the nature of the physical world but rather is a consequence of Einstein’s metatheoretical commitments. But if measures are only names or signs arbitrarily chosen by physicists, how is it possible to know anything about physical reality that is supposed to exist outside physics, a system of human theories? Lajos Jánossy answers this problem by introducing the concept of distinguished measures. While a physical quantity can be described by an infinite number of systems of measures, the majority of the possible descriptions do not contain any information about the quantity in question. Distinguished measures are particular classes of measures which “reflect clearly certain properties of quantities” [Jánossy 1971, 72] Therefore, one of the most important tasks of theoretical research is to find distinguished measures for the quantities under scrutiny, that is, to attempt to find for the description of particular quantities numbers which reflect adequately certain physical properties [ibid.]. To elucidate the concept in more detail, in Chapter III of his monograph Jánossy analyses the measurement of electric charges and then (taking into account that relativity theory is strongly connected to the so called space and time coordinates) in Chapter IV he works out distinguished measures for space and time. According to his analysis distinguished measures are characterized by the fact that in general both their sum and product (or in certain special cases at least their sum) express significant physical quantities, that is, their sum and product also appear in our measurements and/or in the established physical laws. For example, the sum of the usual measures of two electric charges E1 and E2 (say measures e1 and e2) will be equal to the measure we receive measuring the joint charge, while the product of e1 and e2 appears in Columb’s Law. (In fact, Jánossy designates physical quantities with Gothic letters while their measures with Roman letters, so he designates a physical charge with a Gothic e while its measures with a Roman e. For technical reasons we do not follow his notation here.) A physicist used to the usual notation and language of physics may find this terminology rather curious, since physical texts do not usually distinguish the charge and its measure but designate both by the same symbol (say e). However, in the metatheoretical context established by Jánossy it is clear that the charges as objective physical entities do not determine directly the measures to be constructed in the process of measurement and hence it is not at all evident that the measure of joint charges should be the sum of the measures of the two original ones. In Jánossy’s words, “[i]n practice there seems to be no point in introducing nonadditive scales for quantities if there is a possibility of introducing also additive representations. It must be emphasized, however, that it is not trivial that for certain quantities additive measures can be introduced. Whether or not such measures can

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be introduced in a particular case is a question which can be decided experimentally….” [Jánossy 1971, 78]. Of course, the question of measurement is a very complex topic and in his monograph on relativity theory Jánossy could only briefly outline his respective ideas. A more detailed presentation can be found in his earlier monograph Theory and Practice of Evaluation of Measurements which contains a comprehensive presentation of his theory of measurement. That book should be consulted by those interested in this aspect of Jánossy’s theory. [Jánossy 1965] What follows is a brief sketch of Jánossy’s reformulation of relativity theory based on the metatheoretical commitments outlined above. We will attempt to reproduce the logic and the conceptual structure of his theory and will set aside the technical-mathematical details that are essentially the same as the well known textbook formulation of Maxwellian electrodynamics, the formulae of Lorentz transformation and the Einsteinian formalism of the special and general theory of relativity. 3.2. Measures and relativity 3.2.1. Measures of space and time based on rigid rods and physical laws. The definition of ideal clocks. While in his famous paper Einstein firstly introduces a scale of length with the help of rigid rods and then “defines time” (ie, in Jánossy’s terms, “introduces distinguished temporal measures“) with the help of clocks and light signals and so he establishes a “hybrid” scale of space and time, Jánossy separates the rigid rod method from the light signal method and introduces two independent systems of measures: one based on rigid rods, another on light signals. As we have seen, for Jánossy it is not at all trivial that additive length measures can be introduced. The use of additive length scales in everyday practice is based on the fact that with the help of rods considered in every day life as “solid” additive length measures can be obtained. According to Jánossy, science can introduce the term of ideal solid rods only because we are given this experience and he defines a rod to be an ideal solid rod if with its help an additive scale of length can be obtained. [Jánossy 1971, 79] On the other hand, Jánossy emphasizes that with the help of periodical processes (such as mechanical clocks, planetary motions etc) we can complete our system of length measures to set up a combined system of length and temporal measures in terms of which physical phenomena obey certain rules. As measures in general, temporal measures in particular can be obtained in several ways and there is no a priori guarantee that these ways will all result in the same measures (or that measures arrived at in different ways will coincide). However, considering that the

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aim of physics is to discover rules in the behaviour of the physical world and formulate them as physical laws, from the point of view of science it is rational to attempt to complete our length scale with a temporal scale in such a way that certain fundamental and in the practice well confirmed laws, for example, Newton’s first law be fulfilled. At first sight, perhaps, this approach may seem to be logically circular, since physical rules may appear only if we have already a joint scale of length and time, while Jánossy want to complete the length scale with a temporal scale with the help of already known laws. Is this not a vicious circle? Taking a closer look at the issue reveals that the approach is correct. In the history of physics we are given physical rules (for example, Newton’s first law) which seem to work if we use our everyday length and time measures or measures established in the history of physics. These rules appear in terms of measures, which are intuitive and without reflection (or are based on metaphysical commitments as for example in Newton’s case) and therefore it cannot be excluded that they are to a certain extent consequences of our choice of measures. To enlighten the nature of these rules we need an a priori analysis of the applied measures and in this analysis (while suspending the validity of the concerned rules regarding physical reality) we may introduce a hypothetical world in which these rules are assumed to be fulfilled, and Jánossy follows this methodology. Thus we may assume a region where Newton’s first law is valid in terms of a given (but yet unknown) system of measures. Provided that we already have a length scale, in such a region we no longer need Einstein’s radar method to synchronize clocks: it will suffice to observe the motion of free particles and to adjust the local measures of time showed on the local clocks in such a way that Newton’s first law be fulfilled. (To observe the path of a particle we need not use light signals: every observer can measure with the help of his own clock and make a note of the time when his own position is crossed by a moving particle and then the notes can be collected and analysed in order to synchronize the clocks.) Exploiting this a priori possibility, Jánossy introduces the term of „ideal clock”. According to his definition a clock is ideal when it gives immediately (without correction) the distinguished temporal measures based on Newton’s first law. [Jánossy 1971, 95-96] The rate of an ideal clock is by definition constant and our physical practice definitely shows that there are regions in the real world which allow us to introduce good approximations of a system of measures based on ideal solids and ideal clocks. (Otherwise Newton’s first law would not be applicable in practice.) Similarly, we may introduce temporal measures using planetary motions or the rotation of the Earth around its axis and assuming the validity of the law of gravitation and it is also possible to use atoms as clocks and taking into account the physical theories of atoms. Of course, it is not evident that all these scales will correspond to the first, mechanical or ‘ideal’ temporal scale; neither is it evident that the non-mechanical (planetary, sideric or atomic) scales will be adjustable to each

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other. In this respect Jánossy’s definition of ideal clocks is a metatheoretical norm requesting a physical explanation in any case when an applied time scale deviates from the ideal one. (Incidentally, since Newton’s first law is deducible from Leibniz’s principle of sufficient reason, Jánossy’s definition of ideal clocks may be deduced from this fundamental Leibnizian thesis. On the other hand, it can be also shown that the Einsteinian version of special relativity does not fulfil the Leibnizian principle. Thus Jánossy’s version of relativity theory - despite its empirical orientation - can be seen as a reformulation of the original Einsteinian theory, with the aim of satisfying Leibnitz’s principle. Furthermore, Jánossy’s method of definitions of ideal solid rods and ideal clocks, a beautiful example of the application of everyday experiences in physics, follows – unconsciously – the logic of the so called “hermeneutic circle” emphasized by Heideggers’ philosophy and indicates how promising a possible Heideggerian metatheory of physics may be.) 3.2.2. Measures by radar method without rods Jánossy also shows that it is possible to attempt to introduce length and time scales using only light signals, provided that we assume that light is propagated isotropically and with a constant velocity relative to a given reference system, say K. It is clear that similarly to the rigid rod scale, we do not have any a priori guarantee of success in this case either. It is a matter of practice whether a coherent system of space and time coordinates can be constructed in such a way and if we succeed and a system of coordinates introduced by this method passes the test of coherence, then this fact „can be taken to support the hypothesis about the mode of propagation of light in K”. [Jánossy 1971, 99] The introduction of such a scale follows the same logic as the rod scale without the radar method: first an ideal region is assumed where light is propagated isotropically and the measures are defined for this ideal region, then, as the second step, experience will show whether these measures can or cannot be applied in the real world. 4. Lorentz transformations and Jánossy’s theorem 4.1. Lorentz trasformations as transformations of measures Applying the conceptual basis introduced above, Jánossy demonstrates that: if there is a system of coherent measures M of length and time in terms of which light appears to be propagated isotropically and with the velocity c relative to a reference system S, then there exists a group of mathematical transformations of that system of measures with the following characteristic: - each members of the group transforms the system of measures M into another system of measures M’ in whose terms light appears to be propagated isotropically

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and with the velocity c relative to another reference system S’ which is in rectilinear and even motion relative to the original reference system S; - vice versa, for any reference system S’ in rectilinear and even motion relative to the original reference system S there exist a member of the group of transformation above, which transforms the system of measures M into a system of measures M’ so that in the reference system S’ light will appear to be propagated isotropically in terms of M’. [Jánossy 1971, 100-105] Anyone familiar with relativity theory will see that the group of transformations which Jánossy found is the well known group of the Lorentz transformations. That is, he did not discover transformation of a new kind but deduced the famous ones in a new way different from both the Einsteinian and the Lorentzian deductions. However, what is important for us is not simply the new deduction but the new meaning of the transformations. Whereas in Einstein Lorentz transformations are deduced as transformations which connects inertial reference systems so that Einstein’s two axioms be satisfied, in Jánossy they emerge in an investigation of the propagation of light in terms of various systems of measures without referring to the concept of inertia and their existence are stated in the form of an a piori, mathematical theorem. We will refer to this theorem as “Jánossy’s theorem” and (following his terminology) call the reference systems relative to which light appears to be propagated isotropically in terms of a particular system of measures “Lorentz systems”. Notice, that Jánossy’s theorem is not about inertial systems: it is valid independently of whether Lorentz systems are inertial or not. 4.2. The analysis of Jánossy’s theorem Jánossy’s theorem imposes two a priori constraints upon physical reality. A) On the one hand, if rods and clocks are never deformed when in motion with respect to any Lorentz system (that is they preserve their shape and pace), then i) (on simple geometrical grounds) there will be only one Lorentz system in which the system’s own Lorentz measures (that is, the measures in terms of which light appears to be propagated isotropically relative to the system) and measures based on rods and clocks without light signals will coincide; consequently ii) the relative velocity of any other Lorentz system with respect to this special system will be determinable with the help of rods and clocks and light signals, since in terms of measures established with the help of these rods and clocks light will not appear to be propagated isotropically relative to these systems. (This simply follows from the fact that Lorentz measures are connected with Lorentz transformations which change the measures of length and time, while the unchanged rods and clocks will establish the same system of measures independently of their motion relative to any Lorentz system.)

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B) On the other hand, if we observe that Lorentz measures and measures based on rods and clocks co-moving with the systems will always coincide, then this observation will indicate that i) there is a definite Lorentz system in physical reality which may be called as the basic system, and ii) rods and clocks moving relatively to this basic system suffer deformation according to the formulae of the Lorentz transformations. The observed relativistic effects (that is, the relativistic contraction of lengths and the slowing down of physical processes according to Lorentz’s formulae) show that in physical reality the second possibility is the case, thus on the basis of Jánossy’s theorem as an a priori theorem these effects necessarily imply the existence of a basic physical system in which rods and clocks at rest are not deformed, while in motion relative to this system they suffer deformation according to Lorentz’s formulae. 4.3. The hidden epistemological and logical background of Einstein’s special theory The a priori analysis of Jánossy’s theorem makes clear that Einstein’s special theory of relativity is based on two mathematical “boundary-conditions”. On the one hand, special relativity is only possible because Jánossy’s theorem is valid, that is, Lorentz transformations exist and they transform a system of measures in term of which light appears to be propagated isotropically into another system of measures with the same characteristic. On the other hand, the Einsteinian version of the theory, that is, the version in which – in contrast to the implication of Jánossy’s theorem –, the existence of any privileged systems is rejected, can only escape logical contradiction because Einstein implicitly rejects that the spatial relations of the physical entities of a given region form a definite, consistent spatial configuration. To enlighten the latter moment of Einstein’s theory, let us recall that the claim about the isotropic propagation of light in any inertial system is perhaps the most paradoxical ingredient of the Einsteinian theory of special relativity. Namely, if a physical effect is propagated in a given reference system isotropically, then it cannot (on geometrical grounds) be propagated in a similar way in other systems moving rectilinearly and evenly with respect to the former. How is it possible that Einstein succeeded in working out a consistent theory incorporating this geometrically impossible characteristic of the propagation of light? Jánossy’s theorem helps to explore the hidden conceptual background which makes possible for Einstein to avoid the contradiction. Namely, geometry excludes the simultaneous isotropic propagation of light relative two different (physical) reference systems in motion at a constant velocity relative to each other only if the following two premises are fulfilled: i) the space and time relations of the physical entities of the concerned region define a common, definite space in which the investigated systems move, and

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ii) length and time are measured in both systems with the same measures. Consequently, we can construct a consistent physical theory in which light appears to be propagated isotropocilly with respect to two different reference systems in motion relative to each other only if we reject at least one of these premises. Now, Jánossy explains relativistic phenomena with the help of the assumption that rods and clocks in motion relative to the basic system are deformed according to Lorentz’s formulae. Consequently, in the case of two Lorentz systems in motion at different rate relative to the basic system these measuring tools will suffer different deformations and so the systems of measures introduced with their help will be also different. Thus in Jánossy’s conceptual framework it is premise ii) that is not fulfilled, and the function of the assumed deformations of rods and clocks are exactly to give an explanation of the change of measures that takes place despite the use of the same rods and clocks when we change the systems. Of course this explanation – as any Lorentzian kind approach – breaks the ontological symmetry of the relativistic effects: in its context the contraction of rods observed from a system moving faster relative to the basic system than the observed rods is only an apparent phenomenon since the latter suffer smaller contraction than the measuring rod of the observer and hence in reality they are longer than the observer’s rod. Since Einstein’s theory excludes the existence of any basic system and assumes relativistic effects to be symmetric that does not allow to speak about real, physical deformations of measuring tools, his theory can be consistent only if the first premise is rejected. However, if we assume that physical entities are definite entities with definite spatial relations, then these relations will form a definite physical space in which these entities exist and move. So the rejection of premise i) amounts to rejecting that physical entities have definite spatial relations independently of the applied measures and in Einstein’s special relativity this really is the case. Due to Einstein’s neopositivist attitude, in his theory physical entities exist and move not in a common physical space but inside relative co-ordinate spaces, that is, (using Jánossy’s term) inside spaces of different systems of measures and it can’t be introduced any common system of spatial relations that could be independent of our measures. Put differently, Einstein’s axiom of special relativity by exclusion of the existence of any privileged reference system also excludes the possibility of any definite physical configuration formed by the spatial relations of the physical entities, and so, in the words of Hungarian philosopher Melchior Palágyi, it fragments physical reality into an infinite number of reference systems. [Palágyi 1914, 59-60; see also: Székely 1996]

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5. Ether and Lorentz principle 5.1. Ether and Lorentz deformations Jánossy calls the deformations of clocks and rods in motion relative to the basic system “Lorentz deformations”. Taking into account that these deformations emerge when rods and clocks are in motion with respect to the basic system, it is natural to assume that the basic system is connected to some physical entity (such as a background physical field) and the deformation is somehow caused by this entity. Furthermore, considering that the classical concepts of the ether have a function similar to that of this entity, the latter can be called “ether” without any commitment to the notions of the classical ether theories. However, it is not necessary to use this term. What is important is only that if one distinguishes measures as representations from the measured things as parts of physical reality, then the observed relativistic phenomena discussed in the special theory of relativity will imply the existence of such a background entity as well as the Lorentz deformations of clocks and rods in motion relative to it. Now Jánossy identifies this background entity with the electromagnetic ether which he introduces on common sense grounds. According to him “From Maxwell’s theory it follows that light in particular and all electromagnetic action in general is propagated with a velocity c= c’, where c’ is the critical velocity. …. The question cannot be avoided relative to what are electromagnetic waves propagated with velocity c?….A simple answer to this question could be obtained claiming that light is propagated with the velocity c relative to its source. The latter assumption contradicts, however, the well established theory of Maxwell and seems also to be contradicted directly by experiments…. Electromagnetic perturbation once it has left its source is propagated thus with a velocity c independently of how the perturbation comes about. The only reasonable interpretation of this is to assume that the perturbation moves with a velocity c relative to its carrier. The carries may be denoted using Maxwell’s terminology, ether. We shall in accord with the ideas of Maxwell also assume that light is propagated with a velocity c relative to the ether.” [Jánossy 1971, 48] That is, for him the existence of a basic system is granted in advance, independently of the Lorentz transformation and an analysis of relativistic phenomena, on the basis of Maxwell’s theory. So the logic of his presentation does not follow strictly our a priori analysis above. We have made a small change in the

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presentation of his ideas and deduced the existence of a basic system from the observed relativistic phenomena with the help of his theorem just to indicate the heuristic power of his approach. 5.2. The Lorentz principle Relying on the null results of the experiments aiming to determine the translation velocity of the Earth relative to the ether (such as the Michelson-Morley and the Kennedy-Thorndike experiments) and on the observation of the perpendicular Doppler effect, Jánossy finds it reasonable to introduce the following general principle which he calls “the Lorentz principle”: “The law of nature is such that provided S is a real physical system, then the Lorentz deformed systems S* are possible systems obeying the same laws as S.” [Jánossy 1971, 120] It is evident that this is a reformulation of Einstein’s principle of special relativity in physical terms, implying the same observational predictions and the same modifications of classical physics as Einstein’s principle does. It is a frequently repeated argument against Lorentzian-type interpretations that they are ad hoc in contrast to Einstein’s beautiful axiomatic theory. Now Jánossy has definitely showed that this is not the case. On the one hand, the Lorentz transformation can be deduced in a train of thought of simple considerations about measurements and measures. On the other hand, the Lorentz principle as a simple idea based on observational data completely substitutes Einstein’s axiom of the equivalence of inertial systems and predicts the relativistic phenomena in a similarly simple and coherent way as Einstein’s axiom does. Furthermore, if we want to compare the two approaches using the term “ad hoc”, we must conclude that it is Einstein’s theory and not Jánossy’s reformulation that is ad hoc in the particular sense that it states the equivalence of inertial systems as an unexplainable and non-deducible axiom, while Jánossy’s Lorentz principle and the concept of Lorentz deformations are based on an analysis of physical measurement and measures, a problem that Einstein’s positivist attitude prevents even to address. 6. Jánossy’s general relativity Jánossy does not stop at the reformulation of the special theory of relativity, but also reconsiders the general one. His notion of general relativity is based on two ideas: i) the concept of measures applied in the reformulation of the special theory and the term of rigid bodies defined with the help of these measures;

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ii) the generalization of the Lorentz principle originally introduced by him in the context of the special theory. 6.1. Ideal solid rods and the exclusion of the space-time metaphysics of general relativity As we have seen, Jánossy defines ideal solid rods as rods with the help of which a consistent additive length scale may be obtained. In a further step Jánossy introduces a system of space co-ordinate vectors (that is, the usual space coordinate system) with the help of measures determined by rigid measuring rods and defines the distance of two points in this co-ordinate space by the formula (Ri-Rk)G(Ri-Rk)=Rik2 (formula F) where Ri and Rk are the coordinate vectors of points Pi and Pk, G is a positive definite symmetric matrix, and Rik is the distance. It is clear that according to this definition for any N+1 points P0, P1 ….. P(N+1) we are given N(N+1)/2 equations for the 3xN components of the co-ordinate vectors, thus we will have an overdetermined system of equations which does not have necessary solutions. Jánossy applies this fact to an extended definition of ideal solid rods: if in a system of space co-ordinates which has been established by measuring rods the distance formula above will work for any number of points (that is, the system of equations defined by the formula F for any points P0, P1…Pk …..Pn will have solutions), then we may consider our rods to behave as ideal solid rods. [Jánossy 1971, 81] Consequently, if we observe that the system of equations according to the formula F does not have a solution at any set of points, then this fact will indicate that the rods we have used in the construction of our co-ordinate system have been deformed in the process of measurement (that is, they are not ideal solid rods). It is clear that this extended notion of ideal solid rods introduced by Jánossy aims to exclude any word usage about non-Euclidean physical spaces and is in full agreement with Poincare’s idea of the relation of physics to geometry. Our hypothesis is neither on geometry nor on physics in itself but on geometry and physics together, Poincaré emphasizes, and Jánossy commits himself to a connection of physics and geometry in which the structure of space (at least in the Einstenian sense) loses its meaning. If formula F does not work consistently (that is, our co-ordinate space is not Euclidean), that will only inform us about the behaviour of measuring rods but will have nothing to do with the “structure of physical space”: “The above statements can also be formulated in another way. If the measured distances rik between the points of a set can be expressed by a quadratic form (F), then one might conclude the space in which the points are situated is ‘Euclidean’. Or if no consistent co-ordinate measures can be obtained one might conclude that the space is ‘non- Euclidean‘.

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We do not think, however, that such a conclusion has any meaning. The fact that the overdetermined system (F) poses solutions Rk k=0,1, 2 …… n seems to us to reflect upon the method of measurement of the distances rik and in particular upon the measuring rods used. Roughly speaking one may conclude from the consistency of measures that the measuring rods made use of are behaving like rigid bodies, i.e. if the measuring rods are turned or shifted they do not change their length.” [Jánossy 1971, 86. Italics mine: Sz. L.] It is to be noted that this conceptual scheme (whereas it radically opposes Einstein’s view of the relation between geometry and experience presented in his paper of 1921 [Einstein, 1921]) is more than a clever trick to prevent any talk about non-Euclidean physical spaces. On the contrary, it is based on a correct epistemological presentation of the practice of physics and the relations among measures, the measured characteristics of physical entities and measuring tools. Jánossy’s concept of the ideal solid rod makes it once again clear that non-Euclidean spaces in Einstein’s theory are only implications of Einstein’s positivist philosophical commitment which neglects that measures and theoretical spaces built up of them are only human constructs which do not correspond directly to physical entities or their characteristics. On the other hand, this positivist washing away of the difference between physical reality and human representations may turn into its opposite and result in a metaphysics of space-time if (as is the general case in the university teaching of relativity theory) we assume that the metric of space-time appearing in the general theory is the cause of the gravitational phenomena. Namely, in this interpretation the structures and characteristics of co-ordinate spaces will appear as objective properties of the physical world and thus Einstein’s positivist starting point will results in a theory of “objective” curvature of space-time which determines the behaviour of physical phenomena. If we have a feeling that in Einstein the cart is put before the horses [see: Balashov and Jansen 2003, 340; Brown 2005, 133-134]), then Jánossy’s analysis will explain the reason for this feeling. Measures as human constructions are numbers, so co-ordinate systems, coordinate spaces etc. constructed with their help are necessarily of a mathematical-geometrical nature. Washing away the difference between these human constructions and physical reality necessarily transforms physical reality into mathematics. 6.2. The extension of the Lorentz principle from homogeneous regions to inhomogeneous ones On the face of it Jánossy’s notion of general relativity may seem disturbing. While Einstein introduces the principle of special relativity as the equivalence of

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inertial systems and arrives at the general theory by extending that principle to arbitrary systems, in Jánossy’s reformulation the special theory deals with the propagation of light and not with inertial systems. However, this apparent difference can be easily resolved, since (as Jánossy shows) the Lorentz principle implies that Lorentz systems are inertial systems and vice versa. This implication is eventually equivalent to the claim that the two independent systems of measures introduced by Jánossy (that is, the system of measures based on rods without light signals and that established with the help of light signals by means of the radar method) are equivalent. So Jánossy would also be able to introduce the general theory as the extension of the special theory from inertial to arbitrary systems. Nevertheless, he does not follow this path, but continues to investigate the problem in terms of measures and the propagation of light. While he demonstrates that Lorentz systems and inertial systems coincide and thus a Lorentz system can be identified with the help of inertial phenomena, in his approach inertia remains only a secondary characteristic of these systems. The primary characteristic of a Lorentz system is for him the existence of a special system of measures in whose terms light appears to be propagated isotropically and with constant velocity relative to the system itself. Since such systems can only exist in physical regions where light appears to be propagated homogeneously, Lorentz systems are connected to such regions and Jánossy formulates the problems of general relativity with the help of this fact: “In the special theory of relativity only such regions are considered in which light is propagated homogeneously. The laws governing the motion of physical systems inside such regions obey symmetries which can be expressed by the Lorentz principle. In reality light can nowhere be assumed to be propagated strictly homogeneously, as we have reason to believe that the propagation of light is affected by gravitation and regions entirely free of gravitation do not exist. The Lorentz principle can be therefore taken to be valid only to such an approximation as gravitational effects can be neglected. The question arises, how the Lorentz principle should be generalized so as to apply to regions containing not negligible gravitational fields.” [Jánossy 1971, 214] Although his terminology considerably differs from that of Einstein’s, Jánossy’s train of thought is mathematically parallel to the consideration of the German physicist. So he shows that the sufficient and necessary condition of the homogeneous propagation of light in a given physical region is the existence of a straight (that is Euclidean) representation of the region established with the help of light signals, a criterion which is mathematically equivalent to the criterion that the

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Riemann-Christoffel tensor formed of the propagation tensor of light expressed in any measures of coordinate is equal to zero. [Jánossy 1971, 218-220] “We see thus that using signals of light only we are in a position to examine whether or not light is propagated homogeneously in the region we are investigating, and if the propagation of light proves to be homogeneous, we are in a position to construct a straight system of reference with the help of the signals of light.” [Jánossy 1971, 222] The Lorentz principle implies that homogeneous regions obey the same physical laws even if a system is Lorentz deformed, so physical laws of homogeneous regions are Lorentz invariant. Jánossy generalizes this fact along the following train of thoughts: i) From a mathematical point of view the laws valid for homogeneous regions may have several generalizations for inhomogeneous regions even if a) we restrict the possibilities of generalizations by requiring that the Lorentz principle originally valid for homogenous regions should also be valid for sufficiently small inhomogeneous regions [Jánossy 1971, 230], and b) we prescribe that the laws of homogenous regions should be contained as limiting cases by the generalized laws. [Ibid 264] ii) Since i) allows an unlimited number of possibilities for generalization, we ought to seek further restrictions and it seems that the most rational and heuristically most fruitful restriction is to seek only generalizations that can be expressed in tensors and covariant operators. Jánossy introduces the latter requirement as the extended (that is, generalized) Lorentz principle. [Ibid.] Thus in contrast to Einstein’s general principle of relativity which forms a definite claim on the nature of physical reality, his general theory of relativity is based on a methodological principle involving only a vague ontological element: namely the conjecture, that physical reality is such that this principle can be successfully applied to it. There is no place and perhaps it is not even necessary to give a more detailed presentation of Jánossy’s development of the general theory, since it is easy to see that it mathematically corresponds to Einstein’s considerations. What is important for us is the physical meaning of his presentation which considerably differs from Einstein’s. a) Firstly, Jánossy’s version of general relativity primarily is about the propagation of light. As a consequence, in his presentation the metric tensor of the general theory primarily appears as the propagation tensor of light. It emerges only in a later phase of the development of the theory that this tensor coincides with the metric tensor of the gravitation field [Jánossy 1971, 242-256, 266] (a coincidence which requires an explanation since the extended Lorentz principle as a heuristic

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principle cannot explain anything). Similarly, the equivalence of the gravitational and inertial masses (on which Einstein’s theory is based) loses its fundamental role and appears only as a secondary implication of the theory [241, 263]. b) Secondly, since the generalized Lorentz principle is for Jánossy only a heuristic principle, it is not at all granted that laws constructed with its help really are natural laws: “Sometimes suggestions are made to the effect as if the generalizations of the laws of nature which lead to the forms of the laws in gravitational fields could be obtained in a priori considerations. According to this view the laws thus obtained are logically more or less the only possible ones …. Such considerations are at fault; we shall show in the following that relativistic laws are based on well-defined physical hypotheses concerning the structure of matter and gravitation. It is a question of fact as to what extent these hypotheses give a correct description of real nature. ” [Jánossy 1971, 213-214] “So as to find the form of various physical laws in inhomogeneous regions it is useful to see how the mathematical form of such laws, valid in homogeneous regions, can be generalized. It is a question of experiment to find out whether or not the generalizations which suggest themselves are in accord with experiment.” [Jánossy 1971, 235] In any particular case it remains thus to be decided by experiment which of the generalized form of the physical law describes correctly the observed phenomenon. However, we have to go further: it is also a question to be decided by experiment whether or not the law describing a particular phenomenon correctly is an invariant one?” [Jánossy 1971, 264] (Notice that in Jánossy’s terminology a law is characterized as ‘invariant’ if it can be expressed in terms of tensors and covariant operators.) c) Thirdly, while non-Euclidean spaces appear in both Einstein’s and Jánossy’s theory, Jánossy argues that they are only spaces of measures, that is, theoretical spaces constructed by human beings to represent physical reality. The primary physical terms for Jánossy are homogeneous and inhomogeneous regions of the propagation of light, of which the first can but the second cannot be represented with straight coordinates. As a consequence, with Jánossy straight coordinates always indicate homogeneous regions and so regarding such regions they should be

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considered as privileged representations which directly characterize the regions, while in Einstein there are no privileged representations. d) Lastly, in Jánossy the four dimensional space-time is only a construction built up of measures, that is of representations constructed by human beings. So the interpretation that real physical bodies move on their geodetic paths in the four dimensional space-time is meaningless. “… it seems to us that it is a play with words if we suppose the geodetic line to be a ‘straight line in four dimensions’. ….. the solutions [of Einstein’s field equations]…. include among others Kepler’s ellipses along which planets move. – If we call those orbits ‘straight‘ then we lose completely the meaning of what is usually called straight.” [Jánossy 1971, 241] 7. The nature of Jánossy’s ether. 7.1. The antenna problem. The erroneous claim on the simplicity of Einstein’s theory Jánossy’s reformulation of relativity theory in terms of measurement arrives at a mathematical formalism equivalent to that of Einstein’s theory. Jánossy might as well stop at this point since his version of the theory does everything that the original version does. However, since in his conceptual framework the term „structure of space-time” as an entity existing “out there” in the physical world is devoid of meaning, he is firmly opposed to using it to explain relativistic phenomena. In his interpretation it is not the metric of space-time that determines the behaviour of other physical entities but the latter imply the former: the relations and rules of the physical world are such as to permit theoretical representation with the help of this term. As a consequence, in Jánossy‘s conceptual framework the original version of relativity theory appears as a merely phenomenological theory, while the ether-based interpretation of its mathematical formalism serves as its completion with a second, explanatory level. And at this point we arrive at the heart of any ether-based issue: what is the nature of the ether and what is the mechanism by which it impacts physical phenomena? It is often argued in favour of Einstein that his theory needs no such mechanism and so it is incomparably simpler and more elegant as any ether-based approach. A common counterargument (present also in Jánossy) is that simplicity and elegance are no criteria of truth since nature does not have to respect these human qualifications. As a matter of fact, even this counterargument is unnecessary since Einstein does not give us any explanation of how the assumed mathematical properties of space-time can influence physical phenomena or, put differently, how it is possible

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that physical entities follow geodetic lines. Is there an influence, a constraint exercised by the space-time (or the ether) on physical entities determining their behaviour or do the latter have an innate inclination to follow geodesics? Here we face the so called “antenna problem” well known in the literature. [Nerlich 1976, 264; DiSalle 1994; Brown 2005, 24-25] The main point is not, however, the problem, but the fact that the classical formulation of Einstein’s theory does not even attempt to answer the problem. Now, a theory that ignores and fails to address a crucial point of its subject and is, in this respect, incomplete, is highly likely to be simpler than another theory, which not only deals with the issues addressed by the first theory but also confronts problems ignored by the other one. The claim that Einstein’s theory is simpler and more elegant than the ether-based approaches is mere tautology. (Put ironically, the null theory is the simplest theory as it sees no problem and thus only declares that there is nothing to be solved.) Consequently, the ether-based explanation of relativistic phenomena is not an unnecessary and clumsy alternative to the original, Einsteinian explanation but, on the contrary, is a completion of the latter proposing a physical-causal explanation of the phenomena described mathematically by the original one. 7.2. The nature of Jánossy’s ether and Jánossy’s hypothesis on the mechanism of the Lorentz deformation Turning to Jánossy’s views on the physical nature of the ether, it should be emphasized that the main objective of Jánossy’s monograph on relativity theory is to reformulate Einstein’s theory on correct epistemological and physical grounds and to elucidate the logical and physical place of a privileged physical system in the context of a theory of relativistic phenomena. As such, the work does not aim at a complete theory of the ether. It lays down only the basic principles and outlines a few provisional hypotheses in order to assist and orientate further work on the topic. So the Hungarian physicist emphasizes that using the term “ether” he does not want to commit himself to any traditional theory: “ [...] as to avoid misconceptions we wish to emphasize that we regard the ether merely as the carrier of electromagnetic waves and possibly the waves associated with other fields and of elementary particles.” [Jánossy 1971, 48] He also rejects the macroscopic-mechanical models of the ether and its notion as a reference frame at absolute rest: “Einstein’s polemic against the ether concerned mainly the assumption that the ether is at ‘absolute rest’. Thus Einstein

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denied the existence of a system K0 which is at ‘absolute rest’. ” [Jánossy 1971, 49] “We think that the assumption that electromagnetic waves possess a carrier has nothing to do with the question of absolute rest. The concept of ‘absolute rest’ is a metaphysical concept which must be rejected. However, the concept of the ether as the carrier of electromagnetic and other phenomena is quite a different one….. Whether or not the ether, i.e. the carrier of electromagnetic waves, is at rest or at ‘absolute rest’ is a question which does not arise here and certainly has no significance in relation to our problems…. For our consideration it is also immaterial whether or not various parts of the ether move relative to each other. It seems quite plausible that considered on a cosmic scale distant parts of the ether are streaming with various velocities ….” [Jánossy 1971, 49-50] On the other hand, as an affirmative feature of his concept, besides being primarily the carrier of electromagnetic interactions, the ether also appears as an entity causing the Lorentz deformations. Jánossy assumes that the deformation emerge when physical entities accelerate relative to the ether. If the acceleration is slow enough and proceeds step by step, then the accelerated physical system will have time after all consecutive phases to settle down into newer and newer configurations. However, if the acceleration is continuous, the system will lag behind the configuration corresponding to the achieved velocity, and it will settle down into the latter only after a certain small temporal interval following the acceleration. So in the latter case the process of the deformation is (at least theoretically) observable, since there is a minor temporal interval during which the deformation has not yet taken place and thus the states of measuring tools do not coincide with the states expected according to the Lorentz transformation. [Jánossy, 127-128] Jánossy illustrates this hypothesis with the help of a practically solid rod. A rod is a configuration of its atoms and these latter are in a state of dynamic equilibrium. The forces causing the acceleration disturb this state, but after the acceleration has ceased, the atoms - now moving relatively to the ether - will establish a new equilibrium [Ibid, 127]. (Of course, in the case of deceleration inverse processes occur.) These hypothetical processes also constitute a physical explanation for the Lorentz principle. Whereas the principle declares the form of possible physical systems, the mechanism of deformations caused by the acceleration relative to the ether explains how and when such systems come to exist. To express the connection between these processes and the Lorentz principle, Jánossy formulates a dynamic

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version of the principle, which he considers to be one “compatible with the originally formulated Lorentz principle and […] an addition to it” [Jánossy 1971, 126]: “If a connected physical system is carefully accelerated [with respect to the ether] then, as a result of the acceleration, it suffers a Lorentz deformation” [Ibid.]. In contrast with Jánossy, Harvey Brown opines that this principle is only a simple implication of the original formulation of the Lorentz principle [Brown 123124]. However, the original formulation leaves open the question about the concrete physical cause of the Lorentz deformations, since from an a priori point of view it is not necessary to connect these deformations to the acceleration. So one may assume that the deformations are caused by permanent pressure of the ether during the rectilinear and even motion. By connecting the Lorentz deformations to the process of acceleration the dynamic principle excludes the alternative explanations and hence it really contains an additional element with respect to the original formulation. In the context of the general theory Jánossy assigns other characteristics to the ether. So it may have different physical states, it contains inhomogeneous structures, strains, etc. and functions as the seat of different physical fields (such as the field of gravitation). While Einstein explains the phenomena of the general theory with the help of the metric of space-time, for Jánossy the clue is the state of the ether which is represented with the metric tensor: “G (the metric tensor) represents some physical field which appears when observing very different physical phenomena – the propagation of light is only one of many such phenomena. The usually accepted interpretation of G is that it represents the ‘metric of the space-time continuum’. We do not think the latter interpretation to be a fortunate one. We would rather suggest that G represents the state of the ether which is the carrier of all physical fields.” [Jánossy 1971, 266] Before closing this section, we would like to make two brief remarks. The first concerns the above citation which shows some ambiguity. Namely, Jánossy speaks here about the tensor G as a representation but uses a Gothic G rather than a Roman G to denote it, which seems to run contrary to the convention introduced earlier in his book, according to which representations are notated by Roman Gs. However, from the context it is clear, that the word “representation” here does not mean the representation in our theories but a characterization of the state of the ether by physical quantities as, for example, the quantities of volume, temperature, pressure etc. “represent” – that is characterize – the state of a gas cloud. Jánossy’s refers here with the term „metric tensor” to a complex of physical quantities

27

(or briefly to a “tensor quantity”) which is not a mathematical entity and, therefore, does not consist of numerical values or mathematical functions but may be considered as a “tensor” only in the sense that we need mathematical tensors to represent it. Consequently, its notation by a Gothic G is correct and the ambiguity of Jánossy’s text follows not from the notation but from the fact that he uses the word “represent” in two different senses. Our second remark is about a critical reflection by Harvey Brown on the relation between Jánossy’s Lorentz principle and the Lorentz covariance of the laws of physics. “[The] ambiguity in the formulation of the principle would be removed if Jánossy just equated it with the Lorentz covariance of the fundamental laws of physics, and it is hard to see why he didn’t. It is almost as if Jánossy intends the Lorentz principle to stand over Lorentz covariance. At the start of the mentioned discussion of Maxwell’s equations, he announces that ‘Physically new statements are obtained if we apply the Lorentz principle to Maxwell’s equations’. But of course what emerges in the discussion is simply the Lorentz covariance of these equations.” [Brown 2005, 123] Brown is formally right. Lorentz covariance really may substitute Jánossy’s Lorentz principle. However, the requirement of Lorentz covariance in itself is only a formal requirement which allows different physical interpretations. Whereas in the original Einsteinian theory Lorentz convariance relates to our representations depending of the chosen reference system and later is connected to the‘structure of space-time’, in Jánossy the term ‘structure of space-time’ denotes only an element of our theoretical representation and Lorentz covariance is rather connected to physical deformations emerging independent of our reference frames. The function of Jánossy’s Lorentz principle is to exclude the Einsteinian interpretation and to give a physical interpretation to the relativistic phenomena; a function that cannot be served by the formal requirement of the Lorentz covariance of physical laws. (May be I am wrong but it seems to me that Brown overlooks this important moment of Jánossy’s notion since he – despite his excellent analysis of the relation between the mathematical formalism and the physical meaning of relativity theory and his valuable reflections on Jánossy’s theory – ignores the measurement theory aspect of Jánossy’s investigations.) 7.3. The emergence of gravitation forces Jánossy closes his reformulation of general relativity with an interesting hypothesis on the emergence of gravitational forces. According to his hypothesis

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closed physical systems are kept together by internal forces which are propagated with the velocity of light in the ether. When a gravitation field is present, it disturbs the homogeneous propagation of these forces (as it also disturbs the propagation of light). Due to this disturbance a new force emerges which tends to accelerate the system just like the Newtonian gravitational force is expected to do. Consequently, “[t]he gravitational force observed phenomenologically is equal to the self force with which a closed system acts upon itself, if the propagation of the internal forces is made inhomogeneous by the gravitational field.” [Jánossy 1971, 263] In this context we also receive an physical explanation of the equivalence of inertial and free gravitational motion, which forms a basis pillar of Einstein’s general relativity: “in a free falling particle the propagation of inner forces is nearly homogeneous relative to the particle itself, therefore in the free fall no resultant self force is present” [Ibid.]. Jánossy illustrates the applicability of his hypothesis on the example of an electric charge, but he unfortunately does not proceed further in this direction. However, the author of the present paper think that his hypothesis is of heuristic value and worth for further consideration even in this preliminary form and even if one agrees with Brown that several aspects of Jánossy’s idea of the ether are too traditional. If one feels similar to Brown then one must be aware of that we face a problem here characteristic for any ether-based approach. Namely, it is not so easy to find how to satisfy all the requirements we expect from a modern theory of the ether without turning it into a ‘mathematical ghost’ as Walter Ritz had characterized yet not the Einsteinian but the Lorentzian ether 100 years ago [Ritz 1908], which, in turn, was considered by Einstein several years later as still too mechanical. 7.4. Einstein’s and Jánossy’s ether It is well known that after publishing his general theory of relativity, Einstein made several metatheoretical assertions which shed new light on the problem of relativity. So he reintroduced the concept of the ether in the interpretation of the metric field of the general theory, and (as a more far reaching change with respect to his early ideas) he also indicated that his special theory needed a completion concerning the dynamical mechanism of the deformation of rods and clocks. These assertions clearly indicated that after the publication of his theory Einstein had a feeling that it was not sufficiently complete but needed a second, physical level. That is, in contrast to many current representatives of Einstein’s theory at universities and research institutes (who are more Einsteinian in this respect than the German

29

physicists himself was) Einstein did not consider the published version of his theory to be necessarily final and did not exclude an ether-based interpretation of relativistic phenomena and a physical-dynamical theory of the deformation of rods and clocks. [See e.g. Einstein 1920; 1921, 127; 1924; 1949, 22-23; Kostro 2000; 2008; Brown 2005, 113-114] So it is not at all an unfounded reference to Einstein’s authority when at the beginning of Chapter II of his discussed book Jánossy cites an important fragment of Einstein on the ether and insists that his reformulation of the relativity theory is based on similar ideas as the ideas expressed there by the German physicist. The key assertion of the citation runs as follows: “Dass es in der allgemeinen Relativitaetstheorie keine bevorzugten, mit der Metrik eindeutig verknüpften raumzeitlichen Koordinaten gibt, ist mehr für die matematischen Form dieser Theorie als für ihren physikalishen Gehalt charakteristisch.”[Jánossy 1971, 49; the original: Einstein 1924, 90-91] In Jánossy’s translation: “The fact, that in the framework of the general theory of relativity, there are no distinguished space-time representations connected in an unambiguous manner with metric - is rather a characteristic of the mathematical methods of the theory than a characteristic of its physical contents.” [Ibid.] Although this study deals with Jánossy and not with Einstein, it seems necessary to emphasize that this is a very serious statement, which seems to withdraw the principle of general relativity, or more adequately, to degrade it to a mere consequence of the applied methodology. If taken seriously, the assertion will imply that there is a definite, privileged (bevorzugt) metric structure of physical reality (the structure carried by the ether) while the metric structures of a given system of “raumzeitlichen Koordinaten” (and so the “relativity” of the possible systems of coordinates) are only an implication of the “matematischen Form” of the theory. Now it is easy to see that Jánossy translates Einstein’s German terms into his own English terminology (so “bevorzugten … Koordinaten” into “distinguished representations”) but even in the absence of his tendentious translation the German original allows us to perceive a definite parallelism between Einstein’s view expressed here and Jánossy’s ether based notion of relativity. If both the original Einsteinian formulation and the received view of the relativity theory may be characterized by a washing away of the difference between the theoretical-mathematical representation as a human product and the represented physical reality, then here, in this discussion of the problem of the

30

ether Einstein recaptures this difference and represents a view similar to that of Jánossy. So despite the contrast between the original formulation of relativity theory and Jánossy’s reformulation, Einstein’s out-of-theory reflections seem to be near to Jánossy’s view. However, at a closer look it will be also clear that the parallelism between Einstein and Jánossy is limited. Firstly, whereas Jánossy’s ether is the carrier both of the electromagnetic waves and the gravitational field, for Einstein the ether is only the “gravitational ether”. [See for example: Kostro 2008. 52-53] Secondly, for Jánossy the united space-time or the space-time continuum is only a human construction, a human representation of physical reality. Consequently, he opposes the view according to which the difference between space and time is a mere appearance due to the shortcomings of our senses, as it is claimed in Minkowski’s paper introducing the concept of the four dimensional space-time [Minkowski 1909] and then many times endorsed by Einstein. Whereas Einstein claims that the separation of space and time is without ‘objective meaning’ [i.e. Einstein 1949, 22; 1949a, 99-100; Kostro 2008, 57] for Jánossy it is an evident and “objective” physical fact appearing in the radical difference between the measuring tools of length and time. As a consequence, Jánossy’s ether is definitely a three dimensional spatial entity, while in the case of Einstein it is hard to see how his ether could be imagined otherwise than a mystical four dimensional space-time continuum. It also can be easily seen that the problem of four dimensional space-time is closely connected to Jánossy’s thesis on the importance of common sense regarding physical theories. Taking seriously the ontological priority of the EinsteinianMinkowskian space-time, a temporal interval, for example, that between the birth and the death of a person will appear of the same nature as the spatial distance between, say, Budapest and London, and the difference between translational motion and aging will disappear. Considering these consequences we may see that Jánossy’s thesis on the role of common sense is considerably more than a naive insistence on our accustomed everyday habits and judgments; it concerns our most ultimate ontological experiences, such as our experience of life and death. But these consequences also show that Einstein’s claim on the ontological, “objective” priority of the four dimensional space-time has significant metaphysical implications and transforms Einstein’s positivist starting point into a metaphysics of a four dimensional space time. Lastly, whereas Einstein verbally acknowledges that the ether has physical properties and speaks only about its deprivation of “mechanical” characteristics, it is clear that with the term “mechanical” he refers to all traditional physical properties including pressure, strain, density etc. In contrast with Einstein, Jánossy characterizes the state of the ether with the help of these terms. Of course, in doing so he is using the latter only in a metaphorical sense and he does not mean to claim that the ether has exactly the same properties as macroscopic entities. However, the application of

31

these terms definitely indicates that in his view the ontological nature of the ether is basically similar to the macroscopic physical entities. The difference between Einstein’s and Jánossy’s notions of the ether cannot be reduced even if we assume that in the context of physics Einstein also uses the term “geometry of space-time” metaphorically. The metaphorical use does not change the fact that Jánossy’s terms come from physics and they attribute to the ether physical characteristics even if they are used metaphorically, while Einstein’s term is transferred into physics from mathematics and hence its application necessarily results in a mathematization of physical reality. Therefore the conversion of the German physicist to the concept of the ether does not cure the epistemologically inverted relation between mathematics and physics characterizing his theory. In this respect it is often argued that the Einsteinian turn of physics brought about not only a theory change but also transformed the conceptual framework of physics and, as part of this transformation, it gave a new meaning to the word “physical”. The properties of Einstein’s ether are not “physical” if we use the old meaning of the word but in the new conceptual framework they become definitely physical. However, this argument is invalid since the point is exactly whether one accepts or refuses the conceptual change. The new meaning of “physical” is a consequence of the mathematization of physics by the Einsteinian version of relativity theory, the main target of Jánossy’s reformulation of the theory. Dubbing mathematical terms and properties as physical will not change their real nature. On the contrary, physical reality should first be attributed a mathematical nature in order to characterize such properties as physical. And conversely, if we really think that the latter are truly “physical”, then this will amount to transforming the nature of physical reality from physical to mathematical. Or is it possible that the mathematization of physical reality, criticized so vehemently by Lajos Jánossy (and more recently by H. Brown) regarding the theory of relativity, but also present in quantum mechanics, is more than a pure consequence of a methodological mistake? Is it possible that in its ultimate ontology the world around us is not of a physical but a mathematical nature? Maybe the cart is put before the horses not only by the received interpretation of relativity theory but also in physical reality? These are far reaching metaphysical questions that surely do not belong to relativity theory, and especially not to the topic of the present review of Jánossy’s interpretation of relativity theory, but still concern so intensively the whole interpretational problem of the theory that they must be raised at the close of this paper. 8. Summary We have seen that Jánossy’s theory of relativity consists of two levels. At the first level he reformulates Einstein’s theory in terms of measurement, while at the second level he outlines an ether-based explanation of relativistic effects. His

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reformulation of the relativity theory not only elucidates the relation between the mathematical formalism of the theory and physical reality and establishes an etherbased interpretation of relativistic phenomena, but also gives a deep insight into the hidden conceptual background of the Einsteinian version of the theory. In our days when the relation between physics and mathematics in relativity theory has become a topical issue again, Jánossy’s analysis of the relativistic phenomena and his deduction of the formalism of the theory in terms of measurement are especially significant both from a physical and a philosophical point of view. We have seen furthermore that his consideration about the role of the ether in the explanation of relativistic phenomena as well as his hypotheses about the nature of this entity are of high heuristic value and may give significant stimulation for further research in the direction of a dynamical theory of the ether. Acknowledgement The author expresses his gratefulness to the Hungarian Scientific Research Fund (OTKA) for the support granted to his research (Project Number T 046261)

References Balashov, Y. and Janssen M. 2003, “Critical Notices: Presentism and Relativity”, British Journal for the Philosophy of Science, 54. 327-346. J. S. Bell 2001, The Foundation of Quantum Mechanics, M. Bell, K. Gottfried and M. Veltman (eds.) World Scientific. J. S. Bell 1987, Speakable and Unspeakable in Quantum Mechanics. (1. Edition) Cambridge University Press, Cambridge. J. S. Bell 1976, How to Teach Special Relativity, Progress in Scientific Culture, Vol. 1, No. 2, reprinted in [Bell 1987] and [Bell 2001] H. Brown 2005, Physical Relativity. Clarendon Press, Oxford. DiSalle, R. 1995, “On Dynamics, Indiscernability and Space-Time Ontology”, British Journal for the Philosophy of Science, 45, No 1 (Mart 1994) 265-287 Duffy, M.C. 2008, “Ether as a Disclosing Model.” In: Duffy M. C. and Levy J. 2008. Duffy M. C. and Levy J. (eds.) 2008, Ether, Space-Time and Cosmology. Volume 1. Liverpool. Duffy, M. C. (ed.) 1988-2006, Physical Interpretations of Relativity Theory IXIV. Proceedings of the Conferences 1988, 1990, 1992, 1994,1996, 1998, 2000, 2002, 2004, 2006. The British Society for the Philosophy of Science, London.

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Einstein, A. 1920, Äther und Relativitätstheorie. Verlag von J. Springer, Berlin. Einstein, A. 1921, „Geometrie und Erfahrung.” Sitzungsberichte der königlichen Preussische Akademie der Wissenschaften. 123-130. o. Einstein, A. 1924, „Über den Äther.” Verhandlungen der Schweizerischen Naturforschenden Gesellschaft. 105. Teil 2. 85-93. Einstein, A.1949, „Autobiographisches“. In: Schilpp 1949, 1-35. Einstein, A. – Infeld L. 1949a, Die Physik als Abenteur der Erkenntnis. Sijthofs Witgeversmaatschappij, Leiden. Graham, L. R. 1972, Science and Philosophy in the Soviet Union. Knopf, New York. Jánossy L. 1948, 1950, Cosmic Rays. Clarendon Press, Oxford. Jánossy L. 1963, „Foreword 1.” In: Jánossy, L. and Elek, T, A relativitáselmélet filozófiai problémái. (The Philosophical Problems of Relativity Theory.) Budapest: Akadémiai Kiadó. 9-11. (In Hungarian) Jánossy L. 1965, Theory and Practice of the Evaluation of Measurements. Clarendon Press, Oxford. Jánossy, L. 1971, Theory of Relativity Based on Physical Reality. Akadémiai Kiadó, Budapest. Kostro, L. 2000, Einstein and Ether. Aperion, Montreal. Kostro, L. 2008, „Einstein’s New Ether 1916 – 1955.” In: Duffy M. C. and Levy J. 2008. Minkowski, H. 1909, „Raum und Zeit“, Physicalishce Zeitschrift 10. 104-111. Nerlich G. 1976, Cambridge.

The

Shape of Space. Cambridge University Press,

Palágyi, M. 1914, Die Relativitaetstheorie in der modernen Physik. Berlin: 1914. (Also in Palágyi 1925. pp 34-83.) Palágyi, M. 1925, Ausgewaehlte Werke, Band III. Zur Weltmechanik (Beitraege zur Metaphysik der Physik). Leipzig: Johann Ambrosius Barth. Ritz, W. 1908, „Du rôle de l'éther en physique“ Scientia 1908, Vol 3. Nr. VI. 260-274. Republished in Karl Dürr’s German translation as „Über die Rolle des Aethers in Physik” in Ritz 1963. Ritz, W. Theorien Über Aether, Gravitation, Relativitaet und Elektrodynamik.” Bern und Badisch-Reihnfelden: Schritt Verlag 1963.

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Schilpp P. A. (ed.) 1949, Albert Einstein als Philosoph und Naturforscher. Kohlhammer Verlag, Stuggart. Székely, L. 1987, „Physical Theory and Philosophical Values.” Doxa 9. (Published by The Institute for Philosophy of the Hungarian Academy of Sciences.) 159-181. Székely, L. 1988, „A Hungarian Interpretation of Relativity Theory.” In: Duffy (ed.) 1988. Székely, L. 1996, "Melchior Palágyi's Space-Time” Ultimate Reality and Meaning. (Interdisciplinary Studies in the Philosophy of Understanding.) Vol. 19. No. 1. 3-15.

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AETHER THEORY CLOCK RETARDATION vs. SPECIAL RELATIVITY TIME DILATION Joseph Levy 4 Square Anatole France, 91250, St Germain-lès-Corbeil, France E-mail: [email protected]

ABSTRACT Assuming a model of aether non-entrained by the motion of celestial bodies, one can provide a rational explanation of the experimental processes affecting the measurement of time when clocks are in motion. Contrary to special relativity, aether theory does not assume that the time itself is affected by motion; the reading displayed by the moving clocks results from two facts: 1/ Due to their movement through the aether, they tick at a slower rate than in the aether frame. 2/ The usual synchronization procedures generate a synchronism discrepancy effect. These facts give rise to an alteration of the measurement of time which, as we shall show, exactly explains the experimental results. In particular, they enable to solve an apparent paradox that special relativity cannot explain (see chapter 4). When the measurement distortions are corrected, the time proves to be the same in all co-ordinate systems moving away from one another with rectilinear uniform motion. These considerations strongly support the existence of a privileged aether frame. The consequences concern special relativity (SR) as well as general relativity (GR) which is an extension of SR. We should note that Einstein himself became conscious of the necessity of the aether from 1916, in contrast with conventional relativity. Yet the model of aether presented here differs from Einsein’s in that it assumes the existence of an aether drift, in agreement with the discoveries of G.F. Smoot and his co-workers listed in Smoot’s Nobel Lecture, December 8th 2006. Although it makes reference to previous studies, this text remains self-sufficient.

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1. INTRODUCTION In the present text, the points of view of special relativity and aether theory regarding the measurement of time in moving co-ordinate systems are successively presented and compared. The measurement concerns the two way transit time of light along a rod perpendicular to the direction of motion. We show that the approach of aether theory we have developed in Ref [1-4], can give a rational explanation of the experimental processes affecting this measurement, but, contrary to special relativity, these processes do not result from time dilation, but rather from the slowing down of clocks moving through the aether and from the synchronism discrepancy effect caused by the standard synchronization procedures. After correction of these measurement distortions the true value of time in moving co-ordinate systems is rediscovered. This study gives an illustration, in a specific example, of the differences existing between special relativity and aether theory. (We should bear in mind for the reader not informed of our approach, that the concept of aether assumed in this text conforms to the Lorentz views: it is associated with a privileged aether frame and is not entrained by the motion of bodies. It is this approach that we shall refer to as “aether theory” all through the text). This study does not question the experimental results brought about by relativity theory since, as we shall see, at least in the cases studied here, it predicts the same clock readings as SR provided that we use the standard measurement procedures. It nevertheless gives another interpretation of the experimental data (demonstrating that the procedures used entail measurement distortions and that the results obtained conceal hidden variables). This different interpretation and the disclosing of hidden variables should have important consequences for the future development of physics insofar as it concerns not only SR, but also GR. An important argument supporting our approach is that it solves an apparent paradox related to reciprocity that SR cannot explain (see chapter 4). Let us bear in mind that, contrary to what is often believed, Einstein did not definitively reject the concept of aether. He assumed, no later than 1916, that the consistency of general relativity needed recognition of the aether, an opinion which he recorded in an address he delivered on May 5th 1920 in the University of Leyden [5]. But as Einstein declared at the end of this address, “the idea of motion may not be applied to this model of aether…”, and, therefore, it cannot explain the discoveries of G.F. Smoot who, in a report done at the university of California, declared: “The motion of the Earth with respect to the distant matter (“aether drift”) was measured, and the homogeneity and isotropy of the universe (“the cosmological principle”) was probed. This recognition of an aether drift was confirmed in his Nobel lecture, December 8th 2006 [6, 7]. On the contrary, our model assumes an aether drift in agreement with the experimental studies performed by Smoot, Gorenstein and their co-workers.

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2. TIME DILATION ACCORDING TO RELATIVITY THEORY Let us consider two inertial coordinates systems S0 (x,, y, z) and S1 (x’, y’ ,z’) receding from one another along the x-axis of the co-ordinate system S0 , and suppose that a light ray starts from a point M fixed to the coordinate system S1 , and travels along a rod L=MB, perpendicular to the x’-axis (see fig 1). After reflection in a mirror placed in B, the signal returns to point M. In the coordinate system S1 , the two-way transit time of light along the rod is 2 t1 = 2L/C. But, viewed from S0 , the light ray starts from a point A in this co-ordinate system, and after reflection in B returns to point A’. The total duration of the cycle in S0 will be labelled 2t0 .According to relativity, the speed of light is C in all inertial frames and in all directions of space. Let v01 refers to the real relative speed separating S0 and S1 (measured with non contracted standards). When the light ray has covered the distance AB, S1 has moved away from S0 a distance AM = v01t0

S0

O

B

S1

O’

A M

A’

x, x’

FIG 1. In the coordinate system S1 , the light ray travels from point M to the mirror B and, after reflection, returns to M. In S0 the signal starts from point A, and after reflection in B returns to A’. According to an observer attached to S0 the transit time of light t0 along AB is given 2 2 by: C 2t02 − v01 t0 = L2 , therefore:

t0 =

L 2 C 1 − v01 / C2

.

Replacing L/C by its value t1 this expression reduces to:

t0 =

t1 2 1 − v01 /C2

39

.

This classical formula is interpreted as time dilation by special relativity, (an expression obtained because the position of the clock in S1 remains fixed relative to this co-ordinate system) 3. CLOCK RETARDATION ACCORDING TO AETHER THEORY The theory on which this study is based, assumes the existence of a preferred frame in which the aether is at rest. The one-way speed of light is C in the aether frame, and different from C in all other co-ordinate systems moving with respect to the aether frame. Yet, as we saw in Ref [1-4] and [8], due to measurement distortions (that will be evoked in the text which follows), it appears to be of magnitude C in all ‘inertial’ frames and in all directions of space. Contrary to relativity theory, the motion of bodies does not affect the time, but the motion through the aether causes a slowing down of the moving clocks. The real two-way transit time of light, along a rod attached to a certain ‘inertial’ frame, is the same for the observers of all frames, but, due to clock retardation, the reading displayed by clocks moving relative to the rod will depend on their speed with respect to the rod [2, 8, 9]. Although the variables used in this study should be difficult to determine experimentally, our approach, as we shall see, allows an exact theoretical comparison of the concepts of time assumed by the two theories. In this section we shall study successively two different cases: in section 3.1. the clock reading in a moving ‘inertial’ co-ordinate system is compared to the time in the aether frame; this case introduces to the section 3.2. which puts forward exhaustively the differences between aether theory and relativity. The paradox inherent in conventional relativity when we assume a complete symmetry between frames will be examined in section 4. 3.1. Comparison of the clock readings displayed in frames S 0 and S1 In this section we shall compare the clock readings in two ‘inertial’ co-ordinate systems as we did in section 2, but from the point of view of aether theory. The only difference is that the co-ordinate system S 0 is assumed to be at rest in the aether frame where the clock reading is not altered by motion (and which can be regarded as the basic time or, by definition, the real time) (Fig 1). Since the line AB is the path of the light signal in the aether frame, the speed of light is C along this line. Referring to the transit time of light along AB, that would be displayed by a clock attached to frame S0 , as t0 , we have: 2 2 C 2t02 − v01 t0 = L2 ,

and therefore:

t0 =

L 2 C 1 − v01 / C2

40

.

According to the aether theory under consideration, clock retardation is defined with respect to the aether frame; the ratio between the time in the aether frame and the reading displayed by clocks moving at absolute speed v is assumed to be equal to (1 − v 2 / C 2 ) −1 / 2 . This assumption will be justified a posteriori; its experimental implications will be studied in the text that follows. Therefore, the clocks attached to the co-ordinate system S1 tick at a slower rate and display the reading t1app = L/C. Thus,

t0 =

t1app 2 1 − v01 / C2

,

(1)

where the suffix ‘app’ means apparent. This formula assumes the same mathematical form as the time dilation formula of special relativity; yet its meaning is quite different because, contrary to special relativity, t1app is not the true time in the co-ordinate system S1 , it is the clock reading displayed by clocks slowed down by motion. (We bear in mind that, if we assume the existence of a preferred aether frame, then, real frames attached to bodies, even if they are not submitted to physical influences other than the aether drift, are never perfectly inertial. The term ‘inertial’ is an approximation which must be limited to the cases where the absolute speed of the frames under consideration is low compared to the speed of light. See Ref [2]). 3.2. Case of two co-ordinate systems moving away from the aether frame We now propose to study a different case: we shall determine the clock retardation formula between two co-ordinate systems S1 and S 2 receding with rectilinear uniform motion with respect to the co-ordinate system S0 which is attached to the aether frame. The direction of motion is the x-axis (see fig 2). This case is that to which we usually deal with in practice. In relativity, there is no preferred frame, therefore the co-ordinate system S0 is inexistent and the time dilation formula between the systems S1 and S 2 takes the form:

T1 =

t2 1 − v122 / C 2

,

(2)

where v12 refers to the relative speed between the coordinate systems S1 and S 2 . In aether theory, things are very different. Let v01 , v02 and v12 refer to the real relative speeds between the three co-ordinate systems (obtained in the absence of measurement distortions). The rod MB perpendicular to the x”-axis is firmly fixed to the co-ordinate system S 2 . We propose to compare the apparent times (displayed by the clocks attached to S1 and S 2 ) which are needed by the light signal to achieve a

41

cycle (from M to B and to M again in the system S 2 and from A to B and to A’ in the system S1 ).

S0

O

S1

O’

B

S2

O’’

A

M

A’

x, x’, x’’

FIG 2. The co-ordinate systems S1 and S 2 recede from S0 along the common x-axis. The light ray travels along the rod MB which is at rest in S 2 (from M to B and to M again). With respect to S1 it starts from point A, is reflected in B and then returns to A’, (where A and A’ are two points at rest in the co-ordinate system S1 ). During a cycle of the signal, S 2 has moved with respect to S1 a distance AA’. The real value of the one-way speed of light along AB is not equal to C since the coordinate system S1 is not at rest with respect to the aether frame. 3.2.1. We shall first assume that the clocks placed at points A and A’ are exactly synchronized. Let us label as 2 t0 the two-way transit time of the light signal that would be displayed by clocks attached to the co-ordinate system S0 . Due to clock retardation the clock readings in S1 and S 2 are related to t0 as follows: 2 t1app = t0 1 − v01 /C2 ,

(3)

2 t2 app = t0 1 − v02 /C2 .

(4)

and (Note nevertheless that the true time, needed for half a cycle, measured with clocks not slowed down by motion, is t0 for all observers). From (3) and (4) we infer:

t1app = t2 app

2 1 − v01 / C2 2 1 − v02 / C2

.

Assuming that v02 / C << 1 , this expression reduces to first order, to:

42

(5)

t2 app

.

(6)

1v 1 − 122 (v12 + 2v01 ) 2C Noting that (as in section 3.1.) t2 = t2 app = L / C , this expression is different from the relativistic formula (2) which reduces to:

t2 . 1 v122 1− 2 C2

(7)

Therefore if clocks were exactly synchronized, there would be an obvious difference between the two theories. 3.2.2. Practical consequences of the clock synchronization procedures used. We should note that, in practice, in order to determine the duration of a cycle in S1 we must subtract the reading displayed by clock A when the signal starts from this clock, from the reading displayed (after reflection in B) by clock A’ when the signal reaches this clock, and therefore we must synchronize the clocks A and A’ beforehand. According to aether theory if the synchronization of clocks was perfect we would have obtained formula (5). Yet, synchronizing the clocks perfectly is a difficult problem, and, with the standard synchronization procedures, (Einstein-Poincaré method (E. P) or slow clock transport), we make an unavoidable systematic error in measuring the time, (synchronism discrepancy effect) [2, 9, 10] The apparent duration of a cycle measured in S1 is therefore equal to the difference between 2 t1app and the synchronism discrepancy effect (SDE) that will be derived in the text which follows. (The SDE, which was defined by Prokhovnik for the first time, enables to resolve a number of paradoxes in physics). Referring to the SDE that would affect the clocks if they were not slowed down by motion as Δ , the SDE affecting the clocks attached to the coordinate system S1 is:

δ = Δ 1 − v012 / C 2 . The apparent (measured) two-way transit time of the signal (from A to B and to A’) is therefore: 2 2T1app = 2t1app − Δ 1 − v01 /C2 .

It is this apparent time which is in fact measured when a SDE between the clocks A and A’ exists. (a) Derivation of the synchronism discrepancy effect, and clock synchronization. The Einstein-Poincaré method (E. P) consists in sending a light signal from clock A to

43

clock A’ along the x’-axis, at an arbitrary instant where the reading of clock A is set at t=zero. After reflection in A’, the signal comes back to A. The clocks are considered synchronous if upon reception of the signal by clock A’, this clock displays a reading equal to half the reading displayed by clock A upon return of the signal. (The alternative synchronization method, referred to as the slow clock transport procedure, has been shown to be equivalent to the former by different authors [2, 10]). Although the measurement should be difficult to perform with our today technology, it is possible to carry out a theoretical evaluation, as we shall see, of how the SDE modifies the reading of the time (in comparison with the clock readings displayed by clocks exactly synchronized given by formula (5)). The result will then be compared to the time dilation of special relativity, a comparison that will enable to check the theory. Let us label as l 0 the length that would be assumed by the segment AA’ if it was at rest in the aether frame. Due to its motion with respect to S0 it is reduced to 2 l = l 0 1 − v01 / C2 ,

(8)

which, according to aether theory, is the real length in the co-ordinate system S1 . The real time needed by the light signal to travel from A to A’ along the x’-axis is therefore:

traa ' =

2 l 0 1 − v01 / C2

C − v01

,

where C - v01 is assumed to be the real speed of light in S1 along the x’-axis. Here the suffix r (for real) means that the determination of the speed is made without measurement distortions. This formula was the expression used by Lorentz to explain the Michelson experiment. (According to aether theory, real speeds measured along a straight line are simply additive. Only apparent speeds (whose measurement is altered by the systematic measurement distortions) obey the relativistic law of composition of velocities, as we shall see in formula (14). (See also Refs [1, 2]). In the reverse direction we have:

tra 'a =

2 l 0 1 − v01 / C2

C + v01

.

Half the two way transit time of the light signal along the x’-axis (from A to A’ and to A again) measured with clocks not slowed down by motion is therefore:

1 / 2(traa ' + tra 'a ) =

44

l0 2 C 1 − v01 /C2

.

In the absence of clock retardation, the synchronism discrepancy Δ between the clocks A and A’ would be equal to the difference between the exact transit time traa ' of the signal from A to A’ and the apparent (measured) time 1 / 2(traa ' + tra ' a ) :

Δ=

2 l 0 1 − v01 / C2

C − v01

−

l0 C 1− v /C 2 01

2

=

v01l 0 C

2

2 1 − v01 C2

.

Due to clock retardation in the co-ordinate system S1 the SDE is reduced to

δ=

v01l 0 . C2

(b) Apparent transit time of light along the rod in the co-ordinate system S1 . In the absence of SDE, the apparent transit time of light from A to B and to A’ again measured with clocks slowed down by motion would be: 2 2t1app = 2t0 1 − v01 / C2 .

If one takes account of the SDE, the clock reading becomes

2T1app = 2t1app −

v01l 0 . C2

(9)

Important remark Writing this expression in the form

v01l 0 , (10) C2 and taking account of the fact that the measurements in S1 are made with a meter stick which is also contracted, the length AA’ is erroneously found equal to l 0 . We 2 2T1app = 2t0 1 − v01 /C2 −

shall therefore refer to l 0 as X 1app . From (10) we obtain:

2t0 =

2T1app + v01 X 1app / C 2 2 1 − v01 / C2

.

(11)

This expression assumes the same mathematical form as the conventional transformation relative to time, yet its meaning is quite different since it demonstrates that the variables 2 T1app and X 1app which are obtained experimentally are different from the true values. The experiment is altered by measurement distortions [1]. (Such a mathematical form is obtained because, contrary to the case studied in section 3.1., the measurement of time in S1 is made in two different points of the x’axis).

45

-Taking account of expression (8) and of the fact that l = (v02 − v01 ) × 2t0 where

v02 and v01 are the real speeds of S 2 and S1 with respect to S 0 (obtained in the absence v l of measurement distortions) we can express δ = 01 2 0 in the form C 2v01t1app (v02 − v01 ) δ = . 2 C 2 − v01 The apparent (measured) transit time of the signal in S1 (from A to B and to A’ again) is therefore:

2T1app = 2t1app − δ = 2t1app

C 2 − v01v02 . 2 C 2 − v01

Now our objective is to compare the apparent (measured) transit times of the signal in frames S1 and S 2 . From formula (5) we have:

2T1app = 2t2 app

2 1 − v01 / C 2 C 2 − v01v02 . 2 2 2 1 − v02 / C 2 C − v01

(12)

From formula (12) we obtain successively:

2T1app = 2t2 app

= 2t2 app = 2t2 app = 2t2 app

2T1app =

C 2 − v01v02 2 2 C 2 (1 − v02 / C 2 )(1 − v01 /C2)

C 2 − v01v02 2 2 2 2 C 4 − v02 C 2 − v01 C 2 + v01 v02

C 2 − v01v02 2 2 2 2 C 4 + v01 v02 − 2C 2v01v02 − v02 C 2 − v01 C 2 + 2C 2v01v02

C 2 − v01v02 (C 2 − v01v02 ) 2 − C 2 (v02 − v01 ) 2 2t2 app (v02 − v01 ) 2 1− v v C 2 (1 − 01 202 ) 2 C

.

2t2 app

= 1−

C 2 (v02 − v01 ) 2 (C 2 − v01v02 ) 2 (13)

This result is different from formula (5) which was the exact clock retardation formula according to aether theory. Yet, it is the experimental formula obtained in practice due to the measurement distortions.

46

We recognize in the denominator the experimental composition of velocities law. Since it has been derived from the Galilean law which has been submitted to measurement distortions its apparent character is highlighted. We can thus write:

v12 app =

v02 − v01 . 1 − v01v02 / C 2

(14)

With this notation formula (13) becomes:

T1app =

t2 app 1 − v122 app / C 2

.

(15)

This formula has the same mathematical form as the time dilation formula (2) of relativity theory, but obviously its meaning is quite different. In particular we have not assumed the invariance of the one-way speed of light in all inertial frames. This surprising result cannot be the effect of chance. (Note that this formula assumes a mathematical form different from formula (11). This difference results from the fact that in the co-ordinate system S 2 the time is measured at the same point at the beginning and at the end of a cycle). We should note that, when the co-ordinate system S1 is at rest in the preferred aether frame, v01 = 0 and v12 app reduces to v02 . This demonstrates that contrary to what is often claimed, the aether frame can be theoretically distinguished from the other frames. This result, which is in accordance with the experimental facts, strongly supports the existence of a privileged aether frame in a state of absolute rest. 4. THE QUESTION OF RECIPROCITY (RESOLUTION OF A PARADOX). The resolution of the paradoxes inherent in reciprocity which affect special relativity have been first suggested by Builder and Prokhovnik [9]. We will consider here the paradox affecting the measurement of time from the device described in the previous chapters. Let us return to the figure 1. In the case studied there, the rod MB was at rest with respect to the co-ordinate system S1 and was moving relative to the co-ordinate system S0 .We shall now consider the opposite case: i.e. the rod is at rest with respect to S0 and the co-ordinate system S1 moves relative to S0 in the left direction, as the figure 3 shows.

47

S1

O

B

S0

O’

A M

A’

x, x’

FIG 3. The rod L=MB is at rest with respect to the co-ordinate system S0 , and the co-ordinate system S1 is moving relative to S0 in the left direction. According to conventional relativity, contrary to aether theory, nothing differentiates the co-ordinate systems S0 and S1 , because there is no preferred inertial frame; in other words, motion is only relative, and one can consider that S0 moves relative to S1 , in the same way as S1 is moving relative to S0 . Therefore SR predicts a complete symmetry between the frames: for example, a clock in S1 slows down with respect to a clock standing in S0 , but conversely a clock in S0 is subjected to slow down with respect to a clock in S1 . Of course this result appears paradoxical. It defies logic and cannot be rationally explained if this total equivalence between frames is assumed. Yet, as we shall see, the paradox can be solved if we assume the existence of a preferred aether frame in which case the measurements are affected by systematic distortions, and the complete symmetry proves only apparent. The apparent identity of the two opposite situations results from these measurement distortions as the following demonstration will show. In agreement with aether theory, let us assume that S0 is a co-ordinate system at rest in the preferred frame, and S1 a co-ordinate system moving at uniform speed in the left direction, see Fig 3. Since it is S1 which moves, the clocks in S1 tick slower than in S0 and we should have: 2 t1 = t0 1 − v01 / C2 =

L 2 1 − v01 / C2 . C

(16)

Where v01 is the real speed between S0 and S1 , (measured with non-contracted meter sticks and clocks non slowed down by motion)

48

Yet, this result supposes that the measurement has been made with clocks exactly synchronized, and as we saw, the exact synchronization is an objective difficult to achieve and which is not practiced today. Let us describe theoretically the method which therefore should be used by an observer at rest in S1 in order to measure the two way transit time of light along the rod MB, by means of the E. P procedure. Viewed from the co-ordinate system S0 , the light ray travels along the rod MB (from M to B and to M again), but viewed from S1 , it starts from A, is reflected in B and comes back to A’, where A and A’ are two points at rest with respect to the coordinate system S1 . In the absence of length contraction let us suppose that AA’ measures l 0 . Taking account of the reduction of size, we have: 2 2 AA’= l = l 0 1 − v01 / C = 2v01t0 .

According to the E. P procedure, the synchronization requires two clocks placed in A and A’. The clocks are considered synchronous if, when a light ray starting from A at the instant zero and travelling along the x, x’-axis strikes A’, this clock displays the reading: 1/2 l (

l l 1 1 2 + = 0. ) 1 − v01 / C2 = 2 2 C + v01 C − v01 C C 1 − v01 / C

(In this expression we have taken account of the slowing down of the clocks standing in the co-ordinate system S1 , and we have made use of the Galilean composition of velocities law which applies in aether theory to real speeds). Note Let us remark that in S1 the standard used to measure the lengths is contracted in the same ratio as the segment AA’. Therefore the length AA’ is erroneously found equal to l 0 and therefore the light speed is erroneously found equal to C in conformity with the experiment. Yet the real transit time of the light ray from A to A’ along the x, x’-axis is: 2 l 0 1 − v01 /C2

C + v01

.

But due to the slowing down of clocks in the co-ordinate system S1 , the reading in the absence of synchronism discrepancy would be: 2 l 1 − v01 /C2

C + v01

.

The clock A’ will therefore be ahead of the clock A by an amount equal to:

49

l 2 C 1 − v01 /C2

-

2 l 1 − v01 /C2

C + v01

=

2

v01l 2 C 2 1 − v01 / C2

=

2v01 t0 2 C 2 1 − v01 / C2

.

. The measured time of light transit from A to B and to A’ again being made with the clocks A and A’, the result of the measurement will give (instead of 2 2t1 = 2t0 1 − v01 / C2 2

2t1app = 2t0 1 − v / C + 2 01

Which yields

t1app =

2v01 t0

2

2 C 2 1 − v01 / C2

t0 2 1 − v01 / C2

, (17)

This clock reading is the apparent time resulting from the synchronism discrepancy effect. This result enables to explain rationally the paradoxical effect which is anticipated by special relativity without being explained. Yet special relativity regards this result as the true time, while it is in fact an apparent time resulting from the measurement distortions. 5. CONCLUSION The comparison of formulas (6) and (7) demonstrates that relativity and aether theory are fundamentally different. Nevertheless, paradoxically, due to the systematic measurement distortions mentioned above, aether theory leads to a clock reading given by formula (15), which presents a mathematical form identical to formula (2); yet for relativity, the formula is regarded as exact, while for aether theory it results from the measurement distortions. Aether theory provides also an explanation of why formulas (1) and (17) can be both rationally justified, although at first sight they appear incompatible. Aether theory explains that due to the synchronism discrepancy effect formula (17) is observed instead of formula (16), an explanation which solves the paradox. Special relativity obtains the same result but cannot give a rational explanation of it. In conclusion, the choice of one theory rather than the other is not simply a question of philosophical preference. The ideas expressed in this article are identical to those which were developed in the previous version published in arXiv (Physics/0611077). We have only given further explanations and added the chapter 4. ACKNOWLEDGEMENTS I would like to thank Pr. Gianfranco Spavieri and Dr. Dan Wagner for interesting and helpful comments.

50

REFERENCES [1] J. Levy, Aether theory and the principle of relativity, in Ether Space-Time and Cosmology Volume 1, Michael C. Duffy and Joseph Levy Editors (PD Publications Liverpool UK, March 2008) p 125-138. Arxiv: physics/0607067 [2] J. Levy, Basic concepts for a fundamental aether theory in Ether Space-Time and Cosmology, volume 1, Michael C. Duffy and Joseph Levy Editors (PD Publications Liverpool UK, March 2008) p 69-123. Arxiv: physics/0604207 [3] J. Levy, Extended space-time transformations for a fundamental aether theory, Proc. Int. conf. Physical interpretations of relativity theory VIII” (Imperial college, London, 6-9 September 2002) p 257. [4] J. Levy, From Galileo to Lorentz and beyond, (Apeiron, Montreal, 2003) URL, http://redshift.vif.com. [5] A. Einstein, Ether and the theory of relativity, Address delivered in the University of Leyden, May 5th 1920, in Sidelights on relativity (Dover publications, Inc, New York). [6] G.F. Smoot, Cosmic microwave background radiation anisotropies, their discovery and utilization. Nobel Lecture December 8th 2006. Aether drift and the isotropy of the universe: a measurement of anisotropies in the primordial blackbody radiation, Final report 1 November 1978 - 31 Oct 1980, University of California, Berkeley. [7] G.F. Smoot, M.V. Gorenstein and R.A. Muller, Detection of anisotropy in the cosmic blackbody radiation Phys. Rev. Lett, vol 39, p 898-901, (1977). M.V. Gorenstein and G.F. Smoot, Astrophys.J, 244, 361, (1981). [8] J. Levy, Two-way speed of light and Lorentz-Fitzgerald’s contraction in aether theory, ArXiv: physics/0603267 [9] S.J. Prokhovnik, The Logic of special relativity (Cambridge University press, 1967). Light in Einstein’s Universe (Reidel, Dordrecht, 1985). [10] J. Levy, Synchronization procedures and light velocity, Proc. Int. conf. Physical interpretations of relativity theory VIII (Imperial college, London 6-9 September 2002) p 271.

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52

RELATIVITY AND AETHER THEORY A CRUCIAL DISTINCTION Joseph Levy 4 Square Anatole France, 91250 St Germain-lès-Corbeil, France E-mail: [email protected]

ABSTRACT We study the case of two rockets which meet at a point O of an ‘inertial coordinate system’ S, and are scheduled to move at constant speed, in opposite directions, toward two targets placed at equal distances from point O. At the instant they meet, the clocks inside the rockets are set to zero. When they reach the targets the rockets meet two clocks A and B whose reading is identical. This question which was tackled in ref [1] is studied here in depth. Assuming the existence of a preferred aether frame S0 in which the one-way speed of light is isotropic, and the anisotropy of this speed in the other frames, we show that, if the equal reading of the clocks A and B results from an exact synchronization, the clocks inside the rockets will display different readings when they reach A and B in contradiction with the relativity principle. Conversely, if the clocks A and B, which display an equal reading, have been synchronized by means of the Einstein-Poincaré procedure, the inboard clocks will also display the same reading, a fact which seems in agreement with the relativity principle. But this synchronization method presupposes the invariance of the one-way speed of light, in contradiction with the assumptions made, and, therefore, introduces a measurement error. This demonstrates that if we assume the existence of an aether frame, the apparent relativity principle is not a fundamental principle; it depends on an arbitrary synchronization. In any case, this is an example of an experimental measurement which can be explained by aether theory without the assumption of the invariance of the one-way speed of light in all ‘inertial frames’.

53

I. INTRODUCTION A number of arguments today lend support to the existence of a preferred aether frame in which the one-way speed of light is isotropic [1] and to the anisotropy of this speed in the other frames, and it is of the utmost importance to know whether such a preferred frame is compatible with the application of the relativity principle in the physical world. Physicists remain divided about this question. Einstein was convinced that the existence of a preferred frame is at variance with relativity. In the original formulation of his theory [2], he definitely regarded the existence of aether as superfluous. Later he changed his mind in order to formulate the theory of general relativity. But, the aether of Einstein is not associated with a preferred frame. In his little book “Sidelights on relativity” [3], he expressed his views in the following terms: “..according to the theory of general relativity, space is endowed with physical qualities. In this sense, therefore there exists an aether… But this aether may not be thought of as endowed with the quality of ponderable media, as consisting of parts which may be tracked through time. The idea of motion may not be applied to it”. On the contrary, Poincaré acknowledged the Lorentz assumptions which assume the existence of a preferred aether frame and in which length contraction and clock retardation are real processes depending on the velocity of the rods and clocks relative to the aether frame. The agreement of Poincaré with the approach of Lorentz is expressed in the following sentence: “The results I have obtained agree with those of Mr. Lorentz in all important points. I was led to complete and modify them in a few points of detail” [4]. His belief in the aether was expressed in the citations that follow: Does aether really exist? The reason why we believe in aether is simple. If light comes from a distant star and takes many years to reach us, it is (during its travel) no longer on the star, but not yet near the Earth. Nevertheless, it must be somewhere and supported by a material medium; (La science et l’hypothèse chapter 10 p 180 of the French edition “Les theories de la physique moderne” [5]). And: “Let us remark that an isolated electron moving through the aether, generates an electric current, that is to say an electromagnetic field. This

54

field corresponds to a certain quantity of energy localized in the aether rather than in the electron” [6]. But, at the same time, Poincaré acknowledged the relativity principle, as the following sentence shows: “It seems that the impossibility of observing the absolute motion of the Earth is a general law of nature. We are naturally inclined to admit this law that we shall call the relativity postulate and to admit it without restriction” [7]. In this text, we propose to check these different opinions starting from a simple experimental test. II. OVERVIEW OF THE PROBLEM Let us consider two rockets moving uniformly in opposite directions along a straight line of an ‘inertial co-ordinate system’1 S. At the initial instant (0) the rockets meet at a point O and their clocks are set to zero. The rockets are scheduled2 to move at constant speed toward two points A and B placed at equal distances from point O where they meet two clocks whose reading is identical (see Fig 1). When the rockets reach points A and B, their inboard clocks are stopped and then compared. There is neither acceleration nor deceleration during the process. According to Einstein’s special relativity, the inboard clocks should display the same reading when they stop; indeed, since the speed of light is regarded as isotropic in all ‘inertial’ frame, it is assumed that no obstacles are opposed so that an exact synchronization is carried out. Therefore the equal reading displayed by the clocks A and B is regarded as the real time. Due to the complete symmetry of the transit of the two rockets, their inboard clocks must display the same reading, which is equal to the reading of the clocks A and B multiplied by the 1 / γ factor. This is a condition so that the relativity principle is obeyed. As we shall see, a completely different explanation is provided by aether theory. According to Poincaré’s theory, as we have seen, there is no assumed incompatibility between the existence of a privileged frame and the principle of relativity. Is this really the case? This test will enable us to answer this question in the following chapters. (We must bear in mind that, in aether theory, clock

1

Let us remember that perfect inertial frames don’t exist in the physical world. The concept must be regarded as a limit case, which real frames approximate more or less. 2 Note that, as we shall see in the following chapters, even though the rockets are scheduled to move symmetrically, the symmetry will be only apparent if the clocks used to measure the speeds are affected by a synchronism discrepancy effect.

55

retardation is defined with respect to the privileged aether frame which is represented here by the co-ordinate system S0 ).

S0

S O

A

Fig 1: The two rockets are scheduled to move, at constant speed toward two clocks placed in A and B at equal distances from point O. When they reach points A and B, the reading of these clocks is identical. III. MEASUREMENT AND CLOCK SYNCHRONIZATION Assuming the existence of a preferred aether frame, this issue needs to be considered successively from two different points of view. III. 1 The first point of view presumes that one can exactly measure the transit time of the rockets from point O to points A and B . This implies that the identical reading τ of the clocks placed at points A and B, when the rockets reach them, which was assumed by definition, translates the identity of the real time. This fact implies a perfect synchronization of the clocks placed at points A and B with the clock placed at point O. (Yet we know that synchronizing clocks exactly is not an easy process [8]). In any case an exact synchronization can be considered, even if we cannot do it exactly nowadays and it is justified to estimate the implications of such a procedure3 Let us therefore first suppose, for our purpose, that this exact synchronization of clocks has been carried out. Assuming in agreement with aether theory that clock retardation results from the motion of the rockets with respect to the aether frame, the resolution of the problem is easy. Insofar as the rockets do not have the same speed with respect to the aether frame, the slowing down of their inboard clocks will be different and they will display different readings. (Only if the co-ordinate system S was at rest with respect to the aether frame, the clocks inside the rockets would display the same reading 3

Notice that an accurate synchronization of clocks is not impossible knowing that different experiments and astronomical observations have permitted estimation of the absolute speed of the Earth frame and, therefore, of the magnitude of the one-way speed of light [1]. Most probably, in the near future, a more accurate determination of this speed will enable us to synchronize the clocks almost exactly

56

B

irrespective of their direction of motion). Of course, this would inform us whether S is at rest or in motion relative to S0 , in contradiction with the principle of relativity. Therefore insofar as the rockets’ speeds are determined exactly, Poincaré’s relativity principle is shown to be at variance with the existence of a preferred aether frame. III. 2 We shall now study what happens when the transit time of the rockets is measured, in the co-ordinate system S, with clocks synchronized by means of the usual Einstein-Poincaré procedure. In order to synchronize the clocks placed at points A and B, we shall make use of the Einstein-Poincaré synchronization procedure (E. P synchronization) which assumes that the speed of light is equal to C in all inertial frames. To this end, we send a light signal at time t0 = 0 from clock O to clock A (or B). After reflection, the signal returns to O. The clock is supposed to be synchronous with clock O if, at the instant of reflection, it displays the reading t = T/2, where T is the reading displayed by clock O at the instant when the signal comes back to it. Insofar as the one-way speed of light is not isotropic in co-ordinate systems which are not at rest with respect to the aether frame, the use of this method introduces an unavoidable systematic error that must be corrected, as we shall see below. For convenience, we shall assume that the segment AB is aligned and moves along the x-axis of the co-ordinate system S0 which is at rest with respect to the aether frame. Let us refer to the length of the segment AB when it is at rest in the aether frame as 2 l . Since it is moving with respect to the aether frame at speed v, half of its length (measured with a non-contracted standard) will be l 1 − v 2 / C 2 where C is the speed of light in the aether frame. Actually, according to the aether theory considered in this text, the real speed of light relative to the co-ordinate system S along the direction AÆB is equal to C – v, and in the opposite direction to C + v. Even though the magnitude of v is not exactly known, this assumption will be helpful for our purpose. (These formulas were the expressions used by Lorentz to explain the Michelson experiment). As we shall see in appendix 2, for the present case, and in ref [1], for the general cases, only speeds whose measurements are altered by the systematic measurement distortions obey the relativistic law of composition of velocities. The real time needed by the light signal to travel from point O to point B is therefore:

trB =

l 1 − v2 / C 2 C −v

(where the suffix r means real)

57

trB is the time that, in the absence of clock retardation, the clock placed in B and exactly synchronized with clock O would display when the signal which starts at instant zero from point O reaches point B. Taking account of clock retardation in S, the clock reading in the absence of synchronism discrepancy effect would be:

trB 1 − v 2 / C 2 = l

(1 − v 2 / C 2 ) C −v

But, what we measure by means of the E. P synchronization procedure is half the reading displayed by clock O at the instant when the signal returns to it. With clocks not slowed down by motion, the apparent time needed by the light signal to travel from point O to point B would be:

t Bapp = 1 / 2l 1 − v 2 / C 2 (

1 1 l + )= C −v C +v C 1 − v2 / C 2

And the reading displayed by clock B when one takes account of clock retardation is:

t Bapp 1 − v 2 / C 2 = l / C (this expression is equal to the reading t = T/2 defined above) (We can see that, contrary to special relativity, aether theory does not consider the ratio l / C as the real time of light transit from O to B). Thus, taking account of clock retardation in S, the synchronism discrepancy of clock B with respect to clock O is:

Δ=(

l 1 − v2 / C 2 l vl − ) 1 − v2 / C 2 ) = 2 C −v C C 1 − v2 / C 2

(We can see that the apparent time is shorter than the real time.) We shall now determine the synchronism discrepancy of clock A with respect to clock O. We can easily anticipate that it will be equal to - Δ , but, even so, the calculation deserves to be done. The real time needed by the light signal to travel from point O to point A is:

l 1 − v2 / C 2 trA = C+v It is the time that, in the absence of clock retardation, the clock placed in A and exactly synchronized with clock O would display when the signal reaches point A. Taking account of clock retardation in the co-ordinate system S and using an exact synchronization procedure, the reading displayed by clock A would be:

trA 1 − v 2 / C 2 = l

(1 − v 2 / C 2 ) C+v

58

But, what we measure by means of the E. P synchronization procedure is half the reading displayed by clock O at the instant when the signal returns to it. With clocks not slowed down by motion, the apparent time needed by the signal to travel from point O to point A would be:

t Aapp = 1 / 2l 1 − v 2 / C 2 (

1 1 l + )= C +v C −v C 1 − v2 / C 2

Therefore, the reading displayed by clock A when one takes account of clock retardation is:

t Aapp 1 − v 2 / C 2 = l / C (it is the same as the reading displayed by clock B). Thus, taking account of clock retardation, the synchronism discrepancy of clock A with respect to clock O is:

Δ' = −Δ = (

l 1 − v2 / C 2 l vl − ) 1 − v2 / C 2 ) = − 2 C+v C C 1 − v2 / C 2

We note that, contrary to clock B the apparent time given by clock A is longer than the real time. Let us now study the effect of the synchronism discrepancy on the clocks placed inside the rockets. In the experiment, the apparent transit times of the rockets relative to point O, measured by an observer at rest relative to S using the E. P synchronization procedure, are assumed to be identical by definition; therefore, when the rockets reach points A and B, the clocks A and B will display the same reading τ . But due to the synchronism discrepancy effect this reading is erroneous and must be corrected and, as we shall see in the text that follows the real transit times of the rockets are in fact different, and, of course, their real speed also differ. In fact, in the absence of synchronism error, the reading of clock B would have been:

τ +Δ =τ +

vl C2

And the reading of clock A:

τ −Δ =τ −

vl C2

Let us now determine the real transit times t0 A and t0 B that would be displayed by clocks attached to frame S0 when the rockets reach points A and B. We have:

τ + Δ = t0 B 1 − v 2 / C 2 and

59

τ − Δ = t0 A 1 − v 2 / C 2 Thus:

t0 B =

τ + vl / C 2

(1)

1 − v2 / C 2

And

t0 A =

τ − vl / C 2

(2)

1 − v2 / C 2

We note that these expressions assume the same mathematical form as the conventional transformations, yet, as we saw, since they have been measured with clocks E. P synchronized and contracted meter sticks, τ and l are not the real space and time co-ordinates of the points A and B when the rockets reach these points (see Ref [1]). Readings displayed by the clocks inside the rockets. Since, according to our initial conditions, the apparent transit times of the rockets in S, measured with clocks E. P synchronized, are identical, and equal to τ , the apparent speeds will be vapp = l / τ in both sides, (where 2l is the apparent length of AB in S, measured with a contracted standard). Yet the apparent time corresponds to two different real times t0 A and t0 B and therefore to two different real speeds v’ and v”. Using these values we can determine the apparent transit times displayed by the clocks in the rockets’ frames TA and TB and therefore we shall see that they are identical, although the real times given by formulas (1) and (2) are not. Taking account of clock retardation, the clock present inside the rocket travelling toward point B, at the instant when it reaches this point, displays the reading:

TB = t0 B 1 − (v + v' ) 2 / C 2 =

τ + vl / C 2 1− v /C 2

2

1 − (v + v ' ) 2 / C 2

Where v’ is the real speed relative to point O of the rocket travelling toward point B. And the clock of the rocket travelling toward point A will display:

TA = t0 A 1 − (v − v" ) 2 / C 2 =

τ − vl / C 2 1− v /C 2

2

1 − (v − v" ) 2 / C 2

Where v’’ is the real speed relative to point O of the rocket travelling toward A.

60

We easily verify that

l 1 − v 2 / C 2 l(1 − v 2 / C 2 ) = v' = t0 B τ + vl / C 2

(3)

l 1 − v 2 / C 2 l(1 − v 2 / C 2 ) = t0 A τ − vl / C 2

(4)

and

v" =

Replacing v’ and v’’ with their values in TA and TB we remark that TA and TB are identical. We find:

TA2 = TB2 = γ 2 (τ 2 − l 2 / C 2 − v 2τ 2 / C 2 + v 2l 2 / C 4 ) = τ 2 − l2 / C 2 (See the demonstration in appendix 1) For values of l / τ << C we obtain:

2

vapp l2 TA = TB ≈ τ (1 − 1 / 2 2 2 ) = τ (1 − 1 / 2 2 ) Cτ C

(5)

For the usual transits whose speed is low compared to the speed of light, this expression approximates τ , a result which highlights the equivalence of the slow clock transport synchronization procedure and the Einstein-Poincaré method, and provides a key to understand the GPS measurements. IV. CONCLUSION This result is very enlightening. It demonstrates that, if we assume the existence of an aether frame and if the measurements of the rockets’ transit times from O to A and B, by the observer at rest in S, are exactly determined and are found identical, the clocks inside the rockets will display different readings when they reach points A and B. Therefore the relativity principle does not apply with real speeds. Conversely, if one uses the Einstein-Poincaré procedure in S to determine the ‘transit times’ (and therefore the ‘speeds’) and if the measurement yields the same clock reading τ in both sides, then the clocks inside the rockets will also display the same reading when they reach points A and B. This result is due to the systematic inevitable error made when, assuming the isotropy of the one-way speed of light in all ‘inertial’ frames, one relies on this synchronization procedure with light signals. Therefore the study also verifies the agreement of the slow clock transport

61

synchronization with the E. P procedure in accord with the conclusion of several authors[8]4. Therefore, assuming the existence of a preferred aether frame implies that the relativity principle is not a fundamental postulate of physics; it depends on arbitrary synchronization procedures. We emphasize that, although in this experiment, the use of the synchronization procedures mentioned above, to measure the transit times in S, make sure that the clocks inside the rockets will display the same reading when they reach points A and B, (in agreement with what special relativity asserts), the interpretation of this fact by aether theory is completely different. In particular, this result has been obtained without assuming the isotropy of the one-way speed of light in the co-ordinate system S, a fact which should result in significant consequences for the understanding of physics. V. APPENDIX 1 Identical readings displayed by the clocks present inside the rockets when the Einstein-Poincaré synchronization procedure is used. We have

TB2 = γ 2 (τ + vl / C 2 ) 2 [1 −

v2 vv' v'2 − 2 − ] C2 C2 C2

Where γ = (1 − v 2 / C 2 ) −1 / 2 Replacing v’ by its value given in (3), we obtain:

TB2 = γ 2 (τ + = γ 2 [(τ +

vl 2 v 2 2vl(1 − v 2 / C 2 ) l 2 (1 − v 2 / C 2 ) 2 ) [ 1 ] − − − C2 C 2 C 2 (τ + vl / C 2 ) C 2 (τ + vl / C 2 ) 2

vl 2 v 2 vl 2 2vl l2 2 2 2 l − + − − + − ) ( τ ) ( 1 v / C )( τ v / C ) (1 − v 2 / C 2 ) 2 ] 2 2 2 2 2 C C C C C

= γ 2 (τ 2 − l 2 / C 2 − v 2τ 2 / C 2 + v 2l 2 / C 4 ) 4

A lively debate took place in recent years among physicists about the validity of the relatvity principle. The experiment of Hafele and Keating [9] was presented by the authors as a decisive argument in its favour. Yet the interpretation of the experiment was severely criticized by Kelly [10] and Essen [11]. More recent experiments (including GPS measurements [12, 13]) supported the conclusions of Hafele and Keating. Aether theory provides a key to account for the experimental tests. As shown in this text, the relativity principle seems to apply only with physical data resulting from the measurement distortions. It does not apply any more when the distortions are corrected (see section III. 2 and the appendixes).

62

Therefore

TB2 = τ 2 − l 2 / C 2 TA2 = γ 2 (τ − vl / C 2 ) 2 [1 −

v2 vv" v"2 + 2 − ] C2 C2 C2

Replacing v” by its value given in (4), we obtain:

TA2 = γ 2 (τ −

vl 2 v 2 2vl(1 − v 2 / C 2 ) l 2 (1 − v 2 / C 2 ) 2 ) [ 1 ] − + − C2 C 2 C 2 (τ − vl / C 2 ) C 2 (τ − vl / C 2 ) 2

vl 2 v 2 vl 2 2vl l2 2 2 2 l − − + − − − ) ( τ ) ( 1 v / C )( τ v / C ) (1 − v 2 / C 2 ) 2 ] 2 2 2 2 2 C C C C C 2 2 2 2 2 2 2 2 2 4 = γ (τ − l / C − v τ / C + v l / C ) = γ 2 [(τ −

Therefore

TA2 = τ 2 − l 2 / C 2 VI. APPENDIX 2 Composition of velocities law for apparent speeds According to the aether theory referred to in this text, speeds are simply additive. The relativistic composition of velocities law results from the measurement distortions caused by length contraction, clock retardation and unreliable clock synchronization, as the following demonstration will show. We start from the Galilean law VB = v + v' and V A = v − v" , where VB and

VA refer to the real speeds of the rockets with respect to the aether frame. From formulas (1) and (2) we have:

τ = t0 B 1 − v 2 / C 2 − vl / C 2 = t0 A 1 − v 2 / C 2 + vl / C 2

τ is the apparent transit time of the rockets measured in S with the clocks A and B (E. P synchronized). Since OA and OB are measured with a contracted standard they are found equal to l although their real length is l 1 − v 2 / C 2 .

The apparent speed (relative to point O) of the rocket travelling toward B is therefore:

63

v'app =

l

τ

=

l

tOB 1 − v / C 2 − vl / C 2 We note that by definition v'app = v' 'app where v' 'app refers to the apparent speed 2

(relative to point O) of the rocket travelling toward A; a fact which can be easily verified. (Therefore we may also refer to the apparent speed in both directions as vapp .)

l 1 − v2 / C 2 Replacing t0 B by its value VB − v l l we find: v'app = = 2 τ l(1 − v / C 2 ) − vl / C 2 VB − v V −v VB − v = = B 2 vV v v(VB − v) 1 − 2B 1− 2 − 2 C C C This result shows decisively that the relativistic composition of velocities law applies to apparent speeds and not to the real speeds which as we saw are simply additive. The conclusions drawn in this article are identical to those which were expressed in the previous version submitted to arXiv under the reference Physics/0610067. We have only given further explanations and added other references. REFERENCES [1] J. Levy, in Ether space-time and cosmology, Volume 1, Michael C. Duffy and Joseph Levy editors, PD Publications, Liverpool UK : “Basic concepts for a fundamental aether theory” and “Aether theory and the principle of relativity”, ArXiv physics/0604207 and physics/0607067. From Galileo to Lorentz and beyond, Apeiron, Montreal, Canada, 2003, web site http://redshift.vif.com [2] A. Einstein, “On the electrodynamics of moving bodies”, in The principle of relativity, Dover, New York, 1952, page 38, Translated from Annalen der physik, 17, (1905), page 891. [3] A. Einstein, Sidelights on relativity, Dover, New York, 1983, first chapter: “Ether and the theory of relativity”, an address delivered on May 5th 1920 in the University of Leyden.

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[4] H. Poincaré, ‘‘Sur la dynamique de l’électron’’, in La mécanique nouvelle , Jacques Gabay, Paris, France, 1989, page 19. [5] H. Poincaré, La science et l’hypothèse, chapter 10 of the french edition, Les théories de la physique moderne, Champs, Flammarion 1968. [6] H. Poincaré, Lecture given in Lille France in 1909, in La mécanique nouvelle, Jacques Gabay, Paris, France, 1989, Page 10. [7] H. Poincaré, ‘‘Sur la dynamique de l’électron’’, in La mécanique nouvelle, Jacques Gabay, Paris, France, 1989, Page 18. [8] J. Levy, “Synchronization procedures and light velocity”, Proceedings of the International conference, Physical interpretations of relativity theory VIII (PIRT), Imperial college London, 6-9 September 2002, Page 271. From Galileo to Lorentz and beyond, Apeiron, Montreal, Canada, 2003, web site http://redshift.vif.com [9] J. C. Hafele and R.E. Keating, Science 177, 166-168 and 168-170, 1972 [10] A.G. Kelly http://www.biochem.szote.u-szeged.hu/astrojan/hafele.htm [11] L. Essen, Electron.& Wireless World 94 (1624) 126-127, 1988 [12] Metromnia, Issue 18, Spring 2005 [13] T. Van Flandern, What the global positioning system tells us about relativity, in Open questions in relativistic physics, F. Selleri Editor, Apeiron Montreal 1998 p 81-90.

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The Dynamic Universe Zero-energy balance restores absolute time and space Tuomo Suntola Vasamatie 25, 02630 Espoo, Finland E. mail: [email protected]

Abstract The Dynamic Universe is a holistic model of physical reality starting from whole space as a spherically closed zero-energy system of motion and gravitation. Instead of extrapolating the cosmological appearance of space from locally defined field equations, locally observed phenomena are derived from the conservation of the zeroenergy balance of motion and gravitation in whole space. The energy structure of space is described in terms of nested energy frames starting from hypothetical homogeneous space as the universal reference and proceeding down to local frames in space. Time is decoupled from space – the fourth dimension has a geometrical meaning as the radius of the sphere closing the three-dimensional space. Relativity in the Dynamic Universe is the measure of the locally available share of total energy – clocks in fast motion or in a strong gravitational field do not lose time because of slower flow of time but because more energy is bound into interactions in space. For local observations, the DU predictions are essentially identical to the corresponding predictions derived from the theory of relativity. At the extremes, at cosmological distances and in the vicinity of local singularities in space however, differences become remarkable – e.g. galactic space in the DU appears in Euclidean geometry, and the magnitudes of high redshift supernovae are explained without assumptions of dark energy or accelerating expansion. Black holes in DU space have stable orbits down to the critical radius. Instead of a sudden Big Bang, the energy buildup in Dynamic Universe is seen as a continuous process from infinity in the past to infinity in the future. The Dynamic Universe means a major step in the unification in physics and cosmology. Electromagnetic energy is linked to the rest energy of matter and the effects of motion and gravitation in local frames and in the relevant parent frames are inherently included in the expressions of energy states and characteristic emission frequencies of atomic objects.

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Introduction Newtonian physics is local by its nature. There are no overall limits to space or to physical quantities. Newtonian space is Euclidean until infinity, and velocities in space grow linearly as long as there is constant force acting on an object. The theory of relativity introduces finiteness as finiteness of velocities by postulating the velocity of light as an invariant and the maximum velocity for any observer. In the theory of relativity, finiteness of velocities is described by linking time to space in four-dimensional spacetime, which defines spacetime interval ds = cdτ as an invariant equal to the product of the velocity of light and the proper time τ. In the framework of relativity theory, clocks in high a gravitational field and in fast motion conserve the local proper time but lose coordinate time related to time measured by a clock at rest in zero gravitational field. In the Dynamic Universe, finiteness comes from the finiteness of total energy in space — finiteness of velocities in space is a consequence of the zero-energy balance, which does not allow velocities higher than the velocity of space in the fourth dimension. The velocity of space in the fourth dimension is determined by the zero-energy balance of motion and gravitation of whole space and it serves as the reference for all velocities in space. The total energy is conserved in all interactions in space. Motion and gravitation in space reduce the energy available for internal processes within an object. Atomic clocks in fast motion or in high gravitational field in DU space do not lose time because of slower flow of time but because they use part of their total energy for kinetic energy and local gravitation in space. In his lectures on gravitation in early 1960’s Richard Feynman [1] stated: “If now we compare this number (total gravitational energy MΣ2G/R) to the total rest energy of the universe, MΣ c2, lo and behold, we get the amazing result that GMΣ2/R = MΣ c2, so that the total energy of the universe is zero. — It is exciting to think that it costs nothing to create a new particle, since we can create it at the center of the universe where it will have a negative gravitational energy equal to MΣc2. — Why this should be so is one of the great mysteries—and therefore one of the important questions of physics. After all, what would be the use of studying physics if the mysteries were not the most important things to investigate”. and further [2]

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“...One intriguing suggestion is that the universe has a structure analogous to that of a spherical surface. If we move in any direction on such a surface, we never meet a boundary or end, yet the surface is bounded and finite. It might be that our threedimensional space is such a thing, a tridimensional surface of a four sphere. The arrangement and distribution of galaxies in the world that we see would then be something analogous to a distribution of spots on a spherical ball.” Once we adopt the idea of the fourth dimension with metric nature, Feynman’s findings open up the possibility of a dynamic balance of space: the rest energy of matter is the energy of motion mass in space possesses due to the motion of space in the direction of the radius of the 4-sphere. Such a motion is driven by the shrinkage force resulting from the gravitation of mass in the structure. Like in a spherical pendulum in the fourth dimension, contraction building up the motion towards the center is followed by expansion releasing the energy of motion gained in the contraction. The Dynamic Universe approach is just a detailed analysis of combining Feynman’s “great mystery” of zero-energy space to the “intriguing suggestion of spherically closed space” by the dynamics of a four sphere. By equating the integrated gravitational energy in the spherical structure with the energy of motion created by momentum in the direction of the 4-radius we enter into zero-energy space with motion and gravitation in balance. It may not be a surprise that by assuming the presently relevant estimates of the mass density and Hubble radius of space, we can calculate the velocity of spherically closed space in the direction of the 4-radius as 300,000 km/s, equal to the velocity of light in space. In fact, space as the surface of a four sphere is based on quite an old and original idea of describing space as a closed but endless entity. Spherically closed space was outlined in the 1900th century by Ludwig Schläfli, George Riemann and Ernst Mach. Space as the 3-dimensional surface of a four sphere was also Einstein’s original view of the cosmological picture of general relativity he suggested in 1917 [3]. The problem, however, was that Einstein was looking for a static solution — it was just to prevent the dynamics of spherically closed space that made Einstein to add the cosmological constant to the theory. We also find out that dynamic space requires metric fourth dimension which does not fit to the concept of four-dimensional spacetime the theory of relativity is built on. In Dynamic Universe time and distance, the basic coordinate quantities and the key attributes for human conception, are absolute and universal. The fourth dimension is metric by its nature although inaccessible from three-dimensional space. In a local frame, a rough translation from relativity theory to the Dynamic Universe is given by a physical interpretation of the energy four-vector as the vector sum of momentum p4 = mc4 in a physical fourth dimension and momentum p in a space direction.

69

2 2 Etot = c 2 ⎡( mc ) + p 2 ⎤ ⎣ ⎦

(a)

Relativity in Dynamic Universe is a direct consequence of the conservation of the total energy in interactions in space. It does not rely on the relativity principle, spacetime, the equivalence principle, Lorentz covariance, or the invariance of the velocity of light — but just on the zero-energy balance of space. In a detailed analysis, the locally available rest energy mass object m possesses in the n:th energy frame is n

Erest = c0 p = c0 mc = m0 c02 ∏ (1 − δ i ) 1 − β i2

(b)

i =1

where c0 is the velocity of light in hypothetical homogeneous space, which is equal to the velocity of space in the direction of the 4-radius R4. The factors δi = GMi/c2 and βi = vi/ci are the gravitational factor and the velocity factor relevant to the local frame, respectively. On the Earth, for example, the gravitational factors define the gravitational state of an object on the Earth, the gravitational state of the Earth in the solar frame, the gravitational state of the solar frame in the Milky Way frame, etc. The velocity factors related to an object on Earth comprise the rotational velocity of the Earth and the orbital velocities of each sub-frame in each one’s parent frame. The concept of motion in Dynamic Universe is twofold; velocity as the measure of kinetic energy is related to the state of rest in the energy frame where the velocity is obtained — the observed relative velocity between two objects serves as the measure of the change in the distance between the objects, which does not define the content of kinetic energy each object is carrying. Most important, spacetime symmetries of the special and general theory of relativity are replaced by symmetries resulting from the zero-energy balance of energies. Equation (b) means that the locally available rest energy is a function of the gravitational state, and the velocity of the object studied. Substituting (b) for the rest energy of electron in Balmer’s equation the characteristic frequency related to an energy transition obtains the form n

flocal = f 0 ∏ (1 − δ i ) 1 − β i2 = f n −1 (1 − δ n ) 1 − β n2

(c)

i =1

where frequency fn–1 is the characteristic frequency of the atom at rest far from the local mass center in the local frame. The last form of equation (c) is essentially equal to the expression of coordinate time frequency in Earth, or Earth satellite clocks in the GR framework. The physical message of (c) is that “the greater the share of total

70

energy which used for motions and gravitational interactions in space the less energy is left for running internal processes”. The Dynamic Universe links the energy of any localized object to the energy of whole space. Relativity in Dynamic Universe means relativity of local to whole. The balance of the rest energy and the global gravitational energy means also that antimatter of any localized mass object in space is the mass of the rest of space Erest + Eglobal = 0

(d)

At the cosmological scale an important consequence of the linkage between local space and whole space is that local gravitational systems grow in direct proportion to the expansion of space thus, together with the spherical symmetry, explaining the observed Euclidean appearance and surface brightnesses of galaxies in space. The magnitude redshift relation of a standard candle in the DU framework is in an accurate agreement with observations without assumptions of dark energy or any other free parameters. Moreover, the zero-energy balance in the DU leads to stable orbits down to the critical radius in the vicinity of local singularities in space. In the DU framework the energy of a quantum of radiation appears as the unit energy carried by a cycle of radiation Eλ = c0

h0

λ

c = c0 mλ c = c0 p λ

(e)

where h0 ≡ hc is referred to as the intrinsic Planck constant which is solved from Maxwell’s equation, by observing that a point emitter in DU space which is moving at velocity c in the fourth dimension can be regarded as one-wavelength dipole in the fourth dimension. Such a solution shows also that the fine structure constant α is a purely numerical or geometrical factor without linkage to any physical constant. The quantity h0/λ ≡ mλ [kg] in (e) is referred to as the mass equivalence of radiation. Equally, Coulomb energy is expressed in form EC =

e 2 μ0 h c0 c = α 0 c0 c = c0 mC c 2π r 2π r

(f)

where α is the fine structure constant and the quantity h0/2π r ≡ mC is the mass equivalence of Coulomb energy. Equations (b), (e), and (f) give a unified expression of energies which is essential in a detailed energy inventory in the course of the expansion of space and in interactions within space. The zero-energy concept in the Dynamic Universe follows bookkeeper’s logic — the accounts for the energy of motion and potential energy are kept in balance throughout the expansion and within any local frame in space.

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Historically, the basis of the zero-energy concept was first time expressed by Gottfried Leibniz, the great philosopher, mathematician, and physicist contemporary with Isaac Newton. Leibniz introduced the zero-energy principle by stating that vis viva, the living force mv2 (kinetic energy) is obtained against release of vis mortua, the dead force (potential energy) [4]. Inherently, such an approach defines the state of rest as a property of an energy system where kinetic energy (vis viva) is created. It also looks like Leibniz’s monads as “perpetual, living mirrors of the universe”, reflected the idea wholeness and the complementary nature of local and global in material objects in Dynamic Universe. There is no need to expect antimatter in space; via the zero-energy balance of motion and gravitation, the rest energy of any localized mass object is counterbalanced by the gravitational energy due to all rest of mass in space. The Dynamic Universe is a holistic description of physical reality [5]. The system of nested energy frames in spherically closed space links local structures and phenomena to space as whole. The zero-energy approach in the DU allows the derivation of local and cosmological predictions with a minimum of postulates and by honoring universal time and distance as the basic coordinate quantities for human conception. In a mathematically clear and straightforward way it produces precise predictions for phenomena in relativistic physics, celestial mechanics, and cosmology, and allows a unified expression of energies showing the linkage between electromagnetic quantities and mass objects. The Dynamic Universe means major rethinking of the cosmological structure and development of the universe. Instead of a sudden Big Bang switching on time, energy, and the laws of nature, the buildup and release of energy in Dynamic Universe develops in a contraction and expansion process from emptiness in infinity in the past through singularity to emptiness in infinity in the future.

1. Spherically closed space 1.1 Motion and gravitation in homogeneous space 1.1.1 Postulates and definitions

The Dynamic Universe model assumes that space is spherically closed through the fourth dimension; i.e. space is described as the 3-dimensional surface of a 4dimensional sphere free to contract and expand maintaining a zero-energy balance of

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motion and gravitation in the system. Mass as the substance for the expression of energy is the primary conservable in space. For calculating the zero-energy balance in spherically closed space the inherent forms of the energies of gravitation and motion are defined as follows: 1)

The inherent gravitational energy is defined in homogeneous 3-dimensional space as Newtonian gravitational energy

Eg ( 0) = − ρ mG ∫

dV ( r )

V

(1.1.1:1)

r

where G is the gravitational constant, ρ is the density of mass, and r is the distance between m and dV. Total mass in homogeneous space is

M Σ = ρ ∫ dV = ρV

(1.1.1:2)

V

In spherically closed homogeneous 3-dimensional space the total mass is M Σ = ρ ⋅ 2π 2 R03 , where R0 is the radius of space in the fourth dimension. 2)

The inherent energy of motion is defined in environment at rest as the product of the velocity and momentum Em( 0) = v p = v mv = mv 2

(1.1.1:3)

The last form of the energy of motion in (1.1.1:3) has the form of the first formulation of kinetic energy, vis viva, “the living force” suggested by Gottfried Leibniz in late 1600’s [4]. 1.1.2 Zero-energy balance in hypothetical homogeneous space

The energy of motion mass m at rest in space possesses due to the motion of space in the fourth dimension is referred to as the rest energy of matter. In hypothetical homogeneous space, the rest energy is Erest = c0 p 0 = c0 mc0

(1.1.2:1)

where p0 is the momentum of mass m and c0 is the velocity of space in the direction of the 4-radius. The symbol c is used for the velocity of space because it is shown that the velocity of space in the fourth dimension defines the maximum velocity and the velocity of light in space.

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The energy of gravitation resulting from total mass MΣ on mass m is referred to as global gravitational energy. In spherically closed homogeneous space the global gravitational energy is Eglobal = − ρ mG ∫ V

dV GM " m =− r R0

(1.1.2:2)

where M" = 0.776 ⋅MΣ is the mass equivalence at the center of “hollow“ spherically closed space with radius R0. Obviously, any mass m in homogeneous space is at distance R0 from mass M". For mass m at rest in hypothetical homogeneous space with 4-radius R0 the balance of the energies of motion and gravitation is Erest + Eglobal = mc02 −

GM " m =0 R0

(1.1.2:3)

The rest energy is a local expression of energy of an object. In spherically closed space the rest energy of an object is balanced by the global gravitational energy resulting from all the rest of mass in space. The complementarity of energies — the rest energy and the global gravitational energy — means also complementarity of local and global.

For total mass MΣ the balance of the energies of motion and gravitation in hypothetical homogeneous space is Erest (tot ) + Eglobal ( tot ) = M Σ c02 −

GM " MΣ = 0 R0

(1.1.2:4)

Force in Dynamic Universe is the manifestation of a natural trend towards minimum potential energy. Force is expressed as the negative of the gradient of potential energy or in terms of a change of momentum. 1.1.3 The primary energy buildup

Solved from (1.1.2:4), velocity c0 that maintains the zero-energy balance of motion and gravitation in spherically closed space is

c0 = ±

0.776 ⋅ G ρ 2π 2 R03 GM " =± = ±1.246 ⋅ π R0 G ρ R0 R0

where ρ is the average mass density in space.

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(1.1.3:1)

Spherically closed space is accelerated by its own gravitation in a contraction phase from infinite 4-radius to singularity creating the energy of motion against a release of gravitational energy. In the expansion phase after passing the singularity the energy of motion gained in the contraction is paid back to gravitational energy. In the contraction space releases volume and obtains velocity, in the expansion phase velocity is released to recover the volume. In energy bookkeeping, the rest energy of matter, the energy mass possesses due to the motion of space in the fourth dimension is balanced by an equal energy debt to global gravitation (Fig 1.1.3-1). The contraction and expansion of spherically closed space is the primary energy buildup process creating the rest energy of matter as the complementary counterpart to the global gravitational energy. Based on observations of the Hubble constant, space in its present state is in the expansion phase with radius R0 equal to about 14 billion light years. By applying R0 = 14 billion light years and by setting the mass density equal to ρ = 5.0⋅10–27 [kg/m3], which is about half of the critical density ρ0 in the standard cosmology model, velocity c0 in (1.1.3:1) obtains the value c0 ≈ c = 300 000 [km/s].

Contraction

Expansion

Energy of motion

Em = mc02

time

Eg = m

GM " R0

Energy of gravitation Figure 1.1.3-1. Energy buildup and release in spherical space. In the contraction phase, the velocity of motion increases due to the energy gained from loss of gravitation. In the expansion phase, the velocity of motion gradually decreases, while the energy of motion gained in contraction is returned to gravity.

75

9 8

×10 m/s 6

3

0

0

10

30

20

40 × 109 years

Figure 1.1.3-2. The decreasing expansion velocity of space in the direction of R0. Present deceleration of the expansion velocity, and with it the velocity of light, is about 3.6 % per billion years. The velocity of light will drop to half of the present value in about 65 billion years and to 1 m/s in about 2 ⋅1026 billion years.

When solved as a function of time, the expansion velocity since singularity becomes 1/ 3

dR ⎛ 2 ⎞ c0 = 4 = ⎜ GM " ⎟ t −1 3 dt ⎝ 3 ⎠

(1.1.3:2)

and the time since singularity becomes t=

R4

1

0

4

∫c

dR4 =

2 R4 2 1 = = 9.3 ⋅ 109 3 c4 3 H 0

[l.y.]

(1.1.3:3)

The velocity of expansion and, accordingly, the velocity of light decelerate in the course of expansion as dc0 1c =− 0 3 t dt

(1.1.3:4)

The present deceleration rate of the velocity of light is dc0/c0 ≈ 3.6⋅10–11 /year (Fig 1.1.3-2). A detailed analysis shows that the maximum velocity achievable in space is equal to the velocity of space in the fourth dimension. In zero-energy space the rate of atomic processes, like the characteristic emission and absorption frequencies and radioactive decay occur in direct proportion to the velocity of the expansion and, accordingly, to the velocity of light in space. As a result, the velocity of light is observed as constant at any time during the expansion.

76

t(2) R0

R0(2)

R0 t(1)

O

O (a)

emitting object R0(1) O

(b)

(c)

Figure 1.1.3-3. (a) Dimensions of galaxies and other gravitationally bound systems expand in direct proportion to the expansion of space. (b) Localized objects bound by electromagnetic forces conserve their size. The characteristic wavelength emitted by atomic objects is conserved. (c) The wavelength of electromagnetic radiation propagating in space increases in direct proportion to the expansion of space. As a consequence the observed wavelength is redshifted.

In cosmological observables the faster rate of natural processes is seen, e.g., as a faster rate of radioactive decay in the past – correcting the age estimates of the universe given by radiometric dating. It also means a faster rate of the development of galaxy structures in the early universe. Conservation of the total gravitational energy in space links the radii of gravitationally bound local systems to the 4-radius of space — local systems expand in direct proportion to the expansion of space. As a consequence of the linkage distant space appears Euclidean, appearance in a full agreement with observations. Atomic radii are not subject to expansion with the expansion of space, i.e. material objects conserve their dimensions. As shown by Balmer’s equations, the wavelength of characteristic emission is directly proportional to Bohr radius. Once the Bohr radius is conserved then also the emission wavelength is conserved. The wavelength of electromagnetic radiation propagating in space increases in direct proportion to the expansion of space, which means that the observed characteristic wavelength from distant objects is redshifted relative to the reference wavelength of same transition at the time of observation, Fig. 1.1.3-3. In the DU framework the basic form of matter is unstructured “dark matter” characterized as radiation-like form of matter.

77

1.2 Interactions in space

1.2.1 Global and local frames

The initial condition in space is considered as the state at rest with all mass uniformly distributed within space. The state at rest in space means that any mass m in space has the momentum and velocity given by the expansion of the structure in the direction of the 4-radius. Hypothetical homogeneous space is used as the universal frame of reference for all interactions in space. Hypothetical homogeneous space has perfect spherical symmetry. The barycenter of mass in space is in the center of the four-dimensional sphere defining the three-dimensional space. Mass equivalence M" in the barycenter is a hypothetical mass that results in the same gravitational energy on any mass m in space as does the integrated gravitational energy of all mass in homogeneous spherical space. Due to the expansion of space at velocity c0 in the direction of the 4-radius R0, masses m at rest at distance d = α⋅R0 from each other in spherical space have recession velocity vrecession = α ⋅ c0

(1.2.1:1)

relative to each other. Since masses m are at rest in their location in space they have momentum only in the direction of the R0 radius, and the relative velocity between the masses is not associated with momentum of kinetic energy (Fig. 1.2.1-1). Im 0

m

Em ( 0 )

Im 0

Re0

Em ( 0) = c0 p 0

Eg ( 0)

Re0

m

α M"

Eg ( 0) (a)

Re0

m

Eg ( 0) = −

Em ( 0 )

GM " m R0

(b)

Im 0

Figure 1.2.1-1. (a) Hypothetical homogeneous space has the shape of the 3-dimensional “surface” of a perfect 4-dimensional sphere. Mass is uniformly distributed in the structure and the barycenter of mass in space is in the center of the 4-sphere. Mass m is a test mass in hypothetical homogeneous space. (b) In a local presentation a selected space direction is shown as the Re0 axis, and the fourth dimension which in hypothetical homogeneous space is the direction of R0 is shown as the Im0 axis. The velocity of light in hypothetical homogeneous space is equal to the expansion velocity c0.

78

1.2.2 Unified expression of energy

For a detailed analysis of the symmetries and the conservation of energy in local interactions in space it is necessary to express electromagnetic energy in a form distinguishing the mass equivalence of electromagnetic energy and the velocity of light. The energy of electromagnetic radiation has the form of the energy of motion Erad = c p rad

(1.2.2:1)

where momentum prad has the direction of the propagation of the radiation in space; the momentum of electromagnetic energy has no component in the fourth dimension. In DU space moving at c in the fourth dimension a point emitter can be regarded as one-wavelength dipole in the fourth dimension. Solved from Maxwell’s equation the energy of one cycle of radiation from such a dipole is [5,6] Eλ =

h P = N 2 (1.1046 ⋅ 2π 3e 2 μ0 c0 ) f = N 2 hf = N 2 ⋅ 0 c0 c = c0 mλ c f λ

(1.2.2:2)

where h0 ≡ h/c0 [kg⋅m] is referred to as the intrinsic Planck constant. For a point source, factor 1.1046 related to the radiation geometry of the antenna in the fourth dimension (see Section 2.1.2). The quantity mλ [kg] mλ = N 2

1.1046 ⋅ 2π 3e 2 μ0

λ

= N2 ⋅

h0

λ

mλ ( 0 ) =

;

h0

λ

(1.2.2:3)

is defined the mass equivalence of radiation. In the DU framework, a quantum of electromagnetic radiation is the energy carried by one cycle of radiation emitted by a single transition of a unit charge in a point source, i.e. N =1 in equation (1.2.2:2) Eλ ( 0) = c0 p rad = c0 mλ ( 0 ) c = c0

h0

λ

(1.2.2:4)

c

A unified expression of Coulomb energy is obtained by applying vacuum permeability μ0 and the fine structure constant α [which in the DU framework is a numerical constant independent of any physical constants, see equation (2.1.2:6)] Ec =

q1q2 μ0 h c0 c = N 2α 0 c0 c = c0 mc c 4π r 2π r

(1.2.2:5)

where mc = N1 N 2

e 2 μ0 h = N1 N 2 ⋅ α 0 4π r 2π r

(1.2.2:6)

79

ic

Coulomb energy (1.2.2:5) Ec = N 2α

ic

h0 c0 c = c0 mc c 2π r

Energy of a cycle of electromagnetic radiation (1.2.2:2) E = c0 p = N 2

h0

λ

c

cc0 = c0 mλ c ic

Rest energy of localized energy object

E(0) = c0 p = c0 mc Figure 1.2.2-1. Unified expressions for the Coulomb energy, the energy of a cycle of electromagnetic radiation and the rest energy of a localized mass object.

is the mass equivalence of Coulomb energy (Fig. 1.2.2-1). When distance r between objects with charges N1e and –N2e is reduced, the mass equivalence Δmc ( Δr ) = − N1 N 2

e 2 μ0 ⎛ 1 1 ⎞ ⎜ − ⎟ 4π ⎝ r2 r1 ⎠

(1.2.2:7)

is reduced, i.e. mass Δm is released to the buildup of kinetic energy of the charged object accelerated in Coulomb field Ekin = c0 cΔm = N1 N 2

e 2 μ0 ⎛ 1 1 ⎞ ⎜ − ⎟ c0 c 4π ⎝ r2 r1 ⎠

(1.2.2:8)

1.2.3 The energy vector Energy is traditionally regarded a scalar quantity. For illustrating the fourdimensional symmetries in the Dynamic Universe, it is useful to define the energy vector as a complex presentation of energy. The complex presentation of the energy of motion is Em^ = c0p ^ = E 'm + i E "m = c0 p '+ i c0 p " = c0 ( p '+ i p ")

80

(1.2.3:1)

where the imaginary part means the energy equivalence of momentum in the local fourth dimension. The complex presentation of the energy of gravitation is

Eg^ = E 'g + E "g

(1.2.3:2)

where E”g, the imaginary part, is the global gravitational energy resulting from all mass uniformly in spherically closed space. The scalar value of the energy vector (1.2.3:1) is denoted as

Em = Em^

(1.2.3:3)

The kinetic energy of an object moving at velocity β = v/c in a local frame is the total energy of motion minus the energy of motion the object has at rest in the local frame Ekin = Em( β ) − Em( 0) = c0 Δ p = c0 pβ − p0

(1.2.3:4)

Local gravitational energy is defined as the total energy of gravitation minus the global energy in the local frame. Generally, only the scalar value of local gravitational energy is of interest Eg (local ) = Eg^ − E "g

(1.2.3:5)

The energy vector of electromagnetic radiation is defined as the Poynting vector [W/m2] multiplied by the cycle time and the cross section area of radiation, which gives the total energy carried by a cycle of radiation. Electromagnetic radiation propagates in space directions; the energy vector of radiation has real component only.

1.3 Symmetries in zero-energy space 1.3.1 The zero-energy balance of motion and gravitation Mass at rest in hypothetical homogeneous space has the energies of motion and gravitation in the imaginary direction only

Em^ = i E "m

;

Erest = E "m = c0 p 4 = c0 mc0 = mc02

(1.3.1:1)

where c0 is the velocity of space in the direction of 4-radius R0, and

Eg^ = i E "g

;

E "g = −

GM " m R0

(1.3.1:2)

81

Im 0δ

Im 0 Erest = c0 mc0

Erest = c0 mc φ

Re0

m E "g = −

m

φ

GM " m R0

(a )

E "g = −

GM " m R"

( b)

Figure 1.3.1-1. Rest energy and the global gravitational energy (a) in hypothetical homogeneous space where c = c0 and R” = R0, and (b) in locally tilted space where c < c0 and the apparent distance to mass equivalence M” is increased as R”> R0.

In locally tilted space the velocity of space in the direction of the local imaginary axis is reduced as

c = c0δ cos φ

(1.3.1:3)

and the rest energy is expressed as

Erest = c0 mc

(1.3.1:4)

The global gravitational energy in tilted space is reduced as E "g = −

GM " m GM " m cos φ =− R" R0δ "

(1.3.1:5)

where R” is the apparent distance to the mass equivalence M" (Fig. 1.3.1-1). Quantities c0δ and R”0δ in (1.3.1:3) and (1.3.1:5), respectively, refer to apparent homogeneous space around locally tilted space [also the apparent homogeneous space may be tilted relative to hypothetical homogeneous space (see Section 1.4)]. 1.3.2 Conservation of energy in mass center buildup

For conserving the total energies of motion and gravitation in mass center buildup the momentum of free fall is obtained against reduction of the global gravitational energy resulting from mass homogeneously in space. Such a reduction occurs when mass M is removed from the spherical symmetry of homogeneous space. Buildup of the momentum of free fall in the vicinity of a mass center can be described via tilting of space associated with a reduction in the local gravitational energy (the local imaginary part of gravitational energy)

82

Imδ

Im 0δ

E "m ( 0)

Imδ

E "m (δ )

Reδ

E "m ( 0)

E "m (δ )

Reδ

Re0δ

m

m E "g (δ )

E "g ( 0)

E "g ( 0 )

E "g (δ )

Figure 1.3.2-1. The symmetry of imaginary the energies of motion and gravitation in the vicinity of a mass center in space.

⎛ ΔE "g ⎞ ⎟ = E "g ( 0δ ) (1 − δ ) E "g (δ ) = E "g ( 0δ ) − ΔE "g = E "g ( 0δ ) ⎜1 − ⎜ E "g ( 0 δ ) ⎟ ⎝ ⎠

(1.3.2:1)

The symmetry of the rest energy and global gravitational energy at gravitational state δ in tilted space (Fig. 1.3.2-1) is expressed E "g (δ ) = E "m(δ ) = Erest (δ ) = c0 mc0δ (1 − δ ) = c0 mc = E "g ( 0δ ) (1 − δ )

(1.3.2:2)

where c is the local velocity of light

c = c0δ (1 − δ )

(1.3.2:3)

Assuming that mass M at distance r0 from test mass m is accumulated to mass center M the reduction in the global gravitational energy is

ΔE "g = −

GM r0δ

(1.3.2:4)

and the gravitational factor δ becomes

δ≡

r MR " GM = = c M " r0δ r0δ c0 c0δ r0δ

;

rc ≡

GM c0 c0δ

(1.3.2:5)

where rc is the critical radius corresponding to radius where space is tilted 90°. (Obs. The critical radius in the Schwarzschild space is 2GM/c2, which is twice the critical radius rc in the DU.)

83

1.3.3 Kinetic energy and inertial work The kinetic energy of an object moving at velocity β = v/c in a local frame was defined as the total energy of motion minus the energy of motion the object has at rest in the local frame (1.2.3:4). The total energy of motion of an object in free fall from the state of rest far from the local mass center is, Fig. 1.3.3 (a) Em(total ) = c0 ptotal = c0 pδ ( Re ) + pδ ( Im ) = c0 p 0( Im ) = c0 mc0δ

(1.3.3:1)

The rest energy of an object at gravitational state δ characterized by tilting angle φ is

Erest (δ ) = c0 pδ ( Im ) = c0 mcδ

(1.3.3:2)

and the kinetic energy of free fall from the state of rest far from the local mass center is

Ekin( ff ) = Em(total ) − Erest (δ ) = c0 mc0δ − c0 mcδ = c0 m ( c0δ − cδ ) = c0 mΔc

(1.3.3:3)

Equation (1.3.3:3) means that kinetic energy in free fall is obtained against reduction in the local rest energy via tilting of space and the associated reduction in the local velocity of light. In free fall mass is conserved. Buildup of kinetic energy at constant gravitational potential conserves the velocity of light, but requires inserting of local energy into the object accelerated. Insertion of mass Δm via acceleration to velocity β, e.g. in Coulomb field (1.2.2:8) adds the total energy

Etotal ( β ) = Erest ( 0) + c0 cΔm = c0 c ( m + Δm )

(1.3.3:4)

or in complex form ^ Etotal ( β ) = c0 ⎡ ⎣( m + Δm ) β c + i mc ⎤⎦

(1.3.3:5)

where (m+Δm)cβ = p is the momentum created in Coulomb field in a space direction. Equating the squares of the scalar values of (1.3.3:4) and (1.3.3:5) results 2 2 2 2 2 ⎡ 2 2⎤ Etotal ( β ) = c0 c ( m + Δm ) = c0 c ⎣( m + Δm ) β + m ⎦ 2

2

(1.3.3:6)

Dividing (1.3.3:6) by c02 c 2 m 2 gives

( m + Δm ) m2

2

( m + Δm ) β 2 + 1 = 2 2

(1.3.3:7)

m

84

Im

Ekin = c0 Δp

pφ ( Re )

Im

pφ ( Re )

pφ ( Im ) φ

Ekin = c0 Δp

p 0( Im )

Re

ψ Re

(a)

(b)

Figure 1.3.3-1. (a) Kinetic energy in free fall by change in the local rest momentum via tilting of space. (b) Kinetic energy by insert of excess mass.

that allows the solution of m+Δm

( m + Δm ) m

2

2

(1 − β ) = 1 2

⇒

m + Δm =

m 1− β 2

(1.3.3:8)

The total energy of motion can now be expressed, Fig. 1.3.3-1 (b)

Etot ( β ) = c0 ptot = c0 ( m + Δm ) c =

c0 mc 1− β 2

(1.3.3:9)

and the kinetic energy ⎛ ⎞ 1 − 1⎟ Ekin = Etotal ( β ) − Erest ( 0) = c0 mc ⎜ ⎜ 1− β 2 ⎟ ⎝ ⎠

(1.3.3:10)

Ekin( β ) = c0 ( m + Δm ) c − c0 mc = c0 cΔm

(1.3.3:11)

or

Substitution of (1.3.3:8) for (m+Δm) in (1.3.3:5) gives the complex presentation of the total energy of motion ^ Etotal ( β ) = c0 mc

β 1− β 2

+ i c0 mc

(1.3.3:12)

Buildup of kinetic energy in acceleration in free fall conserves the total energy in the local gravitational frame because the momentum of free fall is obtained against reduction in the local rest momentum (reduction of the local velocity of light via tilt-

85

ing of space). Buildup of kinetic energy via inertial acceleration by insertion of mass adds the total energy in the acceleration frame. Combining the two mechanisms of kinetic energy we get Ekin(δ , β ) = c0 Δp = c0 ( mΔc + cΔm )

(1.3.3:13)

where Δc and Δm are determined relative to m and c at the state of rest in the local frame. In the theory of relativity the difference between the two mechanisms of kinetic energy buildup is ignored by the postulated constancy of the velocity of light and the equivalence principle assuming identity of gravitational and inertial accelerations. ^ Figure 1.3.3-2 illustrates the share of the complex kinetic energy Ekin ( β ) in the total ^ energy of motion Etot ( β ) . Subtraction of the complex kinetic energy from the complex

total energy gives the internal energy of the moving object ^ ^ ^ Eint ( β ) = Etot ( β ) − Ekin( β )

⎡⎛ β ⎛ β ⎞ ⎞ = c0 mc ⎜ + i ⎟ − c0 mc ⎢⎜ − β ⎟ + i 1− 1− β 2 2 ⎜ ⎟ ⎟ ⎢⎣⎝⎜ 1 − β 2 ⎝ 1− β ⎠ ⎠

(

Im

E "kin ( β )

(1.3.3:14)

^ Ekin (β )

E "rest ( 0)

E "rest ( β )

)

⎤ ⎥ ⎥⎦

^ Etotal (β )

ψ

E 'kin ( β )

^ Eint (β )

E "g ( β )

Re E 'total ( β ) = c0 p

E "g ( 0 )

Figure 1.3.3-2. Complex presentation of total energy, kinetic energy, internal energy, rest energy and the global gravitational energy. Real components of energies are marked with single apostrophe (') and the imaginary components with double apostrophe ("). Complex energies comprising the real ^

and imaginary components are marked with superscript ( ).

86

Performing the subtraction separately for the imaginary and real components results the complex form of internal energy

(

^ ^ ^ 2 Eint ( β ) = Etot ( β ) − Ekin ( β ) = c0 mc β + i 1 − β

)

(1.3.3:15)

which shows that the absolute value of the internal energy is equal to the absolute value of the rest energy of the object at rest, Eint(β) = Erest(0), and the phase angle ϕ = π /2–φ is equal to the phase angle of the total energy. The real component of the internal energy, c0mcβ = c0⋅mv contributes to the momentum in space and the real component of the total energy. The imaginary part of the internal energy serves as the rest energy of the moving object " 2 Erest ( β ) = Eint ( β ) = E "tot ( β ) − E "kin( β ) = c0 mc 1 − β = c0 mrest ( β ) c

(1.3.3:16)

The velocity of space in the imaginary direction is c, accordingly, the reduced momentum and rest energy of the moving object is interpreted as reduced rest mass

mrest ( β ) = m 1 − β 2

(1.3.3:17)

The physical explanation of the reduction of the rest mass due to motion in space is that any motion in space is central motion relative to the barycenter, the mass equivalence in center of the 4-sphere. Reduction of rest mass means also reduction in global gravitational energy of the moving object — the imaginary component of kinetic energy created in accelerating an object in space is the work done in reducing the global gravitational energy, the gravitational energy due to all other mass on the object accelerated — the inertial work.

Inertial work is the reduction of the rest energy due to motion in space — giving a quantitative explanation to Mach’s principle. Substitution of (1.3.2:3) for the local velocity of light in (1.3.3:16) the local rest energy can be expressed as

Erest = c0 mc0δ (1 − δ ) 1 − β 2

(1.3.3:18)

and the zero-energy balance of motion and gravitation of an object moving at velocity β in a local frame at gravitational state δ becomes Erest (δ , β ) = c0 mc0δ (1 − δ ) 1 − β 2 =

GM " m (1 − δ ) 1 − β 2 = E "g (δ , β ) R "0δ

(1.3.3:19)

which conserves the zero-energy balance of the energies of motion and gravitation in apparent homogeneous space

87

E = c0 mc0δ −

GM " m =0 R "0δ

(1.3.3:20)

and through the system of nested energy frames in whole space (Section 1.4).

1.3.4 Motion as central motion in spherical space A physical interpretation of the reduced rest mass of moving objects comes from the fact that any motion in space is central motion relative to mass equivalence M" in the barycenter of spherically closed space. The effect of centrifugal force caused by mass meff of an object moving at velocity β in space reduces the effective global gravitational force as F "δ , β

meff (1 − β 2 ) 2 ⎛ c2 v2 ⎞ ˆ c0 c0 (1 − δ ) ˆiδ = − meff ⎜ − ⎟ iδ = − c R" ⎝ R" R" ⎠

(1.3.4:1)

m 1− β 2 2 c0 (1 − δ ) ˆiδ =− R" which is the force acting against the gradient of the global gravitational energy in the local fourth dimension, i.e. the effective global gravitational force on object moving at velocity β in space. An object at rest in a frame moving at velocity β in its parent frame has the reduced rest mass mβ = m 1 − β 2 , i.e. the global gravitational force gravitation force acting on the reduced mass at rest is equal to the global gravitational force acting on the enhanced (effective) mass meff moving at velocity β ˆi = − GMm (1 − δ ) 1 − β F "δ ,0 ( β ) = − δ dR " R"

dEg (δ )

m 1− β 2 2 c0 (1 − δ ) ˆiδ =− R"

2

R" ˆ iδ

(1.3.4:2)

When an object with mass m at rest moves at velocity β in a local frame, it can equally be regarded as mass m 1 − β 2 at rest in a local sub-frame moving at velocity

β in the local frame or mass m

1 − β 2 moving at velocity β in the local frame (Fig.

1.3.4-1).

88

Im

E "m ( β ) =

m

E "g ( β ) = −

c0 mc 1− β

(1 − β ) 2

2

E "m ( β ) = c0 mc 1 − β 2

β

1− β 2

GM " m 1 − β 2 (1 − β 2 ) R"

m 1− β

E "g ( β ) = −

GM " m 1− β 2 R"

Figure 1.3.4-1 The symmetry of the imaginary energies of motion and gravitation for an object with mass meff ( β ) = m

1 − β 2 moving at velocity β and an object with mass mrest ( β ) = m 1 − β 2 at rest.

1.3.5 Motion in parent frame The rest energy of object m moving at velocity βB in frame B is

Erest , B ( β ) = c0 mB ( 0) c 1 − β B2

(1.3.5:1)

When the whole frame B is in motion at velocity βA in frame A which is the parent frame to frame B, the rest energy of m becomes Erest ( β 2 ) = c0 mA( 0 ) c 1 − β A2 1 − β B2

(1.3.5:2)

where mA(0) is the mass of the object at rest in frame A (Fig. 1.3.5-1). Im

Im

Erest , B ( 0 ) = Erest , A( β A )

Erest , A( 0) Erest , A( β A )

Erest , B ( β B ) Re

Re0

E "g , B ( β B )

E "g , A ( β A )

E "g , B ( 0) = E "g , A( β A )

E "g , A( 0)

Frame B

Frame A

Figure 1.3.5-1 Motion of mass m at velocity βB in the local frame B, which is moving at velocity βA in its parent frame A.

89

1.3.6 Emission of radiation quanta Applying the rest energy of equation (1.3.3:18) to the rest energy of electron me in the standard solution of hydrogen atom, the main energy states are expressed 2

EZ , n =

2

e 4 μ02 ⎛ Z ⎞ α2 ⎛ Z ⎞ c m c = ⎜ ⎟ 0 e ⎜ ⎟ c0 me( 0) c0δ (1 − δ ) 1 − β 8h02 ⎝ n ⎠ 2 ⎝n⎠

(1.3.6:1)

By further applying the intrinsic Planck constant h0 defined in (1.2.2:2) in Balmer’s equation, the emission frequencies of hydrogen-like atoms obtain the form f (δ , β ) =

ΔE( n1,n 2 ) h0 c0

⎡1 1 ⎤ α2 = Z2 ⎢ 2 − 2 ⎥ me( 0) c0δ (1 − δ ) 1 − β 2 n n h 2 2 ⎦ 0 ⎣ 1

= f ( 0δ ,0δ ) (1 − δ ) 1 − β

(1.3.6:2)

2

and the corresponding wavelengths

λ(δ , β ) =

c0δ (1 − δ ) f (δ , β )

= λ( 0δ ,0δ ) ⎡(1 − δ ) 1 − β 2 ⎤ ⎣ ⎦

(1.3.6:3)

In (1.3.6:2) and (1.3.6:3) f(0δ,0δ) and λ(0δ,0δ) are the characteristic frequency and wavelength of a particular electron transition in the emitter at rest in apparent homogeneous space of the local frame. The momentum of a quantum of electromagnetic radiation occurs in space directions — the absolute value of the momentum is equal to the imaginary momentum released by the emitter (Fig. 1.3.6-1).

Im

emitter

Em ( 0 )

Im

Re( y ) Re( y )

m

Re( x )

Re( x )

Eg ( 0)

electromagnetic radiation

mass at rest

Figure 1.3.6 -1. Emission of electromagnetic radiation is described as a turn of momentum from the fourth dimension into space directions. Absorption of electromagnetic energy turns the momentum of radiation back to momentum in the fourth dimension.

90

1.4 The system of nested energy frames 1.4.1 The linkage of local and global The linkage of local and global is a characteristic feature of the Dynamic Universe. There are no independent objects in space — local objects are linked to the rest of space. The Dynamic Universe model is a holistic approach to the universe. The whole in the Dynamic Universe is not composed as the sum of elementary units — the multiplicity of elementary units is a result of diversification of whole. Starting from hypothetical homogeneous space, the structure and the energy balances in space are described as a system of nested energy frames constructed by the subsequent buildup of local systems. In the cosmological scale, local systems are typically gravitational systems formed by accumulation of mass into mass centers. Accumulation of mass occurs in several steps finally forming a multilevel system of nested gravitational frames (Fig. 1.4.1-1). In its simplest form a frame is formed around a point-like mass in the center of the frame via free fall of mass. The rest energy of mass object m in the n:th frame is expressed by applying equation (1.3.3:18) characterizing the state of motion and gravitation of the object in its parent frame and in each subsequent parent frames until hypothetical homogeneous space is reached n

Erest ( n ) = m0 c02 ∏ (1 − δ i ) 1 − β i2 i =1

= Eglobal ( n )

GM " m0 n =− ∏ (1 − δ i ) 1 − βi2 R "0 i =1

Apparent homogeneous space to gravitational frame M 1 M2

(1.4.1:1)

Apparent homogeneous space to gravitational frame M 2

M1 Figure 1.4.1-1. Space in the vicinity of a local frame, as it would be without the mass center, is referred to as apparent homogeneous space to the gravitational frame. Accumulation of mass into mass centers to form local gravitational frames occurs in several steps. Starting from hypothetical homogeneous space, the “first-order” gravitational frames, like M1 in the figure, have hypothetical homogeneous space as the apparent homogeneous space to the frame. In subsequent steps, smaller mass centers may be formed within the tilted space around in the “first order” frames. For those frames, like M2 in the figure, space in the M1 frame, as it would be without the mass center M2, serves as the apparent homogeneous space to frame M2.

91

where δi is the gravitational factor and βi the velocity of the object in the i:th frame. Mass m0 is the mass of the object at rest in hypothetical homogeneous space and c0 is the velocity of light in hypothetical homogeneous space. Each gravitational factor and velocity is

δi =

rc( i ) r0δ (i )

;

rc (i ) =

GM i GM i ≈ ; c0 c0δi r(i ) c 2

βi =

vi ci

(1.4.1:2)

where rc(i) is the critical radius of the i:th gravitational frame. The velocity of light in the i:th frame is subject to reduction in each step in the nested chain of frames due to the tilting of the local space relative to the apparent homogeneous space of the frame n

c = c0 ∏ (1 − δ i )

(1.4.1:3)

i =1

The effect of the velocity of each frame in its parent frame appears as a reduction in the locally available rest mass in the n:th frame n

m = m0 ∏ 1 − β i2

(1.4.1:4)

i =1

Substitution of equations (1.4.1:3) and (1.4.1:4) into equation (1.4.1:1) gives the rest energy of mass m in a local frame in form

Erest = c0 mc

(1.4.1:5)

where c0 is the velocity of light in hypothetical homogeneous space (which is equal to the expansion velocity of space in the direction of the 4-radius R0 of space). Mass m is the locally available rest mass and c is the local velocity of light. When related to the velocity of light in apparent homogeneous space of the local frame the local velocity of light is c = c0δ (1 − δ )

(1.4.1:6)

where c0δ is the velocity of light in apparent homogeneous space, the (n–1)th frame n −1

c0δ = c0 ∏ (1 − δ i )

(1.4.1:7)

i =1

The rest mass of an object moving at velocity β in the local frame can be related to the rest mass of the object at rest in the local frame as

m = m0 β 1 − β 2

(1.4.1:8)

92

where m0β is related to the rest mass of the object at rest in hypothetical homogeneous space m0 as n −1

m0 β = m0 ∏ 1 − β i2

(1.4.1:9)

i =1

The system of cascaded energy frames relates the rest energy and the global gravitational energy of an object moving in a local frame to the rest energy and global gravitational energy the object had at rest in hypothetical homogeneous space. 1.4.2 Earth gravitational frame Mass m at rest on the surface of the Earth is subject to the rotational velocity of the Earth βE,rot and gravitational factor δE determined by the mass and radius of the Earth c = c0δ ( Earth ) (1 − δ E )

m = m0 β ( Earth ) 1 − β E2,rot

(1.4.2:1)

where mass m0β is the rest mass as it would be without the rotation of the Earth (like at North or South Pole). Velocity c0δ(Earth) is the velocity of light in apparent homogeneous space of the Earth which is the velocity of light at Earth’s distance from the Sun in the solar gravitational frame (without the presence of the Earth). The effect of the gravitation of the Earth on the velocity of light on the surface of the Earth is about 20 cm/s. At the altitude of GPS (Global Positioning System) satellites the velocity of light is about 15 cm/s higher than the velocity of light on the Earth. The effect of the Sun on the velocity of light at Earth’s distance from the Sun is about 3 m/s (Fig. 1.4.2-1). Figure 1.4.2-2 illustrates chain of nested energy frames of the Earth out to hypothetical homogeneous space. Velocity βE and gravitational factor δE are the velocity and gravitational factors of the Earth in the solar gravitational frame, βS and δS the velocity and gravitational factors of the solar system in the Milky Way frame, etc. Δc m/s -2.9

Distance from the Earth –400 –

200

400

c0δ Moon

-3.0

Sun 150⋅106 km

-3.1 Earth

Figure 1.4.2-1. Effect of the gravitation of the Sun, Earth, and Moon on the velocity of light. The tilted baseline at the top shows the effect of the Sun on the velocity of light, which is the apparent homogeneous space velocity of light for the Earth, c0δ(Earth). The Moon is shown in its “full Moon” position, opposite to the Sun. The curves in the figure are based on equation (1.4.1:6) as separately applied to the Earth and the Sun. The effect of the gravitation of the Milky Way on the velocity of light in the solar system is about Δc ≈ −300 m/s.

-3.2

93

Hypothetical homogeneous space

Erest ( 0) = c0 mc0 Extragalactic space 2 Erest ( XG ) = Erest ( 0) (1 − δ XG ) 1 − β XG

Milky Way in galaxy group frame 2 Erest ( MW ) = Erest ( XG ) (1 − δ MW ) 1 − β MW

Solar system in MW frame Erest ( S ) = Erest ( MW ) (1 − δ S ) 1 − β S2

Earth in Solar frame

Erest ( E ) = Erest ( S ) (1 − δ E ) 1 − β E2

Accelerator in Earth frame

Erest ( A) = Erest ( E ) 1 − β A2

Accelerated ion 2 Erest ( Ion) = Erest ( A) 1 − βIon

n

Erest ( n) = m0c02 ∏(1 − δi ) 1 − βi2 i =0

Figure 1.4.2-2. The rest energy of an object in a local frame is a function of the velocity and gravitational state of the object in the local frame and the velocity and gravitational state of the local frame in the parent frame. The system of nested energy frames relates the rest energy of an object in a local frame to the rest energy of the object in hypothetical homogeneous space.

94

2. Electromagnetic energy and a quantum of radiation 2.1 Electromagnetic energy

2.1.1 The Coulomb energy For a detailed study of the conservation of mass and energy in zero-energy space, it is necessary to express different forms of energy in a way distinguishing between the contributions of mass or mass equivalence [kg] as the conserved part and the velocity of light and the 4-radius of space as the parts subject to change with the expansion of space. Applying the vacuum permeability μ0 and taking into account the difference between c and c0, the Coulomb energy for N1+N2 unit charges can be expressed in form EC =

q1q2 μ0 e 2 μ0 2 c0 c = N1 N 2 c = mC c0 c 4π r 4π r

(2.1.1:1)

where the quantity mC [kg] is referred to the mass equivalence of Coulomb energy mC ≡ N1 N 2

e 2 μ0 = N1 N 2 mC ( 0 ) 4π r

(2.1.1:2)

The buildup of kinetic energy by acceleration in Coulomb field is expressed as gain in the effective mass against release of mass equivalence of the electromagnetic energy Ek = c0 cΔm = c0 c ( meff − m ) =

q1q2 μ0 ⎛ 1 1 ⎞ ⎜ − ⎟ c0 c = c0 cΔmC 4π ⎝ r1 r2 ⎠

(2.1.1:3)

2.1.2 The quantum of radiation The standard solution of Maxwell’s equations for the power density [W/m2] of electromagnetic radiation emitted by a dipole can be written in form P = =

Π2μ ω 4 dE = ∫ c E ave dS = 0 20 2 ∫ sin 2 θ dS s dt 32π r c s N 2 e2 z02 μ0 ( 2π f )

4

12π c

95

(2.1.2:1)

where Π0 = Nez0 is the dipole moment with N electrons oscillating in a dipole of length z0. By regrouping and applying λ = c/f, equation (2.1.2:1) can be solved for the energy flux in one cycle of radiation as 2

Eλ =

P N 2 e 2 z02 μ016π 4 f 4 ⎛z ⎞ = = N 2 ⎜ 0 ⎟ A ⋅ 2π 3e 2 μ0 c ⋅ f 12π cf f ⎝λ⎠

(2.1.2:2)

where A is the radiation geometry factor. For a dipole in space A = 2/3, which relates the average power density to the power density on the normal plane of the dipole. Spherically closed zero-energy space is moving at velocity c in the fourth dimension, which means that a point source at rest in space can be regarded as onewavelength dipole in the fourth dimension with all space directions perpendicular to the dipole. By inserting the radiation geometry factor A = 1.1049 the energy emitted by a point source in one cycle, as one-wavelength dipole in the fourth dimension (z0=λ), is Eλ = N 2 (1.1049 ⋅ 2π 3e 2 μ0 c ) f = N 2 ⋅ hf = N 2

h0

λ

c2

(2.1.2:3)

where h is the Planck constant h = 1.1049 ⋅ 2π 3e 2 μ0 c = 6.6261 ⋅ 10−35 [Js]

(2.1.2:4)

and h0 is defined as the intrinsic Planck constant h0 ≡

h = 1.1049 ⋅ 2π 3e 2 μ0 = 2.210 ⋅ 10−42 [kg m] c

(2.1.2:5)

The intrinsic Planck constant expresses the energy of quantum as the energy emitted by a dipole per a unit charge (N =1) in a cycle Eλ ( N =1) =

h0

λ

c0 c = = 0 k ⋅ c0 c = mλ ( 0) c0 c

(2.1.2:6)

where k = 2π/λ and = 0 = h0 2π . Quantity mλ (0) [kg] in (2.1.2:6) is referred to as the mass equivalence of a quantum of radiation. An important message of equations (2.1.2:2–6) is that a quantum of radiation can be expressed in terms of the energy carried by one cycle of radiation. Another important message of equation (2.1.2:4) is that the velocity of light c is included as a hidden parameter in Planck’s constant h. Applying equation (2.1.2:4) the fine structure constant α obtains the form

α≡

e 2 μ0 e 2 μ0 c e2 1 1 = = = ≈ 3 2 3 2hε 0 c 2h0 2 ⋅ 1.1049 ⋅ 2π e μ0 c 2 ⋅ 1.1049 ⋅ 2π 137.035

96

(2.1.2:7)

illustrating the very basic nature of the fine structure constant as a purely numerical or geometrical factor independent of any physical constant or the velocity light, which is not constant in DU space. Applying (2.1.2:7) in (2.1.1:1) gives the Coulomb energy in terms of the fine structure constant and the intrinsic Planck constant EC = N1 N 2

e 2 μ0 h c0 c = N1 N 2α 0 c0 c = c0 mC c 4π r 2π r

(2.1.2:8)

where the mass equivalence of Coulomb energy is mC = N1 N 2α

h0 2π r

(2.1.2:9)

The physical message of equation (2.1.2:6) is that a quantum of radiation can be described as the nominal energy pumped into one cycle of radiation by a single transition of a unit charge in a unit dipole. Equation (2.1.2:6) can be generalized to the energy of a cycle of electromagnetic radiation from any electric dipole by inserting the intrinsic Planck constant back to equation (2.1.2:2) 2

Eλ

h P ⎛z ⎞ h = = N 2 A ⎜ 0 ⎟ 0 c0 c = B ⋅ 0 c0 c = mλ c0 c f λ ⎝λ⎠ λ

(2.1.2:10)

where constant B is determined by the length and the radiation geometry of the dipole, and the number of unit charges oscillating in the dipole. The difference between c0 and c has been added to (2.1.2:10). Based on the current knowledge of the gravitational environment of the Earth and the solar system, the velocity of light c on the Earth is of the order of one ppm (part per million) lower than the velocity of light c0 in hypothetical homogeneous space. At cosmological distances the velocity of light is approximated as c ≈ c0. Electromagnetic radiation carries energy in the direction of propagation in space only which means that also the mass equivalence of electromagnetic radiation is manifested in the direction of propagation only. The wavelength of electromagnetic radiation propagating in expanding space is subject to lengthening in direct proportion to the expansion. Conservation of the energy of a quantum of radiation, or the energy carried by a cycle of radiation in relation to the total energy in space requires that the mass equivalence of radiation, mλ = h0/λe, created at the mission is conserved in the course of the propagation of radiation in expanding space Eλ = mλ c02

(2.1.2:11)

When the radiation is received, the power density observed is reduced due to the increase of the wavelength and the cycle time with the expansion of space.

97

2.1.3 Mass wave The concept of mass equivalence of the wavelength can be applied in a reversed form as the wavelength equivalence of mass, i.e. mass can be presented as wave-like substance propagating at the velocity of light in space or with space in the fourth dimension. In the complex form the total energy of motion of mass m moving at velocity β in a local frame is expressed ^ Etotal = c0

mβ 1− β

2

c + i c0 mc = c0

m 1− β 2

c

(2.1.3:1)

which is rewritten in form (Fig. 2.1.3-1) ^ Etotal = c0 c= 0 kRe( β ) + i c0 c= 0 kIm( 0) = c0 c= 0 kφ ( β )

(2.1.3:2)

where = 0 = h0 2π and k = 2π /λ

= 0 kRe( β ) =

mβ 1− β

2

, kIm( 0) = 0 = m , and

kφ ( β ) = 0 =

m 1− β 2

(2.1.3:3)

or applyingψ = arc(sinβ)

= 0 kRe( β ) = m tanψ ,

kIm( 0) = 0 = m , and

kφ ( β ) = 0 = m cosψ

(2.1.3:4)

Dividing by c0 c= 0 , equation (2.1.3:2) reduces into

kRe( β ) + i kIm( 0 ) = kφ ( β )

(2.1.3:5)

or in squared form, to the wave number equivalence of the energy four vector 2 2 kφ2( β ) = kIm ( 0 ) + kRe( β )

(2.1.3:6)

The wavelength corresponding to wave number kRe(β) in a space direction in (2.1.3:5) is equal to the deBroglie wavelength

λRe( β ) =

2π kRe( β )

= λde Broglie =

h0 1 − β 2 β m

(2.1.3:7)

In equation (2.1.3:5) the wave number in the imaginary direction is the kIm(0), corresponding to the rest energy of mass m at rest in the local frame. The wave number corresponding to the rest energy of mass m moving at velocity β in the local frame is

98

Im

ERe = c0 p Re( β ) = c0 c= 0 ⋅ kRe ( β ) EIm ( 0 ) = c0 p Im ( 0 ) = c0 c= 0 ⋅ kIm ( 0)

EIm ( β ) = c0 p Im ( β ) = c0 c= 0 ⋅ kIm ( β )

EIm g ( β ) = −

EIm g ( β ) = −

Etotal = c0 pφ ( β ) = c0 c= 0 ⋅ kφ ( β ) ψ Re

GM " = 0 ⋅ kIm ( β ) R"

GM " = 0 ⋅ kIm ( 0) R"

Figure 2.1.3-1. Complex plane presentation of the energy four-vector in terms of mass waves given in equation (2.1.3:2).

kIm( β ) = kIm( 0) 1 − β 2

(2.1.3:8)

and the corresponding wavelength is equal to the Compton wavelength

λIm( β ) =

2π kIm( β )

= λCompton =

h0

(2.1.3:9)

m 1− β 2

2.2 Electromagnetic objects

2.2.1 Hydrogen-like atoms

Insertion of a boundary condition n⋅λRe(β) = L (L = the length of the orbit) for an electron in a circular orbit in a Coulomb frame (2.1.2:8) with Z unit charges at the center we obtain the kinetic energy of the electron in form EZ , n

⎛ = c0 me c ⎜ 1 ⎜ ⎝

2 ⎞ α 2 ⎛ Z ⎞2 ⎛ Zα ⎞ 1− ⎜ ⎟ − 1⎟⎟ ≈ ⎜ ⎟ E 2 ⎝ n ⎠ rest ( e ) ⎝ n ⎠ ⎠

99

(2.2.1:1)

which is equivalent to the standard non-relativistic solution of hydrogen-like atoms. Substitution of the effects of gravitation and motion (1.4.1:1) for Erest(e) in equation (2.2.1:1) gives EZ , n =

α2 ⎛ Z ⎞

2

n

2 2 ⎜ ⎟ m0 c0 ∏ (1 − δ i ) 1 − βi 2 ⎝n⎠ i =1

(2.2.1:2)

Balmer’s equation for characteristic emission and absorption frequencies solved from (2.2.1:2) becomes f( n1,n 2) =

ΔE( n1,n 2) h0 c

= f 0( n1,n 2)

n

∏ (1 − δ ) i =1

i

1 − βi2

(2.2.1:3)

which shows the effect of motion and gravitation on the frequency. For clocks on the Earth, frame i = n is the Earth gravitational frame, i = n–1 is the solar gravitational frame, i = n–2 is the Milky Way gravitational frame, etc. In the Earth gravitational frame velocity βn of a stationary clock is the rotational velocity of the Earth, velocity βn–1 is the orbital velocity of the Earth in the solar frame, βn–2 in the Milky Way frame, etc. Substitution of (1.1.3:2) for c0 shows the development of frequency as a function of time since singularity 2 1/ 3 ⎡1 1 ⎤ α me( 0) ⎛ 2 ⎞ −1 3 " f 0( n1,n 2) = Z 2 ⎢ 2 − 2 ⎥ GM ⎜ ⎟ t ⎠ ⎣ n1 n2 ⎦ 2h0 ⎝ 3

(2.2.1:4)

The characteristic wavelength corresponding to frequency (2.2.1:3) is

λ( n1,n 2 ) =

c f ( n1, n 2)

=

λ0( n1,n 2) n

∏ i =1

(2.2.1:5)

1 − βi2

Applying the standard solution for the Bohr radius and equation (1.4.1:4) for the rest mass, the radius of the hydrogen atom can be expressed as a0 =

h02 = πμ0 e2 me( n )

a0( 0) n

∏ i =1

1− β

(2.2.1:6) 2 i

The emission wavelength λ(n1,n2) in equation (2.2.1:5) can be expressed in terms of the Bohr radius a0(0) as

100

λ( n1,n 2 ) =

4π a0( 0) n

α Z ⎡⎣1 n − 1 n ⎤⎦ ∏ 1 − β 2

2 1

2 2

i =1

= 2 i

4π a0

α Z ⎡⎣1 n12 − 1 n22 ⎤⎦ 2

(2.2.1:7)

which shows that the wavelength emitted is directly proportional to the Bohr radius of the atom. Both the characteristic emission wavelength and the Bohr radius are conserved in the course of the expansion of space. In fact, equation (2.2.1:7) is just another form of Balmer’s formula, which does not require any assumptions tied to DU space. Equation (2.2.1:7) also means that, like the dimensions of an atom, the characteristic emission and absorption wavelengths of an atom are unchanged in the course of the expansion of space but increase with the velocity of the atom. 2.2.2 Electromagnetic resonator as an energy object An electromagnetic plane resonator is a closed energy system (energy frame, or energy object) characterized as a system with plane wave emitters or reflectors at each end. It can be shown that in a closed system the mass equivalence of electromagnetic radiation behaves just like the mass of “conventional” mass objects. When a resonator is put into motion in its parent frame in space the mass equivalence of the standing wave in the resonator shows an increased effective mass equivalence relative to the parent frame and reduced rest mass equivalence relative to the state of rest in the resonator frame. A resonator creates a closed energy object by capturing the radiation of two opposite plane waves between the reflectors at the opposite ends of the resonator cavity. As taught by classical wave mechanics, a resonant superposition of waves in opposite directions produces a standing wave A = 2 A0 sin 2π

r

λ

cos 2π f t = 2 A0 sin kr cos ω t

(2.2.2:1)

with nodes at r = n⋅λ/2. The momenta in a resonator have a zero vector sum but a nonzero scalar sum ptot = ½p ( + ) + ½p ( − ) = 0 ;

½p( + ) + ½p( − ) = ptot = ptot

101

(2.2.2:2)

where ptot = prest (EM) is the rest momentum, the scalar sum of the momenta of the waves in opposite directions. The momenta of the opposite waves are p( + ) =

h0

λ0

p( −) = −

c rˆ ;

h0

λ0

c rˆ

(2.2.2:3)

When a resonator frame moves at velocity β in its parent frame the mass equivalences of the opposite waves are reduced as expressed in equation (2.2.1:5). The wavelength measured in the resonator frame for waves in both directions is

λβ ,int =

λ0

(2.2.2:4)

1− β 2

The wavelengths measured in the parent frame are subject to Doppler shift. The wavelength sent by an endplate against velocity β in the parent frame is reduced

λ( + β ) =

λ0 1− β 2

(1 − β ) = λI (1 − β )

(2.2.2:5)

and increased in the opposite direction

λ( − β ) =

λ0 1− β 2

(1 + β ) = λI (1 + β )

(2.2.2:6)

The sum of the momentums of the Doppler shifted waves in the parent frame now becomes

⎛ 1 h 1− β 2 1 h 1− β 2 ⎞ 0 ⎟c ptot ( β ) = ½p( + β ) + ½p( − β ) = ⎜ 0 − ⎜ 2 λ0 (1 − β ) 2 λ0 (1 + β ) ⎟ ⎝ ⎠

(2.2.2:7)

Multiplication of the nominators and denominators of the terms in parenthesis in equation (2.2.2:7) by the factor 1 − β 2 gives

ptot ( β ) =

h0

λ0 1 − β

2

⎡⎣½ (1 + β ) − ½ (1 − β ) ⎤⎦ c =

h0

λ0 1 − β 2

βc

(2.2.2:8)

or by applying the mass equivalence of electromagnetic radiation as

ptot ( β ) =

mλ ( 0) 1− β 2

β c = mλ ( eff ) v

(2.2.2:9)

102

Im

β

p = c0c

h0

λ

1− β 2

p = c0c p = ± ½⋅c

h0

λ

h0

λ

1 − β 2 ˆi

Re

1 − β 2 rˆ

(a)

(b)

Figure 2.2.2-1. An electromagnetic resonator can be studied as an energy object or closed energy system with rest mass equal to the sum of the mass equivalences of the waves in opposite directions.

Equations (2.2.2:8) and (2.2.2:9) show that motion of a resonator, as a closed electromagnetic energy object in its parent frame, creates momentum through the increase of the “effective mass equivalence” exactly in the same way as does any mass object. As a part of the balance, the internal momentum in the resonator, the momentum in the resonator frame, is reduced due to the reduced rest mass equivalence of the radiation (see Fig. 2.2.2-1). The zero momentum condition in the resonator is ⎛ h ⎞ h p β ,int = ⎜ ½ 0 1 − β 2 − ½ 0 1 − β 2 ⎟ c = 0 λ0 ⎝ λ0 ⎠

(2.2.2:10)

In the resonator frame the reference at rest is the resonator body which also means reference at rest to the velocity of light measured in the resonator frame. The frequencies and wavelengths of the waves in both directions in the resonator frame are the internal frequency and wavelength f β ,int =

c

λβ ,int

=

c

λ0

1− β 2

(2.2.2:11)

where β is the velocity of the resonator frame in its parent frame. The analysis of electromagnetic resonator is of special importance for understanding the early experiments on the velocity of light using Michelson–Morley interferometers. Michelson–Morley interferometer in an Earth laboratory is essentially a resonator moving in its parent frames (due to the rotational and orbital velocities of the Earth, the solar system etc.). The measured quantity in the M–M experiment is the difference in the internal wavelengths in different arms of the interferometer. As given in equation (2.2.2:4) the

1

0

3

internal wavelength in a closed energy system is affected by the square root term 1 − β 2 of the velocity of the resonator frame in its parent frame. The square root term is function of the square of the velocity, which ignores the effect of the direction of the velocity relative to the direction of the waves in the resonator. Such a situation guarantees a zero result in the M-M experiment.

3. Properties of local space 3.1 Celestial mechanics in local gravitational frame 3.1.1 Cylinder coordinate system The gravitational frame around a local mass center in the DU framework corresponds to Schwarzschild space in the GR framework. Due to the metric nature of the fourth dimension, the gravitational frame in DU space has a precise geometrical meaning both in the space directions and in the fourth dimension. Notations used in describing a local gravitational frame are summarized in Figure 3.1.1-1.

Im0δ,

Reδ

Imδ,

φ r0δ

dϕ0δ

rδ r

dsϕ

dsr

rc Figure 3.1.1-1. DU line elements dsr = dr and dsϕ = r0δ dϕ. Distance r0δ is the “flat space distance”, the distance measured in the direction of apparent homogeneous space of the local gravitational frame, which has the direction of the normal plane of the Im0δ -axis.

104

The critical radius rc (1.4.1:2) of a gravitational frame in the DU is half of the critical radius in Schwarzschild black hole rc =

GM c0 c0δ

⎛ GM 1 ⎞ ⎜ ≈ 2 = rc( Schwd ) ⎟ c 2 ⎝ ⎠

(3.1.1:1)

Velocity c0 is the velocity of light in hypothetical homogeneous space and c0δ the velocity of light in apparent homogeneous space of the local gravitational frame. Figure 3.1.1-2 illustrates the cylinder coordinate system applied in celestial mechanics in the DU framework The true geometrical nature of the DU gravitational frame allows the derivation of orbital equations by first deriving the projection of an orbit on the flat space plane, the base plane of the cylinder coordinate system, and then calculating the “depth”, the z-coordinate of the orbit in the fourth dimension as the function of the radius on the base plane. The projection of an orbit on the flat space plane can be solved in closed mathematical form following the procedure used in the derivation of Kepler’s equations. The flat space component of radial acceleration in DU space obtains the form y0δ

m

ϕ x0δ

M

z0δ (Im0δ ) r(2)0δ

x0δ r(1)0δ

orbital plane M

Figure 3.1.1-2. Projections of an elliptic orbit on the x0δ−y0δ and x0δ−z0δ planes in a gravitational frame around mass center M.

105

3

a ( 0δ ) = −

c0δ GM ⎛ r ⎞ 1 − c ⎟ rˆ0δ 2 ⎜ c0 r0δ ⎝ r0δ ⎠

(3.1.1:2)

where the minus sign means the direction towards the local barycenter. The z-coordinate of an orbit can be calculated separately as a function of r0δ z ( r0δ ) = 2 2rc ⎡ r0δ − a0δ (1 − e02δ ) ⎤ ⎢⎣ ⎥⎦

(3.1.1:3)

which gives the z-coordinate as the distance from the base plane (in the flat space direction) intersecting the orbiting surface at ϕ = ±π /2. Expression a0δ (1−e0δ2) in equation (3.1.1:3) is the value of r0δ at ϕ0δ = ±π /2, which is used as the reference value for the z-coordinate. When δ << 1, the depth of a dent in the local gravitational frame is r 02

ΔR "0δ ≈

∫

r 01

⎛ r 2rc r ⎞ dr0δ = 2 2 rc ⎜ 02 − 01 ⎟ ⎜ r r0δ rc ⎟⎠ ⎝ c

(3.1.1:4)

where rc = GM/c0c0δ is the critical radius as defined in equation (1.4.1:2). Equation (3.1.1:4) applies for r0δ >> rc, which is the case for “ordinary” mass centers in space. For example, the critical radius for the mass of the Earth, Me ≈ 6 ⋅1024 [kg], is rc(Earth) ≈ 4.5 mm and the critical radius of the Sun rc(Sun) ≈ 1.5 km. Figure 3.1.1-3 illustrates the actual dimensions of the local curvature of space in the solar system. The calculation is based on equation (3.1.1:4). As can be seen, the Sun dips about 26,000 km further into the fourth dimension than does the Earth, which is about 150,000 km “deeper” than the planet Pluto. 200 ΔR” ×1000 km Pluto

150

Neptune Uranus

100 Saturn Jupiter Mars

50

Earth

Venus Mercury

Sun 0

1

2

3 4 5 6 7 Distance from the Sun (×109 km)

106

Figure 3.1.1-3. Topography of the solar System in the fourth dimension. Observe the different scales in the vertical and horizontal axis.

3.1.2 Orbital velocity and the velocity of free fall The velocity of free fall in the DU space is v ff ( 0), DU c0δ

2

⎛ r ⎞ = 1 ⎜1 − c ⎟ − 1 r⎠ ⎝

(3.1.2:1)

At high values of r (r >> rc), (1.3.2:1) can be approximated v ff ( 0), DU c0δ

( r >> rc ) =

2 ⎛ 2rc r r ⎛r ⎞ ⎞ 1 ⎜1 − 2 c + ⎜ c ⎟ ⎟ − 1 ≈ 1 + 2 c − 1 ≈ ⎜ ⎟ r ⎝r⎠ ⎠ r r ⎝

(3.1.2:2)

Orbital velocity at circular orbit in DU space is vorb( DU ) c0δ

rc ⎛ rc ⎞ 1− ⎟ r ⎜⎝ r⎠

= δ (1 − δ ) = 3

3

(3.1.2:3)

which means orbits are stable down to the critical radius (Fig. 3.1.2-1 (a)). Slow orbits at radii rc < r < 2⋅rc are essential for capturing and maintaining the central mass of a singularity, a black hole, in DU space. In Schwarzschild space the solution of the flat space velocity of free fall (coordinate distance/coordinate time) is given in [7] as v ff ( r0 ), Schwd c

=

2rc r

⎛ 2rc ⎜1 − r ⎝

⎞ ⎛ 2rc ⎞ ⎟ = β Newton ⎜ 1 − r ⎟ ⎠ ⎝ ⎠

(3.1.2:4)

Both the Schwarzschild solution and the DU solution approach the Newtonian velocity at high values of r. The critical radius in Schwarzschild space is twice the critical radius in DU space, rc(Schzd) = 2⋅ rc(DU). The orbital (coordinate) velocity at circular orbit in Schwarzschild space [7] is vorb( r0 ), Schwd c

=

1 − 2δ 1 δ −3

=

1 − 2rc r

(3.1.2:5)

r rc − 3

Comparison of equations (3.1.2:4) and (3.1.2:5) shows that the orbital velocity in Schwarzschild space exceeds the velocity of free fall at r = 3⋅rc(Schwd) (Fig. 3.1.2-1 (b)). As a consequence, stable orbits in Schwarzschild space are possible only for orbital radii larger than r > 3 ⋅ rc ( Schwd ) = 6 ⋅ rc ( DU ) .

107

1

1

β 0δ

0.5

stable orbits

β 0δ

β ff ( Newton )

0.5

β ff , DU

β ff ( Newton ) β ff ( Schwarzschild )

β orb, DU

0

β orb( Schwarzschild )

0 0

10

20 (a)

30

r rc ( DU )

0

40

10

20

30

r rc ( DU )

40

(b)

Figure 3.1.2-1. The velocity of free fall and the orbital velocity at circular orbits: (a) in DU space, (b) in Schwarzschild space. The velocity of free fall in Newtonian space is given as a reference.

In DU space the velocity of free fall reaches the local velocity of light when the tilting angle φ reaches 45°, which happens at radius r ≈ 3.414⋅rc. We may assume that reaching the local velocity of light in space could lead to conversion of matter into electromagnetic radiation and further on into elementary particles. Such processes could also produce mass objects with lower velocity to be captured into to the slow orbits at radii rc < r < 2⋅rc. In binary pulsars, the mass of the emitting neutron stars is typically about 1.5 times the mass of the Sun. The critical radius of such mass center is about rc ≈ 2.3 km, which means that the radius at which the velocity of free fall reaches the local velocity of light, the possible matter to radiation conversion radius, is about 3.414 ⋅rc ≈ 8 km, which is roughly the estimated radius of typical neutron stars — suggesting that the interpretation of a neutron star is that of a local singularity. 3.1.3 Orbital period in the vicinity of local singularity Orbital period for circular orbits in DU space is P=

−3 2 2π rc ⎡⎣δ (1 − δ ) ⎤⎦ c0δ

(3.1.3:1)

The period has minimum at radius r0δ = 2rc (Fig. 3.1.3-1) Pmin =

16π rc 16π GM = c c3

(3.1.3:2)

108

40

Porb 2π rc c 0 δ

30

20

Period in DU space

Period in Schwarzschild space

10

0

0

2

4

6

8 r0δ /rc 10

Figure 3.1.3-1. Orbital period for circular obits with radius r0δ close to the critical radius rc.

In Schwarzschild space the shortest period, the period at minimum stable orbit, r = 6⋅rc is Pmin =

12π rc c 16

≈

29.4π GM c3

(3.1.3:3)

The black hole at the center of the Milky Way, compact radio source Sgr A*, has the estimated mass of about 3.6 times the solar mass which means Mblack hole ≈ 7.2⋅1036 kg. When substituted for M in (3.1.3:2) the prediction for the minimum period in a circular orbit around Sgr A* in DU space is about 14.8 min, which is in line with the observed minimum periodicity, 16.8 ± 2 min [8]. 3.1.4 Perihelion advance Elliptic orbits solved from (3.1.1:2) are subject to perihelion advance, which is obtained in a closed mathematical form. For a full revolution the advance is Δψ 0δ ( 2π ) =

6π G ( M + m )

(3.1.4:1)

c 2 a (1 − e 2 )

which is the same result as derived from the general theory of relativity.

109

Im0δ ra

Ra

rp Rp

Figure 1.3.5-1. The orbital radius r of a local rotational system increases when the local system comes closer to the central mass of its parent frame, i.e. R decreases.

3.1.5 Sub-frame in a gravitational frame As a consequence of the overall energy balance in space, the orbital radii of local subsystems increase when the distance to the central mass of the parent frame decreases (see Figure 1.3.5-1). For example, there is about 13 cm annual variation in the Earth to Moon distance due to the eccentricity of the Earth’s orbit. In the Lunar Laser Raging measurement based on two-way light transmission time the variation is not observable due to simultaneous variations in the velocity of light and Earth clock frequencies due to the changing gravitational state and orbital velocity of the Earth in the solar gravitational frame. 3.1.6 The frequency of atomic oscillators The proper time frequency in Schwarzschild space is

fδ , β ( GR ) = f 0,0 1 − 2δ − β 2 1 1 1 1 ⎛ ⎞ ≈ f 0,0 ⎜1 − δ − β 2 − β 4 − δβ 2 − δ 2 ⎟ 2 8 2 2 ⎝ ⎠

(3.1.6:1)

In a local gravitational frame the corresponding equation in DU space (2.2.1:3) becomes

110

1 fδ,β/f0,0 0,8

Figure 3.1.6-1. The difference in the DU and GR predictions of the frequency of atomic oscillators at extreme conditions when δ = β 2 → 1. Such condition may appear close to a black hole in space. The GR and DU predictions in the figure are based on equations (3.1.6:1) and (3.1.6:2), respectively.

0.6 0.4 GR

0.2 0

0

0.2

DU

0.4

0.6

0.8

1

β2 = δ

fδ , β ( DU ) = f 0,0 (1 − δ ) 1 − β 2 1 1 1 ⎛ ⎞ ≈ f 0,0 ⎜1 − δ − β 2 − β 4 + δβ 2 ⎟ 2 8 2 ⎝ ⎠

(3.1.6:2)

The difference between the GR and DU frequencies in equations (3.1.6:1) and (3.1.6:2) is Δfδ , β ( DU −GR ) ≈ δβ 2 + ½δ 2

(3.1.6:3)

In clocks on Earth and in low orbit Earth satellites the difference between the DU and Schwarzschild predictions is of the order Δf/f ≈ 10–18 which is too small a difference to be detected with present clocks. The difference, however, is essential in extreme conditions where δ and β approach unity (Fig. 3.1.6-1). 3.2 Propagation of light

3.2.1 Shapiro delay The increase of the light propagation time in the vicinity of mass centers is referred to as Shapiro delay. The propagation time of light and a radio signal between points A and B in a local gravitational frame is affected by both the lower velocity of light and the increased distance in the vicinity of the local mass center. In a general form the propagation time can be expressed B

B

A

A

t A, B = ∫ dt = ∫

B dx ( r0δ ) dx =∫ A c c ( r0δ )

(3.2.1:1)

111

Im0δ r0δΒ

X(A–B)0δ

B

r0δΑ

AB A

M

Figure 3.2.1-1. Light path AB from location A to location B follows the shape of the dent in space as a geodesic line in the gravitational frame of mass center M. Point A is at flat space distance r0δA and point B is at flat space distance r0δB from mass center M. Point AB is the flat space projection of point A on the flat space plane crossing point B. Line ABB is the distance between A and B as it would be without the dent. The velocity of light in the dent is reduced in proportion to 1/r0δ, i.e. the velocity of light at A is higher than the velocity of light at B. Distance ABA is the projection of path AB on the flat space plane.

For calculating the effect of the curvature of space on the transmission time, it is useful to divide the expression of the propagation time into two parts, where the first part, X(A–B)0δ /c0δ shows the transmission time as it would be without curvature and the second part shows the effect of the curvature (Fig. 3.2.1-1) B

dx0δ B ⎡ d ( dx0δ ) dc ⎤ dx0δ X ( A− B )0δ + ∫⎢ − = + Δt( A− B ) ⎥ c0δ ⎦ c0δ c0δ A c0δ A ⎣ dx0δ

B

∫ dt = ∫ A

(3.2.1:2)

where Δt(A–B) is the Shapiro delay B ⎡ d ( dx0δ ) ⎤ dx GM Δt( A− B ) = ∫ ⎢ + δ ⎥ 0δ = 3 dx0δ c ⎦ c0δ A⎣

B

∫ ( sin A

2

α + 1)

dα cos α

(3.2.1:3)

The meaning of α is illustrated in Figure 3.2.1-2. The two factors of the Shapiro delay are the effect of the increase in the propagation distance and the effect of the reduction in the velocity of light. The effect of the reduction of the velocity of light is obtained from equation (1.4.1:6) as dc/c0δ = –δ . Integration of (3.2.1:3) and conversion to an algebraic expression gives the Shapiro delay in form Δt( A− B ) =

GM c3

⎧⎪ ⎡ xB + rB ⎤ ⎡ xB xA ⎤ ⎫⎪ ⎨2ln ⎢ ⎥ − ⎢ − ⎥⎬ ⎣ x A + rA ⎦ ⎣ rB rA ⎦ ⎭⎪ ⎩⎪

112

(3.2.1:4)

B,xB

dx

α

βB

(a)

α r

dx x A,xA

dα

βA rA

α M

d

dr

(b)

rB

x=0

Figure 3.2.1-2. (a) Notation of distances in the analysis of the signal delay between points A and B in a gravitational frame around mass M. Distances xA and xB are the distances from A and B to point x = 0, which is the shortest distance between line AB and mass center M. (b) Distance differential dx in the direction of a light beam at an angle α to the tangential direction.

The meaning of distances xA, xB, rA, and rB is shown in Fig. 3.2.1-2(a). The last term in parenthesis in (3.2.1:4) comes from the tangential component in the propagation path, which is not subject to lengthening. In the case of light propagation in the radial direction xA = rA and xA = rA and the last term in (3.2.1:4) is zero. The corresponding solution of the general relativity ignores the effect of the tangential component, i.e. the Shapiro delay is given in form Δt( A− B ) B ,GR =

2GM ⎡ xB + rB ⎤ ln ⎢ ⎥ c3 ⎣ xA + rA ⎦

(3.2.1:5)

When xA = rA and xA = rA both the DU and the GR solution reduces to Δt( A− B ) B ,r =

2GM rB ln c3 rA

(3.2.1:6)

When the passing distance is d, the total signal delay between objects at distances D1 and D2 (D1,D2 >> d ) can be expressed (Fig. 3.2.1-3) as Δt D1, D 2 =

2GM ⎧ ⎡ 4 D1 D2 ⎤ ⎫ − 1⎬ ⎨ln c3 ⎩ ⎢⎣ d 2 ⎥⎦ ⎭

(3.2.1:7)

The GR prediction corresponding to (3.2.1:7) is

113

A

D1

D2 M

B

d

Figure 3.2.1-3. Distances D1 and D2 in equation (3.2.1:7) for calculation of the delay of a signal traveling from A to B. The signal passes mass center M at distance d.

Δt D1, D 2( GR ) =

2GM 4 D1 D2 ln c3 d2

(3.2.1:8)

which, again, ignores the effect of the tangential component of the propagation path. Equation (3.2.1:7) and (3.2.1:8) are applicable in cases like the experiments with Mariner 6 and 7 space crafts. In those experiments the observed quantity is the difference in the delay as a function of passing distance d, which means that the two equations work equally well in the interpretation of the observations. 3.2.2 Bending of light path near a mass center The derivative of the delay in light transmission relative to the shortest distance from mass center M in equation (3.2.1:7) is ∂ ( Δt ) ∂d

=−

4GM c3 d

(3.2.2:1)

which gives the difference in the propagation delay versus a difference in the shortest distance d to mass center M a light ray is passing (Fig. 3.2.2-1).

∂D

φ φ

∂d d M

Figure 3.2.2-1. Light ray passing a mass center in space is bent due to a reduced velocity and increased distance close to a mass center.

114

Extra distance the outer side of the ray travels in time differential Δt is ∂D = c ⋅ ∂ ( Δt ) which can be expressed as the arc ∂D ∂D = φ ⋅ ∂d = c ⋅ ∂ ( Δt ) ⇒

φ=

∂ ( Δt ) ∂d

⋅c

(3.2.2:2)

Substitution of (3.2.2:1) into (3.2.2:2) gives the bending angle φ towards the mass center

φ=

4GM c2d

(3.2.2:3)

The result is the same as the corresponding prediction derived from the general theory of relativity. 3.2.3 Gravitational shift of electromagnetic radiation The frequency of an atomic clock at rest in a local gravitational frame at state δA in DU space is given in equation (2.2.1:3) f A = f 0δ (1 − δ A )

(3.2.3:1)

where f0δ is the frequency of the clock in the apparent homogeneous space of the local frame. At δA the velocity of light is cA = c0δ (1–δA ). The wavelength of radiation sent from A at frequency fA is

λA =

c A c0δ (1 − δ A ) c0δ = = f A f 0δ (1 − δ A ) f 0δ

(3.2.3:2)

A similar clock at rest in the same frame at state δB runs at frequency f B = f 0δ (1 − δ B )

(3.2.3:3)

The wavelength of radiation sent from B at frequency fB is

λB =

cB c0δ (1 − δ B ) c0δ = = = λA f B f 0δ (1 − δ B ) f 0δ

(3.2.3:4)

The wavelength observed at B in the signal sent from A is

λ A( B ) =

cB cB f f = λ A = B λ A = B λB f A cA fA fA

(3.2.3:5)

115

cA

A fA = f0(1−δA)

λrec= fB /fA λB cB

B frec = fA

λB =cB /fB

fB = f0(1−δB)

Figure 3.2.3-1. The velocity of light is lower close to a mass center, cΒ < cΑ, which results in a decrease of the wavelength of electromagnetic radiation transmitted from A to B. Accordingly, the signal received at B is blueshifted relative to the reference wavelength observed in radiation emitted by a similar object in the δB-state. The frequency of the radiation is unchanged during the transmission.

I.e., compared to the wavelength of identical emitter in B the wavelength observed in the signal sent from A is shortened by factor fB /fA which is the gravitational blueshift (Fig. 3.2.3-1). Propagation of electromagnetic radiation from gravitational potential at A to gravitational potential at B is not associated with a frequency shift. However, the wavelength of the radiation is shifted due to the different velocity of light at A and B. The frequencies of identical oscillators at different gravitational states are different but the wavelengths that they emit are equal. In the DU framework there is a clear distinction between the gravitational effects on the frequency and wavelength of atomic oscillators and the gravitational effects on the frequency and wavelength of electromagnetic radiation. 3.2.4 Doppler effect and transmission time In a local gravitational frame the Doppler effect of electromagnetic radiation can be derived following the classical procedure. The characteristic frequencies of the emitter A and a reference oscillator B at the receiver are affected by the motion and gravitational state of the emitter and receiver as given in equation (2.2.1:3) f A = f 0δ (1 − δ A ) 1 − β A2

(3.2.4:1)

116

L = λ0 = Tc

vA

fA vA⋅ rˆ

cr

v

vB

fB vB⋅ rˆ

ΔL = Δλ = Tv = λ0 v/c

(a)

rˆ

(b)

Figure 3.2.4-1. (a) The wavelength of electromagnetic radiation emitted by a moving source is shortened in the direction of the motion by the distance moved by the source during the cycle time, Δλ = λ0 v/c. (b) The Doppler effect combines the effects of the velocities of the source and the receiver in the direction of the signal path.

f B = f 0δ (1 − δ B ) 1 − β B2

(3.2.4:2)

where f0δ is the frequency of the emitter at rest in apparent homogeneous space to the local frame and δA,δB and βA,βB are the gravitational factor and velocity of the emitter and receiver in the local frames, respectively. The motion of the radiation source in the local frame in the direction of the radiation shortens the emitted wavelength and the motion of the receiver in the same direction decreases the observed frequency, respectively. As the result the radiation emitted by A is observed at B as f A( B ) = f A

(1 − β ( ) ) (1 − β ( ) ) B r

(3.2.4:3)

Ar

(Fig. 3.2.4-1) or by expressing fA in terms of fB from equations (3.2.4:1) and (3.2.4:2) as f A( B ) = f B

(1 − δ A ) (1 − δ B )

( ) (1 − β ( ) )

1 − β A2 1 − β B (r ) 1 − β B2

(3.2.4:4)

Ar

Completion of (3.2.4:4) for the system of nested energy frames gives n

1 − β Bj2 1 − β jB (r )

∏ (1 − δ )

1− β

Bj

f A( B ) = f B

(

∏ (1 − δ )

j = k +1 m i = k +1

Ai

2 Ai

)

(1 − β ( ) ) iA r

117

(3.2.4:5)

A(k+3)

βA(k+3)

A(k+2)

βA(k+2)

B(k+1)

βA(k+1)

A(k+1)

βB(k+1)

Mk

Figure 3.2.4-2. Transmission of electromagnetic radiation from the source at rest in frame A(k+3) to the receiver at rest in frame B(k+1). The motions of frames A(k+1) … A(k+3) result in a change of the wavelength in radiation propagating in the Mk frame.

where frame k is the root parent frame common to both source and the receiver. The system of energy frames serves as multilevel transmission media where the connection between local frames occurs in the root parent frame common to both the source and the receiver (Fig. 3.2.4-2). The propagation time is TA( t 0)→ B (t1) = TAB =

rAB( t 0) ⋅ rˆAB m

(

c∏ 1 − β [ j ] B ( r ) j =k

(3.2.4:6)

)

where c is the velocity of light along the propagation path. The velocity of light decreases with the expansion of space; the transmission time from distant objects is shorter than the time obtained by using the c at the time the signal is received. 3.2.5 Sagnac effect At short distances, as in satellite communication, the change in c0 during the transmission is negligible. Satellite communication occurs in the Earth gravitational frame. In the case of a stationary receiver the motion of the receiver comes from the rotational velocity of the Earth and the transmission time becomes (Fig. 3.2.5-1 (a)) TAB =

rAB ( t 0) ⋅ rˆAB

(

c 1 − β B(r )

)

≈

rAB (t 0) c

+

rAB ( t 0 ) c2

ω rθ cosψ

(3.2.5:1)

where the last term, referred to as Sagnac effect, is the correction of the signal time due to the rotation of the Earth. In (3.2.5:1) ψ is the elevation angle of the satellite, ω is the angular velocity of Earth’s rotation, and rθ is the rotational radius of the Earth at the latitude of the receiver. The Sagnac term in (3.2.5:1) is equal to the expression

118

C

At0

A

h

rA(t0)→B(t1)

drr Bt1

O

B

ψ Bt0

ωr

drB

O

ψ L

rθ

(a)

(b)

Figure 3.2.5-1. During the signal transmission from a satellite, the rotation of the Earth results in displacement drB relative to a stationary receiver on the Earth. (a) The lengthening of the signal path due to the rotation is the component drr in the direction of the signal path [see equation (3.2.5:1)]. (b) The GR expression for the Sagnac correction is related to the area of the equatorial plane projection of triangle O, At0, Bt1 [see equation (3.2.5:2)]. Mathematically the two results are identical.

ΔTω ( Earth ) =

2ω AABO c2

(3.2.5:2)

referred to as “the relativistic Sagnac effect” [9]. AABO in (3.2.5:2) is the area of the equatorial plane projection of the triangle drawn by a distance vector from the center of the Earth to a propagating wave front from the satellite to the Earth station (Fig. 3.2.5-1 (b)). The term Sagnac correction is also used in connection with slow transport of clocks in the Earth gravitational frame. “Slow transport” means that the transport velocity of a clock is slow compared to the rotational velocity of the Earth. In the case of east-west transportation at fixed latitude, the effect of gravitation is cancelled and the cumulative reading of a clock during transportation can be expressed as 2 ⎛ v ⎞ N = T ⋅ f 0 = T f 0δ (1 − δ ) 1 − β 2 ≈ T f 0 ⎜1 − ½ 2 ⎟ c ⎠ ⎝

(3.2.5:3)

where v is the rotational velocity of the Earth and T is the time of the transportation. Differentiation of equation (3.2.5:3) gives ΔN = −T

v ⋅ Δv f0 c2

(3.2.5:4)

119

In the case of slow transport, Δv in equation (3.2.5:4) can be interpreted as the transportation velocity. Accordingly, the transportation time T can be expressed in terms of the transportation distance L as T=

L Δv

(3.2.5:5)

Substitution of equation (3.2.5:5) for T in equation (3.2.5:4) gives the difference in the reading of the clock due to the transport of the clock for distance L in the direction of Earth’s rotation as ΔN = −

L v ⋅ Δv Lv f0 = − 2 f0 2 Δv c c

(3.2.5:6)

showing that ΔN is independent of the transportation velocity Δv. By expressing the transportation distance as longitudinal angle and the velocity of Earth’s rotation as angular velocity equation (3.2.5:6) obtains the form

ψ ⋅ rθ ⋅ ω rθ 2ω AABO ΔN =− =− 2 f0 c c2

(3.2.5:7)

where AABO is the area of the equatorial plane projection of the sector defined by arc AB (Fig. 3.2.5-2). Equation (3.2.5:7) gives the seconds “lost” when transporting an atomic clock towards the east by ψ degrees at latitude θ. The result is mathematically identical to the Sagnac delay of light through longitudinal angle ψ at latitude θ over links following the surface of the earth. The physical mechanism of the delay in the reading of the clock, however, is different from the mechanism of the delay in the electromagnetic signal.

A

ψ

rθ O

B

Figure 3.2.5-2. Clock transportation by longitudinal angle ψ from point A to B at latitude θ. The delay due to the additional (slow) velocity is mathematically identical to the Sagnac delay of light transmission from point A to B (through links following the surface of the Earth at the same latitude).

120

4. Cosmological appearance of DU space 4.1 Distances and the observed angular size 4.1.1 Cosmological principle in spherically closed space At the cosmological scale spherically closed space is isotropic and homogeneous; i.e., it looks the same from any point in space. As a major difference from the Friedman-Lemaître-Robertson-Walker (FLRW) cosmology, local gravitational systems in DU space are subject to expansion in direct proportion to the expansion of the R4 radius. Accordingly, e.g., the radii of galaxies are not observed as standard rods but as expanding objects which makes the sizes of galaxies appear in Euclidean geometry to the observer. In the Earth gravitational frame, the linkage of orbital radii to the expansion of the R4-radius means that about 2.8 cm of the 3.8 cm annual increase in the Earth to Moon distance is due to expansion of space and only 1 cm is due to tidal interactions or other mechanisms. As shown by the analysis of the Bohr radius, material objects built of atoms and molecules are not subject to expansion with space. As shown by equations (2.2.1:6) and (2.2.1:7), like the Bohr radius, the characteristic emission wavelengths of atomic objects are likewise unchanged in the course of the expansion of space. When propagating in space, the wavelength of electromagnetic radiation is increased in direct proportion to the expansion. Accordingly, when detected after propagation in space, characteristic radiation is observed redshifted relative to the wavelength emitted by the corresponding transition in situ at the time of observation. 4.1.2 Optical distance and the Hubble law As a consequence of the spherical symmetry and the zero-energy balance in space, the velocity of light is determined by the velocity of space in the fourth dimension. The momentum of electromagnetic radiation has the direction of propagation in space. Although the actual path of light is a spiral in four dimensions, the length of the optical path in the direction of the momentum of radiation in space, is the tangential component of the spiral, which is equal to the increase of the 4-radius, the radial component of the path, during the propagation, Fig. 4.1.2-1 D = R0 − R0( 0)

(4.1.2:1)

The differential of optical distance can be expressed in terms of R0 and the distance angle α as

121

c0

c0

observer

observer

t

c0

object

R0

Dphys R0

c0(0)

t(0)

R0

α

α

O

R0(0)

emitting object

O (a)

(b)

Figure 4.1.2-1. (a) The classical Hubble law corresponds to Euclidean space where the distance of the object is equal to the physical distance, the arc Dphys, at the time of the observation. (b) When the propagation time of light from the object is taken into account the optical distance is the length of the integrated path over which light propagates in space in the tangential direction in the 4-sphere Dopt = D = ∫ dD⊥ . Because the velocity of light in space is equal to the expansion of space in the direction of R4, the optical distance is D = R0–R0(0), the lengthening of the 4-radius during the propagation time.

dD = R0 dα = c0 dt = dR0

(4.1.2:2)

By first solving for the distance angle α

α=

R0

∫

R0( 0 )

dR0 R R0 = ln 0 = ln R0 R0( 0) R0 − D

(4.1.2:3)

the optical distance D obtains the form D = R0 (1 − e −α )

(4.1.2:4)

where R0 means the value of the 4-radius at the time of the observation. The observed recession velocity, the velocity at which the optical distance increases, obtains the form vrec( optical ) =

dD D = c0 (1 − e −α ) = c0 dt R0

(4.1.2:5)

As demonstrated by equation (4.1.2:5) the maximum value of the observed (optical) recession velocity never exceeds the velocity of light, c, at the time of the observation, but approaches it asymptotically when distance D approaches the length of 4radius R0.

122

R0

λ

R0(0)

Figure 4.1.2-2. Increase in the wavelength of electromagnetic radiation propagating in expanding space.

λ0

Atoms conserve their dimensions in expanding space. As shown by Balmer’s equation, the characteristic emission wavelength is directly proportional to the Bohr radius, which means that also the characteristic emission wavelengths of atoms are unchanged in the course of the expansion of space. The wavelength of radiation propagating in expanding space is assumed to be subject to increase in direct proportion to the expansion space (Fig. 4.1.2-2). Accordingly, redshift, the increase of the wavelength becomes z=

λ − λ0 R0 − R0( 0 ) D R0 = = = eα − 1 λ0 R0( 0) 1 − D R0

(4.1.2:6)

where D = R0 – R0(0) is the optical distance of the object given in (4.1.2:4), λ and R0 are the wavelength and the 4-radius at the time of the observation, respectively, and R0(0) is the 4-radius of space at the time the observed light was emitted, see Fig. 4.1.2-3. Observer: R0 = 1 0.67 R0(0)

0.5 0.33 0.17

z = 0.5 z=1

Figure 4.1.2-3. Expansion of space during the propagation time of light from objects at different distances: The length of the 4-radius R0 and the corresponding optical path is indicated for redshifts z = 0.5 to 5.

z=2 z=5

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Solved from (4.1.2:6) the optical distance D can be expressed as D = R0

z 1+ z

(4.1.2:7)

4.1.3 Angular sizes of a standard rod and expanding objects The observation angle of an ideal standard rod or non-expanding object (solid object like a star) is

θ=

d rod d rod (1 + z ) = D R0 z

θ

;

d rod R0

=

(1 + z )

(4.1.3:1)

z

where distance D is the optical distance given in equation (4.1.2:7). As shown by equation (4.1.3:1), the observation angle of a standard rod approaches to the size angle αd = drod /R0 at high redshift (z >> 1). The prediction for the angular size of objects in FLRW space is

θ=

rs r = s DA RH

⎡ z ⎢ 1 ∫ ⎢ (1 + z ) 0 ⎣

⎤ dz ⎥ 2 (1 + z ) (1 + Ωm z ) − z ( 2 + z ) Ωλ ⎥⎦ 1

(4.1.3:2)

which is based on the angular diameter distance DA. Angular diameter distance is related to co-moving distance DM (or proper motion distance [10]) in FLRW space DA =

DM c 1 z = 1 + z RH 1 + z ∫0

1

(1 + z ) (1 + Ω m z ) − z ( 2 + z ) ΩΛ 2

dz

(4.1.3:3)

As shown by (4.1.3:3) the angular diameter distance DA turns to a decreasing trend at redshifts above z > 3 (Fig 4.1.3-1). As a consequence of the conservation of total energy in interactions in DU space, the radii of local gravitational systems expand in direct proportion to the expansion of the 4-radius R0. This is a major difference to FLRW cosmology where the radii of local systems are independent of the expansion of space. The angular diameter of expanding objects can be expressed d ( z) =

dR (1 + z )

(4.1.3:4)

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1 D(DU) (1.1.3:7)

D R0

DA(FLRW) (1.1.4:3) Ωm = 0.3 ΩΛ = 0.7

0.1

DA(FLRW) (1.1.4:3) Ωm = 1 ΩΛ = 0

0.01

0.001 0.001

0.01

0.1

1

10

100

z

1000

Figure 4.1.3-1. Optical distance of objects in DU space (4.1.2:7) (solid line) and the angular diameter distance in FLRW space (4.1.3:3) for Ωm = 1, ΩΛ = 0 and for Ωm = 0.27, ΩΛ = 0.73 corresponding to the Einstein-deSitter condition in FLRW space and the present estimates of mass and dark energy densities in ΛCDM corrected space, respectively (dashed lines).

where dR is the diameter of the object at the time of observation (Fig. 4.1.3-2). Substitution of d(z) in (4.1.3:4) for drod in (4.1.3:1) gives the angular size of an expanding objects

θ=

d (z) D

=

d R (1 + z ) d R 1 α d = = (1 + z ) R0 z R0 z z

;

θ d R R0

=

θ 1 = αd z

(4.1.3:5)

observation point R0(t2)

θ cspace

c4

αd R0(0)

dR

d

125

Figure 4.1.3-2. Observation of the angular size of an expanding object in zero-energy space. For all the propagation in expanding space, the velocity of light cspace is equal to the velocity of expansion in the fourth dimension c4. Accordingly, the optical distance D, the tangential component of the propagation path, is equal to the increase of R0 during the light propagation, D = R0 – R0(0).

FLRW cosmology, eq. (4.1.3:2)

100

θ

Ωm = 1 ΩΛ = 0

r R0

10

Ωm = 0.27 ΩΛ = 0.73

1 DU solid objects, eq. (4.1.3:1) 0.1 DU expanding objects, eq. (4.1.3:5)

0.01 0.01

0.1

1

10

z 100

Figure 4.1.3-3. Angular diameter of objects as the function of redshift in FLRW space and in DU space.

where the ratio dR/R0 = αd means the angular size of the expanding object as seen from the barycenter of space. Equation (4.1.3:5) implies a Euclidean appearance of expanding objects in space. A comparison of equations (4.1.3:1), (4.1.3:2), and (4.1.3:5) is given in Figure 4.1.3-3. The DU prediction for solid objects (standard rod) approaches asymptotically to the angular size of the object as it would appear from the barycenter of space (i.e. from M"). It can be concluded that an essential factor in the Euclidean appearance of galaxy space in the DU is the linkage of the gravitational energies of local systems to the gravitational energy in whole space. Such a linkage is missing in the GR based FLRW cosmology due to the local nature of the general relativity. In Figure 4.1.3-4 the DU prediction (4.1.3:5) and the FLRW prediction (4.1.3:2) are compared to observations of the Largest Angular Size (LAS) of galaxies and quasars in the redshift range 0.001 < z < 3 [11]. In figure 1.1.4-4 (a) the observation data is set between two Euclidean lines of the DU prediction in equation (4.1.3:5). The FLRW prediction is calculated for the conventional Einstein de Sitter case (Ωm= 1 and ΩΛ= 0) shown by the solid curve, and for the recently preferred case with a share of dark energy included as Ωm= 0.27 and ΩΛ = 0.73 (dashed curves). Both FLRW predictions deviate significantly from the Euclidean lines in (a) that enclose the set of data uniformly in the whole redshift range. As shown in figure 4.1.3-4 (b) the effect of the dark energy contribution on the FLRW prediction of the angular size is quite marginal.

126

log(LAS)

0.001

0.01

log(LAS)

0.1

1

z

10

0.001

0.01

0.1

1

10

z

(b) FLRW-prediction

(a) DU-prediction (Euclidean)

Ωm= 1, ΩΛ= 0 Ωm= 0.27, ΩΛ= 0.73

Figure 4.1.3-4. Dataset of observed Largest Angular Size (LAS) of quasars and galaxies in the redshift range 0.001 < z < 3. Open circles are galaxies, filled circles are quasars. (Data collection: K. Nilsson et al., Astrophys. J., 413, 453, 1993). In (a) observations are compared with the DU prediction (4.1.3:5). In (b) observations are compared with the FLRW prediction (4.1.3:2) with Ωm= 0 and ΩΛ = 0 (solid curves), and Ωm= 0.27 and ΩΛ= 0.73 (dashed curves).

4.2 Observation of radiation

4.2.1 Apparent magnitude of standard candle In DU space bolometric power density of electromagnetic radiation dilutes in proportion to the square of the optical distance D (4.1.2:7) and in direct proportion to the increase of the wavelength. When related to the power density from non-redshifted reference source at (non-redshifted) distance d0, the power density observed in redshifted radiation from an object at distance D becomes Fz , D F0, d0

d2 1 d 2 (1 + z ) d 02 (1 + z ) 1 = 02 = 02 = D (1 + z ) R0 z 2 (1 + z ) R02 z 2 2

which converts into apparent magnitude

127

(4.2.1:1)

m = M + 5log

R0 + 5log z − 2.5log (1 + z ) d0

(4.2.1:2)

As the part of the definition of apparent magnitude the reference source is located at 10 pc distance and it is assumed to posses a power spectrum and luminosity identical with those of the object. Equation (4.2.1:2) does not include possible effects of galactic extinction, spectral distortion in Earth atmosphere, or effects due to the local motion and gravitational environment of the object and the observer. Equation (4.2.1:2) applies for direct bolometric observations achievable with multi-bandpass photometry by matching the filters to the redshift of the object. In present practice, apparent magnitudes are expressed as K-corrected magnitudes which, in addition to instrumental factors for bolometric magnitude, include a “correction to the source rest frame” required by the prediction of the apparent magnitude in the standard cosmology model [12]. In multi-bandpass photometry with filters matched to the redshift (using optimum bandpass for each redshift) the K-correction can be approximated K = 5log (1 + z ) + K instr

(4.2.1:3)

where Kinstr includes atmospheric corrections (in terrestrial observations) and the instrumental factors like the transmission coefficients and mismatch of the filters. Adding (4.2.1:3) to the prediction in (4.2.1:2), the DU prediction for K-corrected magnitudes becomes m = M + 5log

R0 + 5log z + 2.5log (1 + z ) + K instr d0

(4.2.1:4)

The corresponding prediction for K-corrected magnitudes in FLRW cosmology is given by equation m = M + 5log

RH + K instr d0

⎡ z +5 log ⎢(1 + z ) ∫ 0 ⎢ ⎣

⎤ dz ⎥ 2 (1 + z ) (1 + Ωm z ) − z ( 2 + z ) ΩΛ ⎥⎦ 1

(4.2.1:5)

Equation (4.2.1:5) has its origin in the works of Hubble, Tolman, Humason, deSitter, and Robertson, in the 1930’s [13–18]. The difference from the DU prediction arises from several factors, including the geometry of space, the interpretation of the effect of the expansion on the energy density of radiation, the aberration factor, and the role of the K-correction.

128

50

DU (4.2.1:4)

μ

FLRW (4.2.1:5)

45

40

35

30 0,001

0,01

0,1

1

10

Figure 4.2.1-1. Distance modulus μ = m – M, vs. redshift for Riess et al. “high-confidence” dataset and the data from the HST for Ia supernovae, Riess [19]. The optimum fit for the FLRW prediction (4.2.1:5) is based on Ωm 0 0.27 and ΩΛ = 0.73. The difference between the DU prediction of (4.2.1:4) [20] (solid curve), and the prediction of the standard model (dashed curve) is very small in the redshift range covered by observations, but becomes meaningful at redshifts above z > 3.

Figure 4.2.1-1 compares the predictions of equations (4.2.1:4) and (4.2.1:5) for the K-corrected magnitudes of Ia supernovae in DU and FLRW space, respectively. The observed magnitudes in the figure are based on Riess et al.’s “high-confidence” dataset and the data from the HST [19]. The consistency of the approximation used in the DU prediction for the K-correction (4.2.1:3) in the data is illustrated in Figure 4.2.1-2. 2.5 KB, 2

Figure 4.2.1-2. Average KB,Xcorrections (black squares) collected from the KB,X data in Table 2 used by Riess et al. [10] for the K-corrected distance modulus data shown in Figure 4.2.1-1. The solid curve gives the K-correction K = 5⋅log(1+z) in equation (4.2.1:3).

1.5 1 0.5 0

0

0.5

1

1.5 z 2

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4.2.2 Surface brightness of expanding objects The Tolman test [15,17,21] is considered as a critical test for an expanding universe model. In expanding space, according to Tolman’s prediction, the observed surface brightness of standard objects decreases by the factor (1+z)4 with the redshift. Following the properties of FLRW space, Tolman’s prediction assumes that galaxies and quasars are non-expanding objects. In DU space, galaxies and quasars are expanding objects. With reference to equation (4.1.3:5) the angular area of expanding objects with radius re = re0/(1+z) is ΩD =

re2 1 R02 z 2

(4.2.2:1)

Applying (4.2.1:1) for the power density the surface brightness of an object at distance D relates to the surface brightness of a reference object at distance d0 ( zdo << 1, Ωd0 = 1/d02 ) as SB( D ) SB( d0 )

2 d 02 (1 + z ) Ω d0 (1 + z ) z = 2 = = (1 + z ) R4 z 2 Ω D z2

(4.2.2:2)

or SB( D ) = SB( d0 ) (1 + z )

(4.2.2:3)

When related to the K-corrected power densities in a multi-bandpass photometry with nominal filter wavelengths matched to the redshifted radiation becomes

SB( D ) = SB( d0 ) (1 + z )

−1

(4.2.2:4)

The predictions of equations (4.2.2:3) and (4.2.2:4) do not include the effects of possible evolutionary factors. 4.2.3 Microwave background radiation The bolometric energy density of cosmic microwave background (CMB) radiation, 4.2⋅10–14 [J/m3], corresponds, with a high accuracy, to the energy density within a closed blackbody source at 2.725 °K. (Obs. As indicated by the Stefan-Boltzmann constant, the energy density within a blackbody source is, by a factor of 4, higher than the integrated energy density of the flux radiated by the source)

130

8π h 3 (ν ν 0 ) ⎡ J ⎤ dν = 4.2 ⋅ 10−14 ⎢ 3 ⎥ ν0 ν ν 3 c e 0 −1 ⎣m ⎦ 0 3

∞

Ebol (T = 2.725 ° K ) = Eν dν = ∫

(

)

(4.2.3:1)

where

ν0 ≡

kT c = h λ0

[ Hz ]

(4.2.3:2)

from which ν0 = 5.691010 Hz is obtained for T = 2.725 °K. The rest energy calculated for the total mass in space is Erest = MΣ⋅c2 ≈ 2⋅1070 [J] corresponding to energy density Erest/(2π 2R43) = 4.6⋅10–10 [J] in DU space. Accordingly, the share of the CMB energy density of the total energy density in space is about 10–4. The Dynamic Universe concept does not give a prediction for the value of the 4-radius R4(e) at the emission of the CMB — or exclude the possibility that the CMB is generated by dark matter now at 2.725 °K effective temperature.

5. Summary

The Dynamic Universe model is a detailed analysis of zero-energy condition in spherically closed dynamic space where time is universal and the fourth dimension has a geometrical nature. The Dynamic Universe is a holistic approach covering the energy balance from whole space to local energy structures through a chain of nested energy frames. In DU space, the rest energy of matter as a local expression of energy is balanced by global gravitational energy arising from the rest of mass in space. The complementarity of the energies of motion and gravitation preserves the zero energy condition in any local energy frame as expressed by equation Erest = c0 mc = Eglobal

n

= m0 c02 ∏ (1 − δ i ) 1 − β i2 i =0

GM " m0 n =− ∏ (1 − δ i ) 1 − βi2 R "0 i = 0

(5.1)

Relativity in DU space does not rely on the Lorentz transformation, the relativity principle, or the equivalence principle. In the Dynamic Universe manifestations of relativity are direct consequences of the conservation of the zero-energy condition in space characterized by absolute time and distances. A local state of rest in DU space is determined by the zero-momentum and zero-angular momentum state of the local

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energy frame studied. The chain of nested energy frames in space relates any local energy state to the state of rest in hypothetical homogeneous space. As a consequence of the zero-energy balance in space, any closed energy system has the zero-momentum state as the local reference at rest. Objects in a local frame are associated with a reduced rest energy due to the effects of gravitation and motion of local frame in its parent frames. The DU approach makes a definite distinction between motion as an expression of momentum and kinetic energy and motion as kinematic velocity, which describes the rate of change in the distance between objects. Velocity as an expression of kinetic energy is relative to the state of rest in the frame the kinetic energy is obtained. Kinematic velocities, as the rates of changes in the distances between objects, can be summed up using the Galilean transformation but the resulting relative velocity has very little to do with the kinetic energy of the moving objects. Instead of being derived from field equations as in FLRW space, the local geometry (instead of metrics) of space is described as an equipotential surface in terms of an algebraic, complex presentation of the total energy. Mathematically, this means a major simplification to the field equation based metrics of FLRW space. The DU approach avoids the infinity problem of the field equations at local singularities in space — local singularities in zero-energy space allow circular orbits down to the critical radius, where the orbital velocities approach zero. Due to the linkage of gravitational energy in local frames to the gravitational energy in whole space, local gravitational systems expand in direct proportion to the expansion of whole space. As a consequence, together with the spherical symmetry of space, galactic space is observed in Euclidean geometry. In atoms, the Bohr radius is conserved in the course of expansion, which means that the dimensions of material objects are conserved. As a consequence of the conserved Bohr radius, the wavelengths of characteristic radiation emitted by atoms are unchanged in the course of the expansion and, accordingly, radiation from distant objects, due to the increase of the wavelength during propagation, is observed redshifted. As shown by the analysis of Maxwell’s equations for the emission of electromagnetic radiation by an electric dipole, the energy of a quantum is linked to the energy and mass equivalence carried by a cycle of radiation. Planck constant is expressed in terms of unit charge, vacuum permeability and the velocity of light, which – links mass to the wave number of radiation, and – discloses the essence of the fine structure constant α as a purely numerical constant without any connection to physical constants. As a part of the conservation of mass, the mass equivalence of a cycle of radiation is conserved in expanding space. The power density observed in redshifted radiation, however, is diluted due the increased wavelength and the optical distance affected by the expansion.

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Instead of a sudden appearance of mass, and energy in a Big Bang, singularity in DU space is seen as the turning point of a contraction phase into the ongoing expansion phase. With regard to the wave nature of mass we may assume a quantum limit to the 4-radius at passing the singularity. Such a limit could work as a measurement rod to structures maintaining their dimensions in expanding space. The basic form of matter in hypothetical homogeneous space is considered to have non-structured homogeneous radiation-like appearance with momentum in the direction of the 4-radius. At infinity in the past, like at infinity in the future, the 4-radius of space is infinite. Mass as the substance for the expression of energy exists, but as it is not energized it is not detectable. The energy of motion built up in the primary energy buildup is gained from the structural energy, the energy of gravitation. Space loses size and gains motion. At infinity in the future, all motion gained from gravity in the contraction will have been returned back to the gravitational energy of the structure in the expansion. Mass will no longer be observable because the energy excitation of matter will have vanished along with the cessation of motion. The energy of gravitation will also become zero owing to the infinite distances — completing the cycle of physical existence from emptiness in the past to emptiness in the future. Acknowledgements

I can recognize my friend and former colleague Heikki Kanerva as an important early inspirer in the thinking paving the way for the Dynamic Universe theory in late 1960’s. After many years of maturing, the active development of the theory was triggered by stimulus from my late colleague Jaakko Kajamaa in the early 1990s. I express my sincere gratitude to my early inspirers. Since 2004 my main channel for scientific discussions and publications on the Dynamic Universe has been the PIRT (Physical Interpretations of Relativity Theory) conference biannually organized in London and occasionally in Moscow, Calcutta and Budapest. I like to express my respect to the organizers of PIRT for keeping up scientifically sound critical discussion on the basis of physics and pass my sincere gratitude to Michael Duffy, Peter Rowlands, Mogens Wegener and many conference participants. At national level The Finnish Society for Natural Philosophy has organized seminars and lectures of the Dynamic Universe concept. I express my gratitude to the Society and many individuals for the active encouragement and fruitful and inspiring discussions. I am exceedingly grateful to Ari Lehto, Heikki Sipilä, Tarja KallioTamminen, Paul Talvio, Jouko Seppänen, Johannes Hopiavuori, Jyrki Tyrkkö, Heikki Mäntylä, Seppo Haarala, Antti Lange, Janne Karimäki and my late colleague Pentti Passiniemi, for their continuous encouragement and many insightful discussions on the laws of nature, the structure of physics, and the theoretical basis of the Dynamic

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Universe model. I also like to express my sincere thanks to Bob Day for his activity in finding and analyzing experimental data for testing the DU predictions. My many good friends and colleagues are thanked for their encouragement during the years of my treatise. The unfailing support of my wife Soilikki and my daughter Silja and her family has been of special importance and I am deeply grateful to them. References

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.

R. Feynman, W. Morinigo, and W. Wagner, Feynman Lectures on Gravitation (during the academic year 1962-63), Addison-Wesley Publishing Company, p. 10 (1995) R. Feynman, W. Morinigo, and W. Wagner, Feynman Lectures on Gravitation (during the academic year 1962-63), Addison-Wesley Publishing Company, p. 164 (1995) A. Einstein, Sitzungsberichte der Preussischen Akad. d. Wissenschaften (1917) G. W. Leibniz, Matematischer Naturwissenschaflicher und Technischer Briefwechsel, Sechster band (1694) T. Suntola, “Theoretical Basis of the Dynamic Universe”, Suntola Consulting Ltd., Helsinki, ISBN 952-5502-01-5 (2004) T. Suntola, Proceedings of SPIE Vol. 5866, 18 (2005) J. Foster, J.D. Nightingale, A Short Course in General Relativity, 2nd edition, Springer-Verlag, ISBN 0-387-94295-5 (2001) R. Genzel, et al., Nature 425, 934 (2003) N. Ashby, Living reviews in relativity, http://relativity.livingreviews.org/ Articles/lrr-2003-1/ (2003) S. M. Carroll, W. H. Press, and E. L. Turner, ARA&A, 30, 499 (1992) K. Nilsson et al., Astrophys. J., 413, 453, 1993 Kim, A., Goobar, A., & Perlmutter, S. 1996, PASP, 190–201 Tolman, PNAS 16, 511-520 (1930) E. Hubble, M. L. Humason, Astrophys.J., 74, 43 (1931) W. de Sitter, B.A..N., 7, No 261, 205 (1934) E. Hubble and R. C. Tolman, ApJ, 82, 302 (1935) E. Hubble, Astrophys. J., 84, 517 (1936) Robertson, H.P., Zs.f.Ap., 15, 69 (1938) A. G. Riess, et al., Astrophys. J., 607, 665 (2004) T. Suntola and R. Day, arXiv/astro-ph/0412701 (2004) A. Sandage, J-M. Perelmuter, Astrophys.J., 370, 455 (1991)

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.

DYNAMICAL 3-SPACE: A REVIEW Reginald T. Cahill School of Chemistry, Physics and Earth Sciences, Flinders University, Adelaide 5001, Australia

For some 100 years physics has modelled space and time via the spacetime concept, with space being merely an observer dependent perspective effect of that spacetime - space itself had no observer independent existence - it had no ontological status, and it certainly had no dynamical description. In recent years this has all changed. In 2002 it was discovered that a dynamical 3-space had been detected many times, including the Michelson-Morley 1887 light-speed anisotropy experiment. Here we review the dynamics of this 3-space, tracing its evolution from that of an emergent phenomena in the information-theoretic Process Physics to the phenomenological description in terms of a velocity field describing the relative internal motion of the structured 3-space. The new physics of the dynamical 3-space is extensively tested against experimental and astronomical observations, including the necessary generalisation of the Maxwell, Schr¨odinger and Dirac equations, leading to a derivation and explanation of gravity as a refraction effect of the quantum matter waves. Phenomena now explainable include the bore hole anomaly, the systematics of black hole masses, the flat rotation curves of spiral galaxies, gravitational light bending and lensing, and the supernova and gamma-ray bursts magnitude-redshift data, for the dynamical 3-space possesses a Hubble expanding 3-space solution. Most importantly none of these phenomena now require dark matter nor dark energy. The flat and curved spacetime formalism is derived from the new physics, so explaining the apparent many successes of those formalisms, but which have now proven to be ontologically and experimentally flawed.

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1 Introduction We review here some of the new physics emerging from the discovery that there exists a dynamical 3-space. This discovery changes all of physics. While at a deeper level this emerges from the information-theoretic Process Physics [1, 4, 5, 6, 7, 8, 9] here we focus on the phenomenological description of this 3space in terms of the velocity field that describes the internal dynamics of this structured 3-space. It is straightforward to construct the minimal dynamics for this 3-space, and it involves two constants: G - Newton’s gravitational constant, and α - the fine structure constant. G quantifies the effect of matter upon the flowing 3-space, while α describes the self-interaction of the 3-space. Bore hole experiments and black hole astronomical observations give the value of α as the fine structure constant to within observational errors. A major development is that the Newtonian theory of gravity [10] is fundamentally flawed - that even in the non-relativistic limit it fails to correctly model numerous gravitational phenomena. So Newton’s theory of gravity is far from being ‘universal’. The Hilbert-Einstein theory of gravity (General Relativity - GR), with gravity being a curved spacetime effect, was based on the assumption that Newtonian gravity was valid in the non-relativistic limit. The ongoing effort to save GR against numerous disagreements with experiment and observation lead to the invention first of ‘dark matter’ and then ‘dark energy’. These effects are no longer required in the new physics. The 3-space velocity field has been directly detected in at least eight experiments including the Michelson-Morley experiment [2] of 1887, but most impressively by the superb experiment by Miller in 1925/1926 [3]. The Miller experiment was one of the great physics experiments of the 20th century, but has been totally neglected by mainstream physics. All of these experiments detected the dynamical 3-space by means of the light speed anisotropy - that the speed of light is different in different directions, and the anisotropy is very large, namely some 1 part in a 1000. The existence of this 3-space as a detectable phenomenon implies that a generalisation of all the fundamental theories of physics be carried out. The generalisation of the Maxwell equations leads to a simple explanation for gravitational light bending and lensing effects, the generalisation of the Schr¨odinger equation leads to the first derivation of gravity - as a refraction effect of the quantum matter waves by the time dependence and inhomogeneities of the 3-space, leading as well to a derivation of the equivalence principle. This generalised Schr¨odinger equation also explains the Lense-Thirring effect as being caused by vorticity in the flowing 3-space. This effect is being studied by the Gravity Probe B (GP-B) gyroscope precession experiment. The generalisation of the Dirac equation to take account of the interaction of the spinor with the

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dynamical 3-space results in the derivation of the curved spacetime formalism for the quantum matter geodesics, but without reference to the GR equations for the induced spacetime metric. What emerges from this derivation is that the spacetime is purely a mathematical construct - it has no ontological status. That discovery completely overturns this paradigm of 20th century physics. The dynamical equation for the 3-space has black hole solutions with properties very different from the putative black holes of GR, leading to the verified prediction for the masses of the minimal black holes in spherical star systems. That same dynamics has an expanding 3-space solution - the Hubble effect for the universe. That solution has the expansion mainly determined by space itself. This expansion gives a extremely good account of the supernovae/Gamma-Ray Burst redshift data without the notion of ‘dark energy’ or an accelerating universe. This review focuses on the phenomenological modelling of the 3-space dynamics and its experimental checking. Earlier reviews are available in [1](2005) and [4](2003). Page limitations mean that some developments have not been discussed herein.

2 Dynamics of 3-Space At a deeper level an information-theoretic approach to modelling reality, Process Physics [1], leads to an emergent structured quantum foam ‘space’ which is 3dimensional and dynamic, but where the 3-dimensionality is only approximate, in that if we ignore non-trivial topological aspects of the space, then it may be embedded in a 3-dimensional geometrical manifold. Here the space is a real existent discrete but fractal network of relationships or connectivities, but the embedding space is purely a mathematical way of characterising the gross 3-dimensionality of the network. This is illustrated in Fig.1. Embedding the network in the embedding space is very arbitrary; we could equally well rotate the embedding or use an embedding that has the network translated or translating. These general requirements then dictate the minimal dynamics for the actual network, at a phenomenological level. To see this we assume at a coarse grained level that the dynamical patterns within the network may be described by a velocity field v(r, t), where r is the location of a small region in the network according to some arbitrary embedding. The 3-space velocity field has been observed in at least 8 experiments [2, 3, 16, 17, 18, 19, 20, 21, 22, 23, 24]. For simplicity we assume here that the global topology of the network is not significant for the local dynamics, and so we embed in an E 3 , although a generalisation to an embedding in S 3 is straightforward and might be relevant to cosmology. The minimal dynamics is then obtained by writing down the lowest-order zero-rank tensors, of dimension

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Figure 1: This is an iconic representation of how a quantum foam dynamical network (left), see [1] for details of the Quantum Homotopic Field Theory, has its inherent approximate 3dimensional connectivity displayed by an embedding in a mathematical space, such as an E 3 or an S 3 as shown on the right. The embedding space is not real; it is purely a mathematical artifact. Nevertheless this embeddability helps determine the minimal dynamics for the network, as in (1). The dynamical space is not an ether model, as the embedding space does not exist. 1/t2 , that are invariant under translation and rotation, giving α β 1 ∂v ∇. + (v.∇)v + (trD)2 + tr(D2 ) = −4πGρ; Dij = ∂t 8 8 2 



∂vi ∂vj + ∂xj ∂xi

!

(1) where ρ(r, t) is the matter and EM energy densities expressed as an effective matter density. The embedding space coordinates provide a coordinate system or frame of reference that is convenient to describing the velocity field, but which is not real. In Process Physics quantum matter are topological defects in the network, but here it is sufficient to give a simple description in terms of an effective density. We see that there are only four possible terms, and so we need at most three possible constants to parametrise the dynamics of space: G, α and β. G turns out to be Newton’s gravitational constant, and describes the rate of non-conservative flow of space into matter. To determine the values of α and β we must, at this stage, turn to experimental data. However most experimental data involving the dynamics of space is observed by detecting the so-called gravitational acceleration

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of matter, although increasingly light bending is giving new information. Now the acceleration a of the dynamical patterns in space is given by the Euler convective expression a(r, t) = lim

∆t→0

v(r + v(r, t)∆t, t + ∆t) − v(r, t) ∂v = + (v.∇)v ∆t ∂t

(2)

and this appears in one of the terms in (1). As shown in [11] and discussed later herein the acceleration g of quantum matter is identical to this acceleration, apart from vorticity and relativistic effects, and so the gravitational acceleration of matter is also given by (2). Outside of a spherically symmetric distribution of matter, of total mass M , we find that one solution of (1) is the velocity in-flow field given by s

v(r) = −ˆr

2GM (1 + r

α 2

+ ..)

(3)

but only when β = −α, for only then is the acceleration of matter, from (2), induced by this in-flow of the form g(r) = −ˆr

GM (1 + α2 + ..) r2

(4)

which is Newton’s Inverse Square Law of 1687 [10], but with an effective mass M (1 + α2 + ..) that is different from the actual mass M . So the success of Newton’s law in the solar system informs us that β = −α in (1). But we also see modifications coming from the α-dependent terms. In general because (1) is a scalar equation it is only applicable for vorticityfree flows ∇ × v = 0, for then we can write v = ∇u, and then (1) can always be solved to determine the time evolution of u(r, t) given an initial form at some time t0 . The α-dependent term in (1) (with now β = −α) and the matter acceleration effect, now also given by (2), permits (1) to be written in the form ∇.g = −4πGρ − 4πGρDM , where

(5)

α ((trD)2 − tr(D2 )), (6) 32πG which is an effective matter density that would be required to mimic the αdependent spatial self-interaction dynamics. Then (5) is the differential form for Newton’s law of gravity but with an additional non-matter effective matter ρDM (r, t) ≡

139

density. So we label this as ρDM even though no matter is involved [39, 40]. This effect has been shown to explain the so-called ‘dark matter’ effect in spiral galaxies, bore hole g anomalies, and the systematics of galactic black hole masses. The spatial dynamics is non-local. Historically this was first noticed by Newton who called it action-at-a-distance. To see this we can write (1) as an integrodifferential equation ∂v v2 = −∇ ∂t 2

!

Z

+ G d3 r 0

ρDM (r0 , t) + ρ(r0 , t) (r − r0 ) |r − r0 |3

(7)

This shows a high degree of non-locality and non-linearity, and in particular that the behaviour of both ρDM and ρ manifest at a distance irrespective of the dynamics of the intervening space. This non-local behaviour is analogous to that in quantum systems and may offer a resolution to the horizon problem. However (1) needs to be further generalised [1] to include vorticity, and also the effect of the motion of matter through this substratum via vR (r0 (t), t) = v0 (t) − v(r0 (t), t),

(8)

where v0 (t) is the velocity of an object, at r0 (t), relative to the same frame of reference that defines the flow field; then vR is the velocity of that matter relative to the substratum. One possible generalisation of the flow equation (1) is, with d/dt = ∂/∂t + v.∇ the Euler fluid or total derivative, dDij δij trD δij δij α + tr(D2 ) + (Dij − trD) + ((trD)2 − tr(D2 )) dt 3 2 3 3 8 vi vj δij +(ΩD − DΩ)ij = −4πGρ( + R 2R + ..), i, j = 1, 2, 3. 3 2c 8πGρ vR , c2 1 1 ∂vi ∂vj 1 Ωij = ( − ) = − ijk ωk = − ijk (∇ × v)k , 2 ∂xj ∂xi 2 2 ∇ × (∇ × v) =

(9) (10) (11)

and the vorticity vector field is ω ~ = ∇ × v. For zero vorticity and vR  c (9) reduces to (1). We obtain from (10) the Biot-Savart form for the vorticity 2G ω ~ (r, t) = 2 c

Z

d3 r 0

ρ(r0 , t) vR (r0 , t) × (r − r0 ). |r − r0 |3

(12)

Eqn.(12) has been applied to the precession of gyroscopes in the GP-B satellite experiment, see Sect.10.5.

140

3 Generalised Schr¨odinger Equation and Emergent Gravity Let us consider what might be regarded as the conventional ‘Newtonian’ approach to including gravity in the Schr¨odinger equation [11]. There gravity is described by the Newtonian potential energy field Φ(r, t), such that g = −∇Φ, and we have for a ‘free-falling’ quantum system, with mass m, i¯ h

∂ψ(r, t) ¯h2 2 =− ∇ ψ(r, t) + mΦ(r, t)ψ(r, t) ≡ H(t)Ψ, ∂t 2m

(13)

where the hamiltonian is in general now time dependent. The classical-limit trajectory is obtained via the usual Ehrenfest method [12]: we first compute the time rate of change of the so-called position ‘expectation value’ d i i i d ≡ (ψ, rψ) = (Hψ, rψ) − (ψ, rHψ) = (ψ, [H, r]ψ), dt dt ¯h ¯h ¯h

(14)

which is valid for a normalised state ψ. The norm is time invariant when H is hermitian (H † = H) even if H itself is time dependent, d i i i i (ψ, ψ) = (Hψ, ψ) − (ψ, Hψ) = (ψ, H † ψ) − (ψ, Hψ) = 0. dt ¯h ¯h ¯h ¯h

(15)

Next we compute the matter ‘acceleration’ from (14). d2 dt2

i d (ψ, [H, r]ψ), ¯h dt  2 i i ∂H(t) (ψ, [H, [H, r]]ψ) + (ψ, [ , r]ψ), = ¯h ¯h ∂t = −(ψ, ∇Φψ) = (ψ, g(r, t)ψ) =< g(r, t) > . =

(16)

In the classical limit ψ has the form of a wavepacket where the spatial extent of ψ is much smaller than the spatial region over which g(r, t) varies appreciably. Then we have the approximation < g(r, t) >≈ g(< r >, t), and finally we arrive at the Newtonian 2nd-law equation of motion for the wavepacket, d2 ≈ g(< r >, t). dt2

(17)

In this classical limit we obtain the equivalence principle, namely that the acceleration is independent of the mass m and of the velocity of that mass. But of

141

course that followed by construction, as the equivalence principle is built into (13) by having m as the coefficient of Φ. In Newtonian gravity there is no explanation for the origin of Φ or g. In the new theory gravity is explained in terms of a velocity field, which in turn has a deeper explanation within Process Physics. The key insight is that conventional physics has neglected the interaction of various systems with the dynamical 3-space. Here we generalise the Schr¨odinger equation to take account of this new physics. Now gravity is a dynamical effect arising from the time-dependence and spatial inhomogeneities of the 3-space velocity field v(r, t), and for a ‘free-falling’ quantum system with mass m the Schr¨ odinger equation now has the generalised form 

i¯h

∂ 1 ¯h2 2 + v.∇ + ∇.v ψ(r, t) = − ∇ ψ(r, t), ∂t 2 2m 

(18)

which we write as i¯ h

∂ψ(r, t) = H(t)ψ(r, t), where ∂t

1 ¯h2 2 H(t) = −i¯h v.∇ + ∇.v − ∇ 2 2m 



(19)

This form for H specifies how the quantum system must couple to the velocity field, and it uniquely follows from two considerations: (i) the generalised Schr¨ odinger equation must remain form invariant under a change of observer, i.e. with t → t, and r → r + Vt, where V is the relative velocity of the two observers. ∂ 1 ∂ 1 Then we compute that + v.∇ + ∇.v → + v.∇ + ∇.v, i.e. that it is an ∂t 2 ∂t 2 invariant operator, and (ii) require that H(t) be hermitian, so that the wavefunction norm is an invariant of the time evolution. This implies that the 12 ∇.v term must be included, as v.∇ by itself is not hermitian for an inhomogeneous v(r, t). Then the consequences for the motion of wavepackets are uniquely determined; they are fixed by these two quantum-theoretic requirements. Then again the classical-limit trajectory is obtained via the position ‘expectation value’, first with d dt

d i i¯h (ψ, rψ) = (ψ, [H, r]ψ) = (ψ, (v(r, t) − ∇)ψ) dt ¯h m i¯h = − <∇>, (20) m on evaluating the commutator using H(t) in (19), and which is again valid for a normalised state ψ. Then for the ‘acceleration’ we obtain from (20) that1 vO ≡

=

i¯ h d2 d = (ψ, (v − ∇)ψ) 2 dt dt m 1

Care is needed to identify the range of the various ∇’s.

142

∂v(r, t) i¯ h i + [H, (v − ∇)] ψ), ∂t ¯h m    ∂v(r, t) i¯h 1 i¯h 2 = (ψ, v − ∇ ψ) − ψ) + (ψ, v.∇ + ∇.v − ∇ ∂t 2 2m m    i¯h 1 i¯h 2 (ψ, v − ∇ v.∇ + ∇.v − ∇ ψ), m 2 2m   ∂v(r, t) i¯h = (ψ, + ((v.∇)v) − (∇ × v) × ∇ ψ) + ∂t m i¯ h +(ψ, (∇ × (∇ × v))ψ), 2m   d ∂v i¯h ≈ + (v.∇)v + (∇ × v) × −v + (∇ × (∇ × v)), ∂t dt 2m   d ∂v = + (v.∇)v + (∇ × v) × −v ∂t dt ∂v = + (v.∇)v + (∇ × v) × vR (21) ∂t 



= (ψ,

where in arriving at the 3rd last line we have invoked the small-wavepacket approximation, and also used (20) to identify vR ≡ −

i¯ h <∇>= vO − v, m

(22)

where vO is the velocity of the wavepacket or object ‘O’ relative to the observer, so then vR is the velocity of the wavepacket relative to the local 3-space. Then all velocity field terms are now evaluated at the location of the wavepacket. Note that the operator −

i¯h i¯ h (∇ × v) × ∇ + (∇ × (∇ × v)) m 2m

(23)

is hermitian, but that separately neither of these two operators is hermitian. Then in general the scalar product in (21) is real. But then in arriving at the last line in (21) by means of the small-wavepacket approximation, we must then self-consistently use that ∇ × (∇ × v) = 0, otherwise the acceleration acquires a spurious imaginary part. This is consistent with (10) outside of any matter which contributes to the generation of the velocity field, for there ρ = 0. These observations point to a deep connection between quantum theory and the velocity field dynamics, as already argued in [1]. We see that the test ‘particle’ acquires the acceleration of the velocity field, as in (2), and as well an additional vorticity induced acceleration which is the

143

analogue of the Helmholtz acceleration in fluid mechanics. Then ω ~ /2 is the instantaneous angular velocity of the local 3-space, relative to a distant observer. Hence we find that the equivalence principle arises from the unique generalised Schr¨ odinger equation and with the additional vorticity effect. This vorticity effect depends on the absolute velocity vR of the object relative to the local space, and so requires a change in the Galilean or Newtonian form of the equivalence principle. The vorticity acceleration effect is the origin of the Lense-Thirring so-called ‘frame-dragging’ 2 effect [43] discussed later. While the generation of the vorticity is a relativistic effect, as in (12), the response of the test particle to that vorticity is a non-relativistic effect, and follows from the generalised Schr¨odinger equation, and which is not present in the standard Schr¨odinger equation with coupling to the Newtonian gravitational potential, as in (13). Hence the generalised Schr¨odinger equation with the new coupling to the velocity field is more fundamental. The Helmholtz term in (21) is being explored by the Gravity Probe B gyroscope precession experiment, however the vorticity caused by the motion of the earth is extremely small, as discussed later in Sect.10.5. An important insight emerges from the above: the generalised Schr¨odinger equation involves two fields v(r, t) and ψ(r, t), where the coordinate r is merely a label to relate the two fields, and is not itself the 3-space. In particular while r relates to the embedding space, the 3-space itself has time-dependence and inhomogeneities, and as well in the more general case will exhibit vorticity ω ~ = ∇ × v. Only in the unphysical case does the description of the 3-space become identified with the coordinate system r, and that is when the velocity field v(r, t) becomes uniform and time independent. Then by a suitable choice of observer we may put v(r, t) = 0, and the generalised Schr¨odinger equation reduces to the usual ‘free’ Schr¨odinger equation. As we discuss later the experimental evidence is that v(r, t) is fractal and so cannot be removed by a change to a preferred observer. Hence the generalised Schr¨odinger equation in (19) is a major development for fundamental physics. Of course in general other non-3-space potential energy terms may be added to the RHS of (19). A prediction of this new quantum theory, which also extends to a generalised Dirac equation, is that the fractal structure of space implies that even at the scale of atoms etc there will be timedependencies and inhomogeneities, and that these will affect transition rates of quantum systems. These effects are probably those known as the Shnoll effects [13]. 2

In the spacetime formalism it is mistakenly argued that it is ‘spacetime’ that is ‘dragged’.

144

-

? @ X @ R'$ XXδX zX @ XX @  @ @ &% @  I @ 6

Figure 2:

Shows bending of light through angle δ by the inhomogeneous spatial in-flow, according to the minimisation of the travel time in (30). This effect permits the in-flow speed at the surface of the sun to be determined to be 615km/s. The in-flow speed into the sun at the distance of the earth from the sun has been extracted from the Miller data, giving 50 ± 10km/s [1]. Both speeds are in agreement with (3).

4 Generalised Dirac Equation and Relativistic Gravity An analogous generalisation of the Dirac equation is also necessary giving the coupling of the spinor to the actual dynamical 3-space, and again not to the embedding space as has been the case up until now, ∂ψ 1 = −i¯h cα.∇ ~ + v.∇ + ∇.v ψ + βmc2 ψ ∂t 2 

i¯ h



(24)

where α ~ and β are the usual Dirac matrices. Repeating the Schr¨odinger equation analysis for the space-induced acceleration we obtain ∂v g= + (v.∇)v + (∇ × v) × vR − ∂t

vR 1 d v2 2 dt 1− R c2

2 vR c2

!

(25)

which generalises (21) by having a term which limits the speed of the wave packet relative to space to be < c. This equation specifies the trajectory of a spinor wave packet in the dynamical 3-space.

5 Generalised Maxwell Equations and Light Lensing One of the putative key tests of the GR formalism was the gravitational bending of light. This also immediately follows from the new space dynamics once we also generalise the Maxwell equations so that the electric and magnetic fields are excitations of the dynamical space. The dynamics of the electric and magnetic fields must then have the form, in empty space, ∂H ∂E ∇ × E = −µ + v.∇H , ∇ × H =  + v.∇E , ∇.H = 0, ∇.E = 0 ∂t ∂t (26) 





145



which was first suggested by Hertz in 1890 [14], but with v being a constant vector field. Suppose we have a uniform flow of space with velocity v wrt the embedding space or wrt an observer’s frame of reference. Then we can find plane wave solutions for (26): E(r, t) = E0 ei(k.r−ωt)

H(r, t) = H0 ei(k.r−ωt)

with ω(k, v) = c|~k| + v.k

where

(27)

√ c = 1/ µ

(28)

Then the EM group velocity is ˆ+v ~ k ω(k, v) = ck vEM = ∇

(29)

So the velocity of EM radiation vEM has magnitude c only with respect to the space, and in general not with respect to the observer if the observer is moving through space, as experiment has indicated again and again, as discussed in Sect.9. These experiments show that the speed of light is in general anisotropic, as predicted by (29). The time-dependent and inhomogeneous velocity field causes the refraction of EM radiation. This can be computed by using the Fermat leasttime approximation. Then the EM ray paths r(t) are determined by minimising the elapsed travel time: Z

sf

τ= si

dr | ds |cˆ vR (s) + v(r(s), t(s)| ds|



with vR =

dr − v(r(t), t) dt



(30)

by varying both r(s) and t(s), finally giving r(t). Here s is a path parameter, and vR is a 3-space tangent vector for the path. As an example, the in-flow in (3), which is applicable to light bending by the sun, gives the angle of deflection δ=2

4GM (1 + α2 + ..) v2 = + ... c2 c2 d

(31)

where v is the in-flow speed at distance d and d is the impact parameter. This agrees with the GR result except for the α correction. Hence the observed deflection of 8.4 × 10−6 radians is actually a measure of the in-flow speed at the sun’s surface, and that gives v = 615km/s. These generalised Maxwell equations also predict gravitational lensing produced by the large in-flows associated with the new ‘black holes’ in galaxies, see [15]. So again this effect permits the direct observation of the these black hole effects with their non inverse-square-law accelerations.

146

6 Free-Fall Minimum Proper-Time Trajectories The acceleration in (25) also arises from the following argument, which is the analogue of the Fermat least-time formalism for the quantum matter waves. Consider the elapsed time for a comoving clock. Then taking account of the Lamour time-dilation effect that time is given by Z

τ [r0 ] =

v2 dt 1 − R c2

!1/2

(32)

with vR given by (22) in terms of vO and v. Then this time effect relates to the speed of the clock relative to the local 3-space, and that c is the speed of light relative to that local 3-space. Under a deformation of the trajectory r0 (t) → r0 (t) + δr0 (t), v0 (t) → v0 (t) +

dδr0 (t) , dt

(33)

v(r0 (t) + δr0 (t), t) = v(r0 (t), t) + (δr0 (t).∇)v(r0 (t), t) + ... Evaluating the change in proper travel time to lowest order δτ

= τ [r0 + δr0 ] − τ [r0 ] + ... = − Z

=

Z

1 v2 dt 2 vR .δvR 1 − R c c2

1 vR .(δr0 .∇)v − vR . s dt 2 c v2 1− R c2

!−1/2

+ ... d(δr0 ) dt + ...



 Z

=

1 dt 2 c

    vR .(δr0 .∇)v d vR  + ...  s s + δr0 .   dt 2 2   vR vR

1−

1−

c2

Z

=

c2 



 (vR .∇)v + vR × (∇ × v)  vR 1 d  + ... s dt 2 δr0 .  + s   c dt 2 2  vR vR  1− 2 1− 2 c c 



147

(34)

Hence a trajectory r0 (t) determined by δτ = 0 to O(δr0 (t)2 ) satisfies vR d (vR .∇)v + vR × (∇ × v) s s =− . dt 2 2 vR vR 1− 2 1− 2 c c

(35)

Substituting vR (t) = v0 (t) − v(r0 (t), t) and using ∂v dv(r0 (t), t) = + (v0 .∇)v, dt ∂t

(36)

we obtain dv0 ∂v = + (v.∇)v + (∇ × v) × vR − dt ∂t

vR 1 d v2 2 dt 1− R c2

2 vR c2

!

.

(37)

which is (25). Then in the low speed limit vR  c we may neglect the last term, and we obtain (21). Hence we see a close relationship between the geodesic equation, known first from General Relativity, and the 3-space generalisation of the Schr¨odinger equation, at least in the non-relativistic limit. So in the classical limit, i.e when the wavepacket approximation is valid, the wavepacket trajectory is specified by the least proper-time geodesic. The relativistic term in (37) is responsible for the precession of elliptical orbits and also for the event horizon effect. Hence the trajectory in (21) is a nonrelativistic minimum travel-time trajectory, which is Fermat’s Principle.

7 Deriving the Special Relativity Formalism The detection of absolute motion is not incompatible with Lorentz symmetry; the contrary belief was postulated by Einstein, and has persisted for over 100 years, since 1905. So far the experimental evidence is that absolute motion and Lorentz symmetry are real and valid phenomena; absolute motion is motion relative to some substructure to space, whereas Lorentz symmetry parametrises dynamical effects caused by the motion of systems through that substructure. Motion through the structured space, it is argued, induces actual dynamical time dilations and length contractions in agreement with the Lorentz interpretation of special relativistic effects. Then observers in uniform motion ‘through’ the space will, on measurement of the speed of light using the special but misleading Einstein measurement protocol, obtain always the same numerical value c. To

148

see this explicitly consider how various observers P, P 0 , . . . moving with different speeds through space, measure the speed of light. They each acquire a standard rod and an accompanying standardised clock. That means that these standard rods would agree if they were brought together, and at rest with respect to space they would all have length ∆l0 , and similarly for the clocks. Observer P and accompanying rod are both moving at speed vR relative to space, with the rod longitudinal to that motion. P then measures the time ∆tR , with the clock at end A of the rod, for a light pulse to travel from end A to the other end B and back again to A. The light travels at speed c relative to space. Let the time taken for the light pulse to travel from A → B be tAB and from B → A be tBA , as measured by a clock at rest with respect to space3. The length of the rod moving at speed vR is contracted to s

1−

∆lR = ∆l0

2 vR . c2

(38)

In moving from A to B the light must travel an extra distance because the end B travels a distance vR tAB in this time, thus the total distance that must be traversed is ctAB = ∆lR + vR tAB , (39) similarly on returning from B to A the light must travel the distance ctBA = ∆lR − vR tBA .

(40)

Hence the total travel time ∆t0 is ∆t0 = tAB + tBA =

∆lR ∆lR 2∆l0 + == s . c − vR c + vR 2 vR c 1− 2 c

(41)

Because of the time dilation effect for the moving clock s

∆tR = ∆t0

1−

2 vR . c2

(42)

Then for the moving observer the speed of light is defined as the distance the observer believes the light travelled (2∆l0 ) divided by the travel time according to the accompanying clock (∆tR ), namely 2∆l0 /∆tR = c, from above, which is thus the same speed as seen by an observer at rest in the space, namely c. So 3

Not all clocks will behave in this same “ideal” manner.

149

the speed vR of the observer through space is not revealed by this procedure, and the observer is erroneously led to the conclusion that the speed of light is always c. This follows from two or more observers in manifest relative motion all obtaining the same speed c by this procedure. Despite this failure this special effect is actually the basis of the spacetime Einstein measurement protocol. That this protocol is blind to the absolute motion has led to enormous confusion within physics. To be explicit the Einstein measurement protocol actually inadvertently uses this special effect by using the radar method for assigning historical spacetime coordinates to an event: the observer records the time of emission and reception of radar pulses (tr > te ) travelling through space, and then retrospectively assigns the time and distance of a distant event B according to (ignoring directional information for simplicity) 1 c TB = (tr + te ) , DB = (tr − te ) , (43) 2 2 where each observer is now using the same numerical value of c. The event B is then plotted as a point in an individual geometrical construct by each observer, known as a spacetime record, with coordinates (DB , TB ). This is the same as an historian recording events according to some agreed protocol. Unlike historians, who don’t confuse history books with reality, physicists do so. We now show that because of this protocol and the absolute motion dynamical effects, observers will discover on comparing their historical records of the same events that the expression 1 2 2 2 τAB = TAB − 2 DAB , (44) c is an invariant, where TAB = TA − TB and DAB = DA − DB are the differences in times and distances assigned to events A and B using the Einstein measurement protocol (43), so long as both are sufficiently small compared with the scale of inhomogeneities in the velocity field. To confirm the invariant nature of the construct in (44) one must pay careful attention to observational times as distinct from protocol times and distances, and this must be done separately for each observer. This can be tedious. We now demonstrate this for the situation illustrated in Fig. 3. 0 = v0 , By definition the speed of P 0 according to P is v00 = DB /TB and so vR 0 where TB and DB are the protocol time and distance for event B for observer P P )2 = T 2 − 1 D 2 since according to (43). Then using (44) P would find that (τAB B c2 B v 02

P )2 = (1 − R )T 2 = (t0 )2 where the both TA = 0 and DA =0, and whence (τAB B B c2 last equality follows from the time dilation effect on the P 0 clock, since t0B is the

150

P (v0 = 0)

Figure 3:

P 0 (v00 )

tr H Y HHγ T TB

HH

γ

H H 0 *   B (tB )

  te 

A

DB

Here T − D is the spacetime construct (from the Einstein measurement protocol) of a special observer P at rest wrt space, so that v0 = 0. Observer P 0 is moving with speed v00 as determined by observer P , and therefore with speed 0 vR = v00 wrt space. Two light pulses are shown, each travelling at speed c wrt both P and space. Event A is when the observers pass, and is also used to define zero time for each for convenience.

D

time of event B according to that clock. Then TB is also the time that P 0 would 0 compute for event B when correcting for the time-dilation effect, as the speed vR 0 0 of P through the quantum foam is observable by P . Then TB is the ‘common time’ for event B assigned by both observers. For P 0 we obtain directly, also from P 0 )2 = (T 0 )2 − 1 (D 0 )2 = (t0 )2 , as D 0 = 0 and T 0 = t0 . (43) and (44), that (τAB B B B B B B c2 Whence for this situation P 2 P0 2 ) , (45) (τAB ) = (τAB and so the construction (44) is an invariant. While so far we have only established the invariance of the construct (44) when one of the observers is at rest in space, it follows that for two observers P 0 and P 00 both in absolute motion it follows that they also agree on the invariance of (44). This is easily seen by using the intermediate step of a stationary observer P: P0 2 P 2 P 00 2 (τAB ) = (τAB ) = (τAB ) . (46) Hence the protocol and Lorentzian absolute motion effects result in the construction in (44) being indeed an invariant in general. This is a remarkable and subtle result. For Einstein this invariance was a fundamental assumption, but here it is a derived result, but one which is nevertheless deeply misleading. Explicitly indicating small quantities by ∆ prefixes, and on comparing records retrospectively, an ensemble of nearby observers agree on the invariant ∆τ 2 = ∆T 2 −

1 ∆D2 , c2

(47)

for any two nearby events. This implies that their individual patches of spacetime records may be mapped one into the other merely by a change of coordinates, and that collectively the spacetime patches of all may be represented by one

151

pseudo-Riemannian manifold, where the choice of coordinates for this manifold is arbitrary, and we finally arrive at the invariant ∆τ 2 = gµν (x) ∆xµ ∆xν ,

(48)

with xµ = {D1 , D2 , D3 , T }. Eqn. (48) is invariant under the Lorentz transformations x0µ = Lµν xν , (49) where, for example for relative motion in the x direction, Lµν is specified by x − vt , 1 − v 2 /c2

x0 = p

y 0 = y,

z 0 = z,

t − vx/c2 t0 = p 1 − v 2 /c2

(50)

So absolute motion and special relativity effects, and even Lorentz symmetry, are all compatible: a possible preferred frame is hidden by the Einstein measurement protocol. The experimental question is then whether or not a supposed preferred frame actually exists or not — can it be detected experimentally? The answer is that there are now eight such consistent experiments. The notion that the special relativity formalism requires that the speed of light be isotropic, that it be c in all frames, has persisted for most of the last century. The actual situation is that it only requires that the round trip speed be invariant. This means that the famous Einstein light speed postulate is actually incorrect. This is discussed in [29, 30, 31, 32, 33].

8 Deriving the General Relativity Formalism As discussed above the generalised Dirac equation gives rise to a trajectory determined by (25), which may be obtained by extremising the time-dilated elapsed time (32). Z

τ [r0 ] =

v2 dt 1 − R c2

!1/2

(51)

This happens because of the Fermat least-time effect for quantum matter waves: only along the minimal time trajectory do the quantum waves remain in phase under small variations of the path. This again emphasises that gravity is a quantum effect. We now introduce a spacetime mathematical construct according to the metric ds2 = dt2 − (dr − v(r, t)dt)2 /c2 = gµν dxµ dxν (52)

152

Then according to this metric the elapsed time in (51) is s

Z

τ=

dt gµν

dxµ dxν , dt dt

(53)

and the minimisation of (53) leads to the geodesics of the spacetime, which are thus equivalent to the trajectories from (51), namely (25). Hence by coupling the Dirac spinor dynamics to the 3-space dynamics we derive the geodesic formalism of General Relativity as a quantum effect, but without reference to the HilbertEinstein equations for the induced metric. Indeed in general the metric of this induced spacetime will not satisfy these equations as the dynamical space involves the α-dependent dynamics, and α is missing from GR. So why did GR appear to succeed in a number of key tests where the Schwarzschild metric was used? The answer is provided by identifying the induced spacetime metric corresponding to the in-flow in (3) outside of a spherical matter system, such as the earth. Then (52) becomes 1 ds2 = dt2 − 2 (dr + c

s

2GM (1 + r

α 2

+ ..)

dt)2 −

1 2 2 r (dθ + sin2 (θ)dφ2 ), c2

(54)

Making the change of variables t → t0 and r → r0 = r with 2 t0 = t − c

s

2GM (1+ α2 + . . .)r 4 GM (1+ α2 + . . .) + tanh−1 c2 c3

s

2GM (1+ α2 + . . .) c2 r (55)

this becomes (and now dropping the prime notation) 2GM (1 + α2 + ..) 1 1− dt2 − 2 r2 (dθ2 + sin2 (θ)dφ2 ) 2 c r c !

ds2 =

− c2

dr2 !. 2GM (1 + α2 + ..) 1− c2 r

(56)

which is one form of the the Schwarzschild metric but with the α-dynamics induced effective mass shift. Of course this is only valid outside of the spherical matter distribution, as that is the proviso also on (3). As well the above particular change of coordinates also introduces spurious singularities at the event horizon4 , 4

The event horizon of (56) is at a different radius from the actual event horizon of the black hole solutions that arise from (1).

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but other choices do not do this. Hence in the case of the Schwarzschild metric the dynamics missing from both the Newtonian theory of gravity and General Relativity is merely hidden in a mass redefinition, and so didn’t affect the various standard tests of GR, or even of Newtonian gravity. Note that as well we see that the Schwarzschild metric is none other than Newtonian gravity in disguise, except for the mass shift. While we have now explained why the GR formalism appeared to work, it is also clear that this formalism hides the manifest dynamics of the dynamical space, and which has also been directly detected in gas-mode interferometer and coaxial-cable experiments. Nevertheless we now show [1] that in the limit α → 0 the induced metric in (52), with v from (1), satisfies the Hilbert-Einstein equations so long as we use relativistic corrections for the matter density on the RHS of (1). This means that (1) is consistent with for example the binary pulsar data - the relativistic aspects being associated with the matter effects upon space and the relativistic effects of the matter in motion through the dynamical 3-space. The agreement of GR with the pulsar data is implying that the α-dependent effects are small in this case, unlike in black holes and spiral galaxies. The GR equations are 1 8πG Gµν ≡ Rµν − Rgµν = 2 Tµν , 2 c

(57)

where Gµν is the Einstein tensor, Tµν is the energy-momentum tensor, Rµν = α Rµαν and R = g µν Rµν and g µν is the matrix inverse of gµν . The curvature tensor is ρ Rµσν = Γρµν,σ − Γρµσ,ν + Γρασ Γαµν − Γραν Γαµσ , (58) where Γαµσ is the affine connection Γαµσ

1 = g αν 2



∂gνµ ∂gνσ ∂gµσ + − σ µ ∂x ∂x ∂xν



.

(59)

Let us substitute the metric in (52) into (57) using (58) and (59). The various components of the Einstein tensor are then found to be X

G00 =

vi Gij vj − c2

i,j=1,2,3

Gi0 = −

X

X

G0j vj − c2

j=1,2,3 2

Gij vj + c Gi0 ,

X

vi Gi0 + c2 G00 ,

i=1,2,3

Gij = Gij ,

i, j = 1, 2, 3.

j=1,2,3

where the Gµν are given by G00 =

1 1 ((trD)2 − tr(D2 )), Gi0 = G0i = − (∇ × (∇ × v))i , i = 1, 2, 3. 2 2

154

(60)

Gij

=

d 1 1 (Dij − δij trD) + (Dij − δij trD)trD − δij tr(D2 ) + (ΩD − DΩ)ij , dt 2 2 i, j = 1, 2, 3. (61)

In vacuum, with Tµν = 0, we find from (57) and (60) that Gµν = 0 implies that Gµν = 0. We see that the Hilbert-Einstein equations demand that (trD)2 − tr(D2 ) = 0

(62)

but it is these terms in (1) that explain the various gravitational anomalies. This simply corresponds to the fact that GR does not permit the ‘dark matter’ effect, and this happens because GR was forced to agree with Newtonian gravity, in the appropriate limits, and that theory also has no such effect. As well in GR the energy-momentum tensor Tµν is not permitted to make any reference to absolute linear motion of the matter; only the relative motion of matter or absolute rotational motion is permitted, contrary to the experiments. It is very significant to note that the above exposition of the GR formalism for the metric in (52) is exact. Then taking the trace of the Gij equation in (61) we obtain, also exactly, and in the case of zero vorticity, and outside of matter so that Tµν = 0, ∂ (∇.v) + ∇.((v.∇)v) = 0 (63) ∂t which is the Newtonian ‘velocity field’ formulation of Newtonian gravity outside of matter, as in (1) but with α = β = 0. So GR turns out to be Newtonian gravity in a grossly overstructured mathematical formalism.

9 Experimental and Observational Phenomena I We now briefly review the extensive range of light speed experiments that have detected that the speed of light is not isotropic - the speed is different in different directions when measured in a laboratory experiment on earth, as predicted by the generalised Maxwell equations, Sect.5. The most famous of these experiments was that of Michelson and Morley in 1887. Contrary to often repeated claims, this experiment decisively detected the anisotropy. The cause of the misunderstanding surrounding this experiment is that the Newtonian based theory Michelson used for the calibration of the experiment is simply wrong, and of course not unexpectedly. Clearly as the Michelson interferometer is a 2nd order v/c experiment its calibration requires a ‘relativistic’ analysis, in particular one must take account of arm contractions and also the Fresnel drag effect. The

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Michelson-Morley fringe shift data then gives a speed in excess of 300km/s, as first discovered by Cahill and Kitto in 2002 [17].

9.1 Anisotropy of the Speed of Light That the speed of light in vacuum is the same in all directions, i.e. isotropic, for all observers has been taken as a critical assumption in the standard formulation of fundamental physics, and was introduced by Einstein in 1905 as one of his key postulates when formulating his interpretation of Special Relativity. The need to detect any anisotropy has challenged physicists from the 19th century to the present day, particularly following the Michelson-Morley experiment of 1887. The problem arose when Maxwell in 1861 successfully computed the speed of light c from his unified theory of electric and magnetic fields: but what was the speed c relative to? There have been many attempts to detect any supposed light-speed anisotropy and there have so far been 8 successful and consistent such experiments, and as well numerous unsuccessful experiments, i.e. experiments in which no anisotropy was observed. The reasons for these different outcomes is now understood: any light-speed anisotropy produces not only an expected ‘direct’ effect, being that which is expected to produce a ‘signal’, but also affects the very physical structure of the apparatus, and with this effect usually overlooked in the design of some detectors. In some designs these effects exactly cancel. As already stated there is overwhelming evidence from 8 experiments that the speed of light is anisotropic, and with a large anisotropy at the level of 1 part in 103 : so these experiments show that a dynamical 3-space exists, and that the spacetime concept was only a mathematical construct - it does not exist as an entity of reality, it has no ontological significance. These developments have lead to a new physics in which the dynamics of the 3-space have been formulated, together with the required generalisations of the Maxwell equations (as first suggested by Hertz in 1890 [14]), and of the Schr¨odinger and Dirac equations, which have lead to the new emergent theory and explanation of gravity, with numerous confirmations of that theory from the data from black hole systematics, light bending, spiral galaxy rotation anomalies, bore hole anomalies, etc. This data has revealed that the coupling constant for the self-interaction of the dynamical 3-space is none other than the fine structure constant ≈ 1/137 [39, 40, 41, 42], which suggests an emerging unified theory of quantum matter and a quantum foam description of the dynamical 3-space. The most influential of the early attempts to detect any anisotropy in the speed of light was the Michelson-Morley experiment of 1887, [2]. Despite that, and its influence on physics, its operation was only finally understood in 2002 [16,

156

17, 18]. The problem has been that the Michelson interferometer has a major flaw in its design, when used to detect any light-speed anisotropy effect5 . To see this requires use of Special Relativity effects. The Michelson interferometer compares the round-trip light travel time in two orthogonal arms, by means of interference fringe shifts measuring time differences, as the device is rotated. However if the device is operated in vacuum, any anticipated change in the total travel times caused by the light travelling at different speeds in the outward and inward directions is exactly cancelled by the Fitzgerald-Lorentz mirror-supporting-arm contraction effect - a real physical effect. Of course this is precisely how Fitzgerald and Lorentz independently arrived at the idea of the length contraction effect. In vacuum this means that the round-trip travel times in each arm do not change during rotation. This is the fatal design flaw that has confounded physics for over 100 years. However the cancellation of a supposed change in the round-trip travel times and the Lorentz contraction effect is merely an incidental flaw of the Michelson interferometer. The critical observation is that if we have a gas in the light path, the round-trip travel times are changed, but the Lorentz arm-length contraction effect is unchanged, and then these effects no longer exactly cancel. Not surprisingly the fringe shifts are now proportional to n − 1, where n is the refractive index of the gas. Of course with a gas present one must also take account of the Fresnel drag effect, because the gas itself is in absolute motion. This is an important effect, so large in fact that it reverses the sign of the time differences between the two arms, although in operation that is not a problem. As well, since for example for air n = 1.00029 at STP, the sensitivity of the interferometer is very low. Nevertheless the Michelson-Morley experiment as well as the Miller interferometer experiment of 1925/1926 [3] were done in air, which is why they indeed observed and reported fringe shifts. As well Illingworth [19] and Joos [20] used helium gas in the light paths in their Michelson interferometers; taking account of that brings their results into agreement with those of the air interferometer experiment, and so confirming the refractive index effect. Jaseja et al. [21] used a He-Ne gas mixture of unknown refractive index, but again detected fringe shifts on rotation. A re-analysis of the data from the above experiments, particularly from the enormous data set of Miller, has revealed that a large lightspeed anisotropy had been detected from the very beginning of such experiments, where the speed is some 430 ± 20km/s - this is in excess of 1 part in 103 , and the Right Ascension and Declination of the direction was determined by Miller [3] long ago. We also briefly review the RF coaxial cable speed experiments of 5

Which also severely diminishes its use in long-baseline terrestrial interferometers built to detect gravitational waves.

157

Torr and Kolen [22], DeWitte [23] and Cahill [24], which agree with the gas-mode Michelson interferometer experiments.

9.2 Michelson Gas-mode Interferometer Let us first consider the new understanding of how the Michelson interferometer works. This brilliant but very subtle device was conceived by Michelson as a means to detect the anisotropy of the speed of light, as was expected towards the end of the 19th century. Michelson used Newtonian physics to develop the theory and hence the calibration for his device. However we now understand that this device detects 2nd order effects in v/c to determine v, and so we must take account of relativistic effects. However the application and analysis of data from various Michelson interferometer experiments using a relativistic theory only occurred in 2002, some 97 years after the development of Special Relativity by Einstein, and some 115 years after the famous 1887 experiment. As a consequence of the necessity of using relativistic effects it was discovered in 2002 that the gas in the light paths plays a critical role, and that we finally understand how to calibrate the device, and we also discovered, some 76 years after the 1925/26 Miller experiment, what determines the calibration constant k that Miller had determined using the Earth’s rotation speed about the Sun to set the calibration. This, as we discuss later, has enabled us to now appreciate that gas-mode Michelson interferometer experiments have confirmed the reality of the Fitzgerald-Lorentz length contraction effect: in the usual interpretation of Special Relativity this effect, and others, is usually regarded as an observer dependent effect, an illusion induced by the spacetime. But the experiments are to the contrary showing that the length contraction effect is an actual observer-independent dynamical effect, as Fitzgerald and Lorentz had proposed. The Michelson interferometer compares the change in the difference between travel times, when the device is rotated, for two coherent beams of light that travel in orthogonal directions between mirrors; the changing time difference being indicated by the shift of the interference fringes during the rotation. This effect is caused by the absolute motion of the device through 3-space with speed v, and that the speed of light is relative to that 3-space, and not relative to the apparatus/observer. However to detect the speed of the apparatus through that 3-space gas must be present in the light paths for purely technical reasons. The post relativistic-effects theory for this device is remarkably simple. Consider here only the case where the arms are parallel/anti-parallel to the direction of absolute motion. The relativistic Fitzgerald-Lorentz contraction effect causes the arm AB

158

C ?

L (a)

(b) 6

-

-

A ? D



L

-

B

C C  C  C v  C  CW  C α C  C-  B A1 A2C C D

Figure 4: Schematic diagrams of the Michelson Interferometer, with beamsplitter/mirror at A and mirrors at B and C on arms from A, with the arms of equal length L when at rest. D is a screen or detector. In (a) the interferometer is at rest in space. In (b) the interferometer is moving with speed v relative to space in the direction indicated. Interference fringes are observed at the detector D. If the interferometer is rotated in the plane through 90o , the roles of arms AC and AB are interchanged, and during the rotation shifts of the fringes are seen in the case of absolute motion, but only if the apparatus operates in a gas. By measuring fringe shifts the speed v may be determined. parallel to the absolute velocity to be physically contracted to length (see Fig.4) s

L|| = L

1−

v2 . c2

(64)

The time tAB to travel AB is set by V tAB = L|| + vtAB , while for BA by V tBA = L|| − vtBA , where V = c/n is the speed of light, with n the refractive index of the gas present. For simplicity we ignore here the Fresnel drag effect, an effect caused by the gas also being in absolute motion, see [1]. The Fresnel drag effect is actually large, and results in a change of sign in (67) and (68). For the total ABA travel time we then obtain tABA = tAB + tBA

2LV = 2 V − v2

s

1−

v2 . c2

(65)

For travel in the AC direction we have, from the Pythagoras theorem for the right-angled triangle in Fig.4 that (V tAC )2 = L2 + (vtAC )2 and that tCA = tAC . Then for the total ACA travel time 2L tACA = tAC + tCA = √ . (66) V 2 − v2 Then the difference in travel time is (n2 − 1) L v 2 v4 ∆t = + O c c2 c4

159

!

.

(67)

after expanding in powers of v/c. This clearly shows that the interferometer can only operate as a detector of absolute motion when not in vacuum (n = 1), namely when the light passes through a gas, as in the early experiments (in transparent solids a more complex phenomenon occurs). A more general analysis [1] with the arms at angle θ to v gives ∆t = k 2

LvP2 cos(2(θ − ψ)), c3

(68)

where ψ specifies the direction of v projected onto the plane of the interferometer relative to the local meridian, and where k 2 ≈ n(n2 − 1). Neglect of the relativistic Fitzgerald-Lorentz contraction effect gives k 2 ≈ n3 ≈ 1 for gases, which is essentially the Newtonian theory that Michelson used. However the above analysis does not correspond to how the interferometer is actually operated. That analysis does not actually predict fringe shifts for the field of view would be uniformly illuminated, and the observed effect would be a changing level of luminosity rather than fringe shifts. As Miller knew, the mirrors must be made slightly non-orthogonal with the degree of non-orthogonality determining how many fringe shifts were visible in the field of view. Miller experimented with this effect to determine a comfortable number of fringes: not too few and not too many. Hicks [27] developed a theory for this effect – however it is not necessary to be aware of the details of this analysis in using the interferometer: the non-orthogonality reduces the symmetry of the device, and instead of having period of 180◦ the symmetry now has a period of 360◦ , so that to (68) we must add the extra term a cos(θ − β) in ∆t = k 2

L(1 + eθ)vP2 cos(2(θ − ψ)) + a(1 + eθ) cos(θ − β) + f c3

(69)

The term 1 + eθ models the temperature effects, namely that as the arms are uniformly rotated, one rotation taking several minutes, there will be a temperature induced change in the length of the arms. If the temperature effects are linear in time, as they would be for short time intervals, then they are linear in θ. In the Hick’s term the parameter a is proportional to the length of the arms, and so also has the temperature factor. The term f simply models any offset effect. Michelson and Morley and Miller took these two effects into account when analysing his data. The Hick’s effect is particularly apparent in the Miller and Michelson-Morley data. The interferometers are operated with the arms horizontal. Then in (69) θ is the azimuth of one arm relative to the local meridian, while ψ is the azimuth of the absolute motion velocity projected onto the plane of the interferometer,

160

with projected component vP . Here the Fitzgerald-Lorentz contraction is a real dynamical effect of absolute motion, unlike the Einstein spacetime view that it is merely a spacetime perspective artifact, and whose magnitude depends on the choice of observer. The instrument is operated by rotating at a rate of one rotation over several minutes, and observing the shift in the fringe pattern through a telescope during the rotation. Then fringe shifts from six (Michelson and Morley) or twenty (Miller) successive rotations are averaged to improve the signal to noise ratio, and the average sidereal time noted.

9.3 Michelson-Morley Experiment 1887 Page 340 of the Michelson-Morley 1887 paper reporting the observed fringe shifts is reproduced in Fig.5. Each row of the table is the average from six successive rotations. In the graphs Michelson and Morley are noting that the fringe shifts are much smaller than expected. But they were using Newtonian physics to calibrate the device. We now know that the detector is nearly 2000 times less sensitive than given by that calibration, and that these fringe shifts correspond to a speed in excess of 300km/s. Michelson and Morley implicitly assumed the Newtonian value k=1, while Miller used an indirect method to estimate the value of k, as he understood that the Newtonian theory was invalid, but had no other theory for the interferometer. Of course the Einstein postulates, as distinct from Special Relativity, have that absolute motion has no meaning, and so effectively demands that k = 0. Using k = 1 gives only a nominal value for vP , being some 8–9 km/s for the Michelson and Morley experiment, and some 10 km/s from Miller; the difference arising from the different latitudes of Cleveland and Mt. Wilson, and from Michelson and Morley taking data at limited times. The results from fitting the form in (69) to the data is shown in Fig.6. Most significantly we see that the projected speed and direction vary considerably for the same times on successive days. This effect was seen in later experiments. These are the ‘gravitational waves’ of the induced metric in (52). So we now understand that Michelson and Morley in 1887 detected a dynamical 3-space, and one in which the 3-space velocity fluctuations, the ‘gravitational waves’, were indeed apparent.

9.4 Miller Experiment 1925/26 The Michelson and Morley air-mode interferometer fringe shift data was based upon a total of only 36 rotations in July 1887, revealing the nominal speed of some 8–9 km/s when analysed using the prevailing but incorrect Newtonian theory which has k = 1 in (69), and this value was known to Michelson and Morley.

161

Figure 5: Page 340 from the 1887 Michelson-Morley paper [2] showing the table of observed fringe shifts, measured here in divisions of the telescope screw thread, and which is analysed using (69) with the results shown in Fig.6.

162

Figure 6: Analysis of the Michelson-Morley fringe shift data from the table in Fig.5. The plots are for the sidereal times and days indicated, and each plot arises from averaging six successive rotations, i.e. only 36 rotations were performed in July 1887. The data was fitted with (69) by a 6 parameter least-squares-fit by varying vP , ψ, a, β, e and f . Only vP and ψ are of physical interest, and are shown in each plot. ψ is measured clockwise from North. After these parameters have been determined the Hicks and temperature terms were subtracted from
the data, and plotted above together with the cos 2(θ − ψ) expression. This makes the fringe shifts more easily seen. We see that four of the plots show a good fit to the expected form, while the other two give a poor fit. We also see that at the same time on successive days the speed and direction are significantly different. These are ‘gravitational wave’ effects, and were seen in later experiments as well.

163

Figure 7: Typical Miller rotation-induced fringe shifts from average of 20 rotations, measured every 22.5◦ , in fractions of a wavelength ∆λ/λ, vs arm azimuth θ(deg), measured clockwise from North, from Cleveland Sept. 29, 1929 16:24 UT; 11:29 hrs average local sidereal time. The curve is the best fit using the form in (69), and then subtracting the Hick’s cos(θ − β) and temperature terms from the data. Best fit gives ψ = 158◦ , or 22◦ measured from South, and a projected speed of vP = 315 km/s. This plot shows the high quality of the Miller fringe shift observations. In the 1925/26 run of observations the rotations were repeated some 8,000 times.

Including the Fitzgerald-Lorentz dynamical contraction effect as well as the effect of the gas present as in (69) we find that nair = 1.00029 gives k 2 = 0.00058 for air, which explains why the observed fringe shifts were so small. They rejected their own data on the sole but spurious ground that the value of 8 km/s was smaller than the speed of the Earth about the Sun of 30km/s. What their result really showed was that (i) absolute motion had been detected because fringe shifts of the correct form, as in (69), had been detected, and (ii) that the theory giving k 2 = 1 was wrong, that Newtonian physics had failed. Michelson and Morley in 1887 should have announced that the speed of light did depend of the direction of travel, that the speed was relative to an actual physical 3-space. However contrary to their own data they concluded that absolute motion had not been detected. This has had enormous implications for fundamental theories of space and time over the last 100 years. It was Miller [3] who recognised that in the 1887 paper the theory for the Michelson interferometer must be wrong. To avoid using that theory Miller introduced the scaling factor k, even though he had no theory for its value. He then used the effect of the changing vector addition of the Earth’s orbital velocity and the absolute galactic velocity of the solar system to determine the numerical value of k, because the orbital motion modulated the data, as shown in Fig.8. By making some 8,000 rotations of the interferometer at Mt. Wilson in 1925/26

164

August

40

April

40 20 0 - 20 - 40

20 0 - 20 - 40 0

5

10

15

20

25

0

40 September

40

20

20

0

0

- 20

- 20

- 40

- 40 0

5

10

15

20

5

10

15

20

15

20

February

0

5

10

Figure 8: Miller azimuths ψ, measured from south and plotted against sidereal time in hrs, showing both data and best fit of theory giving v = 433 km/s in the direction (α = 5.2hr , δ = −670 ), using n = 1.000226 appropriate for the altitude of Mt. Wilson. The variation form month to month arises from the orbital motion of the earth about the sun: in different months the vector sum of the galactic velocity of the solar system with the orbital velocity and sun in-flow velocity is different. As shown in Fig.9 DeWitte using a completely different experiment detected the same direction and speed.

Miller determined the first estimate for k and for the absolute linear velocity of the solar system. Fig.7 shows typical data from averaging the fringe shifts from 20 rotations of the Miller interferometer, performed over a short period of time, and clearly shows the expected form in (69). In Fig.7 the fringe shifts during rotation are given as fractions of a wavelength, ∆λ/λ = ∆t/T , where ∆t is given by (69) and T is the period of the light. Such rotation-induced fringe shifts clearly show that the speed of light is different in different directions. The claim that Michelson interferometers, operating in gas-mode, do not produce fringe shifts under rotation is clearly incorrect. But it is that claim that lead to the continuing belief, within physics, that absolute motion had never been detected, and that the speed of light is invariant. The value of ψ from such rotations together lead to plots like those in Fig.8, which show ψ from the 1925/1926 Miller [3] interferometer data for four different months of the year, from which the RA = 5.2 hr is readily apparent. While the orbital motion of the Earth about the Sun slightly affects the RA in each month, and Miller used this effect to determine the value of

165

k, the new theory of gravity required a reanalysis of the data , revealing that the solar system has a large observed galactic velocity of some 420±30 km/s in the direction (RA = 5.2 hr, Dec =−67◦ ). This is different from the speed of 369 km/s in the direction (RA = 11.20 hr, Dec =−7.22◦ ) extracted from the Cosmic Microwave Background (CMB) anisotropy, and which describes a motion relative to the distant universe, but not relative to the local 3-space. The Miller velocity is explained by galactic gravitational in-flows [1]. An important implication of the new understanding of the Michelson interferometer is that vacuum-mode resonant cavity experiments should give a null effect, as is the case [28].

9.5 Other Gas-mode Michelson Interferometer Experiments Two old interferometer experiments, by Illingworth [19] and Joos [20], used helium, enabling the refractive index effect to be recently confirmed, because for helium, with n = = 1.000036, we find that k 2 = 0.00007. Until the refractive index effect was taken into account the data from the helium-mode experiments appeared to be inconsistent with the data from the air-mode experiments; now they are seen to be consistent. Ironically helium was introduced in place of air to reduce any possible unwanted effects of a gas, but we now understand the essential role of the gas. The data from an interferometer experiment by Jaseja et al. [21], using two orthogonal masers with a He-Ne gas mixture, also indicates that they detected absolute motion, but were not aware of that as they used the incorrect Newtonian theory and so considered the fringe shifts to be too small to be real, reminiscent of the same mistake by Michelson and Morley. While the Michelson interferometer is a 2nd order device, as the effect of absolute motion is proportional to (v/c)2 , as in (69), but 1st order devices are also possible and the coaxial cable experiments described next are in this class.

9.6 Coaxial Cable Speed of EM Waves Anisotropy Experiments Rather than use light travel time experiments to demonstrate the anisotropy of the speed of light, another technique is to measure the one-way speed of radio waves through a coaxial electrical cable. While this not a direct ‘ideal’ technique, as then the complexity of the propagation physics comes into play, it provides not only an independent confirmation of the light anisotropy effect, but also one which takes advantage of modern electronic timing technology.

166

20 15 10

ns

5 0 -5 - 10 - 15 0

10

20

30 40 Sidereal Time

50

60

70

Figure 9: (a) Variations in twice the one-way travel time, in ns, for an RF signal to travel 1.5 km through a coaxial cable between Rue du Marais and Rue de la Paille, Brussels. An offset has been used such that the average is zero. The cable has a North-South orientation, and the data is ± difference of the travel times for NS and SN propagation. The sidereal time for maximum effect of ∼ 5 hr and ∼ 17 hr (indicated by vertical lines) agrees with the direction found by Miller. Plot shows data over 3 sidereal days and is plotted against sidereal time. The fluctuations are evidence of turbulence of gravitational waves. (b) Shows the speed fluctuations, essentially ‘gravitational waves’ observed by De Witte in 1991 from the measurement of variations in the RF coaxial-cable travel times. This data is obtained from that in (a) after removal of the dominant effect caused by the rotation of the Earth. Ideally the velocity fluctuations are three-dimensional, but the De Witte experiment had only one arm. This plot is suggestive of a fractal structure to the velocity field. This is confirmed by the power law analysis in [11, 23].

9.7 Torr-Kolen Coaxial Cable Anisotropy Experiment The first one-way coaxial cable speed-of-propagation experiment was performed at the Utah University in 1981 by Torr and Kolen. This involved two rubidium clocks placed approximately 500 m apart with a 5 MHz radio frequency (RF) signal propagating between the clocks via a buried EW nitrogen-filled coaxial cable maintained at a constant pressure of 2 psi. Torr and Kolen found that, while the round-trip speed time remained constant within 0.0001% c, as expected from Sect.7, variations in the one-way travel time were observed. The maximum effect occurred, typically, at the times predicted using the Miller galactic velocity, although Torr and Kolen appear to have been unaware of the Miller experiment. As well Torr and Kolen reported fluctuations in both the magnitude, from 1–3 ns, and the time of maximum variations in travel time. These effects are interpreted as arising from the turbulence in the flow of space past the Earth.

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9.8 De Witte Coaxial Cable Anisotropy Experiment During 1991 Roland De Witte performed a most extensive RF coaxial cable travel-time anisotropy experiment, accumulating data over 178 days. His data is in complete agreement with the Michelson-Morley 1887 and Miller 1925/26 interferometer experiments. The Miller and De Witte experiments will eventually be recognised as two of the most significant experiments in physics, for independently and using different experimental techniques they detected essentially the same velocity of absolute motion. But also they detected turbulence in the flow of space past the Earth — none other than gravitational waves. The De Witte experiment was within Belgacom, the Belgium telecommunications company. This organisation had two sets of atomic clocks in two buildings in Brussels separated by 1.5 km and the research project was an investigation of the task of synchronising these two clusters of atomic clocks. To that end 5MHz RF signals were sent in both directions through two buried coaxial cables linking the two clusters. The atomic clocks were caesium beam atomic clocks, and there were three in each cluster: A1, A2 and A3 in one cluster, and B1, B2, and B3 at the other cluster. In that way the stability of the clocks could be established and monitored. One cluster was in a building on Rue du Marais and the second cluster was due south in a building on Rue de la Paille. Digital phase comparators were used to measure changes in times between clocks within the same cluster and also in the one-way propagation times of the RF signals. At both locations the comparison between local clocks, A1-A2 and A1-A3, and between B1-B2, B1-B3, yielded linear phase variations in agreement with the fact that the clocks have not exactly the same frequencies together with a short term and long term phase noise. But between distant clocks A1 toward B1 and B1 toward A1, in addition to the same linear phase variations, there is also an additional clear sinusoidal-like phase undulation with an approximate 24 hr period of the order of 28 ns peak to peak, as shown in Fig. 9. The experiment was performed over 178 days, making it possible to measure with an accuracy of 25 s the period of the phase signal to be the sidereal day (23 hr 56 min). Changes in propagation times were observed over 178 days from June 3 to November 27, 1991. A sample of the data, plotted against sidereal time for just three days, is shown in Fig.9. De Witte recognised that the data was evidence of absolute motion but he was unaware of the Miller experiment and did not realise that the Right Ascensions for minimum/maximum propagation time agreed almost exactly with that predicted using the Miller’s direction (RA = 5.2 hr, Dec =−67◦ ). In fact De Witte expected that the direction of absolute motion should have been in the CMB direction, but that would have given the data a to-

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tally different sidereal time signature, namely the times for maximum/minimum would have been shifted by 6 hrs. The declination of the velocity observed in this De Witte experiment cannot be determined from the data as only three days of data are available. The De Witte data is analysed in [24] and assuming a declination of 60◦ S a speed of 430 km/s is obtained, in good agreement with the Miller speed and Michelson-Morley speed. So a different and non-relativistic technique is confirming the results of these older experiments. This is dramatic. De Witte reported the sidereal time of the ‘zero’ cross-over time, that is in Fig.9 for all 178 days of data. That showed that the time variations are correlated with sidereal time and not local solar time. A least-squares best fit of a linear relation to that data gives that the cross-over time is retarded, on average, by 3.92 minutes per solar day. This is to be compared with the fact that a sidereal day is 3.93 minutes shorter than a solar day. So the effect is certainly galactic and not associated with any daily thermal effects, which in any case would be very small as the cable is buried. Miller had also compared his data against sidereal time and established the same property, namely that the diurnal effects actually tracked sidereal time and not solar time, and that orbital effects were also apparent, with both effects apparent in Fig.8. The dominant effect in Fig.9 is caused by the rotation of the Earth, namely that the orientation of the coaxial cable with respect to the average direction of the flow past the Earth changes as the Earth rotates. This effect may be approximately unfolded from the data leaving the gravitational waves shown in Fig.9, [11, 23]. This is the first evidence that the velocity field describing the flow of space has a complex structure, and is indeed fractal. The fractal structure, i. e. that there is an intrinsic lack of scale to these speed fluctuations, is demonstrated by binning the absolute speeds and counting the number of speeds within each bin, as discussed in [11, 23]. The Miller data also shows evidence of turbulence of the same magnitude. So far the data from four experiments, namely Miller, Torr and Kolen, De Witte and Cahill, show turbulence in the flow of space past the Earth. This is what can be called gravitational waves. This can be understood by noting that fluctuations in the velocity field induce ripples in the mathematical construct known as spacetime, as in (52). Such ripples in spacetime are known as gravitational waves.

9.9 Cahill Coaxial Cable Anisotropy Experiment During 2006 Cahill [24] performed another RF coaxial cable anisotropy experiment. This detector uses a novel timing scheme that overcomes the limitations associated with the two previous coaxial cable experiments. The intention in such

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Figure 10: Top: De Witte data, with sign reversed, from the first sidereal day in Fig.9. This data gives a speed of approximately 430km/s. The data appears to have been averaged over more than 1hr, but still shows wave effects. Middle: Absolute projected speeds vP in the Miller experiment plotted against sidereal time in hours for a composite day collected over a number of days in September 1925. Maximum projected speed is 417 km/s. The data shows considerable fluctuations. The dashed curve shows the non-fluctuating variation expected over one day as the Earth rotates, causing the projection onto the plane of the interferometer of the velocity of the average direction of the space flow to change. If the data was plotted against solar time the form is shifted by many hours. Note that the min/max occur at approximately 5 hrs and 17 hrs, as also seen by De Witte and in the Cahill experiment. Bottom: Data from the Cahill experiment [24] for one sidereal day on approximately August 23, 2006. We see similar variation with sidereal time, and also similar wave structure. This data has been averaged over a running 1hr time interval to more closely match the time resolution of the Miller experiment. These fluctuations are believed to be real wave phenomena of the 3-space. The new experiment gives a speed of 418 km/s. We see remarkable agreement between all three experiments.

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experiments is simply to measure the one-way travel time of RF waves propagating through the coaxial cable. To that end one would apparently require two very accurate clocks at each end, and associated RF generation and detection electronics. However the major limitation is that even the best atomic clocks are not sufficiently accurate over even a day to make such measurements to the required accuracy, unless the cables are of order of a kilometre or so in length, and then temperature control becomes a major problem. The issue is that the time variations are of the order of 25 ps per 10 meters of cable. To measure that requires time measurements accurate to, say, 1 ps. But atomic clocks have accuracies over one day of around 100 ps, implying that lengths of around 1 kilometre would be required, in order for the effect to well exceed timing errors. Even then the atomic clocks must be brought together every day to resynchronise them, or use De Witte’s method of multiple atomic clocks. The new experiment is based on the notion that optical fibers respond differently to coaxial cable with respect to the speed of propagation of EM radiation. Some results are shown in Fig.10 (bottom), and show the earth rotation and wave effects.

9.10 Cahill Optical Fiber Anisotropy Experiment To measure v(r, t) more easily and more accurately a new optical-fiber detector design has been developed by Cahill [25]. The device is very small, very cheap and easily assembled from readily available opto-electronic components. The schematic layout of the detector is given in Fig.11. The detector relies on the phenomenon where the 3-space velocity v(r, t) affects differently the light travel times in the optical fibers, depending on the projection of v(r, t) along the fiber directions. The differences in the light travel times are measured by means of the interference effects in the beam joiner. The difference in travel times is given by ∆t = k 2

LvP2 cos(2θ) c3

where

(70)

(n2 − 1)(2 − n2 ) n is the instrument calibration constant, obtained by taking account of the three key effects: (i) the different light path trajectories, (ii) Lorentz contraction of the fibers, an effect depending on the angle of the fibers to the flow velocity, and (iii) the refractive index effect, including the Fresnel drag effect. Only if n 6= 1 is there a net effect, otherwise when n = 1 the various effects actually cancel. So in this regard the Michelson interferometer has a serious design flaw. This problem has k2 =

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o

ARM 1 He-Ne laser

2x2 beamsplitter --

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data photodiode 2x2 beamlogger detector joiner

ARM 2   o o



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100mm

Figure 11:

Schematic layout of the interferometric optical-fiber light-speed anisotropy/gravitational wave detector. This is essentially an optical-fiber version of the Michelson interferometer, see Fig.4. Coherent 633nm light from the a He-Ne Laser is split into two lengths of single-mode polarisation preserving fibers by the 2x2 beam splitter. The two fibers take different directions, ARM1 and ARM2, after which the light is recombined in the 2x2 beam joiner, in which the phase differences lead to interference effects that are indicated by the outgoing light intensity, which is measured in the photodiode detector/amplifier, and then recorded in the data logger. The length of one straight section is 100mm, which is the center to center spacing of the plastic turners. The relative travel times, and hence the output light intensity, are affected by the varying speed and direction of the flowing 3-space, by affecting differentially the speed of the light, and hence the net phase difference between the two arms.

been overcome by using optical fibers. Here n = 1.462 at 633nm is the effective refractive index of the single-mode optical fibers (Fibercore SM600, temperature coefficient 5 × 10−2 fs/mm/C). Here L ≈ 200mm is the average effective length of the two arms, and vP (r, t) is the projection of v(r, t) onto the plane of the detector, and the angle θ is that of the projected velocity onto the arm. The interferometer operates by detecting the travel time difference between the two arms as given by (70). The cycle-averaged light intensity emerging from the beam joiner is given by I(t) ∝



2

Re(E1 + E2 eiω(τ +∆t )

ω(τ + ∆t) = 2|E| cos 2 ≈ a + b∆t 2



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2

(71)

Figure 12: D1 photodiode output voltage data (mV), recorded every 5 secs, from 5 successive days, starting September 22, 2007, plotted against local Adelaide time (UT= local time + 9.5hrs). Day sequence may be determined by identifying identical values at 0 and 24hrs. Dominant minima and maxima is earth rotation effect. Fluctuations from day to day are evident as are fluctuations during each day - these are caused by wave effects in the flowing space. Changes in RA cause changes in timing of min/max, while changes in magnitude are caused by changes in declination and/or speed. Blurring effect is caused by laser noise. These day plots correspond to the plots in Fig.10, there plotted against local sidereal time, and also inverted.

Here Ei are the electric field amplitudes and have the same value as the fiber splitter/joiner are 50%-50% types, and having the same direction because polarisation preserving fibers are used, ω is the light angular frequency and τ is a travel time difference caused by the light travel times not being identical, even when ∆t = 0, mainly because the various splitter/joiner fibers will not be identical in length. The last expression follows because ∆t is small, and so the detector operates in a linear regime, in general, unless τ has a value equal to modulo(T ), where T is the light period. The main temperature effect in the detector, so long as a temperature uniformity is maintained, is that τ will be temperature dependent. The temperature coefficient for the optical fibers gives an effective fractional fringe shift error of 3 × 10−2 /mm/C, for each mm of length difference. The photodiode detector output voltage V (t) is proportional to I(t), and so finally linearly related to ∆t. The detector calibration constants a and b depend on k, τ and the laser intensity and are unknown at present.

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Figure 13:

Photodiode data (mV) on October 4, 2007, from detectors D1 and D2 operating simultaneously with D2 located 1.1km due north of D1. A low-pass FFT filter (f ≤ 0.25mHz, Log10 [f(mHz)] ≤ -0.6) was used to remove laser noise. D1 arm is aligned 50 anti-clockwise from local meridian, while D2 is aligned 110 anti-clockwise from local meridian. The alignment offset between D1 and D2 causes the dominant earth-rotation induced minima to occur at different times, with that of D2 at t = 7.6hrs delayed by 0.8hrs relative to D1 at t = 6.8hrs. This is a fundamental test of the detection theory and of the phenomena. As well the data shows a simultaneous sub-mHz gravitational wave correlation at t ≈ 8.8hrs and of duration ≈ 1hr. This is the first observed correlation for spatially separated gravitational wave detectors. Two other wave effects (at t ≈ 6.5hrs in D2 and t ≈ 7.3hrs in D1) seen in one detector are masked by the stronger earth-rotation induced minimum in the other detector.

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Two detectors were used with each detector located inside a sealed air-filled bucket located inside an insulated container containing some 90kg of water for temperature stabilisation. Using two detectors enabled the confirmation of expected phenomena, as a test of the detector theory, and also enabled the simultaneous observations of wave phenomena. Detector D1 was in the School of Chemistry, Physics and Earth Sciences, with an arm orientation of 50 anticlockwise to the local meridian. Detector D2 was located 1.1km North of D1 in the Australian Science and Mathematics School. This detector had an arm orientation of 110 anti-clockwise to the local meridian. Fig.12 shows data from D1 over 5 days. Fig.13 shows an effect caused by D1 and D2 having different arm orientations and, as well, a simultaneous sub-mHz gravitational wave correlation at t ≈ 8.8hrs and of duration ≈ 1hr. This is the first observed correlation for spatially separated gravitational wave detectors. There are now at least 11 detections of the velocity field v(r, t) and of these 6 have observed the 3-space wave/turbulence effect.

10 Experimental and Observational Phenomena II 10.1 Gravitational Phenomena We have shown above that the dynamics of 3-space involves two constants: G and α. When generalising the Schr¨odinger and Dirac equations to take account of this 3-space we discovered that we arrive at an explanation for the phenomenon of gravity including the equivalence principle, as well as an explanation for the spacetime formalism. Here we explore various consequences of this new explanation for gravity particularly those effects which reveal the effects of the α-dependent dynamics, in particular the bore hole anomaly which gives us the best estimate for the value of α from several bore hole experiments. The dynamical 3-space also gives a completely new account of black holes; an account completely different from the putative black holes of GR. In particular these new black holes generate an acceleration g that varies essentially as 1/r, rather than as 1/r2 as in Newtonian gravity (NG) and GR. This is a dramatic difference. It explains immediately the rotation of spiral galaxies, for which the rotation speed is essen√ tially constant at the outer limits, whereas NG and GR predict a 1/ r Keplerian form. It was this dramatic failure of NG and GR, and also in galactic clusters, that lead to the introduction of ‘dark matter’ - to generate a greater gravitational acceleration. The new theory of 3-space does not need this ‘dark matter’. The black hole phenomena is complex, with minimal black holes induced by matter, to primordial black holes that attract matter. In the former case, and where the

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matter, in the form of stars and so on, has an essentially spherically symmetric distribution, it is possible to compute the effective mass of the induced minimal black holes. Observational data from these systems confirms the prediction. Other effects discussed are the gyroscope precession effect caused by the vorticity of the flow of 3-space past the earth. Finally we also discuss the cosmological Hubble expansion that arises from the 3-space dynamics. This gives an excellent parameter-free account of the redshift data from supernovae and gamma-ray bursts. GR requires ‘dark energy’ to fit that data, so here we see that the new 3-space dynamics does away with the need for ‘dark energy’. Not discussed herein are anomalies in the Cavendish-like experiments to determine G [38], the gravitational lensing effects predicted by the generalised Maxwell equations, and also a re-analysis of the precession of elliptical orbits, particularly that of Mercury, and various other gravitational effects, see [1].

10.2 Bore Hole Anomaly and the Fine Structure Constant We now show that the Airy method [34] originally proposed for measuring G actually gives a technique for determining the value of α from earth based bore hole gravity measurements. For a time-independent velocity field (7) may be written in the integral form |v(r)|2 = 2G

Z

d3 r 0

ρ(r0 ) + ρDM (r0 ) . |r − r0 |

(72)

When the matter density of the earth is assumed to be spherically symmetric, and that the velocity field is now radial6 (72) becomes v(r)2 =

8πG r

Z

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s2 [ρ(s) + ρDM (s)] ds + 8πG

Z

0

∞

s [ρ(s) + ρDM (s)] ds,

(73)

r

where, with v 0 = dv(r)/dr, v2 vv 0 + 2r2 r

α ρDM (r) = 8πG

!

.

(74)

Iterating (73) once we find to 1st order in α that α ρDM (r) = 2 2r 6

∞

Z

sρ(s)ds + O(α2 ),

r

This in-flow is additional to the observed velocity of the earth through 3-space.

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(75)

so that in spherical systems the ‘dark matter’ effect is concentrated near the centre, and we find that the total ‘dark matter’ is Z ∞ Z 4πα ∞ 2 α 2 r ρDM (r)dr = r ρ(r)dr + O(α2 ) = M + O(α2 ) (76) MDM ≡ 4π 2 2 0 0 where M is the total amount of (actual) matter. Hence to O(α) MDM /M = α/2 independently of the matter density profile. This turns out to be a very useful property as complete knowledge of the density profile is then not required in order to analyse observational data. As seen in Fig.14 the singular behaviour of both v and g means that there is a black hole7 singularity at r = 0. From (2), which is also the acceleration of matter [11], the gravity acceleration8 is found to be, to 1st order in α, and using that ρ(r) = 0 for r > R, where R is the radius of the earth,  α  (1 + )GM    2  , r > R, rZ2 ! g(r) = (77) Z R Z r  4πG 2παG r  0 2 0 0   s ρ(s )ds ds, r < R. s ρ(s)ds +  r2 r2 s 0 0 This gives Newton’s ‘inverse square law’ for r > R, even when α 6= 0, which explains why the 3-space self-interaction dynamics did not overtly manifest in the analysis of planetary orbits by Kepler and then Newton. However inside the earth (77) shows that g(r) differs from the Newtonian theory, corresponding to α = 0, as in Fig.14, and it is this effect that allows the determination of the value of α from the Airy method. Expanding (77) in r about the surface, r = R, we obtain, to 1st order in α and for an arbitrary density profile, but not retaining any density gradients at the surface,

g(r) =

 GN M 2GN M   (r − R),  2 −   R R3

r > R, (78)

    GN M 2GN M α   − 4π(1 − )GN ρ (r − R),  2 − 3

R

R

2

r
where ρ is the matter density at the surface, M is the total matter mass of the earth, and where we have defined α GN ≡ (1 + )G. (79) 2 7

These are called black holes because there is an event horizon, but in all other aspects differ from the black holes of General Relativity. 8 We now use the convention that g(r) is positive if it is radially inward.

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Upper plot shows speeds from numerical iterative solution of (73) for a solid sphere with uniform density and radius r = 0.5 for (i) upper curve the case α = 0 corresponding to Newtonian gravity, and (ii) lower curve with α = 1/137. These solutions ony differ significantly near r = 0. Middle plot shows matter density and ‘dark matter’ density ρDM , from (74), with arbitrary scales. Lower plot shows the acceleration from (2) for (i) the Newtonian in-flow from the upper plot, and (ii) from the α = 1/137 case. The difference is only significant near r = 0. The accelerations begin to differ just inside the surface of the sphere at r = 0.5, according to (81). This difference is the origin of the bore hole g anomaly, and permits the determination of the value of α from observational data. This generic singular-g behaviour, at r = 0, is seen in the earth, in globular clusters and in galaxies.

The corresponding Newtonian gravity expression is obtained by taking the limit α → 0,  2GN M GN M   − (r − R), r > R,   2  R R3 gN (r) = (80)     GN M 2GN M   − 4πGN ρ (r − R), r < R  2 − R R3 Assuming Newtonian gravity (80) then means that from the measurement of difference between the above-ground and below-ground gravity gradients, namely 4πGN ρ, and also measurement of the matter density, permit the determination of GN . This is the basis of the Airy method for determining GN [34]. When analysing the bore hole data it has been found [35, 36] that the observed difference of the gravity gradients was inconsistent with 4πGN ρ in (80), in that it was not given by the laboratory value of GN and the measured matter density. This is known as the bore hole g anomaly and which attracted much interest in the 1980’s. The bore hole data papers [35, 36] report the discrepancy, i.e. the anomaly or the gravity residual as it is called, between the Newtonian prediction and the measured below-earth gravity gradient. Taking the difference between (78) and (80), assuming the same unknown value of GN in both, we obtain an expression for the gravity residual (

∆g(r) ≡ gN (r) − g(r) =

0, r > R, 2παGN ρ(r − R), r < R.

(81)

When α 6= 0 we have a two-parameter theory of gravity, and from (78) we see that measurement of the difference between the above ground and below ground gravity gradients is 4π(1− α2 )GN ρ, and this is not sufficient to determine both GN and α, given ρ, and so the Airy method is now understood not to be a complete measurement by itself, i.e. we need to combine it with other measurements. If we now use laboratory Cavendish experiments to determine GN , then from the bore hole gravity residuals we can determine the value of α, as already indicated in [39, 40]. These Cavendish experiments can only determine GN up to corrections of order α/4, simply because the analysis of the data from these experiments assumed the validity of Newtonian gravity [1]. So the analysis of the bore hole residuals will give the value of α up to O(α2 ) corrections, which is consistent with the O(α) analysis reported above. Gravity residuals from a bore hole into the Greenland Ice Shelf were determined down to a depth of 1.5 km by Ander et al. [35] in 1989. The observations were made at the Dye 3 2033 m deep bore hole, which reached the basement

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Figure 15:

The data shows the gravity residuals for the Greenland Ice Shelf [35] Airy measurements of the g(r) profile, defined as ∆g(r) = gN ewton −gobserved , and measured in mGal (1mGal = 10−3 cm/s2 ) and plotted against depth in km. The bore hole effect is that Newtonian gravity and the new theory differ only beneath the surface, provided that the measured above surface gravity gradient is used in both theories. This then gives the horizontal line above the surface. Using (81) we obtain α−1 = 137.9 ± 5 from fitting the slope of the data, as shown. The non-linearity in the data arises from modelling corrections for the gravity effects of the irregular sub ice-shelf rock topography.

Figure 16: Gravity residuals from two of the Nevada bore hole experiments [36] that give a best fit of α−1 = 136.8 ± 3 on using (81). Some layering of the rock is evident.

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rock. This bore hole is 60 km south of the Arctic Circle and 125 km inland from the Greenland east coast at an elevation of 2530 m. It was believed that the ice provided an opportunity to use the Airy method to determine GN , but now it is understood that in fact the bore hole residuals permit the determination of α, given a laboratory value for GN . Various steps were taken to remove unwanted effects, such as imperfect knowledge of the ice density and, most dominantly, the terrain effects which arises from ignorance of the profile and density inhomogeneities of the underlying rock. The bore hole gravity meter was calibrated by comparison with an absolute gravity meter. The ice density depends on pressure, temperature and air content, with the density rising to its average value of ρ = 920 kg/m3 within some 200 m of the surface, due to compression of the trapped air bubbles. This surface gradient in the density has been modelled by the author, and is not large enough the affect the results. The leading source of uncertainty was from the gravitational effect of the bedrock topography, and this was corrected for using Newtonian gravity. The correction from this is actually the cause of the non-linearity of the data points in Fig.15. A complete analysis would require that the effect of this rock terrain be also computed using the new theory of gravity, but this was not done. Using GN = 6.6742 × 10−11 m3 s−2 kg−1 , which is the current CODATA value, we obtain from a least-squares fit of the linear term in (81) to the data points in Fig.15 that α−1 = 137.9±5, which equals the value of the fine structure constant α−1 = 137.036 to within the errors, and for this reason we identify the constant α in (81) as being the fine structure constant. The first analysis [39, 40] of the Greenland Ice Shelf data incorrectly assumed that the ice density was 930 kg/m3 which gave α−1 = 139 ± 5. However trapped air reduces the standard ice density to the ice shelf density of 920 kg/m3 , which brings the value of α immediately into better agreement with the value of α = e2 /¯hc known from quantum theory. Thomas and Vogel [36] performed another bore hole experiment at the Nevada Test Site in 1989 in which they measured the gravity gradient as a function of depth, the local average matter density, and the above ground gradient, also known as the free-air gradient. Their intention was to test the extracted Glocal and compare with other values of GN , but of course using the Newtonian theory. The Nevada bore holes, with typically 3 m diameter, were drilled as a part of the U.S. Government tests of its nuclear weapons. The density of the rock is measured with a γ − γ logging tool, which is essentially a γ-ray attenuation measurement, while in some holes the rock density was measured with a coreing tool. The rock density was found to be 2000 kg/m3 , and is dry. This is the density used in the analysis herein. The topography for 1 to 2 km beneath the surface is dominated by

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Figure 17: The data shows Log10 [MBH ] for the black hole masses MBH for a variety of spherical matter systems with masses M , plotted against Log10 [M ], in solar masses M0 . The straight line is the prediction from (83) with α = 1/137. See [42] for references to the data.

a series of overlapping horizontal lava flows and alluvial layers. Gravity residuals from two of the bore holes are shown in Figs.16. All gravity measurements were corrected for the earth’s tide, the terrain on the surface out to 168 km distance, and the evacuation of the holes. The gravity residuals arise after allowing for, using Newtonian theory, the local lateral mass anomalies but assumed that the matter beneath the holes occurs in homogeneous ellipsoidal layers. We see in Fig.16 that the gravity residuals are linear with depth, where the density is the average value of 2000 kg/m3 , but interspersed by layers where the residuals show non-linear changes with depth. It is assumed here that these non-linear regions are caused by variable density layers. So in analysing this data we have only used the linear regions, and a simultaneous least-squares fit of the slope of (81) to the slopes of these four linear regions gives α−1 = 136.8 ± 3, which again is in extraordinary agreement with the value of 137.04 from quantum theory. Here we again used GN = 6.6742 × 10−11 m3 s−2 kg−1 , as for the Greenland data analysis. Zumberge et al. [37] performed an extensive underwater Airy experiment, but failed to measure the above water g, so their results cannot be analysed in the above manner.

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10.3 Black Hole Masses and the Fine Structure Constant Equation (1) (with β = −α) has ‘black hole’ solutions. The generic term ‘black hole’ is used because they have a compact closed event horizon where the inflow speed relative to the horizon equals the speed of light, but in other respects they differ from the putative black holes of General Relativity - in particular their gravitational acceleration is not inverse square law. The evidence is that it is these new ‘black holes’ from (1) that have been detected. There are two categories: (i) an in-flow singularity induced by the flow into a matter system, such as, herein, a spherical galaxy or globular cluster. These black holes are termed minimal black holes, as their effective mass is minimal, (ii) primordial naked black holes which then attract matter. These result in spiral galaxies, and the effective mass of the black hole is larger than required merely by the matter induced in-flow. These are therefore termed non-minimal black holes. These explain the rapid formation of structure in the early universe, as the gravitational acceleration is approximately 1/r rather than 1/r2 . This is the feature that also explains the so-called ‘dark matter’ effect in spiral galaxies. We consider now the minimal black holes. Equation (1) has exact analytic ‘black hole’ solutions where ρ = 0 (actually a one-parameter family - but we write in this form for comparison with the next section)  1/2  α 1 Rs 2  1 v(r) = K  + (82)  r Rs r where the 1/r term can only arise if matter is present, and the 2nd term is the ‘black hole’ effect. The consequent ‘black hole’ contribution to the total acceleration can be attributed to an effective mass MDM , which we now also call MBH . To O(α) this effective mass is independent of the matter density profile, and is given by (76), Z

MBH = MDM = 4π

∞

r2 ρDM (r)dr =

0

α M + O(α2 ) 2

(83)

This solution is applicable to the black holes at the centre of spherical star systems, where we identify MDM as MBH . So far black holes in 19 spherical star systems have been detected and together their masses are plotted in Fig.17 and compared with (83) [41, 42]. This result applies to any spherically symmetric matter distribution. This means that the bore hole anomaly is indicative of an in-flow singularity at the centre of the earth. This contributes some 0.4% of the effective mass of the earth,

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20

25

Data shows the non-Keplerian rotation-speed curve vO for the spiral galaxy NGC 3198 in km/s plotted against radius in kpc/h. Lower curve is the rotation curve from the Newtonian theory for an exponential disk, √ which decreases asymptotically like 1/ r. The upper curve shows the asymptotic form from (85), with the decrease determined by the small value of α. This asymptotic form is caused by the primordial black holes at the centres of spiral galaxies, and which play a critical role in their formation. The spiral structure is caused by the rapid in-fall towards these primordial black holes.

as defined by Newtonian gravity. However in star systems this minimal black hole effect is more apparent, and we label MDM as MBH . Essentially even in the nonrelativistic regime the Newtonian theory of gravity, with its ‘universal’ Inverse Square Law, is deeply flawed.

10.4 Spiral Galaxies and the Rotation Anomaly Equation (82) gives also a direct explanation for the spiral galaxy rotation anomaly. For a non-spherical system numerical solutions of (1) are required, but sufficiently far from the centre we find an exact non-perturbative two-parameter class of analytic solutions as in (82). There K and Rs are arbitrary constants in the ρ = 0 region, but whose values are determined by matching to the solution in the matter region. Here Rs characterises the length scale of the non-perturbative part of this expression, and K depends on α, G and details of the matter distribution. From (4) and (82) we obtain a replacement for the Newtonian ‘inverse square law’,    α 2 K 1 α Rs 2  g(r) = (84)  2 + , 2 r 2rRs r

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in the asymptotic limit. The centripetal acceleration relation for circular orbits p vO (r) = rg(r) gives a ‘universal rotation-speed curve’ 

vO (r) =

K 1 α  + 2 r 2Rs



Rs r

α

1/2

2 

(85)

Because of the α dependent part this rotation-velocity curve falls off extremely slowly with r, as is indeed observed for spiral galaxies. An example is shown in Fig.18. It was the inability of the Newtonian gravity and GR to explain these observations that led to the notion of ‘dark matter’. So ‘dark matter’ is not a part of reality. For the spatial flow in (82) we may compute the effective ‘dark matter’ density from (74)   (1 − α)α K 2 Rs 2+α/2 ρDM (r) = (86) 16πG Rs3 r We see the standard 1/r2 behaviour usually attributed to ‘dark matter’ in spiral galaxies. It should be noted that the Newtonian component of (82) does not contribute, and that ρDM (r) is exactly zero in the limit α → 0. So supermassive black holes and the spiral galaxy rotation anomaly are all α-dynamics phenomena.

10.5 Lense-Thirring Effect and the GPB Gyroscope Experiment The Gravity Probe B (GP-B) satellite experiment was launched in April 2004. It has the capacity to measure the precession of four on-board gyroscopes to unprecedented accuracy [44, 45, 46, 47]. Such a precession is predicted by GR, with two components (i) a geodetic precession, and (ii) a ‘frame-dragging’ precession known as the Lense-Thirring effect. The latter is particularly interesting effect induced by the rotation of the earth, and described in GR in terms of a ‘gravitomagnetic’ field. According to GR this smaller effect will give a precession of 0.042 arcsec per year for the GP-B gyroscopes. Here we show that GR and the new theory make very different predictions for the ‘frame-dragging’ effect, and so the GP-B experiment will be able to decisively test both theories. While predicting the same earth-rotation induced precession, the new theory has an additional much larger ‘frame-dragging’ effect caused by the observed translational motion of the earth. As well the new theory explains the ‘frame-dragging’ effect in terms of vorticity in a ‘substratum flow’. Herein the magnitude and signature of this

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Figure 19: Shows the earth (N is up) and vorticity vector field component ω ~ induced by the rotation of the earth, as in (87). The polar orbit of the GP-B satellite is shown, S is the gyroscope starting spin orientation, directed towards the guide star IM Pegasi, RA = 22h 530 2.2600 , Dec = 160 500 28.200 , and VE is the vernal equinox.

S VE

new component of the gyroscope precession is predicted for comparison with data from GP-B when it becomes available. Here we consider one difference between the two theories, namely that associated with the vorticity part of (12), leading to the ‘frame-dragging’ or LenseThirring effect. In GR the vorticity field is known as the ‘gravitomagnetic’ field B = −c ω ~ . In both GR and the new theory the vorticity is given by (10) but with a key difference: in GR vR is only the rotational velocity of the matter in the earth, whereas in the 3-space dynamics vR is the vector sum of the rotational velocity and the translational velocity of the earth through the substratum. First consider the common but much smaller rotation induced ‘frame-dragging’ or vorticity effect. Then vR (r) = w × r in (12), where w is the angular velocity of the earth, giving G 3(r.L)r − r2 L ω ~ (r) = 4 2 , (87) c 2r5 where L is the angular momentum of the earth, and r is the distance from the centre. This component of the vorticity field is shown in Fig.19. Vorticity may be detected by observing the precession of the GP-B gyroscopes. The vorticity term in (87) leads to a torque on the angular momentum S of the gyroscope, Z

~τ =

d3 rρ(r) r × (~ ω (r) × vR (r)),

(88)

where ρ is its density, and where vR is used here to describe the rotation of the gyroscope. Then dS = ~τ dt is the change in S over the time interval dt. In the

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S VE

v

Figure 20: Shows the earth (N is up) and the much larger vorticity vector field component ω ~ induced by the translation of the earth, as in (90). The polar orbit of the GP-B satellite is shown, and S is the gyroscope starting spin orientation, directed towards the guide star IM Pegasi, RA = 22h 530 2.2600 , Dec = 160 500 28.200 , VE is the vernal equinox, and V is the direction RA = 5.2h , Dec = −670 of the translational velocity vc .

above case vR (r) = s × r, where s is the angular velocity of the gyroscope. This gives 1 ~ ×S (89) ~τ = ω 2 ~ /2 is the instantaneous angular velocity of precession of the gyroscope. and so ω This corresponds to the well known fluid result that the vorticity vector is twice the angular velocity vector. For GP-B the direction of S has been chosen so that this precession is cumulative and, on averaging over an orbit, corresponds to some 7.7 × 10−6 arcsec per orbit, or 0.042 arcsec per year. GP-B has been superbly engineered so that measurements to a precision of 0.0005 arcsec are possible. However for the unique translation-induced precession if we use vR ≈ vC = 430 km/s in the direction RA = 5.2hr , Dec = −670 , namely ignoring the effects of the orbital motion of the earth, the observed flow past the earth towards the sun, and the flow into the earth, and effects of the gravitational waves, then (12) gives 2GM vC × r ω ~ (r) = . (90) c2 r3 This much larger component of the vorticity field is shown in Fig.20. The maximum magnitude of the speed of this precession component is ω/2 = gvC /c2 = 8 × 10−6 arcsec/s, where here g is the gravitational acceleration at the altitude of the satellite. This precession has a different signature: it is not cumulative, and is detectable by its variation over each single orbit, as its orbital average is zero, to first approximation. Fig.21 shows ∆Θ = |∆S(t)|/|S(0)| over one orbit, where,

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0.016 0.014

arcsec

0.012 0.01 0.008 0.006 0.004 0.002 0

20

40 60 orbit - minutes

80

Figure 21: Predicted variation of the precession angle ∆Θ = |∆S(t)|/|S(0)|, in arcsec, over one 97 minute GP-B orbit, from the vorticity induced by the translation of the earth, as given by (91). The orbit time begins at location S. Predictions are for the months of April, August, September and February, labeled by increasing dash length. The ‘glitches’ near 80 minutes are caused by the angle effects in (91). These changes arise from the effects of the changing orbital velocity of the earth about the sun. The GP-B expected angle measurement accuracy is 0.0005 arcsec.

as in general, Z

∆S(t) = 0

t

1 dt0 ω ~ (r(t0 )) × S(t0 ) ≈ 2

Z 0

t

1 dt0 ω ~ (r(t0 )) × S(0). 2 

(91)

Here ∆S(t) is the integrated change in spin, and where the approximation arises because the change in S(t0 ) on the RHS of (91) is negligible. The plot in Fig.21 shows this effect to be some 30× larger than the expected GP-B errors, and so easily detectable, if it exists as predicted herein. This precession is about the instantaneous direction of the vorticity ω ~ (r((t)) at the location of the satellite, and so is neither in the plane, as for the geodetic precession, nor perpendicular to the plane of the orbit, as for the earth-rotation induced vorticity effect. Because the yearly orbital rotation of the earth about the sun slightly effects vC [16], predictions for four months throughout the year are shown in Fig.21. Such yearly effects were first seen in the Miller [3] experiment, see Fig.8.

10.6 Cosmology: Expanding 3-Space and the Hubble Effect We now examine the predictions for the global expansion of the 3-space that follows from (1) (with β = −α). We shall see that the solution gives an excellent

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parameter-free fit to the supernovae and gamma-ray bursts magnitude - redshift data [48]. This implies that there is no need to have a cosmological constant or ‘dark energy’, which are required by GR in order to fit this data. These also lead to the prediction that the universe expansion will accelerate in the future. This effect is also not required by the new 3-space dynamics. So, like ‘dark matter’, ‘dark energy’ is an unnecessary and spurious notion. Let us now explore the expanding 3-space from (1). Critically, and unlike the GR-FLRW model, the 3-space expands even when the energy density is zero. Suppose that we have a radially symmetric effective density ρ(r, t), modelling EM radiation, matter, cosmological constant etc, and that we look for a radially symmetric time-dependent flow v(r, t) = v(r, t)ˆr from (1) (with β = −α). Then ∂v(r, t) v(r, t) satisfies the equation, with v 0 = , ∂r ∂ ∂t



2v vv 0 α + v 0 + vv 00 + 2 + (v 0 )2 + r r 4 

v 2 2vv 0 + r2 r

!

= −4πGρ(r, t)

(92)

Consider first the zero energy case ρ = 0. Then we have a Hubble solution v(r, t) = H(t)r, a centreless flow, determined by α H˙ + 1 + H2 = 0 4 



(93)

dH with H˙ = . We also introduce in the usual manner the scale factor a(t) dt 1 da according to H(t) = . We then obtain the solution a dt 1 t0 t = H0 ; a(t) = a0 α (1 + 4 )t t t0 

H(t) =

4/(4+α)

(94)

where H0 = H(t0 ) and a0 = a(t0 ). Note that we obtain an expanding 3-space even where the energy density is zero - this is in sharp contrast to the GR-FLRW model for the expanding universe, as shown below. We can write the Hubble function H(t) in terms of a(t) via the inverse function t(a), i.e. H(t(a)) and finally as H(z), where the redshift observed now, t0 , relative to the wavelengths at time t, is z = a0 /a − 1. Then we obtain H(z) = H0 (1 + z)1+α/4

(95)

To test this expansion we need to predict the relationship between the cosmological observables, namely the relationship between the apparent energy-flux magnitudes and redshifts. This involves taking account of the reduction in photon

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count caused by the expanding 3-space, as well as the accompanying reduction in photon energy. To that end we first determine the distance travelled by the light from a supernova or GRB event before detection. Using a choice of embeddingspace coordinate system with r = 0 at the location of a supernova/GRB event the speed of light relative to this embedding space frame is c + v(r(t), t), i.e. c wrt the space itself, as noted above, where r(t) is the embedding-space distance from the source. Then the distance travelled by the light at time t after emission at time t1 is determined implicitly by Z

t

r(t) =

dt0 (c + v(r(t0 ), t0 ),

(96)

t1

which has the solution on using v(r, t) = H(t)r Z

t

r(t) = ca(t) t1

dt0 . a(t0 )

(97)

This distance gives directly the surface area 4πr(t)2 of the expanding sphere and so the decreasing photon count per unit of that surface area. However also because of the expansion the flux of photons is reduced by the factor 1/(1 + z), simply because they are spaced further apart by the expansion. The photon flux is then given by LP FP = (98) 4πr(t)2 (1 + z) where LP is the source photon-number luminosity. However usually the energy flux is measured, and the energy of each photon is reduced by the factor 1/(1 + z) because of the redshift. Then the energy flux is, in terms of the source energy luminosity LE , LE LE FE = ≡ (99) 4πr(t)2 (1 + z)2 4πrL (t)2 which defines the effective energy-flux luminosity distance rL (t). Expressed in terms of the observable redshift z this gives an energy-flux luminosity effective distance Z z dz 0 rL (z) = (1 + z)r(z) = c(1 + z) (100) 0 0 H(z ) The dimensionless ‘energy-flux’ luminosity effective distance is then given by Z

dL (z) = (1 + z) 0

190

z

H0 dz 0 H(z 0 )

(101)

Figure 22: Plot of the scale factor R(t) vs t, with t = 0 being ‘now’ with R(0) = 1, for the four cases discussed in the text, and corresponding to the plots in Figs.23 and 24: (i) the upper curve is the ‘dark energy’ only case, resulting in an exponential acceleration at all times, (ii) the bottom curve is the matter only prediction, (iii) the 2nd highest curve (to the right of t = 0) is the best-fit ‘dark energy’ plus matter case showing a past deceleration and future exponential acceleration effect. The straight line plot is the dynamical 3-space prediction showing a slightly older universe compared to case (iii). We see that the best-fit ‘dark energy’ - matter curve essentially converges on the dynamical 3-space result. All plots have the same slope at t = 0, i.e. the same value of H0 . If the age of the universe is inferred to be some 14Gyrs for case (iii) then the age of the universe is also some 14Gyr for case (iv).

and the theory distance modulus is defined by µ(z) = 5 log10 (dL (z)) + m.

(102)

Using the Hubble expansion (95) in (101) and (102) we obtain the curve shown in Figs.23 and 24, yielding an excellent agreement with the supernovae and GRB data. Note that because α/4 is so small it actually has negligible effect on these plots. Hence the dynamical 3-space gives an immediate account of the universe expansion data, and does not require the introduction of a cosmological constant or ‘dark energy’, but which will be nevertheless discussed next. When the energy density is not zero we need to take account of the dependence of ρ(r, t) on the scale factor of the universe. In the usual manner we thus write ρ(r, t) =

ρm ρr + +Λ 3 R(t) R(t)4

(103)

for matter, EM radiation and the cosmological constant or ‘dark energy’ Λ, respectively, where the matter and radiation is approximated by a spatially uniform

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Figure 23: Hubble diagram showing the combined supernovae data from Davis et al. [49] using several data sets from Riess et al. (2007)[50] and Wood-Vassey et al. (2007)[51] (dots without error bars for clarity - see Fig.24 for error bars) and the Gamma-Ray Bursts data (with error bars) from Schaefer [52]. Upper curve is ‘dark energy’ only ΩΛ = 1, lowest curve is matter only Ωm = 1. Two middle curves show best fit of ‘dark energy’-matter and dynamical 3space prediction, and are essentially indistinguishable. However the theories make very different predictions for the future and for the age of the universe. We see that the best-fit ‘dark energy’ - matter curve essentially converges on the dynamical 3-space prediction.

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Figure 24:

Hubble diagram as in Fig.23 but plotted logarithmically to reveal details for z < 2, and without GRB data. Upper curve is ‘dark energy’ only ΩΛ = 1. Next curve is best fit of ‘dark energy’-matter. Lowest curve is matter only Ωm = 1. 2nd lowest curve is dynamical 3-space prediction.

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(i.e independent of r) equivalent matter density. We argue here that Λ - the dark energy density, like dark matter, is an unnecessary concept. Then (92) becomes for R(t)   ¨ α R˙ 2 R 4πG ρm ρr =− + 4 +Λ + R 4 R2 3 R3 R

(104)

giving 8πG R˙ 2 = 3



ρm α ρr + 2 + ΛR2 − R R 2 

Z

R˙ 2 dR R

(105)

In terms of R˙ 2 this has the solution !

8πG ΛR2 ρm ρr R = + + +bR−α/2 α α 3 (1 − 2 )R (1 − 4 )R2 (1 + α4 ) ˙2

(106)

which is easily checked by substitution into (105), and where b is an arbitrary integration constant. Finally we obtain from (106) Z

R

t(R) = R0

dR s

8πG 3



ρm ρr + 2 + ΛR2 + bR−α/2 R R

(107) 

where now we have re-scaled parameters ρm → ρm /(1 − α2 ), ρr → ρr /(1 − α4 ) and Λ → Λ/(1 + α4 ). When ρm = ρr = Λ = 0, (107) reproduces the expansion in (94), and so the density terms in (107) give the modifications to the dominant purely spatial expansion dynamics, which we have noted above already gives an excellent account of the data. From (107) we then obtain H(z)2 = H0 2 (Ωm (1 + z)3 + Ωr (1 + z)4 + ΩΛ + Ωs (1 + z)2+α/2 )

(108)

with Ωm + Ωr + ΩΛ + Ωs = 1.

(109)

Using the Hubble function (108) in (101) and (102) we obtain the plots in Figs.23 and 24 for four cases: (i) Ωm = 0, Ωr = 0, ΩΛ = 1, Ωs = 0, i.e a pure ‘dark energy’ driven expansion, (ii) Ωm = 1, Ωr = 0, ΩΛ = 0, Ωs = 0 showing that a matter only expansion is not a good account of the data, (iii) from a least squares fit with Ωs = 0 we find Ωm = 0.28, Ωr = 0, ΩΛ = 0.68 which led to the suggestion that ‘dark energy’ effect was needed to fix the poor fit from (ii), and finally (iv) Ωm = 0, Ωr = 0, ΩΛ = 0, Ωs = 1, as noted above, that the spatial expansion dynamics alone gives a good account of the data. Of course the EM

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radiation term Ωr is non-zero but small and determines the expansion during the baryogenesis initial phase, as does the spatial dynamics expansion term because of the α dependence. If the age of the universe is inferred to be some 14Gyrs for case (iii) then, as seen in Fig.22, the age of the universe is also some 14Gyr for case (iv). We see that the best-fit ‘dark energy’ - matter curve essentially converges on the dynamical 3-space result. The induced effective spacetime metric in (52) is for the Hubble expansion ds2 = gµν dxµ dxν = dt2 − (dr − H(t)r)dt)2 /c2

(110)

The occurrence of c has nothing to do with the dynamics of the 3-space - it is related to the geodesics of relativistic quantum matter, as shown in Sect.8. Changing variables r → R(t)r we obtain ds2 = gµν dxµ dxν = dt2 − R(t)2 dr2 /c2

(111)

which is the usual Friedman-Robertson-Walker (FRW) metric in the case of a flat spatial section. However when solving for R(t) using the Hilbert-Einstein GR equations the Ωs term (with α → 0) is usually only present when the spatial curvature is non-zero. So some problem appears to be present in the usual GR analysis of the FRW metric. However above we see that that term arises in fact even when the embedding space is flat.

11 Conclusions We have briefly reviewed the extensive evidence for a dynamical 3-space, with the minimal dynamical equation now known and confirmed by numerous experimental and observational data. This 3-space has been repeatedly detected since the Michelson-Morley experiment of 1887, and they also detected ‘gravitational waves’, which are just 3-space velocity fluctuations. As well the dynamical 3-space has been indirectly detected by means of the dynamical equation explaining diverse phenomena. We have shown that this equation has a Hubble expanding 3-space solution that in a parameter-free manner manifestly fits the recent supernovae and gamma-ray bursts redshift data. All of these successes imply that ‘dark energy’ and ‘dark matter’ are unnecessary notions. This Hubble solution leads to a uniformly expanding universe, and so without acceleration: the claimed acceleration is merely a spurious artifact related to the unnecessary ‘dark energy’ notion. This result gives an age for the universe of some 14Gyr, and resolves as well various problems such as the fine turning problem, the horizon problem and

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other difficulties in the current modelling of the universe. We have also shown why the spacetime formalism appeared to be so successful, despite having no ontological status. One key discovery has been that Newton’s theory of gravity is flawed, except in the very special case of planets in orbit about a sun, which is of course the restricted manifestation of gravity that was available to Newton. At a deeper l