[Michael C. Duffy, Joseph Levy] Ether Space-time a(BookZZ.org)

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: researches un-dertaken during the last 20 years have confirmed the existence of physical properties within space, even where it is devoid of ordinary matter. In addition to the well known properties of permittivity, permeability and the ability to transmit electromagnetic waves, other features have been more recently associated with the concept of space. These include the Casimir Effect and a significant amount of energy. 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. Ether theory plays a creative role even if given different names such as physical vacuum, fundamental plenum or cosmic sub-stratum. This second volume, as the first did, presents articles written by experienced physicists dealing with different aspects of the ether concept. One of the objectives of this series of books is to progressively disclose its properties. 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. 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, in-cluding names and addresses of contacts and sponsors, are given on the PIRT web site www.physicsfoundations.org

“Horsehead nebula”

Apeiron

Apeiron

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

------------------------------ 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

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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

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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 Missouri-Columbia, 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

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Contents of Volume 2 -1- Introduction Michael C. Duffy & Joseph Levy -3- 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] -37-

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]

-53- RELATIVITY AND AETHER THEORY A CRUCIAL DISTINCTION Joseph Levy 4 Square Anatole France, 91250 St Germain lès Corbeil, France

E-mail: : [email protected]

-67- The Dynamic Universe Zero-energy balance restores absolute time and space Tuomo Suntola Vasamatie 25, 02630 Espoo, Finland

E-mail: [email protected]

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-135- DYNAMICAL 3-SPACE: A REVIEW Reginald T. Cahill School of Chemistry, Physics and Earth Sciences,

Flinders University, Adelaide 5001, Australia

-201- Relativistic physics from paradoxes

to good sense - 1

F. Selleri Dipartimento di Fisica, Università di Bari

INFN, Sezione di Bari

-267- Photon-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 -323- Cosmological 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

-355-

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

James E. Beichler

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

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-417- On 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]

-433- CAN PHYSICS LAWS BE DERIVED FROM MONOGENIC FUNCTIONS

José B. Almeida Universidade do Minho, Physics Department Braga, Portugal, e-mail:[email protected] -473- 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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2

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 Machian-positivist 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 non-additive 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)=Rik

2 (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 123-124]. 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

28

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 co-ordinates) 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 Einsteinian-Minkowskian 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

32

reformulation of the relativity theory not only elucidates the relation between the mathematical formalism of the theory and physical reality and establishes an ether-based 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

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Einstein, A. 1920, Äther und Relativitätstheorie. Verlag von J. Springer, Berlin.

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Jánossy L. 1965, Theory and Practice of the Evaluation of Measurements. Clarendon Press, Oxford.

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36

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.

37

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.

38

2. TIME DILATION ACCORDING TO RELATIVITY THEORY Let us consider two inertial coordinates systems 0S (x,, y, z) and 1S (x’, y’ ,z’) receding from one another along the x-axis of the co-ordinate system 0S , and suppose that a light ray starts from a point M fixed to the coordinate system 1S , 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 1S , the two-way transit time of light along the rod is 2 1t = 2L/C. But, viewed from 0S , 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 0S will be labelled 02t .According to relativity, the speed of light is C in all inertial frames and in all directions of space. Let 01v refers to the real relative speed separating 0S and 1S (measured with non contracted standards). When the light ray has covered the distance AB, 1S has moved away from 0S a distance AM = 001tv

FIG 1. In the coordinate system 1S , the light ray travels from point M to the mirror B and, after reflection, returns to M. In 0S the signal starts from point A, and after reflection in B returns to A’. According to an observer attached to 0S the transit time of light 0t along AB is given

by: 220

201

20

2 LtvtC =− , therefore:

22

01

0/1 CvC

Lt−

= .

Replacing L/C by its value 1t this expression reduces to:

22

01

10

/1 Cvtt

−= .

S1 S0 B

x, x’ O O’ A M A’

39

This classical formula is interpreted as time dilation by special relativity, (an expression obtained because the position of the clock in 1S 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 0S and 1S 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 0S 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 0S , as 0t , we have:

220

201

20

2 LtvtC =− , and therefore:

22

01

0/1 CvC

Lt−

= .

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 2/122 )/1( −− Cv . 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 1S tick at a slower rate and display the reading appt1 = L/C. Thus,

22

01

10

/1 Cv

tt app

−= , (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, appt1 is not the true time in the co-ordinate system 1S , 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 1S and 2S receding with rectilinear uniform motion with respect to the co-ordinate system 0S 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 0S is inexistent and the time dilation formula between the systems 1S and 2S takes the form:

22

12

21

/1 CvtT

−= , (2)

where 12v refers to the relative speed between the coordinate systems 1S and 2S . In aether theory, things are very different. Let 01v , 02v and 12v 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 2S . We propose to compare the apparent times (displayed by the clocks attached to 1S and 2S ) which are needed by the light signal to achieve a

41

cycle (from M to B and to M again in the system 2S and from A to B and to A’ in the system 1S ).

FIG 2. The co-ordinate systems 1S and 2S recede from 0S along the common x-axis. The light ray travels along the rod MB which is at rest in 2S (from M to B and to M again). With respect to 1S 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 1S ). During a cycle of the signal, 2S has moved with respect to 1S a distance AA’. The real value of the one-way speed of light along AB is not equal to C since the coordinate system

1S 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 0t the two-way transit time of the light signal that would be displayed by clocks attached to the co-ordinate system 0S . Due to clock retardation the clock readings in 1S and 2S are related to 0t as follows:

220101 /1 Cvtt app −= , (3)

and

220202 /1 Cvtt app −= . (4)

(Note nevertheless that the true time, needed for half a cycle, measured with clocks not slowed down by motion, is 0t for all observers). From (3) and (4) we infer:

22

02

2201

21/1

/1

Cv

Cvtt appapp

−

−= . (5)

Assuming that 1/02 <<Cv , this expression reduces to first order, to:

S2 S1 B

x, x’, x’’ O’ O’’ A M A’

S0

O

42

)2(

211 01122

12

2

vvCv

t app

+− . (6)

Noting that (as in section 3.1.) CLtt app /22 == , this expression is different from the relativistic formula (2) which reduces to:

2

212

2

211

Cv

t

−. (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 1S 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 1S is therefore equal to the difference between 2 appt1 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 asΔ , the SDE affecting the clocks attached to the coordinate system 1S is:

2201 /1 Cv−Δ=δ .

The apparent (measured) two-way transit time of the signal (from A to B and to A’) is therefore:

220111 /122 CvtT appapp −Δ−= .

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 0l 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 0S it is reduced to

l = 22010 /1 Cv−l , (8)

which, according to aether theory, is the real length in the co-ordinate system 1S . The real time needed by the light signal to travel from A to A’ along the x’-axis is therefore:

01

22010

'

/1vC

Cvtraa −

−=l

,

where C - 01v is assumed to be the real speed of light in 1S 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:

01

22010

'

/1vC

Cvt ara +

−=l

.

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:

)(2/1 '' araraa tt + =22

01

0

/1 CvC −

l.

44

In the absence of clock retardation, the synchronism discrepancyΔ between the clocks A and A’ would be equal to the difference between the exact transit time

'raat of the signal from A to A’ and the apparent (measured) time )(2/1 '' araraa tt + :

=Δ01

22010 /1vC

Cv−−l

2201

0

/1 CvC −−

l=

2201

2001

1 CvCv−

l.

Due to clock retardation in the co-ordinate system 1S the SDE is reduced to

=δ 2001

Cv l

.

(b) Apparent transit time of light along the rod in the co-ordinate system 1S . 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:

220101 /122 Cvtt app −= .

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

2001

11 22C

vtT appappl

−= . (9)

Important remark Writing this expression in the form

200122

0101 /122C

vCvtT appl

−−= , (10)

and taking account of the fact that the measurements in 1S are made with a meter stick which is also contracted, the length AA’ is erroneously found equal to 0l . We shall therefore refer to 0l as appX1 . From (10) we obtain:

22

01

21011

0/1

/22

Cv

CXvTt appapp

−

+= . (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 appT1 and appX1 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 1S is made in two different points of the x’-axis).

45

-Taking account of expression (8) and of the fact that 00102 2)( tvv ×−=l where

02v and 01v are the real speeds of 2S and 1S with respect to 0S (obtained in the absence

of measurement distortions) we can express =δ 2001

Cv l

in the form

201

20102101 )(2

vCvvtv app

−

−=δ .

The apparent (measured) transit time of the signal in 1S (from A to B and to A’ again) is therefore:

201

20201

2

111 222vCvvCttT appappapp −

−=−= δ .

Now our objective is to compare the apparent (measured) transit times of the signal in frames 1S and 2S . From formula (5) we have:

201

20201

2

2202

2201

21/1

/122

vCvvC

Cv

CvtT appapp −

−

−

−= . (12)

From formula (12) we obtain successively:

)/1)(/1(22

2201

2202

20201

2

21CvCvC

vvCtT appapp−−

−=

202

201

2201

2202

40201

2

22vvCvCvC

vvCt app+−−

−=

0201222

0122

02020122

02201

40201

2

222

2vvCCvCvvvCvvC

vvCt app+−−−+

−=

20102

220201

20201

2

2)()(

2vvCvvC

vvCt app−−−

−=

20201

2

20102

2

2

)()(1

2

vvCvvC

t app

−−

−

=

appT12

2202012

20102

2

)1(

)(1

2

CvvC

vv

t app

−

−−

= . (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:

20201

010212 /1 Cvv

vvv app −−

= . (14)

With this notation formula (13) becomes:

appT1 2212

2

/1 Cv

t

app

app

−= . (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 2S 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 1S is at rest in the preferred aether frame, 01v = 0 and appv12 reduces to 02v . 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 1S and was moving relative to the co-ordinate system 0S .We shall now consider the opposite case: i.e. the rod is at rest with respect to 0S and the co-ordinate system 1S moves relative to 0S in the left direction, as the figure 3 shows.

47

FIG 3. The rod L=MB is at rest with respect to the co-ordinate system 0S , and the co-ordinate system 1S is moving relative to 0S in the left direction. According to conventional relativity, contrary to aether theory, nothing differentiates the co-ordinate systems 0S and 1S , because there is no preferred inertial frame; in other words, motion is only relative, and one can consider that 0S moves relative to 1S , in the same way as 1S is moving relative to 0S . Therefore SR predicts a complete symmetry between the frames: for example, a clock in 1S slows down with respect to a clock standing in 0S , but conversely a clock in 0S is subjected to slow down with respect to a clock in 1S . 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 0S is a co-ordinate system at rest in the preferred frame, and 1S a co-ordinate system moving at uniform speed in the left direction, see Fig 3. Since it is 1S which moves, the clocks in 1S tick slower than in 0S and we should have:

220101 /1 Cvtt −= = 22

01 /1 CvCL

− . (16)

Where 01v is the real speed between 0S and 1S , (measured with non-contracted meter sticks and clocks non slowed down by motion)

S0 S1 B

x, x’ O O’ A M A’

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 1S 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 0S , the light ray travels along the rod MB (from M to B and to M again), but viewed from 1S , 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 co-ordinate system 1S . In the absence of length contraction let us suppose that AA’ measures 0l . Taking account of the reduction of size, we have:

AA’= l = 22010 /1 Cv−l = 0012 tv .

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 lCCvC

CvvCvC

022

01

2201

0101 /1/1)11( ll

=−

=−−

++

.

(In this expression we have taken account of the slowing down of the clocks standing in the co-ordinate system 1S , 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 1S 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 0l 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:

01

22010 /1vC

Cv+−l

.

But due to the slowing down of clocks in the co-ordinate system 1S , the reading in the absence of synchronism discrepancy would be:

01

2201 /1vC

Cv+−l

.

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

49

22

01 /1 CvC −

l-

01

2201 /1vC

Cv+−l

=22

012

01

/1 CvCv−

l=

2201

20

201

/12

CvCtv

−.

. 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

220101 /122 Cvtt −=

220101 /122 Cvtt app −= +

2201

20

201

/12

CvCtv

−,

Which yields 22

01

01

/1 Cvtt app

−= (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.

51

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 co-ordinate 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 0S 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 γ/1 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 0S ).

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 readingτ 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

A

S0 S

O B

56

irrespective of their direction of motion). Of course, this would inform us whether S is at rest or in motion relative to 0S , 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 0t = 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 0S 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 22 /1 Cv−l 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 A B 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:

vCCvtrB −

−=

22 /1l

(where the suffix r means real)

57

rBt 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:

vCCvCvtrB −

−=−

)/1(/122

22 l

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:

22

22

/1)11(/12/1

CvCvCvCCvtBapp

−=

++

−−=

ll

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

CCvtBapp //1 22 l=− (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 C/l 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:

222

22

22

)/1)/1

/1(CvCv

CvCvCCv lll

=−−

−−

−=Δ

(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:

vCCvtrA +

−=

22 /1l

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:

vCCvCvtrA +

−=−

)/1(/122

22 l

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:

22

22

/1)11(/12/1

CvCvCvCCvtAapp

−=

−+

+−=

ll

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

CCvtAapp //1 22 l=− (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:

222

22

22

)/1)/1

/1('CvCv

CvCvCCv lll

−=−−

−+

−=Δ−=Δ

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 readingτ . 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:

2Cvl

+=Δ+ ττ

And the reading of clock A:

2Cvl

−=Δ− ττ

Let us now determine the real transit times At0 and Bt0 that would be displayed by clocks attached to frame 0S when the rockets reach points A and B. We have:

220 /1 Cvt B −=Δ+τ

and

59

220 /1 Cvt A −=Δ−τ

Thus:

22

2

0/1/

CvCvt B

−

+=

lτ (1)

And

22

2

0/1/

CvCvt A

−

−=

lτ (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 appv = l /τ in both sides, (where l2 is the apparent length of AB in S, measured with a contracted standard). Yet the apparent time corresponds to two different real times At0 and Bt0 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 AT and BT 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:

220 /)'(1 CvvtT BB +−=

22

22

2

/)'(1/1/ Cvv

CvCv

+−−

+=

lτ

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:

220 /)"(1 CvvtT AA −−=

22

22

2

/)"(1/1/ Cvv

CvCv

−−−

−=

lτ

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

60

We easily verify that

2

22

0

22

/)/1(/1'

CvCv

tCvv

B l

ll

+−

=−

=τ

(3)

and

2

22

0

22

/)/1(/1"

CvCv

tCvv

A l

ll

−−

=−

=τ

(4)

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

)///( 422222222222 CvCvCTT BA ll +−−== ττγ

222 / Cl−=τ (See the demonstration in appendix 1) For values of C<<τ/l we obtain:

)2/11()2/11( 2

2

22

2

Cv

CTT app

BA −=−≈= ττ

τ l (5)

For the usual transits whose speed is low compared to the speed of light, this expression approximatesτ , 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 readingτ 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

]''21[)/( 2

2

22

22222

Cv

Cvv

CvCvTB −−−+= lτγ

Where 2/122 )/1( −−= Cvγ Replacing v’ by its value given in (3), we obtain:

])/()/1(

)/()/1(21[)( 222

2222

22

22

2

22

222

CvCCv

CvCCvv

Cv

CvTB

l

l

l

ll

+−

−+−

−−+=ττ

τγ

])/1()/)(/1(2)()[( 2222

2222

22

22

22

22 Cv

CCvCv

Cv

Cv

Cv

Cv

−−+−−+−+=l

llll

τττγ

)///( 4222222222 CvCvC ll +−−= ττγ

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

2222 /CTB l−= τ

]""21[)/( 2

2

22

22222

Cv

Cvv

CvCvTA −+−−= lτγ

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

])/()/1(

)/()/1(21[)( 222

2222

22

22

2

22

222

CvCCv

CvCCvv

Cv

CvTA

l

l

l

ll

−−

−−−

+−−=ττ

τγ

])/1()/)(/1(2)()[( 2222

2222

22

22

22

22 Cv

CCvCv

Cv

Cv

Cv

Cv

−−−−+−−−=l

llll τττγ

)///( 4222222222 CvCvC ll +−−= ττγ Therefore

2222 /CTA l−= τ 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 'vvVB += and "vvVA −= , where BV and

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

2220 //1 CvCvt B l−−=τ

2220 //1 CvCvt A l+−=

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 22 /1 Cv−l . The apparent speed (relative to point O) of the rocket travelling toward B is therefore:

63

222 //1'

CvCvtv

OB

appl

ll

−−==

τ

We note that by definition appapp vv ''' = where appv '' refers to the apparent speed (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 appv .)

Replacing Bt0 by its value vV

Cv

B −− 22 /1l

we find:2

22

/)/1('

CvvVCv

v

B

app

ll

ll

−−

−==

τ

22

2 )(1C

vVvCv

vV

B

B

−−−

−=

21CvV

vVB

B

−

−=

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.

64

[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.

65

66

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 equa-tions, locally observed phenomena are derived from the conservation of the zero-energy balance of motion and gravitation in whole space. The energy structure of space is described in terms of nested energy frames starting from hypothetical homo-geneous space as the universal reference and proceeding down to local frames in space. Time is decoupled from space – the fourth dimension has a geometrical mean-ing 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 be-come 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 Uni-verse 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 cosmol-ogy. 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.

67

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 postulat-ing 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 dimen-sion. 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 be-cause 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 MΣ

2G/R) to the total rest energy of the universe, MΣ c2, lo and behold, we get the amazing result that GMΣ

2/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 impor-tant 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]

68

“...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 three-dimensional space is such a thing, a tridimensional surface of a four sphere. The ar-rangement and distribution of galaxies in the world that we see would then be some-thing 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 mat-ter 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 fol-lowed by expansion releasing the energy of motion gained in the contraction.

The Dynamic Universe approach is just a detailed analysis of combining Feyn-man’s “great mystery” of zero-energy space to the “intriguing suggestion of spheri-cally 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 prob-lem, however, was that Einstein was looking for a static solution — it was just to pre-vent the dynamics of spherically closed space that made Einstein to add the cosmo-logical 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 direc-tion.

69

( )22 2 2totE c mc p⎡ ⎤= +⎣ ⎦ (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, space-time, 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

( )2 20 0 0 0

1

1 1n

rest i ii

E c c mc m c δ β=

= = = − −∏p (b)

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 gravita-tional 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 ve-locity 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 relativ-ity 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 gravi-tational 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 tran-sition obtains the form

( ) ( )2 20 1

1

1 1 1 1n

local i i n n ni

f f fδ β δ β−=

= − − = − −∏ (c)

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

0rest globalE E+ = (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 ob-served Euclidean appearance and surface brightnesses of galaxies in space. The mag-nitude 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 en-ergy carried by a cycle of radiation

00 0 0

hE c c c m c cλ λ λλ= = = p (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 radia-tion. Equally, Coulomb energy is expressed in form

20 0

0 0 02 2C Ce hE c c c c c m c

r rμ

απ π

= = = (f)

where α is the fine structure constant and the quantity h0/2π r ≡ mC is the mass equiva-lence 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.

71

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 ma-terial 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 phenom-ena 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 devel-ops 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 4-dimensional sphere free to contract and expand maintaining a zero-energy balance of

72

motion and gravitation in the system. Mass as the substance for the expression of en-ergy 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

( )( )

0gV

dV rE mG

rρ= − ∫ (1.1.1:1)

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

V

M dV Vρ ρΣ = =∫ (1.1.1:2)

In spherically closed homogeneous 3-dimensional space the total mass is 2 3

02M Rρ πΣ = ⋅ , 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

( )2

0mE v v m mv= = =p v (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 ho-mogeneous space, the rest energy is

0 0 0 0restE c c mc= =p (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.

73

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

0

"global

V

dV GM mE mGr R

ρ= − = −∫ (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 dis-tance 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

20

0

" 0rest globalGM mE E mc

R+ = − = (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 result-ing 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 hypo-thetical homogeneous space is

( ) ( )20

0

" 0rest tot global totGME E M c M

RΣ Σ+ = − = (1.1.2:4)

Force in Dynamic Universe is the manifestation of a natural trend towards mini-mum 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

2 30

0 00 0

0.776 2" 1.246G RGMc R GR R

ρ ππ ρ

⋅= ± = ± = ± ⋅ (1.1.3:1)

where ρ is the average mass density in space.

74

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 contrac-tion 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].

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 ve-locity of motion gradually decreases, while the energy of motion gained in contraction is returned to gravity.

20mE mc=

0

"g

GME mR

=

Energy of motion

Energy of gravitation

Expansion

time

Contraction

75

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 be-comes

1/ 31 34

02 "3

dRc GM tdt

−⎛ ⎞= = ⎜ ⎟⎝ ⎠

(1.1.3:2)

and the time since singularity becomes

[ ]4

944

4 4 00

1 2 2 1 9.3 10 l.y.3 3

R Rt dRc c H

= = = = ⋅∫ (1.1.3:3)

The velocity of expansion and, accordingly, the velocity of light decelerate in the course of expansion as

0 013

dc cdt t

= − (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 ra-dioactive decay occur in direct proportion to the velocity of the expansion and, accord-ingly, to the velocity of light in space. As a result, the velocity of light is observed as constant at any time during the expansion.

×108 m/s

0

3

6

9

0 20 40 × 109 years 10 30

76

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 uni-verse 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 gravita-tionally 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 elec-tromagnetic radiation propagating in space increases in direct proportion to the expan-sion 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” char-acterized as radiation-like form of matter.

O

R0

(b)

R0

O

(a)

R0(1)

emitting object

O

R0(2)

t(2)

t(1)

(c)

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 uni-formly 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 homo-geneous 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

0recessionv cα= ⋅ (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).

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.

m

( ) 0 00mE c= p

( )00

"g

GM mER

= −

( )0mE

( )0gE

(a) (b)

"M

0Im

0Re

( )0mE( )0gE

m

m

0Im

0Im0Re

0Re

α

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 distin-guishing the mass equivalence of electromagnetic energy and the velocity of light. The energy of electromagnetic radiation has the form of the energy of motion

rad radE c= p (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]

( )2 3 2 2 2 00 0 0 01.1046 2 hPE N e c f N hf N c c c m c

fλ λπ μλ

= = ⋅ = = ⋅ = (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]

( )

3 22 20 0 0

0

1.1046 2 ;e h hm N N mλ λ

π μλ λ 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)

( ) ( )0

0 0 00 0radhE c c m c c cλ λ λ

= = =p (1.2.2:4)

A unified expression of Coulomb energy is obtained by applying vacuum perme-ability 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)]

21 2 0 0c 0 0 0 c4 2

q q hE c c N c c c m cr rμ

απ π

= = = (1.2.2:5)

where 2

0 01 2 1 24 2c

e hm N N N Nr r

μα

π π= = ⋅ (1.2.2:6)

79

Figure 1.2.2-1. Unified expressions for the Coulomb energy, the energy of a cycle of electromag-

netic 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

( )

20

1 22 1

1 14c r

em N Nr r

μπΔ

⎛ ⎞Δ = − −⎜ ⎟

⎝ ⎠ (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

20

0 1 2 02 1

1 14kin

eE c c m N N c cr r

μπ

⎛ ⎞= Δ = −⎜ ⎟

⎝ ⎠ (1.2.2:8)

1.2.3 The energy vector

Energy is traditionally regarded a scalar quantity. For illustrating the four-dimensional 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

( )0 0 0 0' i " ' i " ' i "m m mE c E E c p c p c p p= = + = + = +p (1.2.3:1)

ic ic

c

ic

Coulomb energy (1.2.2:5)

Energy of a cycle of electromagnetic radiation (1.2.2:2)

Rest energy of localized energy object

2 0c 0 0 c2

hE N c c c m cr

απ

= =

2 00 0 0

hE c N cc c m cλλ= = =p

(0) 0 0E c c mc= =p

80

where the imaginary part means the energy equivalence of momentum in the local fourth dimension.

The complex presentation of the energy of gravitation is

' "g g gE E E= + (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

m mE E= (1.2.3:3)

The kinetic energy of an object moving at velocity β = v/c in a local frame is the to-tal energy of motion minus the energy of motion the object has at rest in the local frame

( ) ( ) 0 0 00kin m mE E E c c p pββ= − = Δ = −p (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

( ) "g gg localE E E= − (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 propa-gates 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

20 4 0 0 0i " ; "m m rest mE E E E c c mc mc= = = = =p (1.3.1:1)

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

0

"i " ; "g g gGM mE E E

R= = − (1.3.1:2)

81

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

0 cosc c δ φ= (1.3.1:3)

and the rest energy is expressed as

0restE c mc= (1.3.1:4)

The global gravitational energy in tilted space is reduced as

0

" "" cos" "g

GM m GM mER R δ

φ= − = − (1.3.1:5)

where R” is the apparent distance to the mass equivalence M" (Fig. 1.3.1-1). Quanti-ties c0δ and R”0δ in (1.3.1:3) and (1.3.1:5), respectively, refer to apparent homogene-ous 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 imagi-nary part of gravitational energy)

m

0 0restE c mc=

0

""gGM mE

R= −

0Im

0Re m

0Im δ

0restE c mc=

"""g

GM mER

= −

( )a ( )b

φ

φ

82

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

a mass center in space.

( ) ( ) ( )( )

( ) ( )0 0 00

"" " " " 1 " 1

"g

gg g g gg

EE E E E E

Eδ δ δ δδ

δ⎛ ⎞Δ⎜ ⎟= − 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

( ) ( ) ( ) ( ) ( ) ( )0 0 0 0" " 1 " 1g m rest gE E E c mc c mc Eδδ δ δ δδ δ= = = − = = − (1.3.2:2)

where c is the local velocity of light

( )0 1c c δ δ= − (1.3.2:3)

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

0

"gGMEr δ

Δ = − (1.3.2:4)

and the gravitational factor δ becomes

0 0 0 0 0 0 0

" ;"

cc

rMR GM GMrM r r c c r c cδ δ δ δ δ

δ ≡ = = ≡ (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.)

( )"gE δ

0Re δReδ

0Im δ

Reδ ( )"mE δ

( )"gE δ

( )"mE δ

mm

ImδImδ

( )0"mE( )0"mE

( )0"gE ( )0"gE

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 de-fined 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)

( ) ( ) ( ) ( )0 0 0 0 0Re Im 0 Imtotalm totalE c c c c mc δδ δ= = + = =p p p p (1.3.3:1)

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

( ) ( )0 0ImrestE c c mcδδ δ= =p (1.3.3:2)

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

( ) ( ) ( ) ( )0 0 0 0 0 0kin ff m total restE E E c mc c mc c m c c c m cδ δ δ δδ= − = − = − = Δ (1.3.3:3)

Equation (1.3.3:3) means that kinetic energy in free fall is obtained against reduc-tion 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

( ) ( ) ( )0 00total restE E c c m c c m mβ = + Δ = + Δ (1.3.3:4)

or in complex form

( ) ( )0 itotalE c m m c mcβ β= ⎡ + Δ + ⎤⎣ ⎦ (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 22 2 2 2 2 2 20 0totalE c c m m c c m m mβ β⎡ ⎤= + Δ = + Δ +⎣ ⎦ (1.3.3:6)

Dividing (1.3.3:6) by 2 2 20c c m gives

( ) ( )2 22

2 2 1m m m m

m mβ

+ Δ + Δ= + (1.3.3:7)

84

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

( ) ( )2

22 2

1 11

m m mm mm

ββ

+ Δ− = ⇒ + Δ =

− (1.3.3:8)

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

( ) ( ) 00 0 21

tottotc mcE c p c m m cβ

β= = + Δ =

− (1.3.3:9)

and the kinetic energy

( ) ( ) 00 2

1 11

kin total restE E E c mcββ

⎛ ⎞⎜ ⎟= − = −⎜ ⎟−⎝ ⎠

(1.3.3:10)

or

( ) ( )0 0 0kinE c m m c c mc c c mβ = + Δ − = Δ (1.3.3:11)

Substitution of (1.3.3:8) for (m+Δm) in (1.3.3:5) gives the complex presentation of the total energy of motion

( ) 0 02 i

1totalE c mc c mcβ

β

β= +

− (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-

0kinE c= Δp

0kinE c= Δp

( )Imφp( )0 Imp

Re

Im

ψ

Im

Reφ

(a) (b)

( )Reφp ( )Reφp

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

( ) ( )0 0,kinE c c m c c mδ β = Δ = Δ + Δp (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 en-ergy 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 ( )kinE β in the total

energy of motion ( )totE β . Subtraction of the complex kinetic energy from the complex

total energy gives the internal energy of the moving object

( ) ( ) ( )

( )20 02 2

i i 1 11 1

int tot kinE E E

c mc c mc

β β β

β β β ββ β

= −

⎡ ⎤⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟= + − − + − −

⎜ ⎟ ⎜ ⎟⎢ ⎥− −⎝ ⎠ ⎝ ⎠⎣ ⎦

(1.3.3:14)

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 ( ).

Re

Im

( )"gE β

( )0"gE

( )"restE β( )'kinE β

( )0"restE

( )"kinE β ( )kinE β

( )totalE β

( ) 0'totalE c pβ =

( )intE β

ψ

86

Performing the subtraction separately for the imaginary and real components re-sults the complex form of internal energy

( ) ( ) ( ) ( )20 i 1int tot kinE E E c mcβ β β β 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 mo-mentum 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

0 0" " 1rest int tot kin restE E E E c mc c m cβ β β β ββ= = − = − = (1.3.3:16)

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

( )21restm mβ β= − (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 equiva-lence 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 en-ergy 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 accel-erated — 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

( ) 20 0 1 1restE c mc δ δ β= − − (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

( ) ( ) ( ) ( )2 2

0 0, ,0

"1 1 1 1 ""rest g

GM mE c mc ERδδ β δ β

δ

δ β δ β= − − = − − = (1.3.3:19)

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

87

0 00

" 0"

GM mE c mcRδ

δ

= − = (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 gravi-tational force as

( )( )

( )

22 220

, 0

220

1ˆ ˆ" 1" " "

1 ˆ1"

effeff

mc c vm cc R R R

mc

R

δ β δ δ

δ

βδ

βδ

−⎛ ⎞= − − = − −⎜ ⎟

⎝ ⎠

−= − −

F i i

i

(1.3.4:1)

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 21m mβ β= − , 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 en-hanced (effective) mass meff moving at velocity β

( ) ( ) ( )

( )

2

,0

220

1 1 "ˆ ˆ"" "

1 ˆ1"

δδ δ δ

δ

δ ββ

βδ

− −= − = −

−= − −

gdE GMm RdR R

mc

R

F i i

i

(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 21m β− at rest in a local sub-frame moving at velocity

β in the local frame or mass 21m β− moving at velocity β in the local frame (Fig. 1.3.4-1).

88

Figure 1.3.4-1 The symmetry of the imaginary energies of motion and gravitation for an object with

mass ( )21effm mβ β= − moving at velocity β and an object with mass ( )

21restm mβ β= − at rest.

1.3.5 Motion in parent frame

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

( ) ( )2

0, 0 1 Brest B BE c m cβ β= − (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

( ) ( )2 2

02 0 1 1A Brest AE c m cβ β β= − − (1.3.5:2)

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

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.

0Re

Im

( ) ( ), 0 ," "Ag B g AE E β=

( ), Brest BE β

( ),"Bg BE β

Re

Im

( ),"Ag AE β

( ), 0"g AE

( ), 0rest AE

( ), Arest AE β

( ) ( ), 0 , Arest B rest AE E β=

Frame A Frame B

( ) ( )202

" 11

mc mcE β β

β= −

−

( ) ( )2

2" 1" 1"g

GM mERβ

ββ

−= − −

21m β− 1m β−β

Im ( )2

0" 1mE c mcβ β= −

( )2"" 1

"gGM mE

Rβ β= − −

8 9

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 24 2 2

0, 0 0 002

0

1 18 2Z n e ee Z ZE c m c c m c

h n n δμ α δ β⎛ ⎞ ⎛ ⎞= = − −⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ (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

( )( )

( ) ( )

( ) ( )

21, 2 2 2

0, 02 20 0 1 2 0

20 ,0

1 1 1 12

1 1

n ne

Ef Z m c

h c n n h

f

δδ β

δ δ

α δ β

δ β

Δ ⎡ ⎤= = − − −⎢ ⎥

⎣ ⎦

= − −

(1.3.6:2)

and the corresponding wavelengths

( )( )( )

( ) ( )0 2, 0 ,0

,

11 1

cf

δδ β δ δ

δ β

δλ λ δ β

− ⎡ ⎤= = − −⎣ ⎦ (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 homo-geneous space of the local frame.

The momentum of a quantum of electromagnetic radiation occurs in space direc-tions — the absolute value of the momentum is equal to the imaginary momentum released by the emitter (Fig. 1.3.6-1).

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.

emitterIm

( )Re x

electromagnetic radiation

( )0mE

Im

( )Re x

mass at rest

( )Re y

( )0gE

( )Re y

m

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 bal-ances 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 typi-cally gravitational systems formed by accumulation of mass into mass centers. Accu-mulation 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 ex-pressed by applying equation (1.3.3:18) characterizing the state of motion and gravita-tion of the object in its parent frame and in each subsequent parent frames until hypo-thetical homogeneous space is reached

( ) ( )

( ) ( )

2 20 0

1

20

10

1 1

" 1 1"

n

i irest ni

n

i iglobal ni

E m c

GM mER

δ β

δ β

=

=

= − −

= = − − −

∏

∏ (1.4.1:1)

Figure 1.4.1-1. Space in the vicinity of a local frame, as it would be without the mass center, is re-

ferred 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 homoge-neous space, the “first-order” gravitational frames, like M1 in the figure, have hypothetical homoge-neous space as the apparent homogeneous space to the frame. In subsequent steps, smaller mass cen-ters 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 ap-parent homogeneous space to frame M2.

1M

2M 2

Apparent homogeneous spaceto gravitational frame M

1

Apparent homogeneous spaceto gravitational frame M

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

( )

( )( )

( )2

0 00

; ;i

c i i i ii ic i

ii i

r GM GM vrr c c r c cδδ

δ β= = ≈ = (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 ho-mogeneous space of the frame

( )01

1n

ii

c c δ=

= −∏ (1.4.1:3)

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

20

1

1n

ii

m m β=

= −∏ (1.4.1:4)

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

0restE c 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

( )0 1c c δ δ= − (1.4.1:6)

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

( )1

0 01

1n

ii

c cδ δ−

=

= −∏ (1.4.1:7)

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

20 1m m β β= − (1.4.1:8)

92

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

12

0 01

1n

ii

m mβ β−

=

= −∏ (1.4.1:9)

The system of cascaded energy frames relates the rest energy and the global gravi-tational 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

( ) ( )0 1 EEarthc c δ δ= − ( )2

,0 1 E rotEarthm m β β= − (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 homogene-ous 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 hypo-thetical 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.

-2.9

-3.0

-3.1

-3.2

Sun 150⋅106 km Moon

Earth

Δc

c0δ

Distance from the Earth

–400 – 200 400 m/s 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 ho-mogeneous 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.

93

Figure 1.4.2-2. The rest energy of an object in a local frame is a function of the velocity and gravi-

tational 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.

Extragalactic space

Accelerator in Earth frame

Milky Way in galaxy group frame

Accelerated ion

Earth in Solar frame

Hypothetical homogeneous space

( ) 0 00restE c mc=

( ) ( ) ( ) 20 1 1XG XGrest XG restE E δ β= − −

( ) ( ) ( ) 21 1MW MWrest MW rest XGE E δ β= − −

( ) ( ) ( ) 21 1S Srest S rest MWE E δ β= − −

( ) ( ) ( ) 21 1E Erest E rest SE E δ β= − −

( ) ( )21 Arest A rest EE E β= −

( ) ( )21 Ionrest Ion rest AE E β= −

Solar system in MW frame

( ) ( )2 20 0

0

1 1n

i irest ni

E m c δ β=

= − −∏

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 be-tween c and c0, the Coulomb energy for N1+N2 unit charges can be expressed in form

221 2 0 0

0 1 2 04 4C Cq q eE c c N N c m c c

r rμ μ

π π= = = (2.1.1:1)

where the quantity mC [kg] is referred to the mass equivalence of Coulomb energy

( )

20

1 2 1 2 04C C

em N N N N mr

μπ

≡ = (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 en-ergy

( ) 1 2 00 0 0 0

1 2

1 14k eff C

q qE c c m c c m m c c c c mr r

μπ

⎛ ⎞= Δ = − = − = Δ⎜ ⎟

⎝ ⎠ (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

( )

2 420 0

ave 2 2

42 2 20 0

sin32

212

s s

dEP c dS dSdt r c

N e z fc

μ ωθ

π

μ ππ

Π= = =

=

∫ ∫E (2.1.2:1)

95

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

22 2 2 4 42 3 20 0 0

016 2

12N e z f zPE N A e c f

f cfλμ π

π μπ 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 dimen-sion, which means that a point source at rest in space can be regarded as one-wavelength 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

( )2 3 2 2 2 2001.1049 2 hE N e c f N hf N cλ π μ

λ= ⋅ = ⋅ = (2.1.2:3)

where h is the Planck constant 3 2 35

01.1049 2 6.6261 10 [Js]h e cπ μ −= ⋅ = ⋅ (2.1.2:4)

and h0 is defined as the intrinsic Planck constant

3 2 420 01.1049 2 2.210 10 [kg m]π μ −≡ = ⋅ = ⋅

hh ec

(2.1.2:5)

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

( ) ( )0

0 0 0 01 0N

hE c c k c c m c cλ λλ= = = ⋅ = (2.1.2:6)

where k = 2π/λ and 0 0 2h π= . 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 impor-tant 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 2 22

0 03 2 3

0 0 0

1 12 2 2 1.1049 2 2 1.1049 2 137.035

e e ceh c h e c

μ μα

ε π μ π≡ = = = ≈

⋅ ⋅ ⋅ ⋅ (2.1.2:7)

96

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

20 0

1 2 0 1 2 0 04 2C Ce hE N N c c N N c c c m c

r rμ

απ π

= = = (2.1.2:8)

where the mass equivalence of Coulomb energy is

01 2 2C

hm N Nr

απ

= (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 transi-tion of a unit charge in a unit dipole. Equation (2.1.2:6) can be generalized to the en-ergy of a cycle of electromagnetic radiation from any electric dipole by inserting the intrinsic Planck constant back to equation (2.1.2:2)

22 0 0 0

0 0 0z h hPE N A c c B c c m c c

fλ λλ 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 hypotheti-cal homogeneous space. At cosmological distances the velocity of light is approxi-mated 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 mani-fested in the direction of propagation only. The wavelength of electromagnetic radia-tion 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

20E m cλ λ= (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 di-mension. In the complex form the total energy of motion of mass m moving at velocity β in a local frame is expressed

0 0 02 2i

1 1total

m mE c c c mc c cβ

β β= + =

− − (2.1.3:1)

which is rewritten in form (Fig. 2.1.3-1)

( ) ( ) ( )0 0 0 0 0 0Re Im 0itotalE c c k c c k c c kβ φ β= + = (2.1.3:2)

where 0 0 2h π= and k = 2π /λ

( ) ( ) ( )0 0 0Re Im 02 2 , , and

1 1m mk k m kβ φ β

β

β β= = =

− − (2.1.3:3)

or applyingψ = arc(sinβ)

( ) ( ) ( )0 0 0Re Im 0tan , , and cosk m k m k mβ φ βψ ψ= = = (2.1.3:4)

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

( ) ( ) ( )Re Im 0ik k kβ φ β+ = (2.1.3:5)

or in squared form, to the wave number equivalence of the energy four vector

( ) ( ) ( )2 2 2

Im 0 Rek k kφ β 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

( )( )

20

ReRe

12de Broglie

hk mβ

β

βπλ λβ−

= = = (2.1.3:7)

In equation (2.1.3:5) the wave number in the imaginary direction is the kIm(0), corre-sponding 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

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).

( ) ( )2

Im Im 0 1k kβ β= − (2.1.3:8)

and the corresponding wavelength is equal to the Compton wavelength

( )( )

0Im 2

Im

21

Comptonh

k mβ

β

πλ λβ

= = =−

(2.1.3:9)

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

( )

22 2

, 0 1 1 12Z n e rest e

ZZE c m c En n

αα⎛ ⎞ ⎛ ⎞⎛ ⎞⎜ ⎟= − − ≈⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠ (2.2.1:1)

Re

Im

ψ

( ) ( )Re 0 0 0Re ReE c c c kβ β= = ⋅p

( ) ( )0 0 0totalE c c c kφ β φ β= = ⋅p( ) ( ) ( )0 0 0Im 0 Im 0 Im 0E c c c k= = ⋅p

( ) ( ) ( )0 0 0Im Im ImE c c c kβ β β= = ⋅p

( ) ( )0Im Im"

"gGME k

Rβ β= − ⋅

( ) ( )0Im Im 0"

"gGME k

Rβ = − ⋅

99

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

( )22

2 2, 0 0

1

1 12

n

Z n i ii

ZE m cn

α δ β=

⎛ ⎞= − −⎜ ⎟⎝ ⎠

∏ (2.2.1:2)

Balmer’s equation for characteristic emission and absorption frequencies solved from (2.2.1:2) becomes

( )( )

( ) ( )1, 2 21, 2 0 1, 2

10

1 1n

n ni in n n n

i

Ef f

h cδ β

=

Δ= = − −∏ (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/ 302 1 3

0 1, 2 2 21 2 0

1 1 2 "2 3

en n

mf Z GM t

n n h

α−⎡ ⎤ ⎛ ⎞= −⎢ ⎥ ⎜ ⎟

⎝ ⎠⎣ ⎦ (2.2.1:4)

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

( )( )

( )0 1, 21, 2

21, 2

1

1

n nn n n

n ni

i

cf

λλ

β=

= =−∏

(2.2.1:5)

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

( )

( )2

0 000 2

20

1

1n

e ni

i

ahae mπμ β

=

= =−∏

(2.2.1:6)

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

( )( )0 0 0

1, 2 2 2 22 2 2 2 1 2

1 21

4 41 11 1 1

n n n

ii

a aZ n nZ n n

π πλ

αα β=

= =⎡ ⎤−⎣ ⎦⎡ ⎤− −⎣ ⎦∏

(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 ve-locity 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 reso-nator frame.

A resonator creates a closed energy object by capturing the radiation of two oppo-site 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

0 02 sin 2 cos2 2 sin cosrA A f t A kr tπ π ωλ

= = (2.2.2:1)

with nodes at r = n⋅λ/2. The momenta in a resonator have a zero vector sum but a non-zero scalar sum

( ) ( ) ( ) ( )½ ½ 0 ; ½ ½tot tot totp+ − + −= + = + = =p p p p p p (2.2.2:2)

101

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

( ) ( )0 0

0 0

ˆ ˆ;h hc cλ λ+ −= = −p r p r (2.2.2:3)

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

0,int 21

βλ

λβ

=−

(2.2.2:4)

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

( ) ( ) ( )02

1 11

Iβ

λλ β λ β

β+ = − = −

− (2.2.2:5)

and increased in the opposite direction

( ) ( ) ( )02

1 11

Iβ

λλ β λ β

β− = + = +

− (2.2.2:6)

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

( ) ( ) ( ) ( ) ( )2 2

0 0

0 0

1 11 1½ ½2 1 2 1tot

h hβ β β

β βλ β λ β+ −

⎛ ⎞− −⎜ ⎟= + = −⎜ ⎟− +⎝ ⎠

p p p c (2.2.2:7)

Multiplication of the nominators and denominators of the terms in parenthesis in equation (2.2.2:7) by the factor 21 β− gives

( ) ( ) ( )0 02 2

0 0

½ 1 ½ 11 1

tot

h hβ β β β

λ β λ β= ⎡ + − − ⎤ =⎣ ⎦

− −p c c (2.2.2:8)

or by applying the mass equivalence of electromagnetic radiation as

( )( )

( )0

21tot eff

mmλ

β λββ

= =−

p c v (2.2.2:9)

102

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 elec-tromagnetic 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

2 20 0,int

0 0

½ 1 ½ 1 0h hβ β β

λ λ⎛ ⎞

= − − − =⎜ ⎟⎝ ⎠

p c (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 frequen-cies and wavelengths of the waves in both directions in the resonator frame are the internal frequency and wavelength

2,int

,int 0

1c cfββ

βλ λ

= = − (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 understand-

ing the early experiments on the velocity of light using Michelson–Morley interfer-ometers. Michelson–Morley interferometer in an Earth laboratory is essentially a reso-nator 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

(a) (b)

Re

Im

20 ˆ½ 1hc βλ

= ± ⋅ −p r

200 1hc c β

λ= −p

β

200

ˆ1hc c βλ

= −p i

1 0 3

internal wavelength in a closed energy system is affected by the square root term 21 β− 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 guar-antees 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 corre-sponds 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 mean-ing both in the space directions and in the fourth dimension. Notations used in describ-ing a local gravitational frame are summarized in Figure 3.1.1-1.

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.

dsϕ

r0δ dϕ0δ

Im0δ,

φ Imδ,

Reδ

rδ

r dsr

rc

104

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

( )20 0

12c c Schwd

GM GMr rc c cδ

⎛ ⎞= ≈ =⎜ ⎟⎝ ⎠

(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 me-chanics 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 mathe-matical 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

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.

ϕ

orbital plane

M

M

y0δ m

z0δ (Im0δ )

x0δ

x0δ

r(2)0δ

r(1)0δ

105

( )

3

000 2

0 0 0

ˆ1 cc rGMc r r

δδδ

δ δ

⎛ ⎞= − −⎜ ⎟

⎝ ⎠a r (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

( ) ( )20 0 0 02 2 1cz r r r a eδ δ δ δ

⎡ ⎤= − −⎢ ⎥⎣ ⎦ (3.1.1:3)

which gives the z-coordinate as the distance from the base plane (in the flat space di-rection) intersecting the orbiting surface at ϕ = ±π /2. Expression a0δ (1−e0δ

2) in equa-tion (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 02

02 010 0

001

2" 2 2r

cc

c cr

r r rR dr rr r rδ δ

δ

⎛ ⎞Δ ≈ = −⎜ ⎟⎜ ⎟

⎝ ⎠∫ (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 cen-

ters 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.

Uranus

Neptune

Pluto

Saturn Jupiter Mars

Venus Earth

Mercury

0 2 1 3 5 4 7 6 Distance from the Sun (×109 km)

Sun

50

100

150

200 ΔR” ×1000 km

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

106

3.1.2 Orbital velocity and the velocity of free fall

The velocity of free fall in the DU space is

( )2

0 ,

0

1 1 1ff DU cv r

rc δ

⎛ ⎞= − −⎜ ⎟⎝ ⎠

(3.1.2:1)

At high values of r (r >> rc), (1.3.2:1) can be approximated

( ) ( )2

0 ,

0

21 1 2 1 1 2 1ff DU c cc cc

v r rr rr rr rc r rδ

⎛ ⎞⎛ ⎞>> = − + − ≈ + − ≈⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠ (3.1.2:2)

Orbital velocity at circular orbit in DU space is

( ) ( )3

3

0

1 1orb DU c cv r r

c r rδ

δ δ ⎛ ⎞= − = −⎜ ⎟⎝ ⎠

(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 (coordi-nate distance/coordinate time) is given in [7] as

( )0 , 2 2 21 1ff r Schwd c c cNewton

v r r rc r r r

β⎛ ⎞ ⎛ ⎞= − = −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

(3.1.2:4)

Both the Schwarzschild solution and the DU solution approach the Newtonian ve-locity at high values of r. The critical radius in Schwarzschild space is twice the criti-cal radius in DU space, rc(Schzd) = 2⋅ rc(DU).

The orbital (coordinate) velocity at circular orbit in Schwarzschild space [7] is

( )0 , 1 21 21 3 3

orb r Schwd c

c

v r rc r r

δδ

−−= =

− − (3.1.2:5)

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 ( ) ( )3 6c Schwd c DUr r r> ⋅ = ⋅ .

107

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

( ) 3 2

0

2 1crPc δ

πδ δ

−= ⎡ − ⎤⎣ ⎦ (3.1.3:1)

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

min 3

16 16cr GMPc cπ π

= = (3.1.3:2)

( )c DUr r

,ff DUβ

,orb DUβ

( )ff Newtonβ

(a)

( )ff Newtonβ

( )ff Schwarzschildβ( )orb Schwarzschildβ

( )c DUr r

0δβ

0

0.5

1

0 10 20 30 400

0.5

1

0 10 20 30 40

stable orbits

(b)

0δβ

108

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

min 3

12 29.41 6

cr GMPcc

π π= ≈ (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 ob-tained in a closed mathematical form. For a full revolution the advance is

( ) ( )( )0 2 2

62

1G M m

c a eδ

πψ π

+Δ =

− (3.1.4:1)

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

02orb

c

Pr c δπ

0

10

20

30

40

0 4 8 r0δ /rc 10 6 2

Period in DU space Period in

Schwarzschild space

109

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 de-creases (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 simul-taneous 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

( )2

0,0,

2 4 2 20,0

1 1 1 1

2 8 2 2

1 2

1

GRf f

f

δ β δ β

δ β β δβ δ

= − −

⎛ ⎞≈ − − − − −⎜ ⎟⎝ ⎠

(3.1.6:1)

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

Im0δ

pr

araR

pR

110

( ) ( ) 20,0,

2 4 20,0

1 1 1

2 8 2

1 1

1

DUf f

f

δ β δ β

δ β β δβ

= − −

⎛ ⎞≈ − − − +⎜ ⎟⎝ ⎠

(3.1.6:2)

The difference between the GR and DU frequencies in equations (3.1.6:1) and (3.1.6:2) is

( )2 2

, ½DU GRfδ β δβ δ−Δ ≈ + (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 differ-ence to be detected with present clocks. The difference, however, is essential in ex-treme 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

( )( )

0,

0

B B B

A B A A A

dx rdxt dtc c r

δ

δ

= = =∫ ∫ ∫ (3.2.1:1)

0

0.2

0.4

0.6

0,8

1

0 0.2 0.4 0.6 0.8

GR DU

β 2 = δ1

fδ,β/f0,0 Figure 3.1.6-1. The difference in the DU and GR predictions of the fre-quency of atomic oscillators at ex-treme 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.

111

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)

( ) ( )( )

000 0

0 0 0 0 0

B B BA B

A BA A A

Xd dxdx dxdcdt tc dx c c c

δδδ δ

δ δ δ δ δ

−−

⎡ ⎤= + − = + Δ⎢ ⎥

⎣ ⎦∫ ∫ ∫ (3.2.1:2)

where Δt(A–B) is the Shapiro delay

( )( ) ( )0 20

30 0

sin 1cos

B B

A BA A

d dx dx GM dtdx c c

δ δ

δ δ

αδ αα−

⎡ ⎤Δ = + = +⎢ ⎥

⎣ ⎦∫ ∫ (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 conver-sion to an algebraic expression gives the Shapiro delay in form

( ) 3 2ln B B B AA B

A A B A

x r x xGMtc x r r r−

⎧ ⎫⎡ ⎤ ⎡ ⎤+⎪ ⎪Δ = − −⎨ ⎬⎢ ⎥ ⎢ ⎥+⎪ ⎪⎣ ⎦ ⎣ ⎦⎩ ⎭ (3.2.1:4)

M

Im0δ

r0δΒ

A

B

r0δΑ

AB

X(A–B)0δ

112

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 propaga-tion 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 tan-gential component, i.e. the Shapiro delay is given in form

( ) , 3

2 ln B BA B B GR

A A

x rGMtc x r−

⎡ ⎤+Δ = ⎢ ⎥+⎣ ⎦

(3.2.1:5)

When xA = rA and xA = rA both the DU and the GR solution reduces to

( ) , 3

2 ln BA B B r

A

rGMtc r−Δ = (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

1 21, 2 3 2

42 ln 1D DD DGMt

c d⎧ ⎫⎡ ⎤Δ = −⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭

(3.2.1:7)

The GR prediction corresponding to (3.2.1:7) is

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 tan-gential direction.

dx

(b)

α

dr

B,xB

A,xA

rA

d

rB

α

dx

r

x=0

dα

α

x

βA

βB

M

(a)

113

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.

( )1 2

1, 2 3 2

42 lnD D GRD DGMt

c dΔ = (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 differ-ence in the delay as a function of passing distance d, which means that the two equa-tions 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

( )3

4t GMd c d

∂ Δ= −

∂ (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).

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

creased distance close to a mass center.

M

d

d∂

φ

φD∂

A D1 D2 B

M d

114

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

( ) ( )tD d c t c

dφ φ

∂ Δ∂ = ⋅ ∂ = ⋅ ∂ Δ ⇒ = ⋅

∂ (3.2.2:2)

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

2

4GMc d

φ = (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)

( )0 1A Af f δ δ= − (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

( )( )

0 0

0 0

11

AAA

A A

c ccf f f

δ δ

δ δ

δλ

δ−

= = =−

(3.2.3:2)

A similar clock at rest in the same frame at state δB runs at frequency

( )0 1B Bf f δ δ= − (3.2.3:3)

The wavelength of radiation sent from B at frequency fB is

( )( )

0 0

0 0

11

BBB A

B B

c ccf f f

δ δ

δ δ

δλ λ

δ−

= = = =−

(3.2.3:4)

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

( )B B B B

A A BA BA A A A

c c f ff c f f

λ λ λ λ= = = = (3.2.3:5)

115

Figure 3.2.3-1. The velocity of light is lower close to a mass center, cΒ < cΑ, which results in a de-

crease of the wavelength of electromagnetic radiation transmitted from A to B. Accordingly, the sig-nal 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 blue-shift (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 gravita-tional state of the emitter and receiver as given in equation (2.2.1:3)

( ) 20 1 1A A Af f δ δ β= − − (3.2.4:1)

λrec= fB /fA λB

cA

frec = fA

A

fB = f0(1−δB) λB =cB /fB

cB

fA = f0(1−δA)

B

116

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 re-ceiver in the direction of the signal path.

( ) 20 1 1B B Bf f δ δ β= − − (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, respec-tively. As the result the radiation emitted by A is observed at B as

( )( )( )( )( )

1

1B

AA B

A

f fβ

β

−=

−

r

r

(3.2.4:3)

(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

( )( )( )

( )( )( )( )

2

2

11 1

11 1

BA ABA B

B B A

f fβδ β

βδ β

−− −=

−− −

r

r

(3.2.4:4)

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

( )

( ) ( )( )( ) ( )( )

2

1

2

1

1 1 1

1 1 1

n

Bj Bj jBj k

BA B m

Ai Ai iAi k

f fδ β β

δ β β

= +

= +

− − −=

− − −

∏

∏

r

r

(3.2.4:5)

L = λ0 = Tc

ΔL = Δλ = Tv = λ0 v/c

v

(a)

vA

vB

r

vA⋅ r

vB⋅ r

cr

fA

fB

(b)

117

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

( ) ( )( )

[ ] ( )( )0

0 1

ˆ

1

ABAB tABA t B t m

j Bj k

T Tc β

→

=

⋅= =

−∏ r

r r (3.2.4:6)

where c is the velocity of light along the propagation path. The velocity of light de-creases 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 trans-mission 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 rota-tional velocity of the Earth and the transmission time becomes (Fig. 3.2.5-1 (a))

( )

( )( )( ) ( )0 0 0

2

ˆcos

1ABAB t AB t AB t

AB

B

r rT r

c ccθω ψ

β

⋅= ≈ +

− r

r r (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

Mk

A(k+3)

A(k+2)

A(k+1) B(k+1)

βB(k+1)

βA(k+3)

βA(k+2)

βA(k+1)

118

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.

( ) 2

2 ABOEarth

ATcω

ωΔ = (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

20 0 0 21 1 1 ½ vN T f T f T f

cδ δ β⎛ ⎞

= ⋅ = − − ≈ −⎜ ⎟⎝ ⎠

(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

02

v vN T fc⋅ Δ

Δ = − (3.2.5:4)

(b) (a)

drB O

At0

Bt1

drr

Bt0

rA(t0)→B(t1)

O

Lωr

h

rθ

B

A C

ψ ψ

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

LTv

=Δ

(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

0 02 2

L v v LvN f fv c c

⋅ ΔΔ = − = −

Δ (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

2 20

2 ABOr r ANf c c

θ θψ ω ω⋅ ⋅Δ= − = − (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.

O

rθ

B

A

ψ

Figure 3.2.5-2. Clock transportation by longitudinal angle ψ from point A to B at latitude θ. The delay due to the ad-ditional (slow) velocity is mathemati-cally 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 Fried-man-Lemaître-Robertson-Walker (FLRW) cosmology, local gravitational systems in DU space are subject to expansion in direct proportion to the expansion of the R4 ra-dius. 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 expan-sion 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 in-teractions 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 propa-gating in space, the wavelength of electromagnetic radiation is increased in direct pro-portion to the expansion. Accordingly, when detected after propagation in space, char-acteristic 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 opti-cal path in the direction of the momentum of radiation in space, is the tangential com-ponent of the spiral, which is equal to the increase of the 4-radius, the radial compo-nent of the path, during the propagation, Fig. 4.1.2-1

( )0 0 0D R R−= (4.1.2:1)

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

121

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

optD D dD⊥= = ∫ . Because the velocity of light in space is equal to the expansion of space in the di-

rection of R4, the optical distance is D = R0–R0(0), the lengthening of the 4-radius during the propaga-tion time.

0 0 0dD R d c dt dRα= = = (4.1.2:2)

By first solving for the distance angle α

( ) ( )

0

0 0

0 0 0

0 00 0

ln lnR

R

dR R RR R R D

α = = =−∫ (4.1.2:3)

the optical distance D obtains the form

( )0 1D R 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 in-

creases, obtains the form

( ) ( )0 00

1rec opticaldD Dv c e cdt R

α−= = − = (4.1.2:5)

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

c0

R0 R0

observer

objectDphys

α

O

c0

c0

R0(0)

observer

emitting object

α

O

c0(0)

R0

t

t(0)

(b) (a)

122

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

Atoms conserve their dimensions in expanding space. As shown by Balmer’s equa-

tion, 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 expan-sion space (Fig. 4.1.2-2). Accordingly, redshift, the increase of the wavelength be-comes

( )

( )

0 0 00 0

0 00 0

11

R R D Rz eR D R

αλ λλ

−−= = = = −

− (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.

z = 5

0.5

Observer: R0 = 1

z = 0.5

z = 2 0.17 z = 1

0.67

0.33 R0(0)

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.

R0(0) λ

λ0

R0

123

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

0 1zD R

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 ob-ject like a star) is

( ) ( )0 0

1 1;rod rod

rod

z zd dD R z d R z

θθ+ +

= = = (4.1.3:1)

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

( ) ( ) ( ) ( )20

1 11 1 1 2

zs s

HA m

r r dzR zD z z z z λ

θ⎡ ⎤⎢ ⎥= =⎢ ⎥+ + + Ω − + Ω⎣ ⎦

∫ (4.1.3:2)

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

( ) ( ) ( )0 2

1 11 1 1 1 2

zMA

H m

D cD dzz R z z z z z Λ

= =+ + + + Ω − + Ω

∫ (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

( ) ( )1Rdd zz

=+

(4.1.3:4)

124

Figure 4.1.3-1. Optical distance of objects in DU space (4.1.2:7) (solid line) and the angular diame-

ter 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). Substi-tution of d(z) in (4.1.3:4) for drod in (4.1.3:1) gives the angular size of an expanding objects

( )( )

( )0 0 0

1 1 1;1

dR R

R d

d z zd dD z R z R z z d R z

α θ θθα

+= = = = = =

+ (4.1.3:5)

0.001

0.01

1000

Ωm = 0.3 ΩΛ = 0.7

D(DU) (1.1.3:7)

0.1

1

0.01 0.1 1 10 100 0.001 z

Ωm = 1 ΩΛ = 0

DA(FLRW) (1.1.4:3)

DA(FLRW) (1.1.4:3) 0

DR

αd

R0(0)

R0(t2)

d

dR

θ

observation point

cspace c4

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 expan-sion 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).

125

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 expand-ing 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 qua-sars 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 predic-tions 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.

0.01 0.1 1 10 100 z

0.1

1

10

100

0.01

0r Rθ

Ωm = 0.27 ΩΛ = 0.73

Ωm = 1 ΩΛ = 0

DU solid objects, eq. (4.1.3:1)

DU expanding ob-jects, eq. (4.1.3:5)

FLRW cosmology, eq. (4.1.3:2)

126

Figure 4.1.3-4. Dataset of observed Largest Angular Size (LAS) of quasars and galaxies in the red-

shift 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 predic-tion (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 pro-portion 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 red-shifted radiation from an object at distance D becomes

( )( )

( )( )

0

22 2 2, 0 0 0

2 2 2 2 20, 0 0

1 11 11 1

z D

d

F z zd d dF D z R z z R z

+ += = =

+ + (4.2.1:1)

which converts into apparent magnitude

(a) DU-prediction (Euclidean) (b) FLRW-prediction

0.001 0.01 0.1 1 10 z

log(LAS)

Ωm= 1, ΩΛ= 0 Ωm= 0.27, ΩΛ= 0.73

0.001 0.01 0.1 1 10 z

log(LAS)

127

( )0

0

5log 5log 2.5log 1Rm M z zd

= + + − + (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 identi-cal 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 bolo-metric observations achievable with multi-bandpass photometry by matching the fil-ters 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 “correc-tion 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

( )5log 1 instrK z K= + + (4.2.1:3)

where Kinstr includes atmospheric corrections (in terrestrial observations) and the in-strumental factors like the transmission coefficients and mismatch of the filters. Add-ing (4.2.1:3) to the prediction in (4.2.1:2), the DU prediction for K-corrected magni-tudes becomes

( )0

0

5log 5log 2.5log 1 instrRm M z z Kd

= + + + + + (4.2.1:4)

The corresponding prediction for K-corrected magnitudes in FLRW cosmology is given by equation

( )( ) ( ) ( )

0

0 2

5log

15 log 11 1 2

Hinstr

z

m

Rm M Kd

z dzz z z z Λ

= + +

⎡ ⎤⎢ ⎥+ +⎢ ⎥+ + Ω − + Ω⎣ ⎦

∫ (4.2.1:5)

Equation (4.2.1:5) has its origin in the works of Hubble, Tolman, Humason, deSit-ter, 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

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 red-shift 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” data-set 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.

30

35

40

45

50

0,001 0,01 0,1 1 10

FLRW (4.2.1:5)

DU (4.2.1:4)μ

0

0.5

1

1.5

2

2.5

0 0.5 1 1.5 z 2

KB,

Figure 4.2.1-2. Average KB,X-corrections (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).

129

4.2.2 Surface brightness of expanding objects

The Tolman test [15,17,21] is considered as a critical test for an expanding uni-verse model. In expanding space, according to Tolman’s prediction, the observed sur-face 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 expand-ing objects. With reference to equation (4.1.3:5) the angular area of expanding objects with radius re = re0/(1+z) is

2

2 20

1eD

rR z

Ω = (4.2.2:1)

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

2 ) as

( )

( )

( ) ( ) ( )0

0

2202 2 24

1 11D d

Dd

SB z z zd zSB R z z

Ω+ += = = +

Ω (4.2.2:2)

or

( ) ( ) ( )0

1D dSB SB 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

( ) ( ) ( )0

11D dSB SB z −= + (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

( )( )

( )0

303 14

02.725 K 3 30

8 J4.2 10m1bol T

hE E d dc eν ν ν

ν νπν ν ν∞

−= °

⎡ ⎤= = = ⋅ ⎢ ⎥− ⎣ ⎦∫ (4.2.3:1)

where

[ ]00

HzkT ch

νλ

≡ = (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. Accord-

ingly, 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 en-ergy 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

( )

( )

2 20 0 0

0

20

00

1 1

" 1 1"

n

rest i ii

n

global i ii

E c mc m c

GM mER

δ β

δ β

=

=

= = − −

= = − − −

∏

∏ (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

131

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. Kine-matic 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 geome-try (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 ma-jor simplification to the field equation based metrics of FLRW space. The DU ap-proach 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 en-ergy 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 wave-lengths 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 electromag-netic 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 con-

stant 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.

132

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 expan-sion 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 direc-tion 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 van-ished 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 trig-gered by stimulus from my late colleague Jaakko Kajamaa in the early 1990s. I ex-press my sincere gratitude to my early inspirers.

Since 2004 my main channel for scientific discussions and publications on the Dy-namic 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 scien-tifically sound critical discussion on the basis of physics and pass my sincere gratitude to Michael Duffy, Peter Rowlands, Mogens Wegener and many conference partici-pants. 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 Kallio-Tamminen, 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

133

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. R. Feynman, W. Morinigo, and W. Wagner, Feynman Lectures on Gravitation (during the academic year 1962-63), Addison-Wesley Publishing Company, p. 10 (1995)

2. R. Feynman, W. Morinigo, and W. Wagner, Feynman Lectures on Gravitation (during the academic year 1962-63), Addison-Wesley Publishing Company, p. 164 (1995)

3. A. Einstein, Sitzungsberichte der Preussischen Akad. d. Wissenschaften (1917)

4. G. W. Leibniz, Matematischer Naturwissenschaflicher und Technischer Briefwechsel, Sechster band (1694)

5. T. Suntola, “Theoretical Basis of the Dynamic Universe”, Suntola Consulting Ltd., Helsinki, ISBN 952-5502-01-5 (2004)

6. T. Suntola, Proceedings of SPIE Vol. 5866, 18 (2005) 7. J. Foster, J.D. Nightingale, A Short Course in General Relativity, 2nd edition,

Springer-Verlag, ISBN 0-387-94295-5 (2001) 8. R. Genzel, et al., Nature 425, 934 (2003) 9. N. Ashby, Living reviews in relativity, http://relativity.livingreviews.org/

Articles/lrr-2003-1/ (2003) 10. S. M. Carroll, W. H. Press, and E. L. Turner, ARA&A, 30, 499 (1992) 11. K. Nilsson et al., Astrophys. J., 413, 453, 1993 12. Kim, A., Goobar, A., & Perlmutter, S. 1996, PASP, 190–201 13. Tolman, PNAS 16, 511-520 (1930) 14. E. Hubble, M. L. Humason, Astrophys.J., 74, 43 (1931) 15. W. de Sitter, B.A..N., 7, No 261, 205 (1934) 16. E. Hubble and R. C. Tolman, ApJ, 82, 302 (1935) 17. E. Hubble, Astrophys. J., 84, 517 (1936) 18. Robertson, H.P., Zs.f.Ap., 15, 69 (1938) 19. A. G. Riess, et al., Astrophys. J., 607, 665 (2004) 20. T. Suntola and R. Day, arXiv/astro-ph/0412701 (2004) 21. 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 spacetimeconcept, with space being merely an observer dependent perspective effectof that spacetime - space itself had no observer independent existence - ithad no ontological status, and it certainly had no dynamical description. Inrecent years this has all changed. In 2002 it was discovered that a dynam-ical 3-space had been detected many times, including the Michelson-Morley1887 light-speed anisotropy experiment. Here we review the dynamics of this3-space, tracing its evolution from that of an emergent phenomena in theinformation-theoretic Process Physics to the phenomenological description interms of a velocity field describing the relative internal motion of the struc-tured 3-space. The new physics of the dynamical 3-space is extensively testedagainst experimental and astronomical observations, including the necessarygeneralisation of the Maxwell, Schrodinger and Dirac equations, leading to aderivation and explanation of gravity as a refraction effect of the quantummatter waves. Phenomena now explainable include the bore hole anomaly,the systematics of black hole masses, the flat rotation curves of spiral galax-ies, gravitational light bending and lensing, and the supernova and gamma-raybursts magnitude-redshift data, for the dynamical 3-space possesses a Hub-ble expanding 3-space solution. Most importantly none of these phenomenanow require dark matter nor dark energy. The flat and curved spacetimeformalism is derived from the new physics, so explaining the apparent manysuccesses of those formalisms, but which have now proven to be ontologicallyand experimentally flawed.

135

1 Introduction

We review here some of the new physics emerging from the discovery that thereexists a dynamical 3-space. This discovery changes all of physics. While ata 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 3-space in terms of the velocity field that describes the internal dynamics of thisstructured 3-space. It is straightforward to construct the minimal dynamics forthis 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 theflowing 3-space, while α describes the self-interaction of the 3-space. Bore holeexperiments and black hole astronomical observations give the value of α as thefine structure constant to within observational errors. A major development isthat the Newtonian theory of gravity [10] is fundamentally flawed - that evenin the non-relativistic limit it fails to correctly model numerous gravitationalphenomena. So Newton’s theory of gravity is far from being ‘universal’. TheHilbert-Einstein theory of gravity (General Relativity - GR), with gravity beinga curved spacetime effect, was based on the assumption that Newtonian gravitywas valid in the non-relativistic limit. The ongoing effort to save GR againstnumerous disagreements with experiment and observation lead to the inventionfirst of ‘dark matter’ and then ‘dark energy’. These effects are no longer requiredin the new physics. The 3-space velocity field has been directly detected in atleast eight experiments including the Michelson-Morley experiment [2] of 1887,but most impressively by the superb experiment by Miller in 1925/1926 [3]. TheMiller experiment was one of the great physics experiments of the 20th century,but has been totally neglected by mainstream physics. All of these experimentsdetected the dynamical 3-space by means of the light speed anisotropy - that thespeed 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 phe-nomenon implies that a generalisation of all the fundamental theories of physicsbe carried out. The generalisation of the Maxwell equations leads to a simpleexplanation for gravitational light bending and lensing effects, the generalisationof the Schrodinger equation leads to the first derivation of gravity - as a refractioneffect of the quantum matter waves by the time dependence and inhomogeneitiesof the 3-space, leading as well to a derivation of the equivalence principle. Thisgeneralised Schrodinger equation also explains the Lense-Thirring effect as beingcaused by vorticity in the flowing 3-space. This effect is being studied by theGravity Probe B (GP-B) gyroscope precession experiment. The generalisationof 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 formalismfor the quantum matter geodesics, but without reference to the GR equationsfor the induced spacetime metric. What emerges from this derivation is thatthe spacetime is purely a mathematical construct - it has no ontological status.That discovery completely overturns this paradigm of 20th century physics. Thedynamical equation for the 3-space has black hole solutions with properties verydifferent from the putative black holes of GR, leading to the verified predictionfor the masses of the minimal black holes in spherical star systems. That samedynamics has an expanding 3-space solution - the Hubble effect for the uni-verse. That solution has the expansion mainly determined by space itself. Thisexpansion gives a extremely good account of the supernovae/Gamma-Ray Burstredshift data without the notion of ‘dark energy’ or an accelerating universe. Thisreview focuses on the phenomenological modelling of the 3-space dynamics andits 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, ProcessPhysics [1], leads to an emergent structured quantum foam ‘space’ which is 3-dimensional and dynamic, but where the 3-dimensionality is only approximate, inthat if we ignore non-trivial topological aspects of the space, then it may be em-bedded in a 3-dimensional geometrical manifold. Here the space is a real existentdiscrete but fractal network of relationships or connectivities, but the embeddingspace is purely a mathematical way of characterising the gross 3-dimensionalityof the network. This is illustrated in Fig.1. Embedding the network in the em-bedding space is very arbitrary; we could equally well rotate the embedding oruse an embedding that has the network translated or translating. These gen-eral requirements then dictate the minimal dynamics for the actual network, ata phenomenological level. To see this we assume at a coarse grained level thatthe dynamical patterns within the network may be described by a velocity fieldv(r, t), where r is the location of a small region in the network according to somearbitrary embedding. The 3-space velocity field has been observed in at least 8experiments [2, 3, 16, 17, 18, 19, 20, 21, 22, 23, 24]. For simplicity we assume herethat the global topology of the network is not significant for the local dynamics,and so we embed in an E3, although a generalisation to an embedding in S3 isstraightforward and might be relevant to cosmology. The minimal dynamics isthen obtained by writing down the lowest-order zero-rank tensors, of dimension

137

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 3-dimensional connectivity displayed by an embedding in a mathematical space, such as an E3

or an S3 as shown on the right. The embedding space is not real; it is purely a mathematicalartifact. 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

∇.(∂v∂t

+ (v.∇)v)

+α

8(trD)2 +

β

8tr(D2) = −4πGρ; Dij =

12

(∂vi

∂xj+∂vj

∂xi

)(1)

where ρ(r, t) is the matter and EM energy densities expressed as an effectivematter density. The embedding space coordinates provide a coordinate systemor frame of reference that is convenient to describing the velocity field, but whichis not real. In Process Physics quantum matter are topological defects in thenetwork, but here it is sufficient to give a simple description in terms of aneffective density.

We see that there are only four possible terms, and so we need at most threepossible constants to parametrise the dynamics of space: G,α and β. G turns outto be Newton’s gravitational constant, and describes the rate of non-conservativeflow of space into matter. To determine the values of α and β we must, at thisstage, turn to experimental data. However most experimental data involving thedynamics of space is observed by detecting the so-called gravitational acceleration

138

of matter, although increasingly light bending is giving new information. Now theacceleration a of the dynamical patterns in space is given by the Euler convectiveexpression

a(r, t) = lim∆t→0

v(r + v(r, t)∆t, t+ ∆t)− v(r, t)∆t

=∂v∂t

+ (v.∇)v (2)

and this appears in one of the terms in (1). As shown in [11] and discussed laterherein the acceleration g of quantum matter is identical to this acceleration,apart from vorticity and relativistic effects, and so the gravitational accelerationof 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

v(r) = −r

√2GM(1 + α

2 + ..)r

(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) = −rGM(1 + α

2 + ..)r2

(4)

which is Newton’s Inverse Square Law of 1687 [10], but with an effective massM(1 + α

2 + ..) that is different from the actual mass M . So the success of New-ton’s law in the solar system informs us that β = −α in (1). But we also seemodifications coming from the α-dependent terms.

In general because (1) is a scalar equation it is only applicable for vorticity-free flows ∇× v = 0, for then we can write v = ∇u, and then (1) can always besolved to determine the time evolution of u(r, t) given an initial form at some timet0. The α-dependent term in (1) (with now β = −α) and the matter accelerationeffect, now also given by (2), permits (1) to be written in the form

∇.g = −4πGρ− 4πGρDM , (5)

whereρDM (r, t) ≡ α

32πG((trD)2 − tr(D2)), (6)

which is an effective matter density that would be required to mimic the α-dependent spatial self-interaction dynamics. Then (5) is the differential formfor Newton’s law of gravity but with an additional non-matter effective matter

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 spiralgalaxies, bore hole g anomalies, and the systematics of galactic black hole masses.

The spatial dynamics is non-local. Historically this was first noticed by New-ton who called it action-at-a-distance. To see this we can write (1) as an integro-differential equation

∂v∂t

= −∇(

v2

2

)+G

∫d3r′

ρDM (r′, t) + ρ(r′, t)|r− r′|3

(r− r′) (7)

This shows a high degree of non-locality and non-linearity, and in particularthat the behaviour of both ρDM and ρ manifest at a distance irrespective of thedynamics of the intervening space. This non-local behaviour is analogous to thatin quantum systems and may offer a resolution to the horizon problem.

However (1) needs to be further generalised [1] to include vorticity, and alsothe 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 ofreference that defines the flow field; then vR is the velocity of that matter relativeto the substratum. One possible generalisation of the flow equation (1) is, withd/dt = ∂/∂t+ v.∇ the Euler fluid or total derivative,

dDij

dt+δij3tr(D2) +

trD

2(Dij −

δij3trD) +

δij3α

8((trD)2 − tr(D2))

+(ΩD −DΩ)ij = −4πGρ(δij3

+viRv

jR

2c2+ ..), i, j = 1, 2, 3. (9)

∇× (∇× v) =8πGρc2

vR, (10)

Ωij =12(∂vi

∂xj− ∂vj

∂xi) = −1

2εijkωk = −1

2εijk(∇× v)k, (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

~ω(r, t) =2Gc2

∫d3r′

ρ(r′, t)|r− r′|3

vR(r′, t)× (r− r′). (12)

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

140

3 Generalised Schrodinger Equation and EmergentGravity

Let us consider what might be regarded as the conventional ‘Newtonian’ approachto including gravity in the Schrodinger equation [11]. There gravity is describedby the Newtonian potential energy field Φ(r, t), such that g = −∇Φ, and we havefor a ‘free-falling’ quantum system, with mass m,

ih∂ψ(r, t)∂t

= − h2

2m∇2ψ(r, t) +mΦ(r, t)ψ(r, t) ≡ H(t)Ψ, (13)

where the hamiltonian is in general now time dependent. The classical-limittrajectory is obtained via the usual Ehrenfest method [12]: we first compute thetime rate of change of the so-called position ‘expectation value’

d<r>dt

≡ d

dt(ψ, rψ) =

i

h(Hψ, rψ)− i

h(ψ, rHψ) =

i

h(ψ, [H, r]ψ), (14)

which is valid for a normalised state ψ. The norm is time invariant when H ishermitian (H† = H) even if H itself is time dependent,

d

dt(ψ,ψ) =

i

h(Hψ,ψ)− i

h(ψ,Hψ) =

i

h(ψ,H†ψ)− i

h(ψ,Hψ) = 0. (15)

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

d2<r>dt2

=i

h

d

dt(ψ, [H, r]ψ),

=(i

h

)2

(ψ, [H, [H, r]]ψ) +i

h(ψ, [

∂H(t)∂t

, r]ψ),

= −(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 ofψ 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 arriveat the Newtonian 2nd-law equation of motion for the wavepacket,

d2<r>dt2

≈ g(<r>, t). (17)

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

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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 explanationfor the origin of Φ or g. In the new theory gravity is explained in terms of avelocity field, which in turn has a deeper explanation within Process Physics.

The key insight is that conventional physics has neglected the interaction ofvarious systems with the dynamical 3-space. Here we generalise the Schrodingerequation to take account of this new physics. Now gravity is a dynamical ef-fect arising from the time-dependence and spatial inhomogeneities of the 3-spacevelocity field v(r, t), and for a ‘free-falling’ quantum system with mass m theSchrodinger equation now has the generalised form

ih

(∂

∂t+ v.∇+

12∇.v

)ψ(r, t) = − h2

2m∇2ψ(r, t), (18)

which we write as

ih∂ψ(r, t)∂t

= H(t)ψ(r, t), where H(t) = −ih(v.∇+

12∇.v

)− h2

2m∇2 (19)

This form for H specifies how the quantum system must couple to the veloc-ity field, and it uniquely follows from two considerations: (i) the generalisedSchrodinger 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.

Then we compute that∂

∂t+ v.∇+

12∇.v → ∂

∂t+ v.∇+

12∇.v, i.e. that it is an

invariant operator, and (ii) require that H(t) be hermitian, so that the wavefunc-tion norm is an invariant of the time evolution. This implies that the 1

2∇.v termmust 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 ‘expec-tation value’, first with

vO ≡d<r>dt

=d

dt(ψ, rψ) =

i

h(ψ, [H, r]ψ) = (ψ, (v(r, t)− ih

m∇)ψ)

= <v(r, t)> − ihm<∇>, (20)

on evaluating the commutator using H(t) in (19), and which is again valid for anormalised state ψ. Then for the ‘acceleration’ we obtain from (20) that1

d2<r>dt2

=d

dt(ψ, (v − ih

m∇)ψ)

1Care is needed to identify the range of the various ∇’s.

142

= (ψ,(∂v(r, t)∂t

+i

h[H, (v − ih

m∇)]

)ψ),

= (ψ,∂v(r, t)∂t

ψ) + (ψ,(v.∇+

12∇.v − ih

2m∇2)(

v − ih

m∇)ψ)−

(ψ,(v − ih

m∇)(

v.∇+12∇.v − ih

2m∇2))

ψ),

= (ψ,(∂v(r, t)∂t

+ ((v.∇)v)− ih

m(∇× v)×∇

)ψ) +

+(ψ,ih

2m(∇× (∇× v))ψ),

≈ ∂v∂t

+ (v.∇)v + (∇× v)×(d<r>dt

− v)

+ih

2m(∇× (∇× v)),

=∂v∂t

+ (v.∇)v + (∇× v)×(d<r>dt

− v)

=∂v∂t

+ (v.∇)v + (∇× v)× vR (21)

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

vR ≡ −ih

m<∇>= vO − v, (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. Thenall velocity field terms are now evaluated at the location of the wavepacket. Notethat the operator

− ihm

(∇× v)×∇+ih

2m(∇× (∇× v)) (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 thelast line in (21) by means of the small-wavepacket approximation, we must thenself-consistently use that ∇ × (∇ × v) = 0, otherwise the acceleration acquiresa spurious imaginary part. This is consistent with (10) outside of any matterwhich contributes to the generation of the velocity field, for there ρ = 0. Theseobservations point to a deep connection between quantum theory and the velocityfield 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 theinstantaneous angular velocity of the local 3-space, relative to a distant observer.Hence we find that the equivalence principle arises from the unique generalisedSchrodinger equation and with the additional vorticity effect. This vorticity effectdepends 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 equivalenceprinciple.

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 vor-ticity is a relativistic effect, as in (12), the response of the test particle to thatvorticity is a non-relativistic effect, and follows from the generalised Schrodingerequation, and which is not present in the standard Schrodinger equation withcoupling to the Newtonian gravitational potential, as in (13). Hence the gener-alised Schrodinger equation with the new coupling to the velocity field is morefundamental. The Helmholtz term in (21) is being explored by the Gravity ProbeB gyroscope precession experiment, however the vorticity caused by the motionof the earth is extremely small, as discussed later in Sect.10.5.

An important insight emerges from the above: the generalised Schrodingerequation involves two fields v(r, t) and ψ(r, t), where the coordinate r is merelya label to relate the two fields, and is not itself the 3-space. In particular while rrelates to the embedding space, the 3-space itself has time-dependence and inho-mogeneities, 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 identifiedwith the coordinate system r, and that is when the velocity field v(r, t) becomesuniform and time independent. Then by a suitable choice of observer we mayput v(r, t) = 0, and the generalised Schrodinger equation reduces to the usual‘free’ Schrodinger equation. As we discuss later the experimental evidence is thatv(r, t) is fractal and so cannot be removed by a change to a preferred observer.Hence the generalised Schrodinger equation in (19) is a major development forfundamental physics. Of course in general other non-3-space potential energyterms may be added to the RHS of (19). A prediction of this new quantumtheory, which also extends to a generalised Dirac equation, is that the fractalstructure of space implies that even at the scale of atoms etc there will be time-dependencies and inhomogeneities, and that these will affect transition rates ofquantum systems. These effects are probably those known as the Shnoll effects[13].

2In the spacetime formalism it is mistakenly argued that it is ‘spacetime’ that is ‘dragged’.

144

-XXXXXXX

XXXXzδ

@@

@@

@@

@@

-

?

6

I

R

&%'$ Figure 2: Shows bending of light

through angle δ by the inhomogeneous spa-tial in-flow, according to the minimisationof the travel time in (30). This effect per-mits the in-flow speed at the surface of thesun to be determined to be 615km/s. Thein-flow speed into the sun at the distance ofthe earth from the sun has been extractedfrom 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 thecoupling of the spinor to the actual dynamical 3-space, and again not to theembedding space as has been the case up until now,

ih∂ψ

∂t= −ih

(c~α.∇+ v.∇+

12∇.v

)ψ + βmc2ψ (24)

where ~α and β are the usual Dirac matrices. Repeating the Schrodinger equationanalysis for the space-induced acceleration we obtain

g =∂v∂t

+ (v.∇)v + (∇× v)× vR −vR

1− v2R

c2

12d

dt

(v2

R

c2

)(25)

which generalises (21) by having a term which limits the speed of the wave packetrelative to space to be <c. This equation specifies the trajectory of a spinor wavepacket 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 bendingof light. This also immediately follows from the new space dynamics once wealso generalise the Maxwell equations so that the electric and magnetic fields areexcitations of the dynamical space. The dynamics of the electric and magneticfields must then have the form, in empty space,

∇×E = −µ(∂H∂t

+ v.∇H),∇×H = ε

(∂E∂t

+ v.∇E),∇.H = 0,∇.E = 0

(26)

145

which was first suggested by Hertz in 1890 [14], but with v being a constantvector field. Suppose we have a uniform flow of space with velocity v wrt theembedding space or wrt an observer’s frame of reference. Then we can find planewave solutions for (26):

E(r, t) = E0ei(k.r−ωt) H(r, t) = H0e

i(k.r−ωt) (27)

withω(k,v) = c|~k|+ v.k where c = 1/

√µε (28)

Then the EM group velocity is

vEM = ~∇kω(k,v) = ck + v (29)

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

τ =∫ sf

si

ds|drds|

|cvR(s) + v(r(s), t(s)|with vR =

(drdt− v(r(t), t)

)(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 = 2v2

c2=

4GM(1 + α2 + ..)

c2d+ ... (31)

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

146

6 Free-Fall Minimum Proper-Time Trajectories

The acceleration in (25) also arises from the following argument, which is theanalogue of the Fermat least-time formalism for the quantum matter waves. Con-sider the elapsed time for a comoving clock. Then taking account of the Lamourtime-dilation effect that time is given by

τ [r0] =∫dt

(1− v2

R

c2

)1/2

(32)

with vR given by (22) in terms of vO and v. Then this time effect relates to thespeed of the clock relative to the local 3-space, and that c is the speed of lightrelative 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) + ... (34)

Evaluating the change in proper travel time to lowest order

δτ = τ [r0 + δr0]− τ [r0] + ...

= −∫dt

1c2

vR.δvR

(1− v2

R

c2

)−1/2

+ ...

=∫dt

1c2

vR.(δr0.∇)v − vR.d(δr0)dt√

1− v2R

c2

+ ...

=∫dt

1c2

vR.(δr0.∇)v√1− v2

R

c2

+ δr0.d

dt

vR√1− v2

R

c2

+ ...

=∫dt

1c2δr0 .

(vR.∇)v + vR × (∇× v)√1− v2

R

c2

+d

dt

vR√1− v2

R

c2

+ ...

147

Hence a trajectory r0(t) determined by δτ = 0 to O(δr0(t)2) satisfies

d

dt

vR√1− v2

R

c2

= −(vR.∇)v + vR × (∇× v)√1− v2

R

c2

. (35)

Substituting vR(t) = v0(t)− v(r0(t), t) and using

dv(r0(t), t)dt

=∂v∂t

+ (v0.∇)v, (36)

we obtain

dv0

dt=∂v∂t

+ (v.∇)v + (∇× v)× vR −vR

1− v2R

c2

12d

dt

(v2

R

c2

). (37)

which is (25). Then in the low speed limit vR c we may neglect the lastterm, and we obtain (21). Hence we see a close relationship between the geodesicequation, known first from General Relativity, and the 3-space generalisation ofthe Schrodinger equation, at least in the non-relativistic limit. So in the classicallimit, i.e when the wavepacket approximation is valid, the wavepacket trajectoryis specified by the least proper-time geodesic.

The relativistic term in (37) is responsible for the precession of elliptical orbitsand also for the event horizon effect. Hence the trajectory in (21) is a non-relativistic 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 100years, since 1905. So far the experimental evidence is that absolute motion andLorentz symmetry are real and valid phenomena; absolute motion is motion rel-ative to some substructure to space, whereas Lorentz symmetry parametrisesdynamical effects caused by the motion of systems through that substructure.Motion through the structured space, it is argued, induces actual dynamical timedilations and length contractions in agreement with the Lorentz interpretationof special relativistic effects. Then observers in uniform motion ‘through’ thespace will, on measurement of the speed of light using the special but misleadingEinstein measurement protocol, obtain always the same numerical value c. To

148

see this explicitly consider how various observers P, P ′, . . . moving with differentspeeds through space, measure the speed of light. They each acquire a standardrod and an accompanying standardised clock. That means that these standardrods would agree if they were brought together, and at rest with respect to spacethey would all have length ∆l0, and similarly for the clocks. Observer P andaccompanying rod are both moving at speed vR relative to space, with the rodlongitudinal to that motion. P then measures the time ∆tR, with the clock atend A of the rod, for a light pulse to travel from end A to the other end B andback again to A. The light travels at speed c relative to space. Let the timetaken for the light pulse to travel from A→B be tAB and from B→A be tBA, asmeasured by a clock at rest with respect to space3. The length of the rod movingat speed vR is contracted to

∆lR = ∆l0

√1− v2

R

c2. (38)

In moving from A to B the light must travel an extra distance because the endB travels a distance vRtAB in this time, thus the total distance that must betraversed 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 =∆lRc− vR

+∆lRc+ vR

==2∆l0

c

√1− v2

R

c2

. (41)

Because of the time dilation effect for the moving clock

∆tR = ∆t0

√1− v2

R

c2. (42)

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

3Not 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 isalways c. This follows from two or more observers in manifest relative motion allobtaining the same speed c by this procedure. Despite this failure this specialeffect is actually the basis of the spacetime Einstein measurement protocol. Thatthis protocol is blind to the absolute motion has led to enormous confusion withinphysics.

To be explicit the Einstein measurement protocol actually inadvertently usesthis special effect by using the radar method for assigning historical spacetimecoordinates to an event: the observer records the time of emission and reception ofradar pulses (tr >te) travelling through space, and then retrospectively assignsthe time and distance of a distant event B according to (ignoring directionalinformation for simplicity)

TB =12

(tr + te) , DB =c

2(tr − te) , (43)

where each observer is now using the same numerical value of c. The event B isthen 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 anhistorian recording events according to some agreed protocol. Unlike historians,who don’t confuse history books with reality, physicists do so. We now show thatbecause of this protocol and the absolute motion dynamical effects, observerswill discover on comparing their historical records of the same events that theexpression

τ2AB = T 2

AB −1c2D2

AB , (44)

is an invariant, where TAB = TA−TB and DAB = DA−DB are the differences intimes and distances assigned to events A and B using the Einstein measurementprotocol (43), so long as both are sufficiently small compared with the scale ofinhomogeneities in the velocity field.

To confirm the invariant nature of the construct in (44) one must pay carefulattention to observational times as distinct from protocol times and distances,and this must be done separately for each observer. This can be tedious. Wenow demonstrate this for the situation illustrated in Fig. 3.

By definition the speed of P ′ according to P is v′0 = DB/TB and so v′R = v′0,where TB and DB are the protocol time and distance for event B for observer Paccording to (43). Then using (44) P would find that (τP

AB)2 = T 2B − 1

c2D2

B since

both TA = 0 and DA=0, and whence (τPAB)2 = (1 − v′2R

c2)T 2

B = (t′B)2 where thelast equality follows from the time dilation effect on the P ′ clock, since t′B is the

150

A

P (v0 = 0)

B (t′B)

DDB

T

P ′(v′0)

*H

HHHHHHY

te

TB

tr

γ

γ

Figure 3: Here T −D is the spacetime con-struct (from the Einstein measurement protocol)of a special observer P at rest wrt space, so thatv0 = 0. Observer P ′ is moving with speed v′0 as de-termined by observer P , and therefore with speedv′R = v′0 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 alsoused to define zero time for each for convenience.

time of event B according to that clock. Then TB is also the time that P ′ wouldcompute for event B when correcting for the time-dilation effect, as the speed v′Rof P ′ through the quantum foam is observable by P ′. Then TB is the ‘commontime’ for event B assigned by both observers. For P ′ we obtain directly, also from(43) and (44), that (τP ′

AB)2 = (T ′B)2 − 1c2

(D′B)2 = (t′B)2, as D′

B = 0 and T ′B = t′B.Whence for this situation

(τPAB)2 = (τP ′

AB)2, (45)

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 ′

and P ′′ both in absolute motion it follows that they also agree on the invarianceof (44). This is easily seen by using the intermediate step of a stationary observerP :

(τP ′AB)2 = (τP

AB)2 = (τP ′′AB)2. (46)

Hence the protocol and Lorentzian absolute motion effects result in the construc-tion in (44) being indeed an invariant in general. This is a remarkable and subtleresult. For Einstein this invariance was a fundamental assumption, but here it isa derived result, but one which is nevertheless deeply misleading. Explicitly indi-cating small quantities by ∆ prefixes, and on comparing records retrospectively,an ensemble of nearby observers agree on the invariant

∆τ2 = ∆T 2 − 1c2

∆D2, (47)

for any two nearby events. This implies that their individual patches of spacetimerecords 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

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pseudo-Riemannian manifold, where the choice of coordinates for this manifoldis arbitrary, and we finally arrive at the invariant

∆τ2 = gµν(x) ∆xµ∆xν , (48)

with xµ = D1, D2, D3, T. Eqn. (48) is invariant under the Lorentz transforma-tions

x′µ = Lµν x

ν , (49)

where, for example for relative motion in the x direction, Lµν is specified by

x′ =x− vt√1− v2/c2

, y′ = y, z′ = z, t′ =t− vx/c2√1− v2/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 measure-ment protocol.

The experimental question is then whether or not a supposed preferred frameactually exists or not — can it be detected experimentally? The answer is thatthere are now eight such consistent experiments.

The notion that the special relativity formalism requires that the speed oflight be isotropic, that it be c in all frames, has persisted for most of the lastcentury. The actual situation is that it only requires that the round trip speed beinvariant. This means that the famous Einstein light speed postulate is actuallyincorrect. 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 de-termined by (25), which may be obtained by extremising the time-dilated elapsedtime (32).

τ [r0] =∫dt

(1− v2

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 phaseunder small variations of the path. This again emphasises that gravity is aquantum effect. We now introduce a spacetime mathematical construct accordingto the metric

ds2 = dt2 − (dr− v(r, t)dt)2/c2 = gµνdxµdxν (52)

152

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

τ =∫dt

√gµν

dxµ

dt

dxν

dt, (53)

and the minimisation of (53) leads to the geodesics of the spacetime, which arethus equivalent to the trajectories from (51), namely (25). Hence by coupling theDirac spinor dynamics to the 3-space dynamics we derive the geodesic formalismof General Relativity as a quantum effect, but without reference to the Hilbert-Einstein equations for the induced metric. Indeed in general the metric of thisinduced spacetime will not satisfy these equations as the dynamical space involvesthe α-dependent dynamics, and α is missing from GR. So why did GR appear tosucceed in a number of key tests where the Schwarzschild metric was used? Theanswer is provided by identifying the induced spacetime metric corresponding tothe in-flow in (3) outside of a spherical matter system, such as the earth. Then(52) becomes

ds2 = dt2 − 1c2

(dr +

√2GM(1 + α

2 + ..)r

dt)2 − 1c2r2(dθ2 + sin2(θ)dφ2), (54)

Making the change of variables t→ t′ and r→ r′ = r with

t′ = t− 2c

√2GM(1+α

2 + . . .)rc2

+4 GM(1+α

2 + . . .)c3

tanh−1

√2GM(1+α

2 + . . .)c2r

(55)this becomes (and now dropping the prime notation)

ds2 =

(1−

2GM(1 + α2 + ..)

c2r

)dt2 − 1

c2r2(dθ2 + sin2(θ)dφ2)

− dr2

c2

(1−

2GM(1 + α2 + ..)

c2r

) . (56)

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

4The event horizon of (56) is at a different radius from the actual event horizon of the blackhole solutions that arise from (1).

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but other choices do not do this. Hence in the case of the Schwarzschild metricthe dynamics missing from both the Newtonian theory of gravity and GeneralRelativity is merely hidden in a mass redefinition, and so didn’t affect the variousstandard tests of GR, or even of Newtonian gravity. Note that as well we seethat the Schwarzschild metric is none other than Newtonian gravity in disguise,except for the mass shift. While we have now explained why the GR formalismappeared to work, it is also clear that this formalism hides the manifest dynamicsof the dynamical space, and which has also been directly detected in gas-modeinterferometer 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 userelativistic 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 aspectsbeing associated with the matter effects upon space and the relativistic effects ofthe matter in motion through the dynamical 3-space. The agreement of GR withthe 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

Gµν ≡ Rµν −12Rgµν =

8πGc2

Tµν , (57)

where Gµν is the Einstein tensor, Tµν is the energy-momentum tensor, Rµν =Rα

µαν and R = gµνRµν and gµν is the matrix inverse of gµν . The curvature tensoris

Rρµσν = Γρ

µν,σ − Γρµσ,ν + Γρ

ασΓαµν − Γρ

ανΓαµσ, (58)

where Γαµσ is the affine connection

Γαµσ =

12gαν

(∂gνµ

∂xσ+∂gνσ

∂xµ− ∂gµσ

∂xν

). (59)

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

G00 =∑

i,j=1,2,3

viGijvj − c2∑

j=1,2,3

G0jvj − c2∑

i=1,2,3

viGi0 + c2G00,

Gi0 = −∑

j=1,2,3

Gijvj + c2Gi0, Gij = Gij , i, j = 1, 2, 3. (60)

where the Gµν are given by

G00 =12((trD)2 − tr(D2)), Gi0 = G0i = −1

2(∇× (∇× v))i, i = 1, 2, 3.

154

Gij =d

dt(Dij − δijtrD) + (Dij −

12δijtrD)trD − 1

2δijtr(D2) + (ΩD −DΩ)ij ,

i, j = 1, 2, 3. (61)

In vacuum, with Tµν = 0, we find from (57) and (60) that Gµν = 0 implies thatGµν = 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 inGR the energy-momentum tensor Tµν is not permitted to make any referenceto absolute linear motion of the matter; only the relative motion of matter orabsolute rotational motion is permitted, contrary to the experiments.

It is very significant to note that the above exposition of the GR formalismfor 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 matterso that Tµν = 0,

∂

∂t(∇.v) +∇.((v.∇)v) = 0 (63)

which is the Newtonian ‘velocity field’ formulation of Newtonian gravity outsideof matter, as in (1) but with α = β = 0. So GR turns out to be Newtoniangravity in a grossly overstructured mathematical formalism.

9 Experimental and Observational Phenomena I

We now briefly review the extensive range of light speed experiments that havedetected that the speed of light is not isotropic - the speed is different in differ-ent directions when measured in a laboratory experiment on earth, as predictedby the generalised Maxwell equations, Sect.5. The most famous of these exper-iments was that of Michelson and Morley in 1887. Contrary to often repeatedclaims, this experiment decisively detected the anisotropy. The cause of the mis-understanding surrounding this experiment is that the Newtonian based theoryMichelson used for the calibration of the experiment is simply wrong, and ofcourse not unexpectedly. Clearly as the Michelson interferometer is a 2nd orderv/c experiment its calibration requires a ‘relativistic’ analysis, in particular onemust 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, asfirst 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, forall observers has been taken as a critical assumption in the standard formulationof fundamental physics, and was introduced by Einstein in 1905 as one of hiskey postulates when formulating his interpretation of Special Relativity. Theneed to detect any anisotropy has challenged physicists from the 19th centuryto the present day, particularly following the Michelson-Morley experiment of1887. The problem arose when Maxwell in 1861 successfully computed the speedof light c from his unified theory of electric and magnetic fields: but what wasthe speed c relative to? There have been many attempts to detect any supposedlight-speed anisotropy and there have so far been 8 successful and consistent suchexperiments, and as well numerous unsuccessful experiments, i.e. experiments inwhich no anisotropy was observed. The reasons for these different outcomes is nowunderstood: any light-speed anisotropy produces not only an expected ‘direct’effect, being that which is expected to produce a ‘signal’, but also affects thevery physical structure of the apparatus, and with this effect usually overlookedin the design of some detectors. In some designs these effects exactly cancel. Asalready stated there is overwhelming evidence from 8 experiments that the speedof light is anisotropic, and with a large anisotropy at the level of 1 part in 103: sothese experiments show that a dynamical 3-space exists, and that the spacetimeconcept was only a mathematical construct - it does not exist as an entity ofreality, it has no ontological significance. These developments have lead to a newphysics in which the dynamics of the 3-space have been formulated, together withthe required generalisations of the Maxwell equations (as first suggested by Hertzin 1890 [14]), and of the Schrodinger and Dirac equations, which have lead to thenew emergent theory and explanation of gravity, with numerous confirmationsof that theory from the data from black hole systematics, light bending, spiralgalaxy rotation anomalies, bore hole anomalies, etc. This data has revealed thatthe coupling constant for the self-interaction of the dynamical 3-space is noneother than the fine structure constant ≈ 1/137 [39, 40, 41, 42], which suggestsan emerging unified theory of quantum matter and a quantum foam descriptionof the dynamical 3-space.

The most influential of the early attempts to detect any anisotropy in thespeed 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 flawin its design, when used to detect any light-speed anisotropy effect5. To see thisrequires use of Special Relativity effects. The Michelson interferometer comparesthe round-trip light travel time in two orthogonal arms, by means of interferencefringe shifts measuring time differences, as the device is rotated. However ifthe device is operated in vacuum, any anticipated change in the total traveltimes caused by the light travelling at different speeds in the outward and inwarddirections is exactly cancelled by the Fitzgerald-Lorentz mirror-supporting-armcontraction effect - a real physical effect. Of course this is precisely how Fitzgeraldand Lorentz independently arrived at the idea of the length contraction effect. Invacuum this means that the round-trip travel times in each arm do not changeduring rotation. This is the fatal design flaw that has confounded physics forover 100 years. However the cancellation of a supposed change in the round-triptravel times and the Lorentz contraction effect is merely an incidental flaw of theMichelson interferometer. The critical observation is that if we have a gas in thelight path, the round-trip travel times are changed, but the Lorentz arm-lengthcontraction 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 therefractive index of the gas. Of course with a gas present one must also takeaccount 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 timedifferences 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 theinterferometer is very low. Nevertheless the Michelson-Morley experiment as wellas the Miller interferometer experiment of 1925/1926 [3] were done in air, which iswhy they indeed observed and reported fringe shifts. As well Illingworth [19] andJoos [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 airinterferometer experiment, and so confirming the refractive index effect. Jaseja etal. [21] used a He-Ne gas mixture of unknown refractive index, but again detectedfringe 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 light-speed 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, andthe 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

5Which also severely diminishes its use in long-baseline terrestrial interferometers built todetect gravitational waves.

157

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

9.2 Michelson Gas-mode Interferometer

Let us first consider the new understanding of how the Michelson interferometerworks. This brilliant but very subtle device was conceived by Michelson as ameans to detect the anisotropy of the speed of light, as was expected towards theend of the 19th century. Michelson used Newtonian physics to develop the theoryand hence the calibration for his device. However we now understand that this de-vice detects 2nd order effects in v/c to determine v, and so we must take accountof relativistic effects. However the application and analysis of data from variousMichelson interferometer experiments using a relativistic theory only occurredin 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 thenecessity of using relativistic effects it was discovered in 2002 that the gas in thelight paths plays a critical role, and that we finally understand how to calibratethe device, and we also discovered, some 76 years after the 1925/26 Miller ex-periment, what determines the calibration constant k that Miller had determinedusing 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 Michelsoninterferometer experiments have confirmed the reality of the Fitzgerald-Lorentzlength contraction effect: in the usual interpretation of Special Relativity thiseffect, and others, is usually regarded as an observer dependent effect, an illusioninduced by the spacetime. But the experiments are to the contrary showing thatthe 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 betweentravel times, when the device is rotated, for two coherent beams of light thattravel in orthogonal directions between mirrors; the changing time differencebeing indicated by the shift of the interference fringes during the rotation. Thiseffect is caused by the absolute motion of the device through 3-space with speedv, and that the speed of light is relative to that 3-space, and not relative to theapparatus/observer. However to detect the speed of the apparatus through that3-space gas must be present in the light paths for purely technical reasons. Thepost relativistic-effects theory for this device is remarkably simple. Consider hereonly the case where the arms are parallel/anti-parallel to the direction of absolutemotion. The relativistic Fitzgerald-Lorentz contraction effect causes the arm AB

158

- -

6

?

?

L

A BL

C

D

- - -

-

CCCCCCCCCC

CCCCCCW

α

A1 A2DB

C

v

(a) (b)

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

parallel to the absolute velocity to be physically contracted to length (see Fig.4)

L|| = L

√1− v2

c2. (64)

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

tABA = tAB + tBA =2LV

V 2 − v2

√1− v2

c2. (65)

For travel in the AC direction we have, from the Pythagoras theorem for theright-angled triangle in Fig.4 that (V tAC)2 = L2 + (vtAC)2 and that tCA = tAC .Then for the total ACA travel time

tACA = tAC + tCA =2L√

V 2 − v2. (66)

Then the difference in travel time is

∆t =(n2 − 1)L

c

v2

c2+ O

(v4

c4

). (67)

159

after expanding in powers of v/c. This clearly shows that the interferometer canonly operate as a detector of absolute motion when not in vacuum (n=1), namelywhen the light passes through a gas, as in the early experiments (in transparentsolids a more complex phenomenon occurs). A more general analysis [1] with thearms at angle θ to v gives

∆t = k2Lv2P

c3cos(2(θ − ψ)), (68)

where ψ specifies the direction of v projected onto the plane of the interfer-ometer relative to the local meridian, and where k2≈n(n2− 1). Neglect of therelativistic Fitzgerald-Lorentz contraction effect gives k2≈n3≈ 1 for gases, whichis essentially the Newtonian theory that Michelson used.

However the above analysis does not correspond to how the interferometer isactually operated. That analysis does not actually predict fringe shifts for thefield of view would be uniformly illuminated, and the observed effect would be achanging level of luminosity rather than fringe shifts. As Miller knew, the mir-rors must be made slightly non-orthogonal with the degree of non-orthogonalitydetermining how many fringe shifts were visible in the field of view. Miller exper-imented with this effect to determine a comfortable number of fringes: not toofew and not too many. Hicks [27] developed a theory for this effect – however itis not necessary to be aware of the details of this analysis in using the interfer-ometer: the non-orthogonality reduces the symmetry of the device, and insteadof 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 = k2L(1 + eθ)v2P

c3cos(2(θ − ψ)) + a(1 + eθ) cos(θ − β) + f (69)

The term 1 + eθ models the temperature effects, namely that as the arms areuniformly rotated, one rotation taking several minutes, there will be a temper-ature induced change in the length of the arms. If the temperature effects arelinear in time, as they would be for short time intervals, then they are linear inθ. 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 offseteffect. Michelson and Morley and Miller took these two effects into account whenanalysing his data. The Hick’s effect is particularly apparent in the Miller andMichelson-Morley data.

The interferometers are operated with the arms horizontal. Then in (69) θis the azimuth of one arm relative to the local meridian, while ψ is the azimuthof the absolute motion velocity projected onto the plane of the interferometer,

160

with projected component vP . Here the Fitzgerald-Lorentz contraction is a realdynamical effect of absolute motion, unlike the Einstein spacetime view thatit is merely a spacetime perspective artifact, and whose magnitude depends onthe choice of observer. The instrument is operated by rotating at a rate ofone rotation over several minutes, and observing the shift in the fringe patternthrough a telescope during the rotation. Then fringe shifts from six (Michelsonand Morley) or twenty (Miller) successive rotations are averaged to improve thesignal 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 shiftsis reproduced in Fig.5. Each row of the table is the average from six successiverotations. In the graphs Michelson and Morley are noting that the fringe shifts aremuch smaller than expected. But they were using Newtonian physics to calibratethe device. We now know that the detector is nearly 2000 times less sensitivethan given by that calibration, and that these fringe shifts correspond to a speedin excess of 300km/s. Michelson and Morley implicitly assumed the Newtonianvalue k=1, while Miller used an indirect method to estimate the value of k, ashe understood that the Newtonian theory was invalid, but had no other theoryfor the interferometer. Of course the Einstein postulates, as distinct from SpecialRelativity, have that absolute motion has no meaning, and so effectively demandsthat k = 0. Using k = 1 gives only a nominal value for vP , being some 8–9 km/sfor the Michelson and Morley experiment, and some 10 km/s from Miller; thedifference arising from the different latitudes of Cleveland and Mt. Wilson, andfrom Michelson and Morley taking data at limited times. The results from fittingthe form in (69) to the data is shown in Fig.6. Most significantly we see that theprojected speed and direction vary considerably for the same times on successivedays. This effect was seen in later experiments. These are the ‘gravitationalwaves’ of the induced metric in (52). So we now understand that Michelsonand Morley in 1887 detected a dynamical 3-space, and one in which the 3-spacevelocity fluctuations, the ‘gravitational waves’, were indeed apparent.

9.4 Miller Experiment 1925/26

The Michelson and Morley air-mode interferometer fringe shift data was basedupon a total of only 36 rotations in July 1887, revealing the nominal speed of some8–9 km/s when analysed using the prevailing but incorrect Newtonian theorywhich 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 observedfringe shifts, measured here in divisions of the telescope screw thread, and which is analysedusing (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. Theplots are for the sidereal times and days indicated, and each plot arises from averaging sixsuccessive rotations, i.e. only 36 rotations were performed in July 1887. The data was fittedwith (69) by a 6 parameter least-squares-fit by varying vP , ψ, a, β, e and f . Only vP and ψ areof physical interest, and are shown in each plot. ψ is measured clockwise from North. Afterthese parameters have been determined the Hicks and temperature terms were subtracted fromthe 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, whilethe other two give a poor fit. We also see that at the same time on successive days the speedand direction are significantly different. These are ‘gravitational wave’ effects, and were seen inlater experiments as well.

163

Figure 7: Typical Miller rotation-induced fringe shifts from average of 20 rotations, measuredevery 22.5, in fractions of a wavelength ∆λ/λ, vs arm azimuth θ(deg), measured clockwise fromNorth, from Cleveland Sept. 29, 1929 16:24 UT; 11:29 hrs average local sidereal time. The curveis the best fit using the form in (69), and then subtracting the Hick’s cos(θ−β) and temperatureterms from the data. Best fit gives ψ = 158, or 22 measured from South, and a projectedspeed 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 effectof the gas present as in (69) we find that nair =1.00029 gives k2 =0.00058 for air,which explains why the observed fringe shifts were so small. They rejected theirown data on the sole but spurious ground that the value of 8 km/s was smallerthan the speed of the Earth about the Sun of 30km/s. What their result reallyshowed was that (i) absolute motion had been detected because fringe shifts ofthe correct form, as in (69), had been detected, and (ii) that the theory givingk2 =1 was wrong, that Newtonian physics had failed. Michelson and Morley in1887 should have announced that the speed of light did depend of the directionof travel, that the speed was relative to an actual physical 3-space. Howevercontrary to their own data they concluded that absolute motion had not beendetected. This has had enormous implications for fundamental theories of spaceand time over the last 100 years.

It was Miller [3] who recognised that in the 1887 paper the theory for theMichelson interferometer must be wrong. To avoid using that theory Miller in-troduced the scaling factor k, even though he had no theory for its value. Hethen used the effect of the changing vector addition of the Earth’s orbital velocityand the absolute galactic velocity of the solar system to determine the numericalvalue 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

0 5 10 15 20

- 40

- 20

0

20

40 September

0 5 10 15 20

- 40

- 20

0

20

40 February

0 5 10 15 20 25

- 40

- 20

0

20

40 April

0 5 10 15 20

- 40

- 20

0

20

40 August

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 formmonth to month arises from the orbital motion of the earth about the sun: in different monthsthe vector sum of the galactic velocity of the solar system with the orbital velocity and sunin-flow velocity is different. As shown in Fig.9 DeWitte using a completely different experimentdetected the same direction and speed.

Miller determined the first estimate for k and for the absolute linear velocity ofthe solar system. Fig.7 shows typical data from averaging the fringe shifts from20 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 ro-tation 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 clearlyshow that the speed of light is different in different directions. The claim thatMichelson interferometers, operating in gas-mode, do not produce fringe shiftsunder rotation is clearly incorrect. But it is that claim that lead to the continuingbelief, within physics, that absolute motion had never been detected, and thatthe speed of light is invariant. The value of ψ from such rotations together leadto plots like those in Fig.8, which show ψ from the 1925/1926 Miller [3] interfer-ometer data for four different months of the year, from which the RA= 5.2 hr isreadily apparent. While the orbital motion of the Earth about the Sun slightlyaffects the RA in each month, and Miller used this effect to determine the value of

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k, the new theory of gravity required a reanalysis of the data , revealing that thesolar system has a large observed galactic velocity of some 420±30 km/s in thedirection (RA =5.2 hr, Dec =−67). This is different from the speed of 369 km/sin the direction (RA =11.20 hr, Dec =−7.22) extracted from the Cosmic Mi-crowave Background (CMB) anisotropy, and which describes a motion relative tothe distant universe, but not relative to the local 3-space. The Miller velocity isexplained by galactic gravitational in-flows [1].

An important implication of the new understanding of the Michelson inter-ferometer is that vacuum-mode resonant cavity experiments should give a nulleffect, as is the case [28].

9.5 Other Gas-mode Michelson Interferometer Experiments

Two old interferometer experiments, by Illingworth [19] and Joos [20], used he-lium, enabling the refractive index effect to be recently confirmed, because forhelium, with n= = 1.000036, we find that k2 =0.00007. Until the refractive in-dex effect was taken into account the data from the helium-mode experimentsappeared to be inconsistent with the data from the air-mode experiments; nowthey are seen to be consistent. Ironically helium was introduced in place of airto reduce any possible unwanted effects of a gas, but we now understand theessential role of the gas. The data from an interferometer experiment by Jasejaet al. [21], using two orthogonal masers with a He-Ne gas mixture, also indicatesthat they detected absolute motion, but were not aware of that as they used theincorrect Newtonian theory and so considered the fringe shifts to be too smallto be real, reminiscent of the same mistake by Michelson and Morley. While theMichelson interferometer is a 2nd order device, as the effect of absolute motionis proportional to (v/c)2, as in (69), but 1st order devices are also possible andthe coaxial cable experiments described next are in this class.

9.6 Coaxial Cable Speed of EM Waves AnisotropyExperiments

Rather than use light travel time experiments to demonstrate the anisotropy ofthe speed of light, another technique is to measure the one-way speed of radiowaves through a coaxial electrical cable. While this not a direct ‘ideal’ technique,as then the complexity of the propagation physics comes into play, it providesnot only an independent confirmation of the light anisotropy effect, but also onewhich takes advantage of modern electronic timing technology.

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0 10 20 30 40 50 60 70Sidereal Time

- 15

- 10

- 5

0

5

10

15

20

ns

Figure 9: (a) Variations in twice the one-way travel time, in ns, for an RF signal to travel1.5 km through a coaxial cable between Rue du Marais and Rue de la Paille, Brussels. Anoffset 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 timefor maximum effect of ∼ 5 hr and ∼ 17 hr (indicated by vertical lines) agrees with the directionfound by Miller. Plot shows data over 3 sidereal days and is plotted against sidereal time. Thefluctuations 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 varia-tions in the RF coaxial-cable travel times. This data is obtained from that in (a) after removalof the dominant effect caused by the rotation of the Earth. Ideally the velocity fluctuations arethree-dimensional, but the De Witte experiment had only one arm. This plot is suggestive of afractal 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 performedat the Utah University in 1981 by Torr and Kolen. This involved two rubidiumclocks placed approximately 500 m apart with a 5MHz radio frequency (RF)signal propagating between the clocks via a buried EW nitrogen-filled coaxialcable 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 expectedfrom Sect.7, variations in the one-way travel time were observed. The maximumeffect 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 interpretedas 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 cabletravel-time anisotropy experiment, accumulating data over 178 days. His data isin complete agreement with the Michelson-Morley 1887 and Miller 1925/26 inter-ferometer experiments. The Miller and De Witte experiments will eventually berecognised as two of the most significant experiments in physics, for indepen-dently and using different experimental techniques they detected essentially thesame velocity of absolute motion. But also they detected turbulence in the flowof space past the Earth — none other than gravitational waves. The De Witte ex-periment was within Belgacom, the Belgium telecommunications company. Thisorganisation had two sets of atomic clocks in two buildings in Brussels separatedby 1.5 km and the research project was an investigation of the task of synchro-nising these two clusters of atomic clocks. To that end 5MHz RF signals weresent 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 eachcluster: 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. Onecluster was in a building on Rue du Marais and the second cluster was due southin a building on Rue de la Paille. Digital phase comparators were used to measurechanges in times between clocks within the same cluster and also in the one-waypropagation times of the RF signals. At both locations the comparison betweenlocal clocks, A1-A2 and A1-A3, and between B1-B2, B1-B3, yielded linear phasevariations in agreement with the fact that the clocks have not exactly the samefrequencies together with a short term and long term phase noise. But betweendistant clocks A1 toward B1 and B1 toward A1, in addition to the same linearphase variations, there is also an additional clear sinusoidal-like phase undulationwith an approximate 24 hr period of the order of 28 ns peak to peak, as shownin Fig. 9. The experiment was performed over 178 days, making it possible tomeasure with an accuracy of 25 s the period of the phase signal to be the siderealday (23 hr 56 min).

Changes in propagation times were observed over 178 days from June 3 toNovember 27, 1991. A sample of the data, plotted against sidereal time for justthree days, is shown in Fig.9. De Witte recognised that the data was evidenceof absolute motion but he was unaware of the Miller experiment and did not re-alise that the Right Ascensions for minimum/maximum propagation time agreedalmost 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 motionshould 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/minimumwould have been shifted by 6 hrs. The declination of the velocity observed in thisDe Witte experiment cannot be determined from the data as only three days ofdata are available. The De Witte data is analysed in [24] and assuming a declina-tion of 60 S a speed of 430 km/s is obtained, in good agreement with the Millerspeed and Michelson-Morley speed. So a different and non-relativistic techniqueis confirming the results of these older experiments. This is dramatic.

De Witte reported the sidereal time of the ‘zero’ cross-over time, that is inFig.9 for all 178 days of data. That showed that the time variations are correlatedwith sidereal time and not local solar time. A least-squares best fit of a linearrelation to that data gives that the cross-over time is retarded, on average, by3.92 minutes per solar day. This is to be compared with the fact that a siderealday is 3.93 minutes shorter than a solar day. So the effect is certainly galacticand not associated with any daily thermal effects, which in any case would bevery small as the cable is buried. Miller had also compared his data againstsidereal time and established the same property, namely that the diurnal effectsactually tracked sidereal time and not solar time, and that orbital effects werealso apparent, with both effects apparent in Fig.8.

The dominant effect in Fig.9 is caused by the rotation of the Earth, namelythat the orientation of the coaxial cable with respect to the average directionof the flow past the Earth changes as the Earth rotates. This effect may beapproximately unfolded from the data leaving the gravitational waves shown inFig.9, [11, 23]. This is the first evidence that the velocity field describing the flowof 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 demonstratedby binning the absolute speeds and counting the number of speeds within eachbin, as discussed in [11, 23]. The Miller data also shows evidence of turbulence ofthe same magnitude. So far the data from four experiments, namely Miller, Torrand Kolen, De Witte and Cahill, show turbulence in the flow of space past theEarth. This is what can be called gravitational waves. This can be understood bynoting that fluctuations in the velocity field induce ripples in the mathematicalconstruct known as spacetime, as in (52). Such ripples in spacetime are knownas gravitational waves.

9.9 Cahill Coaxial Cable Anisotropy Experiment

During 2006 Cahill [24] performed another RF coaxial cable anisotropy experi-ment. This detector uses a novel timing scheme that overcomes the limitationsassociated with the two previous coaxial cable experiments. The intention in such

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0 5 10 15 20

100

200

300

400

500

600

Figure 10: Top: De Witte data, withsign reversed, from the first sidereal dayin Fig.9. This data gives a speed ofapproximately 430km/s. The data ap-pears to have been averaged over morethan 1hr, but still shows wave effects.Middle: Absolute projected speeds vP

in the Miller experiment plotted againstsidereal time in hours for a compositeday collected over a number of days inSeptember 1925. Maximum projectedspeed is 417 km/s. The data shows con-siderable fluctuations. The dashed curveshows the non-fluctuating variation ex-pected over one day as the Earth rotates,causing the projection onto the plane ofthe interferometer of the velocity of theaverage direction of the space flow tochange. If the data was plotted againstsolar time the form is shifted by manyhours. Note that the min/max occur atapproximately 5 hrs and 17 hrs, as alsoseen by De Witte and in the Cahill ex-periment. Bottom: Data from the Cahillexperiment [24] for one sidereal day onapproximately August 23, 2006. Wesee similar variation with sidereal time,and also similar wave structure. Thisdata has been averaged over a running1hr time interval to more closely matchthe time resolution of the Miller experi-ment. These fluctuations are believed tobe real wave phenomena of the 3-space.The new experiment gives a speed of418 km/s. We see remarkable agreementbetween all three experiments.

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experiments is simply to measure the one-way travel time of RF waves propa-gating through the coaxial cable. To that end one would apparently require twovery accurate clocks at each end, and associated RF generation and detectionelectronics. However the major limitation is that even the best atomic clocksare not sufficiently accurate over even a day to make such measurements to therequired 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 thetime variations are of the order of 25 ps per 10 meters of cable. To measure thatrequires time measurements accurate to, say, 1 ps. But atomic clocks have accu-racies over one day of around 100 ps, implying that lengths of around 1 kilometrewould be required, in order for the effect to well exceed timing errors. Even thenthe atomic clocks must be brought together every day to resynchronise them, oruse De Witte’s method of multiple atomic clocks. The new experiment is basedon the notion that optical fibers respond differently to coaxial cable with respectto 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 detectordesign has been developed by Cahill [25]. The device is very small, very cheapand easily assembled from readily available opto-electronic components. Theschematic layout of the detector is given in Fig.11. The detector relies on thephenomenon where the 3-space velocity v(r, t) affects differently the light traveltimes in the optical fibers, depending on the projection of v(r, t) along the fiberdirections. The differences in the light travel times are measured by means of theinterference effects in the beam joiner. The difference in travel times is given by

∆t = k2Lv2P

c3cos(2θ) (70)

where

k2 =(n2 − 1)(2− n2)

n

is the instrument calibration constant, obtained by taking account of the threekey effects: (i) the different light path trajectories, (ii) Lorentz contraction of thefibers, 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 therea net effect, otherwise when n = 1 the various effects actually cancel. So in thisregard the Michelson interferometer has a serious design flaw. This problem has

171

He-Nelaser

2x2 beam-splitter

-

photodiodedetector

datalogger

2x2 beam-joiner

--

ARM 1

ARM 2

-

100mm

oo

o

Figure 11: Schematic layout of the interferometric optical-fiber light-speedanisotropy/gravitational wave detector. This is essentially an optical-fiber version ofthe Michelson interferometer, see Fig.4. Coherent 633nm light from the a He-Ne Laseris split into two lengths of single-mode polarisation preserving fibers by the 2x2 beamsplitter. The two fibers take different directions, ARM1 and ARM2, after which the lightis recombined in the 2x2 beam joiner, in which the phase differences lead to interfer-ence effects that are indicated by the outgoing light intensity, which is measured in thephotodiode detector/amplifier, and then recorded in the data logger. The length of onestraight 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 vary-ing speed and direction of the flowing 3-space, by affecting differentially the speed of thelight, and hence the net phase difference between the two arms.

been overcome by using optical fibers. Here n = 1.462 at 633nm is the effectiverefractive index of the single-mode optical fibers (Fibercore SM600, temperaturecoefficient 5 × 10−2 fs/mm/C). Here L ≈ 200mm is the average effective lengthof the two arms, and vP (r, t) is the projection of v(r, t) onto the plane of thedetector, and the angle θ is that of the projected velocity onto the arm.

The interferometer operates by detecting the travel time difference betweenthe two arms as given by (70). The cycle-averaged light intensity emerging fromthe beam joiner is given by

I(t) ∝(Re(E1 + E2e

iω(τ+∆t))2

= 2|E|2 cos(ω(τ + ∆t)

2

)2

≈ a+ b∆t (71)

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Figure 12: D1 photodiode output voltage data (mV), recorded every 5 secs, from 5successive days, starting September 22, 2007, plotted against local Adelaide time (UT=local time + 9.5hrs). Day sequence may be determined by identifying identical values at0 and 24hrs. Dominant minima and maxima is earth rotation effect. Fluctuations fromday to day are evident as are fluctuations during each day - these are caused by waveeffects in the flowing space. Changes in RA cause changes in timing of min/max, whilechanges in magnitude are caused by changes in declination and/or speed. Blurring effectis caused by laser noise. These day plots correspond to the plots in Fig.10, there plottedagainst local sidereal time, and also inverted.

Here Ei are the electric field amplitudes and have the same value as the fibersplitter/joiner are 50%-50% types, and having the same direction because polari-sation preserving fibers are used, ω is the light angular frequency and τ is a traveltime difference caused by the light travel times not being identical, even when∆t = 0, mainly because the various splitter/joiner fibers will not be identicalin length. The last expression follows because ∆t is small, and so the detectoroperates 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, solong as a temperature uniformity is maintained, is that τ will be temperaturedependent. The temperature coefficient for the optical fibers gives an effectivefractional 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 fi-nally linearly related to ∆t. The detector calibration constants a and b dependon 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 D2operating 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 aligned50 anti-clockwise from local meridian, while D2 is aligned 110 anti-clockwise from localmeridian. The alignment offset between D1 and D2 causes the dominant earth-rotationinduced minima to occur at different times, with that of D2 at t = 7.6hrs delayed by0.8hrs relative to D1 at t = 6.8hrs. This is a fundamental test of the detection theory andof the phenomena. As well the data shows a simultaneous sub-mHz gravitational wavecorrelation at t ≈ 8.8hrs and of duration ≈ 1hr. This is the first observed correlation forspatially separated gravitational wave detectors. Two other wave effects (at t ≈ 6.5hrs inD2 and t ≈ 7.3hrs in D1) seen in one detector are masked by the stronger earth-rotationinduced minimum in the other detector.

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Two detectors were used with each detector located inside a sealed air-filledbucket located inside an insulated container containing some 90kg of water fortemperature stabilisation. Using two detectors enabled the confirmation of ex-pected phenomena, as a test of the detector theory, and also enabled the si-multaneous observations of wave phenomena. Detector D1 was in the Schoolof Chemistry, Physics and Earth Sciences, with an arm orientation of 50 anti-clockwise to the local meridian. Detector D2 was located 1.1km North of D1in the Australian Science and Mathematics School. This detector had an armorientation of 110 anti-clockwise to the local meridian. Fig.12 shows data fromD1 over 5 days. Fig.13 shows an effect caused by D1 and D2 having differentarm orientations and, as well, a simultaneous sub-mHz gravitational wave corre-lation at t ≈ 8.8hrs and of duration ≈ 1hr. This is the first observed correlationfor spatially separated gravitational wave detectors. There are now at least 11detections of the velocity field v(r, t) and of these 6 have observed the 3-spacewave/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 andα. When generalising the Schrodinger and Dirac equations to take account of this3-space we discovered that we arrive at an explanation for the phenomenon ofgravity including the equivalence principle, as well as an explanation for the space-time formalism. Here we explore various consequences of this new explanationfor gravity particularly those effects which reveal the effects of the α-dependentdynamics, in particular the bore hole anomaly which gives us the best estimatefor the value of α from several bore hole experiments. The dynamical 3-spacealso gives a completely new account of black holes; an account completely dif-ferent from the putative black holes of GR. In particular these new black holesgenerate an acceleration g that varies essentially as 1/r, rather than as 1/r2 asin Newtonian gravity (NG) and GR. This is a dramatic difference. It explainsimmediately 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 gravitationalacceleration. The new theory of 3-space does not need this ‘dark matter’. Theblack 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 symmetricdistribution, it is possible to compute the effective mass of the induced mini-mal black holes. Observational data from these systems confirms the prediction.Other effects discussed are the gyroscope precession effect caused by the vorticityof the flow of 3-space past the earth. Finally we also discuss the cosmologicalHubble expansion that arises from the 3-space dynamics. This gives an excel-lent parameter-free account of the redshift data from supernovae and gamma-raybursts. GR requires ‘dark energy’ to fit that data, so here we see that the new3-space dynamics does away with the need for ‘dark energy’. Not discussed hereinare anomalies in the Cavendish-like experiments to determine G [38], the gravi-tational lensing effects predicted by the generalised Maxwell equations, and alsoa 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 Gactually gives a technique for determining the value of α from earth based borehole gravity measurements. For a time-independent velocity field (7) may bewritten in the integral form

|v(r)|2 = 2G∫d3r′

ρ(r′) + ρDM (r′)|r− r′|

. (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 =8πGr

∫ r

0s2 [ρ(s) + ρDM (s)] ds+ 8πG

∫ ∞

rs [ρ(s) + ρDM (s)] ds, (73)

where, with v′ = dv(r)/dr,

ρDM (r) =α

8πG

(v2

2r2+vv′

r

). (74)

Iterating (73) once we find to 1st order in α that

ρDM (r) =α

2r2

∫ ∞

rsρ(s)ds+O(α2), (75)

6This in-flow is additional to the observed velocity of the earth through 3-space.

176

so that in spherical systems the ‘dark matter’ effect is concentrated near thecentre, and we find that the total ‘dark matter’ is

MDM ≡ 4π∫ ∞

0r2ρDM (r)dr =

4πα2

∫ ∞

0r2ρ(r)dr +O(α2) =

α

2M +O(α2) (76)

where M is the total amount of (actual) matter. Hence to O(α) MDM/M = α/2independently of the matter density profile. This turns out to be a very usefulproperty as complete knowledge of the density profile is then not required in orderto analyse observational data. As seen in Fig.14 the singular behaviour of bothv 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 accelera-tion8 is found to be, to 1st order in α, and using that ρ(r) = 0 for r > R, whereR is the radius of the earth,

g(r) =

(1 +

α

2)GM

r2, r > R,

4πGr2

∫ r

0s2ρ(s)ds+

2παGr2

∫ r

0

(∫ R

ss′ρ(s′)ds′

)ds, r < R.

(77)

This gives Newton’s ‘inverse square law’ for r > R, even when α 6= 0, whichexplains why the 3-space self-interaction dynamics did not overtly manifest inthe analysis of planetary orbits by Kepler and then Newton. However inside theearth (77) shows that g(r) differs from the Newtonian theory, corresponding toα = 0, as in Fig.14, and it is this effect that allows the determination of the valueof α from the Airy method.

Expanding (77) in r about the surface, r = R, we obtain, to 1st order in αand for an arbitrary density profile, but not retaining any density gradients atthe surface,

g(r) =

GNM

R2− 2GNM

R3(r −R), r > R,

GNM

R2−(

2GNM

R3− 4π(1− α

2)GNρ

)(r −R), r < R

(78)

where ρ is the matter density at the surface, M is the total matter mass of theearth, and where we have defined

GN ≡ (1 +α

2)G. (79)

7These are called black holes because there is an event horizon, but in all other aspects differfrom the black holes of General Relativity.

8We now use the convention that g(r) is positive if it is radially inward.

177

0 0.2 0.4 0.6 0.8 1r

0

2

4

6

8

10

12

14

g

0 0.2 0.4 0.6 0.8 1r

0

1

2

3

4

dens

ity

0 0.2 0.4 0.6 0.8 1r

- 2.4

- 2.2

- 2

- 1.8

- 1.6

- 1.4

v

Figure 14: Upper plot showsspeeds from numerical iterativesolution of (73) for a solid spherewith uniform density and radiusr = 0.5 for (i) upper curve thecase α = 0 corresponding toNewtonian gravity, and (ii) lowercurve with α = 1/137. Thesesolutions ony differ significantlynear r = 0. Middle plot showsmatter density and ‘dark matter’density ρDM , from (74), with ar-bitrary scales. Lower plot showsthe acceleration from (2) for (i)the Newtonian in-flow from theupper plot, and (ii) from the α =1/137 case. The difference is onlysignificant near r = 0. The ac-celerations begin to differ just in-side the surface of the sphere atr = 0.5, according to (81). Thisdifference is the origin of the borehole g anomaly, and permits thedetermination of the value of αfrom observational data. Thisgeneric singular-g behaviour, atr = 0, is seen in the earth, inglobular clusters and in galaxies.

178

The corresponding Newtonian gravity expression is obtained by taking the limitα→ 0,

gN (r) =

GNM

R2− 2GNM

R3(r −R), r > R,

GNM

R2−(

2GNM

R3− 4πGNρ

)(r −R), r < R

(80)

Assuming Newtonian gravity (80) then means that from the measurement ofdifference between the above-ground and below-ground gravity gradients, namely4πGNρ, and also measurement of the matter density, permit the determinationof 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 observeddifference of the gravity gradients was inconsistent with 4πGNρ in (80), in thatit 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 inthe 1980’s. The bore hole data papers [35, 36] report the discrepancy, i.e. theanomaly or the gravity residual as it is called, between the Newtonian predictionand the measured below-earth gravity gradient. Taking the difference between(78) and (80), assuming the same unknown value of GN in both, we obtain anexpression for the gravity residual

∆g(r) ≡ gN (r)− g(r) =

0, r > R,

2παGNρ(r −R), r < R.(81)

When α 6= 0 we have a two-parameter theory of gravity, and from (78) we seethat measurement of the difference between the above ground and below groundgravity gradients is 4π(1− α

2 )GNρ, 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 completemeasurement by itself, i.e. we need to combine it with other measurements. If wenow use laboratory Cavendish experiments to determine GN , then from the borehole gravity residuals we can determine the value of α, as already indicated in[39, 40]. These Cavendish experiments can only determine GN up to correctionsof order α/4, simply because the analysis of the data from these experimentsassumed the validity of Newtonian gravity [1]. So the analysis of the bore holeresiduals will give the value of α up to O(α2) corrections, which is consistent withthe O(α) analysis reported above.

Gravity residuals from a bore hole into the Greenland Ice Shelf were deter-mined down to a depth of 1.5 km by Ander et al. [35] in 1989. The observationswere 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 resid-uals for the Greenland Ice Shelf [35] Airy mea-surements of the g(r) profile, defined as ∆g(r) =gNewton−gobserved, and measured in mGal (1mGal= 10−3 cm/s2) and plotted against depth in km.The bore hole effect is that Newtonian gravity andthe new theory differ only beneath the surface,provided that the measured above surface grav-ity gradient is used in both theories. This thengives the horizontal line above the surface. Using(81) we obtain α−1 = 137.9 ± 5 from fitting theslope of the data, as shown. The non-linearity inthe data arises from modelling corrections for thegravity effects of the irregular sub ice-shelf rocktopography.

Figure 16: Gravity residuals from two of the Nevada bore hole experiments [36] that give abest 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 fromthe Greenland east coast at an elevation of 2530 m. It was believed that the iceprovided an opportunity to use the Airy method to determine GN , but now itis 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 unwantedeffects, such as imperfect knowledge of the ice density and, most dominantly,the terrain effects which arises from ignorance of the profile and density inho-mogeneities of the underlying rock. The bore hole gravity meter was calibratedby comparison with an absolute gravity meter. The ice density depends on pres-sure, temperature and air content, with the density rising to its average valueof ρ = 920 kg/m3 within some 200 m of the surface, due to compression of thetrapped air bubbles. This surface gradient in the density has been modelled bythe author, and is not large enough the affect the results. The leading source ofuncertainty was from the gravitational effect of the bedrock topography, and thiswas corrected for using Newtonian gravity. The correction from this is actuallythe cause of the non-linearity of the data points in Fig.15. A complete analysiswould require that the effect of this rock terrain be also computed using the newtheory of gravity, but this was not done. Using GN = 6.6742×10−11 m3s−2kg−1,which is the current CODATA value, we obtain from a least-squares fit of thelinear term in (81) to the data points in Fig.15 that α−1 = 137.9±5, which equalsthe value of the fine structure constant α−1 = 137.036 to within the errors, andfor this reason we identify the constant α in (81) as being the fine structureconstant. The first analysis [39, 40] of the Greenland Ice Shelf data incorrectlyassumed that the ice density was 930 kg/m3 which gave α−1 = 139± 5. Howevertrapped 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 ofα = e2/hc known from quantum theory.

Thomas and Vogel [36] performed another bore hole experiment at the NevadaTest Site in 1989 in which they measured the gravity gradient as a function ofdepth, the local average matter density, and the above ground gradient, alsoknown 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 theU.S. Government tests of its nuclear weapons. The density of the rock is measuredwith 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 rockdensity was found to be 2000 kg/m3, and is dry. This is the density used in theanalysis 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 ofspherical matter systems with masses M , plotted against Log10[M ], in solar masses M0. Thestraight 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 residualsfrom two of the bore holes are shown in Figs.16. All gravity measurements werecorrected 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 thematter beneath the holes occurs in homogeneous ellipsoidal layers. We see inFig.16 that the gravity residuals are linear with depth, where the density is theaverage value of 2000 kg/m3, but interspersed by layers where the residuals shownon-linear changes with depth. It is assumed here that these non-linear regionsare caused by variable density layers. So in analysing this data we have onlyused 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 inextraordinary agreement with the value of 137.04 from quantum theory. Here weagain used GN = 6.6742× 10−11 m3s−2kg−1, as for the Greenland data analysis.Zumberge et al. [37] performed an extensive underwater Airy experiment, butfailed to measure the above water g, so their results cannot be analysed in theabove 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 ‘blackhole’ is used because they have a compact closed event horizon where the in-flow speed relative to the horizon equals the speed of light, but in other respectsthey differ from the putative black holes of General Relativity - in particular theirgravitational acceleration is not inverse square law. The evidence is that it is thesenew ‘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, aspherical galaxy or globular cluster. These black holes are termed minimal blackholes, as their effective mass is minimal, (ii) primordial naked black holes whichthen attract matter. These result in spiral galaxies, and the effective mass of theblack hole is larger than required merely by the matter induced in-flow. These aretherefore termed non-minimal black holes. These explain the rapid formation ofstructure in the early universe, as the gravitational acceleration is approximately1/r rather than 1/r2. This is the feature that also explains the so-called ‘darkmatter’ effect in spiral galaxies. We consider now the minimal black holes.

Equation (1) has exact analytic ‘black hole’ solutions where ρ = 0 (actuallya one-parameter family - but we write in this form for comparison with the nextsection)

v(r) = K

1r

+1Rs

(Rs

r

)α2

1/2

(82)

where the 1/r term can only arise if matter is present, and the 2nd term isthe ‘black hole’ effect. The consequent ‘black hole’ contribution to the totalacceleration can be attributed to an effective mass MDM , which we now also callMBH . To O(α) this effective mass is independent of the matter density profile,and is given by (76),

MBH = MDM = 4π∫ ∞

0r2ρDM (r)dr =

α

2M +O(α2) (83)

This solution is applicable to the black holes at the centre of spherical star sys-tems, where we identify MDM as MBH . So far black holes in 19 spherical starsystems have been detected and together their masses are plotted in Fig.17 andcompared with (83) [41, 42].

This result applies to any spherically symmetric matter distribution. Thismeans that the bore hole anomaly is indicative of an in-flow singularity at thecentre of the earth. This contributes some 0.4% of the effective mass of the earth,

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5 10 15 20 25r

25

50

75

100

125

150

175

200

V

Figure 18: Data shows the non-Keplerianrotation-speed curve vO for the spiral galaxyNGC 3198 in km/s plotted against radius inkpc/h. Lower curve is the rotation curve fromthe 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 smallvalue of α. This asymptotic form is caused bythe primordial black holes at the centres of spiralgalaxies, and which play a critical role in theirformation. The spiral structure is caused bythe rapid in-fall towards these primordial blackholes.

as defined by Newtonian gravity. However in star systems this minimal black holeeffect is more apparent, and we label MDM as MBH . Essentially even in the non-relativistic regime the Newtonian theory of gravity, with its ‘universal’ InverseSquare 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 sufficientlyfar from the centre we find an exact non-perturbative two-parameter class of an-alytic solutions as in (82). There K and Rs are arbitrary constants in the ρ = 0region, but whose values are determined by matching to the solution in the mat-ter region. Here Rs characterises the length scale of the non-perturbative part ofthis 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 squarelaw’,

g(r) =K2

2

1r2

+α

2rRs

(Rs

r

)α2

, (84)

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in the asymptotic limit. The centripetal acceleration relation for circular orbitsvO(r) =

√rg(r) gives a ‘universal rotation-speed curve’

vO(r) =K

2

1r

+α

2Rs

(Rs

r

)α2

1/2

(85)

Because of the α dependent part this rotation-velocity curve falls off extremelyslowly with r, as is indeed observed for spiral galaxies. An example is shown inFig.18. It was the inability of the Newtonian gravity and GR to explain theseobservations that led to the notion of ‘dark matter’. So ‘dark matter’ is not apart of reality.

For the spatial flow in (82) we may compute the effective ‘dark matter’ densityfrom (74)

ρDM (r) =(1− α)α

16πGK2

R3s

(Rs

r

)2+α/2

(86)

We see the standard 1/r2 behaviour usually attributed to ‘dark matter’ in spiralgalaxies. It should be noted that the Newtonian component of (82) does notcontribute, and that ρDM (r) is exactly zero in the limit α→ 0. So supermassiveblack holes and the spiral galaxy rotation anomaly are all α-dynamics phenomena.

10.5 Lense-Thirring Effect and the GPB GyroscopeExperiment

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 un-precedented accuracy [44, 45, 46, 47]. Such a precession is predicted by GR, withtwo components (i) a geodetic precession, and (ii) a ‘frame-dragging’ precessionknown as the Lense-Thirring effect. The latter is particularly interesting effectinduced by the rotation of the earth, and described in GR in terms of a ‘gravito-magnetic’ field. According to GR this smaller effect will give a precession of 0.042arcsec per year for the GP-B gyroscopes. Here we show that GR and the newtheory make very different predictions for the ‘frame-dragging’ effect, and so theGP-B experiment will be able to decisively test both theories. While predictingthe same earth-rotation induced precession, the new theory has an additionalmuch larger ‘frame-dragging’ effect caused by the observed translational motionof the earth. As well the new theory explains the ‘frame-dragging’ effect in termsof vorticity in a ‘substratum flow’. Herein the magnitude and signature of this

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S

VE

Figure 19: Shows the earth (N is up) and vor-ticity vector field component ~ω induced by therotation of the earth, as in (87). The polar or-bit of the GP-B satellite is shown, S is the gyro-scope starting spin orientation, directed towardsthe guide star IM Pegasi, RA = 22h 53′ 2.26′′, Dec= 160 50′ 28.2′′, and VE is the vernal equinox.

new component of the gyroscope precession is predicted for comparison with datafrom GP-B when it becomes available.

Here we consider one difference between the two theories, namely that asso-ciated with the vorticity part of (12), leading to the ‘frame-dragging’ or Lense-Thirring effect. In GR the vorticity field is known as the ‘gravitomagnetic’ fieldB = −c ~ω. In both GR and the new theory the vorticity is given by (10) butwith a key difference: in GR vR is only the rotational velocity of the matter inthe earth, whereas in the 3-space dynamics vR is the vector sum of the rotationalvelocity 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 velocityof the earth, giving

~ω(r) = 4G

c23(r.L)r− r2L

2r5, (87)

where L is the angular momentum of the earth, and r is the distance from thecentre. This component of the vorticity field is shown in Fig.19. Vorticity maybe detected by observing the precession of the GP-B gyroscopes. The vorticityterm in (87) leads to a torque on the angular momentum S of the gyroscope,

~τ =∫d3rρ(r) r× (~ω(r)× vR(r)), (88)

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

186

S

VE

v

Figure 20: Shows the earth (N is up) and themuch larger vorticity vector field component ~ω in-duced 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 53′ 2.26′′, Dec = 160 50′ 28.2′′, VE is thevernal 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. Thisgives

~τ =12~ω × S (89)

and so ~ω/2 is the instantaneous angular velocity of precession of the gyroscope.This corresponds to the well known fluid result that the vorticity vector is twicethe angular velocity vector. For GP-B the direction of S has been chosen so thatthis precession is cumulative and, on averaging over an orbit, corresponds to some7.7 × 10−6 arcsec per orbit, or 0.042 arcsec per year. GP-B has been superblyengineered 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 effectsof the orbital motion of the earth, the observed flow past the earth towards thesun, and the flow into the earth, and effects of the gravitational waves, then (12)gives

~ω(r) =2GMc2

vC × rr3

. (90)

This much larger component of the vorticity field is shown in Fig.20. The max-imum magnitude of the speed of this precession component is ω/2 = gvC/c

2 =8× 10−6arcsec/s, where here g is the gravitational acceleration at the altitude ofthe satellite. This precession has a different signature: it is not cumulative, andis 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,

187

0 20 40 60 80orbit - minutes

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016arcsec

Figure 21: Predicted variation of the precession angle ∆Θ = |∆S(t)|/|S(0)|, in arcsec, overone 97 minute GP-B orbit, from the vorticity induced by the translation of the earth, as givenby (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 arecaused by the angle effects in (91). These changes arise from the effects of the changing orbitalvelocity of the earth about the sun. The GP-B expected angle measurement accuracy is 0.0005arcsec.

as in general,

∆S(t) =∫ t

0dt′

12~ω(r(t′))× S(t′) ≈

(∫ t

0dt′

12~ω(r(t′))

)× S(0). (91)

Here ∆S(t) is the integrated change in spin, and where the approximation arisesbecause the change in S(t′) on the RHS of (91) is negligible. The plot in Fig.21shows this effect to be some 30× larger than the expected GP-B errors, and soeasily detectable, if it exists as predicted herein. This precession is about theinstantaneous 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 perpendicularto 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 effectsvC [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 thatfollows from (1) (with β = −α). We shall see that the solution gives an excellent

188

parameter-free fit to the supernovae and gamma-ray bursts magnitude - redshiftdata [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 leadto the prediction that the universe expansion will accelerate in the future. Thiseffect 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 unlikethe 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), modellingEM radiation, matter, cosmological constant etc, and that we look for a radiallysymmetric time-dependent flow v(r, t) = v(r, t)r from (1) (with β = −α). Then

v(r, t) satisfies the equation, with v′ =∂v(r, t)∂r

,

∂

∂t

(2vr

+ v′)

+ vv′′ + 2vv′

r+ (v′)2 +

α

4

(v2

r2+

2vv′

r

)= −4πGρ(r, t) (92)

Consider first the zero energy case ρ = 0. Then we have a Hubble solutionv(r, t) = H(t)r, a centreless flow, determined by

H +(

1 +α

4

)H2 = 0 (93)

with H =dH

dt. We also introduce in the usual manner the scale factor a(t)

according to H(t) =1a

da

dt. We then obtain the solution

H(t) =1

(1 + α4 )t

= H0t0t

; a(t) = a0

(t

t0

)4/(4+α)

(94)

where H0 = H(t0) and a0 = a(t0). Note that we obtain an expanding 3-spaceeven where the energy density is zero - this is in sharp contrast to the GR-FLRWmodel for the expanding universe, as shown below.

We can write the Hubble functionH(t) in terms of a(t) via the inverse functiont(a), i.e. H(t(a)) and finally as H(z), where the redshift observed now, t0, relativeto 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 cosmolog-ical observables, namely the relationship between the apparent energy-flux mag-nitudes and redshifts. This involves taking account of the reduction in photon

189

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

r(t) =∫ t

t1dt′(c+ v(r(t′), t′), (96)

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

r(t) = ca(t)∫ t

t1

dt′

a(t′). (97)

This distance gives directly the surface area 4πr(t)2 of the expanding sphereand so the decreasing photon count per unit of that surface area. However alsobecause 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 fluxis then given by

FP =LP

4πr(t)2(1 + z)(98)

where LP is the source photon-number luminosity. However usually the energyflux 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 energyluminosity LE ,

FE =LE

4πr(t)2(1 + z)2≡ LE

4πrL(t)2(99)

which defines the effective energy-flux luminosity distance rL(t). Expressed interms of the observable redshift z this gives an energy-flux luminosity effectivedistance

rL(z) = (1 + z)r(z) = c(1 + z)∫ z

0

dz′

H(z′)(100)

The dimensionless ‘energy-flux’ luminosity effective distance is then given by

dL(z) = (1 + z)∫ z

0

H0dz′

H(z′)(101)

190

Figure 22: Plot of the scale factor R(t) vs t, with t = 0 being ‘now’ with R(0) = 1, for thefour cases discussed in the text, and corresponding to the plots in Figs.23 and 24: (i) the uppercurve is the ‘dark energy’ only case, resulting in an exponential acceleration at all times, (ii) thebottom curve is the matter only prediction, (iii) the 2nd highest curve (to the right of t = 0) isthe best-fit ‘dark energy’ plus matter case showing a past deceleration and future exponentialacceleration effect. The straight line plot is the dynamical 3-space prediction showing a slightlyolder universe compared to case (iii). We see that the best-fit ‘dark energy’ - matter curveessentially 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 shownin Figs.23 and 24, yielding an excellent agreement with the supernovae and GRBdata. Note that because α/4 is so small it actually has negligible effect on theseplots. Hence the dynamical 3-space gives an immediate account of the universeexpansion data, and does not require the introduction of a cosmological constantor ‘dark energy’, but which will be nevertheless discussed next.

When the energy density is not zero we need to take account of the dependenceof ρ(r, t) on the scale factor of the universe. In the usual manner we thus write

ρ(r, t) =ρm

R(t)3+

ρr

R(t)4+ Λ (103)

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

191

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] (dotswithout 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 ismatter only Ωm = 1. Two middle curves show best fit of ‘dark energy’-matter and dynamical 3-space prediction, and are essentially indistinguishable. However the theories make very differentpredictions 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.

192

Figure 24: Hubble diagram as in Fig.23 but plotted logarithmically to reveal details forz < 2, and without GRB data. Upper curve is ‘dark energy’ only ΩΛ = 1. Next curve is bestfit of ‘dark energy’-matter. Lowest curve is matter only Ωm = 1. 2nd lowest curve is dynamical3-space prediction.

193

(i.e independent of r) equivalent matter density. We argue here that Λ - the darkenergy density, like dark matter, is an unnecessary concept. Then (92) becomesfor R(t)

R

R+α

4R2

R2= −4πG

3

(ρm

R3+ρr

R4+ Λ

)(104)

giving

R2 =8πG

3

(ρm

R+ρr

R2+ ΛR2

)− α

2

∫R2

RdR (105)

In terms of R2 this has the solution

R2 =8πG

3

(ρm

(1− α2 )R

+ρr

(1− α4 )R2

+ΛR2

(1 + α4 )

+bR−α/2

)(106)

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

t(R) =∫ R

R0

dR√8πG

3

(ρm

R+ρr

R2+ ΛR2 + bR−α/2

) (107)

where now we have re-scaled parameters ρm → ρm/(1− α2 ), ρr → ρr/(1− α

4 ) andΛ → Λ/(1 + α

4 ). When ρm = ρr = Λ = 0, (107) reproduces the expansion in(94), and so the density terms in (107) give the modifications to the dominantpurely spatial expansion dynamics, which we have noted above already gives anexcellent account of the data. From (107) we then obtain

H(z)2 = H02(Ωm(1 + z)3 + Ωr(1 + z)4 + ΩΛ + Ωs(1 + z)2+α/2) (108)

withΩm + Ωr + ΩΛ + Ωs = 1. (109)

Using the Hubble function (108) in (101) and (102) we obtain the plots inFigs.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 showingthat a matter only expansion is not a good account of the data, (iii) from a leastsquares fit with Ωs = 0 we find Ωm = 0.28,Ωr = 0,ΩΛ = 0.68 which led to thesuggestion that ‘dark energy’ effect was needed to fix the poor fit from (ii), andfinally (iv) Ωm = 0,Ωr = 0,ΩΛ = 0,Ωs = 1, as noted above, that the spatialexpansion dynamics alone gives a good account of the data. Of course the EM

194

radiation term Ωr is non-zero but small and determines the expansion during thebaryogenesis initial phase, as does the spatial dynamics expansion term becauseof the α dependence. If the age of the universe is inferred to be some 14Gyrsfor case (iii) then, as seen in Fig.22, the age of the universe is also some 14Gyrfor case (iv). We see that the best-fit ‘dark energy’ - matter curve essentiallyconverges on the dynamical 3-space result.

The induced effective spacetime metric in (52) is for the Hubble expansion

ds2 = gµνdxµdxν = dt2 − (dr−H(t)r)dt)2/c2 (110)

The occurrence of c has nothing to do with the dynamics of the 3-space - itis related to the geodesics of relativistic quantum matter, as shown in Sect.8.Changing variables r→ R(t)r we obtain

ds2 = gµνdxµdxν = dt2 −R(t)2dr2/c2 (111)

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

11 Conclusions

We have briefly reviewed the extensive evidence for a dynamical 3-space, withthe minimal dynamical equation now known and confirmed by numerous experi-mental and observational data. This 3-space has been repeatedly detected sincethe Michelson-Morley experiment of 1887, and they also detected ‘gravitationalwaves’, which are just 3-space velocity fluctuations. As well the dynamical 3-spacehas been indirectly detected by means of the dynamical equation explaining di-verse phenomena. We have shown that this equation has a Hubble expanding3-space solution that in a parameter-free manner manifestly fits the recent su-pernovae and gamma-ray bursts redshift data. All of these successes imply that‘dark energy’ and ‘dark matter’ are unnecessary notions. This Hubble solutionleads to a uniformly expanding universe, and so without acceleration: the claimedacceleration 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 aswell various problems such as the fine turning problem, the horizon problem and

195

other difficulties in the current modelling of the universe. We have also shownwhy the spacetime formalism appeared to be so successful, despite having noontological status. One key discovery has been that Newton’s theory of gravityis flawed, except in the very special case of planets in orbit about a sun, which isof course the restricted manifestation of gravity that was available to Newton.

At a deeper level the occurrence of α in (1) suggests that 3-space is actuallya quantum system, and that (1) is merely a phenomenological description of thatat the ‘classical’ level. In which case the α-dependent dynamics amounts to thedetection of quantum space and quantum gravity effects, although clearly not ofthe form suggested by the quantisation of General Relativity. At a deeper levelthe information-theoretic Process Physics has given insights into the possiblenature of reality as a limited self-referential system, in which quantum space andquantum matter are emergent phenomena, with both exhibiting non-local effects.In particular it implies that we have a ‘universal’ process time, as distinct fromthe current prevailing geometrical modelling of time. These results all suggestthat a radically different paradigm for reality is emerging, and in which we see aunification of quantum space and quantum matter, and with gravity an emergentphenomenon.

Thanks to Tim Eastman, Erich Weigold, Igor Bray and Lance McCarthy forongoing support.

References

[1] Cahill R.T. Process Physics: From Information Theory to Quantum Spaceand Matter, Nova Science Pub., New York, 2005.

[2] Michelson A.A. and Morley E.W. Am. J. Sc. 34, 333-345, 1887.

[3] Miller D.C. Rev. Mod. Phys., 5, 203-242, 1933.

[4] Cahill R.T. Process Physics, Process Studies Supplement, Issue 5, 1-131,2003.

[5] Cahill R.T. and Klinger C.M. Bootstrap Universe from Self-referential Noise,Progress in Physics, 2, 108-112, 2005.

[6] Cahill R.T. and Klinger C.M. Self-referential Noise as a Fundamental Aspectof Reality, published in proc. 2nd Intl. Conf. on Unsolved Problems of Noise

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and Fluctuations (UPoN 99), eds Abbott, D. and Kish L. 511, 43, AmericanInstitute of Physics, NY, 2000.

[7] Cahill R.T., Klinger C.M. and Kitto K. Process Physics: Modelling Realityas Self-organising Information, The Physicist, 37(6), 191-195, 2000.

[8] Cahill R.T. and Klinger C.M. Self-referential Noise and the Synthesis ofThree-dimensional Space, Gen. Rel. and Grav. 32(3), 529, 2000.

[9] Cahill R.T. Process Physics: Inertia, Gravity and the Quantum, Gen. Rel.and Grav. 34, 1637-1656, 2002.

[10] Newton I. Philosophiae Naturalis Principia Mathematica, 1687.

[11] Cahill R.T. Dynamical Fractal 3-Space and the Generalised SchrodingerEquation: Equivalence Principle and Vorticity Effects, Progress in Physics,1, 27-34, 2006.

[12] Ehrenfest P. Z. Physik, v.45, 455, 1927.

[13] Shnoll, S.E. et al. Experiments with Radioactive Decay of 239Pu: EvidenceSharp Anisotropy of Space, Progress in Physics, v.1, pp.81-84, 2005, andreferences therein.

[14] Hertz H. On the Fundamental Equations of Electro-Magnetics for Bodiesin Motion, Wiedemann’s Ann. 41, 369, 1890; Electric Waves, Collection ofScientific Papers, Dover Pub., New York, 1962.

[15] Cahill R.T. Dynamical 3-Space: Alternative Explanation of the ‘Dark MatterRing’, arXiv:0705.2846v1, 2007.

[16] Cahill R.T. Absolute Motion and Gravitational Effects, Apeiron, 11(1), 53-111, 2004.

[17] Cahill R.T. and Kitto K. Michelson-Morley Experiments Revisited, Apeiron,10(2),104-117, 2003.

[18] Cahill R.T. The Michelson and Morley 1887 Experiment and the Discoveryof Absolute Motion, Progress in Physics, 3, 25-29, 2005.

[19] Illingworth K.K. Phys. Rev. 3, 692-696, 1927.

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[23] Cahill R.T. The Roland DeWitte 1991 Experiment, Progress in Physics, 3,60-65, 2006.

[24] Cahill R.T. A New Light-Speed Anisotropy Experiment: Absolute Motionand Gravitational Waves Detected, Progress in Physics, 4, 73-92, 2006.

[25] Cahill R.T. Optical-Fiber Gravitational Wave Detector: Dynamical 3-SpaceTurbulence Detected, Progress in Physics, 4, 63-68, 2007.

[26] Cahill R.T. and Stokes F. Correlated Detection of sub-mHz GravitationalWaves by Two Optical-Fiber Interferometers, Progress in Physics, 2, 103-110, 2008.

[27] Hicks W. M.On the Michelson-Morley Experiment Relating to the Drift ofthe Ether. Phil. Mag., v. 3, 9–42, 1902.

[28] Muller, H. et al. Modern Michelson-Morley Experiment using Cryogenic Op-tical Resonators. Phys. Rev. Lett. 91(2), 020401-1, 2003.

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[41] Cahill R.T. Black Holes in Elliptical and Spiral Galaxies and in GlobularClusters, Progress in Physics, 3, 51-56, 2005.

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[47] Cahill R.T. Novel Gravity Probe B Frame-Dragging Effect, Progress inPhysics, 3, 30-33, 2005.

[48] Cahill R.T. Dynamical 3-Space: Supernova and the Hubble Expansion - OlderUniverse and End of Dark Energy, arXiv:0705.1569v1, 2007.

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Relativistic physics from paradoxes

to good sense - 1

F. Selleri Dipartimento di Fisica, Università di Bari

INFN, Sezione di Bari

Abstract The present paper reviews the results obtained in recent years by the author in relativistic physics. Historically the two theories of relativity were born from the clash of positivism and realism. The former current of thought used relativism as a weapon against ideas of realistic inclination, like Lorentz’s. Paradoxes were the consequence in the new relativistic paradigm of emarginating realism. The recent understanding of the role of the conventional definition of simultaneity in relativistic physics has opened the doors to new lines of thought. Epistemologists have stressed that the coefficient of the space variable x in the Lorentz transformation of time (we call it e1 ) has a nonphysical (“conventional“) nature. Therefore, it should be possible to modify e1 without touching the empirical predictions of the theory. Given that Einstein’s principle of relativity leads necessarily to the Lorentz transformations, such a modification implies however a reformulation of the relativistic idea itself. With respect to this ideal picture, the concrete development of the research has produced some exciting surprises. Nature does not seem to be so indifferent about the value of e1 , given that several phenomena, in particular those taking place on a rotating platform (Sagnac effect, and all that) converge in a strong indication of the value e1 = 0 . This implies absolute simultaneity and a new type of space and time transformations which we call "inertial". Today we count on six proofs of absolute simultaneity, which are essentially independent of one another (three are contained in the second part of the paper). The cosmological consequences of the new structure of space and time go against the big bang model. After our results relativism, although weakened, is not dead and keeps proposing itself under milder forms

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1. Einstein positivist/realist Mach’s epistemology had a strong impact at the end of the XIXth and the beginning of the XXth century. One can say that the theory of relativity was formulated by trying to satisfy at least part of the epistemological demands of the Viennese philosopher, which are so described by Kostro: “Notions such as “force,” “matter,” “atom,” “absolute space” are our subjective inventions, not something experimentally tangible. They should therefore be eliminated from physics. After this, one would be left only with “sense impressions,” which Mach preferred to call “elements.” What we call the world is nothing but the system of such “elements”.” [1]

From the theory of special relativity (TSR) Einstein deduced that every clock in motion slows the pace of its time. His 1905 standpoint was the following: ether does not exist, therefore it does not make any sense to consider motion with respect to nothing. Motion has to be described with respect to concrete systems only. The slowing down of clocks is always relative to observers who see them in motion, and a complete physical and philosophical symmetry exists between the conclusions of different inertial observers. Considering a clock in motion relative to the inertial observers O1, O2, . .. On with respective velocities v 1, v 2, . .. v n , its rate should appear slowed by the respective factors R (v 1), R (v 2 ), .. . R(v n ), given by a unique function of relative velocity, in agreement with the relativity principle. A legitimate question seems to remain: “What really happens to the clock, which is it its true rate?” The relativistic answer is that this question does not make sense, and that the conclusions of all the different observers are equally valid. In this way the philosophy of relativism and subjectivism becomes dominant in physics for the observations of the inertial observers. Of course the argument can be generalized by going from the time marked by clocks to any other physical quantity: we will see it done by the English physicist J. Jeans in the third section. Einstein’s relativism clearly originates in positivism and it is surprising that it was never disavowed by the founder of relativity in spite of his sharp break with Mach. “The fact that Mach condemned the theory of relativity was a very unpleasant experience for Einstein. He stopped praising Mach’s achievements and started criticizing him and his epistemological views” [2] To show this, Kostro quotes Einstein’s answer to a question asked by Emil Meyerson during a reception on April 6, 1922 in Paris, organised by the French Philosophical Society in honour of Albert Einstein. : “There does not appear to be a great relation from the logical point of view between the theory of relativity and Mach’s theory. [...] Mach’s system studies the existing relations between data of experience; for Mach science is the totality of these relations. That point of view is wrong, and, in fact, what Mach has done is to make a catalogue, not a system. To the extent that Mach was a good mechanician he was a deplorable philosopher.’ ” [3]

The critical point of view of Mach’s philosophy was kept till the end, as one can see from the 1948 Scientific Autobiography [4] where Einstein expressed an

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appraisal of the Machian philosophy very similar to the previous one. Ten years before, in a letter to M. Solovine Einstein had stated: “In these days the subjective and positivist viewpoint dominates in a most excessive manner. The need for conceiving nature as an objective reality is declared to be an obsolete prejudice, and thus a virtue is made of the necessity of quantum theory. Men are just as subject to suggestion as horses, and each epoch is dominated by a fashion, and the majority do not even see the tyrant who dominates them.” [5] Einstein’s criticism of positivism is surely deep and interesting, nevertheless it is difficult to avoid the impression that he underestimated its impact on his own scientific creations.

It is worth recalling that important epistemologists shared Einstein’s critical evaluation of positivism. E.g., Karl Popper wrote: “Positivists […] are constantly trying to prove that metaphysics by its very nature is nothing but nonsensical twaddle – sophistry and illusion – as Hume says, which we should commit to the flames. […] There is no doubt that what the positivists really want to achieve is not so much a successful demarcation as the final overthrow and the annihilation of metaphysics.” [6] and: “Positivists, in their anxiety to annihilate metaphysics, annihilate natural science with it. For scientific laws ... cannot be logically reduced to elementary statements of experience. If consistently applied, Wittgenstein’s criterion of meaningfulness rejects as meaningless those natural laws the search for which, as Einstein says, is ‘the supreme task of the physicist’ ” [7] Popper ascribed the spreading of positivism in physics to the influence of the young Einstein. A statement that seems to me to go to the core of the problem is the following: “The philosophical impact of Mach’s positivism was largely transmitted by the young Einstein. But Einstein turned away from Machian positivism, partly because he realized with a shock some of its consequences; consequences which the next generation of brilliant physicists, among them Bohr, Pauli and Heisenberg, not only discovered but enthusiastically embraced: they became subjectivists. But Einstein’s withdrawal came too late. Physics had become a stronghold of subjectivist philosophy, and it has remained so ever since.” [8]. In fact Popper could witness Einstein’s radical change of opinion about Mach’s philosophy: “It is an interesting fact that Einstein himself was for years a dogmatic positivist and operationalist. He later rejected this interpretation: he told me in 1950 that he regretted no mistake he ever made as much as this mistake.” [9]

It is fair to add that Einstein described himself as oscillating between different philosophies. He devoted many papers to epistemology. Other famous physicists published articles and books on the same argument (Planck, Schrödinger, Bohr, Heisenberg), but nobody with the richness and the critical skill of Einstein. About the relationship between physics and philosophy he wrote:

“The reciprocal relationship of epistemology and science is of noteworthy kind. They are dependent upon each other. Epistemology without contact with science becomes an empty scheme. Science without epistemology is – insofar as it is thinkable at all - primitive and muddled. ... He [the scientist] accepts gratefully the

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epistemological conceptual analysis; but the external conditions, which are set for him by the facts of experience, do not permit him to let himself be too much restricted in the construction of his conceptual world by the adherence to an epistemological system. He therefore must appear to the systematic epistemologist as a type of unscrupulous opportunist: he appears as realist insofar as he seeks to describe a world independent of the acts of perception; as idealist insofar as he looks upon the concepts and theories as the free inventions of the human spirit (not logically derivable from what is empirically given); as positivist insofar as he considers his concepts and theories justified only to the extent to which they furnish a logical representation of relations among sensory experiences. He may even appear as Platonist or Pythagorean insofar as he considers the viewpoint of logical simplicity as an indispensable and effective tool of his research”. [10]

The philosophical standpoint of physicists is rarely the eclectic one here described. Different scientists embraced different philosophies, but almost always very well defined for every single author. The previous description should rather be understood in the autobiographical sense, as Einstein in different moments of his scientific activity indeed followed different philosophical ideas. I insist to say that he conformed to positivism when the two relativistic theories were formulated and defended their interpretation based on relativism during his whole lifetime. He behaved as a realist, however, in other famous papers of 1905 (on Brownian motion and on the light quanta) and in his long battle against the Copenhagen formulation of quantum mechanics.

The most essential conflict that accompanied and followed the birth of the Copenhagen-Göttingen theory was a philosophical clash around the idea of physical reality. The realists, headed by Einstein, included Planck, Ehrenfest, Schrödinger and de Broglie. Winners were however the antirealists (Bohr, Born, Heisenberg, Pauli, Dirac) not because they could prove the realists’ ideas false, but because they were united in developing a theory coherent with their philosophical choices and able to explain a remarkable number of phenomena. A philosopher who always defended the orthodox position is our expert of relativistic synchronization, Hans Reichenbach, who wrote: “You say that while you are in your office your house stands unchanged in its place. How do you know? [ … ] The trouble is that unless you can find a better answer to that question than is supplied by the arguments of common sense, you will not be able to solve the problem of whether light and matter consist of particles or waves.” [11] The strong idealistic taste of quantum theory was confirmed by many writers, e.g. by Karl Popper: "… the Copenhagen interpretation of quantum mechanics, is about universally accepted. In brief, it says that objective reality has evaporated, an that quantum mechanics does not represent particles, but rather our knowledge, our observations, or our consciousness, of particles. [Popper's italics] [12] The degree of idealism of quantum theory was too much to bear for Einstein, as his relativism was surely a milder form of rejection of the objective reality than provided by the Copenhagen doctrine. Anyway, Einstein never accepted the final formulation

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of quantum mechanics, which he considered at least as incomplete as classical thermodynamics (in so far as not based on atomism). The Copenhagen physicists had the feeling to have faithfully pursued the path, which Einstein had shown adopting positivism and relativism, while he himself stopped at a certain point.

Einstein published comments of sharply realistic mould in the context of quantum theory. For example: “There is such a thing as the ‘real state’ of a physical system, which exists objectively, independently of any observation or measurement, and which can be described, in principle, with the means of description afforded by physics.” A few lines below he added: “All men, the quantum theoreticians included, actually stick steadfastly to this thesis on reality, as long as they do not discuss the foundations of quantum theory.” And right afterwards he wrote: “I am not ashamed to make the ‘real state of a system’ the central concept of my approach.” [13]

Einstein insisted that the physicist should try to form an image of the studied process, almost a hypothetical picture that can acquire validity only after many controls and which must be taken as the basis of the theoretical constructions. In a letter to Born of 1947 he wrote: “Therefore I cannot seriously believe in it [in quantum mechanics], because the theory is incompatible with the idea that physics should describe a reality in time and space without spookish actions at a distance.” [14]

Einstein fought another fundamental battle in the defense of causality: “Even the great initial success of the quantum theory does not make me believe in the fundamental dice-game, although I am well aware that our younger colleagues interpret this as a consequence of senility. No doubt the day will come when we will see whose instinctive attitude was the correct one.” [15] This statement of 1944 joins coherently what Einstein had written twenty years before in a letter to Born: “Bohr’s opinion about radiation is of great interest. But I should not want to be forced into abandoning strict causality without defending it more strongly than I have so far. I find the idea quite intolerable that an electron exposed to radiation should choose [of its own free will], not only its moment to jump off, but also its direction. In that case, I would rather be a cobbler, or even an employee in a gaming-house, than a physicist.” [16]

Thus Einstein defended realism, causality and description in space and time,

against those physicists of Copenhagen and Göttingen who believed to have only continued on the path he had indicated with relativity. From this comes out all the

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richness, but also the complexity, of the Einsteinian conceptions. Reckoning with these ideas means entering in the eye of the epistemological-scientific storm of the XXth century.

2. Relativistic paradoxes Einstein’s theories had great success in explaining many known phenomena and in predicting new ones. Therefore they contain important advances of our knowledge of the physical world and belong forever to the history of the natural sciences, similarly to Newton’s mechanics and Maxwell’s electromagnetism. It is however difficult to believe that they are final forms of knowledge. On the contrary, the lesson to learn from epistemology (Popper, Lakatos, Kuhn) is about the conjectural, provisional, improvable nature of the physical theories of the XXth century.

In March 1949, answering his friend M. Solovine who had sent him an affective letter for the seventieth birthday, Einstein had written: “You imagine that I look backwards on the work of my life with calm satisfaction. But from nearby it looks very different. There is not a single concept of which I am convinced that it will resist firmly.“ [17] Einstein did not hide the transitoriness of his creations. On April 4, 1955, he wrote the last paper of his life. It was a three pages long preface (in German) to a book celebrating the fiftieth anniversary of the theory of relativity. It ended with the following words: “The last, quick remarks must only demonstrate how far in my opinion we still are from possessing a conceptual basis of physics, on which we can somehow rely.” [18] In a way this is a declaration of failure, but one has to admire the ethical dimension of the great scientist who had devoted the superhuman efforts of a lifetime to the attempt of reaching the deepest truths of nature and now, arrived at the end, declares to posterity: “I did not succeed.”

The successes of the relativistic theories are very well known. The reciprocal convertibility of energy and mass, the effects of velocity and gravitation on the pace of clocks, the weight of light and the precession of planetary motions, provide only a partial summary of the great conquests of Einsteinian physics. Nevertheless, it would not be correct to conclude that every comparison of the theoretical predictions with experiments invariably led to a perfect agreement. Physics is a human activity and from us inherits the habit to parade the successes and to hide difficulties and failures. Thus only silence surrounded the Sagnac effect (discovered in 1913) for which there is a veritable explanatory inability of the two relativistic theories, the attempts by Langevin [19], Post [20], Landau and Lifshitz [21] notwithstanding. There are, furthermore, the half explanations of the aberration of star light and of the clock paradox, phenomena for which the mathematical formalism of the theory can reproduce the observations at the price of twisting the meaning of symbols beyond rightfulness.

One should never forget that behind the equations of a theory there is a huge qualitative structure made of empirical results, generalizations, hypotheses,

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philosophical choices, historical conditionings, personal tastes, conveniences. When one becomes aware of this reality and compares it with the little portrait of physics handed down by logical empiricism, which is worth less than a caricature, one easily understands that relativity, not only can present weak points side by side with its undeniable successes, but can also survive some failures. The correctness of the mathematical formalism is not enough to validate a scientific structure as coherent and not contradictory. I add that not even hundreds of physicists unconditionally favorable to a theory can warrant absence of unsolved problems, because much too often their thoughts are oriented since the university studies towards an acritical acceptance of the dominating theory.

In reality the two relativistic theories are crammed with paradoxes. Let us try to make a list, with no claim of completeness, limited to the TSR: 1. The idea that the simultaneity of spatially separated events does not exist in nature and must therefore be established with a human convention; 2. The relativity of simultaneity, according to which two events simultaneous for an observer in general are no more such for a different observer; 3. The velocity of a light signal, considered equal for observers at rest and observers pursuing it with velocity 0.99 c ; 4. and 5. The contraction of moving objects and the retardation of moving clocks, phenomena for which the theory does not provide a description in terms of objectivity; 6. The hyperdeterministic universe of relativity, fixing in the least details the future of every observer; 7. The conflict between the reciprocal transformability of mass and energy and the ideology of relativism, which declares all inertial observers perfectly equivalent so depriving energy of its full reality; 8. The existence of a discontinuity between the inertial reference systems and those endowed with a very small acceleration; 9. The propagations from the future towards the past, generated in the TSR by the possible existence of superluminal signals; 10. The asymmetrical ageing of the twins in relative motion in a theory waving the flag of relativism.

In section 15, after having established the validuty of the IT, we will show that the previous paradoxes are fully overcome, so that they will disappear from the scientific debate as soon as the new IT will be accepted. The substitution of Einstein’s relativity principle with a weaker principle will also be one of the results of our present research.

Herbert Dingle, professor of History and Philosophy of Science in London, in the fifties and early sixties fought a battle against some features of the relativity theory, in particular against the asymmetrical ageing present in the clock paradox argument. He believed that the slowing down of moving clocks was pure fantasy. This idea has of course been demolished by direct experimental evidence, collected after his time. Nevertheless, his work has left posterity a rare jewel: the syllogism bearing his name. Given that syllogism is a technical model of perfect deduction, its consequences are absolutely necessary for any person accepting rational thinking in science. Dingle’s syllogism is the following [22]:

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1. (Main premise) According to the postulate of relativity, if two bodies (for example two identical clocks) separate and reunite, there is no observable phenomenon that will show in an absolute sense that one rather than the other has moved. 2. (Minor premise) If upon reunion, one clock were retarded by a quantity depending on its relative motion, and the other not, that phenomenon would show that the first clock had moved (in an observer independent “absolute” sense) and not the second. 3. (Conclusion) Hence, if the postulate of relativity is true, the clocks must be retarded equally or not at all: in either case, their readings will concord upon reunion if they agreed at separation. If a difference between the two readings were to show up, the postulate of relativity cannot be true. Today it can be said that the asymmetrical behaviour of the two clocks is empirically certain (muons in cosmic rays, experiment with the CERN muon storage ring, experiments with linear beams of unstable particles, Hafele and Keating experiment). Therefore, as a consequence of point 3. above, the postulate of relativity must somehow be negated. Actually, in recent times there are some authors who think that “theory of relativity” is just a name, not to be taken too literally. The total relativism which the theory could seem to embody is now perceived to be only an illusion. One can conclude that not all is relative in relativity, because this theory contains also some features that are observer independent, then features which are absolute! As Dingle wrote: “It should be obvious that if there is an absolute effect which is a function of velocity, then the velocity must be absolute. No manipulation of formulae or devising of ingenious experiments can alter that simple fact.” [22]

How is it possible that respected experts of relativistic physics believe that

those listed above and numbered from 1 to 10 are not real paradoxes? The answer is not difficult and is based on what in Italian is called “buon senso” (literally: good sense). This expression is easily translated in all neo-Latin languages, but is absent in other languages. English speaking authors use sometimes “common sense”, which carries however a very different idea because the common sense is that of the majority and the history of science teaches that in scientific matters the majority is rarely right.

Well, if good sense tells us that a certain prediction of a theory is

unreasonable, there are two possibilities. Firstly, it is possible that the good sense misleads us, secondly that in the theory there are more or less explicit hypotheses contrary to the natural order of things giving its predictions an incorrect meaning. It is well known that many physicists and philosophers of science of the XXth century followed the fashion of declaring good sense obsolete, but we will show that the second road can easily be traveled over and allows one to get rid of all the paradoxes of relativity. Of course one could object that it is not a priori obvious that the

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paradoxes can be eliminated without spoiling the successes of the TSR. Nevertheless, it will be seen that the theory reviewed in the present article, based on the replacement of the Lorentz by the “inertial” transformations and on a radical modification of the philosophical taste of the theory, not only explains all what the TSR does, but succeeds also where the latter does not. It explains the Sagnac effect, for example. 3. Relativism and the nature of energy An important paradox of the relativistic theory arises from the application of the idea of relativism to the physical quantities. When this is done they all seem to lose their concreteness and almost to vanish into nothingness, including the most fundamental one, energy. In the present section we will see why this happens.

Let us start from the idea of relativism, which is best presented with an example. Two inertial reference systems are given, the system S0 of the stationmaster and the system S of the passenger on the train. Let v = 0,6c be the train velocity,

hence the Lorentz factor is R = 1− v2 /c2 = 0,80. It follows in the standard way that if the stationmaster sees a meter immobile on the floor of the train parallel to the rail and measures its length, he finds 80 cm. It is equally clear that if the passenger on the train sees a meter immobile on the floor of the railway station parallel to the rail, and measures its length he finds 80 cm. They will both conclude that a meter moving at a speed 0,6c relative to their rest systems is 80 cm long. Who is right? According to Albert Einstein: they are both right in the same way. The latter statement is the basis of the relativism of the theory of relativity, but is not necessarily true. The TSR could be correct as a scientific theory and, at the same time, relativism could not hold. For example, Lorentz’s reformulation of the theory is experimentally indistinguishable from Einstein’s TSR while admitting the existence of ether and thus of a privileged inertial system. In Lorentz’s approach the opinions of stationmaster and passenger are not necessarily equivalent. For example, if S0 were the privileged system, the stationmaster would be right and the passenger wrong. In general, the observer with smaller absolute velocity would give a better judgment about the true length of the meter.

The theory of relativity led to the conclusion that an arbitrary object, whose quantity of matter is measured by mass, and motion of the same object, measured by energy, have the same properties and can be transformed into one another. This corresponds to a basic reality of energy, which shares all the properties of mass. For example: “If the theory corresponds to the facts, the radiation conveys inertia between the emitting and the absorbing bodies.” [23] Mass and energy have to be considered different forms of a unique reality. The reciprocal transformability of mass and energy has been confirmed in an enormous number of experiments of nuclear and subnuclear physics, so that it can now be considered an irreversible progress of science. The mass-energy equivalence is expressed by the famous formula

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E = m c2 (3.1) The unitary nature of energy and mass was so described in the book by Einstein and Infeld: “A further consequence of the (special) theory of relativity is the connection between mass and energy. Mass is energy and energy has mass. The two conservation laws of mass and energy are combined by the relativity theory into one, the conservation law of mass-energy.” [24]

The mass-energy equivalence had many consequences, for example it predicted a continuity between that form of energy diffused in space which is called “field” and the material sources generating it: “From the relativity theory we know that matter represents vast stores of energy and that energy represents matter. We cannot, in this way, distinguish qualitatively between mass and field, since the distinction between mass and energy is not a qualitative one. We could therefore say: Matter is where the concentration of energy is great, field where the concentration of energy is small. But if this is the case, then the difference between matter and field is a quantitative rather than a qualitative one.” [25]

From the experimental point of view the mass-energy equivalence means that a material object can be transformed into pure motion (that is, into kinetic energy of other objects) and, viceversa, that it is possible to create matter at the expenses of motion. These transformations take place according to the rigorous laws of conservation of energy and momentum in absolutely concrete processes: it is possible to make two protons with high enough kinetic energy collide to produce in the final state the same two protons with identical properties (mass, electric charge, etc.) and, additionally, one or several new pieces of matter, for example π mesons, which were born from nothingness during the collision. Rather, they seem to be born from nothingness to a person observing the phenomenon only superficially. Actually, if one compares the kinetic energies of the initial and final state one finds that exactly the quantity of kinetic energy has disappeared that is necessary to produce the new mass in the final state. The reaction is:

P + P → P + P + π 0 Two colliding protons give rise to a new physical state including two protons and a neutral π meson. The meson π is a quantum of nuclear forces and has a rest mass 264 times that of the electron.

Inverse processes exist as well, in which energy is created at the expenses of mass. Of this type are the uranium fission reactions. In this way one sees how false was the belief of the past that matter can neither be created nor destroyed. In reality there is no law of conservation of matter: what is conserved under all circumstances is energy together with its vectorial daughter, the quantity of motion (momentum). These are the fundamental quantities of reality, whereas the stability of matter is pure

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appearance, due to the fact that we live in a low energy world. If the energy increases, matter can start to disappear! In fact at the center of the Sun there is a temperature of 15-20 million degrees, the kinetic energy of thermal agitation is correspondingly high and every second four million tons of matter are transformed into radiant energy.

It is rather obvious that the achievements of relativity on the just described mass-energy relationship belong to the philosophical field of realism. Positivism, however, did not disappear. On the contrary it extended its domination to the very notion of energy, as we will see next. Energy has all the right properties to be considered a kind of fundamental substance of the universe: it is indestructible, it enters in all dynamical processes and matter itself can be considered a localized form of energy. Naturally this “energetic materialism”, if possible, would be very different from the anti-atomistic energetism proposed by Ostwald towards the end of the XIXth century. However, according to the TSR energy has no fundamental role. Different inertial observers assign different velocities, and thus different energies to any given particle. The relativistic formula of the total energy E (kinetic energy plus rest mass energy) of a particle having rest mass m and velocity u relative to a frame of reference S is

E = m c2

1 − u2 / c2 (3.2)

where c is the velocity of light, as usual. This formula holds in all inertial systems S, ′ S , ′ ′ S , .. . provided one uses the particle velocity u, ′ u , ′ ′ u , .. . relative to each of them. If one asks which is the real value of energy, the TSR answers that all observers are equivalent, so that their calculations are all equally valid. And since each of them attributes to the particle energy a different value, in the impossibility of choosing one of these as “more true”, one is forced to conclude that a well defined value of energy does not exist. In this way energy, possible substratum of the universe, is at once stripped by relativism of its most important property, that of having an objectively well defined value.

In 1943 J. Jeans used a similar argument against the objectivity of forces. For him the essence of a physical explanation, at least classically, is that each particle of a system experiences a real and definite force. This force should be objective as regards both quantity and quality, so that its measure should always be the same, whatever means of measurement are employed to measure it - just as a real object must always weigh the same, whether it is weighed on a spring balance or on a weighing beam. But the TSR shows that if motions are attributed to forces, these forces will be differently estimated, as regards both quantity and quality, by observers who happen to be moving at different speeds, and furthermore that all their estimates have an equal claim to be considered right. “Thus - Jeans concludes - the supposed forces cannot have a real objective existence; they are seen to be mere mental constructs which we make for ourselves in our efforts to understand the workings of nature.” [26]

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Naturally for Jeans it was immediately possible to generalize his argument to all physical quantities: force, energy, momentum, and so on. With his words: “But the physical theory of relativity has now shown ... that electric and magnetic forces are not real at all; they are mere mental constructs of our own, resulting from our rather misguided efforts to understand the motions of the particles. It is the same with the Newtonian force of gravitation, and with energy, momentum and other concepts which were introduced to help us understand the activities of the world - all prove to be mere mental constructs, and do not even pass the test of objectivity. If the materialists are pressed to say how much of the world they now claim as material, their only possible answer would seem to be: Matter itself. Thus their whole philosophy is reduced to a tautology, for obviously matter must be material. But the fact that so much of what used to be thought to possess an objective physical existence now proves to consist only of subjective mental constructs must surely be counted a pronounced step in the direction of mentalism.” [27] After such a striking conclusion it is no surprise that Jeans arrives to the most genuine philosophical idealism: “Today there is a wide measure of agreement, which on the physical side of science approaches almost to unanimity, that the stream of knowledge is heading towards a non-mechanical reality. The universe begins to look more like a great thought than like a great machine. Mind no longer appears as an accidental intruder into the realm of matter. We ought rather to hail it as the creator and governor of the real of matter.” [28]

For avoiding these unpleasant conclusions there is only one possibility, giving up the philosophy of relativism that originates from the space-time symmetry of the Lorentz transformations, atmittedly constituting the most natural interpretation of the Einsteinian theory. The retrieval of the objectivity of energy and of the other physical quantities should rather aim at the inequivalence of the different reference frames. But such lack of equivalence is easily achieved with the inertial transformations (see section 7 below), based on the existence of a privileged system, which give back to the mass-energy equivalence the great conceptual value of a substance leading to the unification of physics. Deduced from the inertial transformations the formula of the total energy E (kinetic energy plus rest mass energy) of a particle having rest mass m and velocity

r u relative to a frame of reference S is

E = m c2 1−

r u ⋅

r v /c2( )

1−r u ⋅

r v /c2( )2

− u2 /c2 (3.3)

if

r v is the velocity of S relative to the privileged frame S0 [29].

After overcoming the philosophy of relativism, energy can take up its fundamental role, its true value being the one calculated in the privileged isotropic inertial system. Notice that if

r v = 0 Eq. (3.3) reduces to (3.2), the latter giving the

true value of energy. After all, one remains with the impression that relativism is only

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an ideological element inserted in a physical reality (e.g., that of energy) that is sound and perfectly capable to run the game of physics. 4. Einstein’s relativistic ether In the 1905 paper introducing the theory of special relativity Einstein wrote that the hypothesis of a luminiferous ether could be considered superfluous, given that the new theory needed neither an absolutely stationary space endowed with particular properties, nor a medium in which electromagnetic processes, such as the propagation of light, could take place.

Einstein started to reconsider the whole question of the ether in the years of his explicit transition from positivism to realism (1916-1924). At this time he admitted that it was still possible to think ether as existing, even if only to designate particular properties of space. The reasoning which promoted the ether idea from superfluous to admissible was more or less the following.

If every ray of light propagates in the vacuum with velocity c relative to the inertial system K, we must imagine this luminiferous ether everywhere at rest with respect to K. But if the laws of propagation of light relative to the different inertial system K’ (moving with respect to K) are the same as relative to K, we must with the same right accept the existence of a luminiferous ether at rest with respect to K’. The standpoint of the 1905 formulation of the TSR was that it is absurd to accept that ether is at rest at the same time in both systems and that one must give up introducing it. After 1916 Einstein modified his position and assumed that ether is somehow at rest both with respect to K and K’, that is to say, given the arbitrariness of K and K’, at rest at the same time with respect to all inertial frames. It was certainly a very unusual idea to deprive a physical entity of the right to be seen in motion, but that was Einstein’s choice. He so described the situation:

“ [...] in 1905, I was of the opinion that it was no longer allowed to speak about the ether in physics. This opinion, however, was too radical [...]. It does remain allowed, as always, to introduce a medium filling all space and to assume that the electromagnetic fields (and matter as well) are its states. But, it is not allowed to attribute to this medium a state of motion in each point, in analogy to ponderable matter. This ether may not be conceived as consisting of particles that can be individually tracked in time.” [30]

The abolishment of the ether cannot be considered a necessary consequence of Einstein’s relativism. This philosophy, embodied in the principle of relativity, demands only that the description of the physical reality be the same in all inertial reference frames and this can be achieved also with an ether deprived of mobility: “More careful reflection teaches us, however, that this denial of the existence of the ether is not demanded by the special principle of relativity. We may assume the existence of an ether; only we must give up ascribing a definite state of motion to it,

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i.e., we must by abstraction take away from it the last mechanical characteristic that Lorentz had still left it.” [31]

If one considers pointlike particles one is bound to conclude that motion is always possible, for a particle has well defined values of the space-time coordinates and the Lorentz transformations can be applied to produce motion from rest. However: “Extended physical objects can be imagined to which the idea of motion cannot be applied. They are not to be thought of as consisting of particles that allow themselves to be separately tracked through time. In Minkowski’s idiom this is expressed as follows: Not every extended conformation in the four-dimensional world can be regarded as composed of lines of Universe.” [32] In this way the ether is postulated to be devoid of motion. Obviously this means that also the notion of “motionlessness” cannot be applied to it, at least because immobility is a particular case of motion with zero velocity. Thus Einstein writes: “As to the mechanical nature of the Lorentzian ether, it may be said of it, in a somewhat playful spirit, that immobility is the only mechanical property of which it has not been deprived by H.A. Lorentz. It may be added that the whole change in the conception of the ether, which the special theory of relativity brought about, consisted of taking away from the ether its last mechanical quality, namely, its immobility.” [33]

Thus Einstein was pushed to admit the limited horizon of his previous research on the physics of space and time: “It would have been more correct if I had limited myself, in my earlier publications, to emphasising only the nonexistence of an ether velocity, instead of arguing the total nonexistence of the ether, for I can see that with the word ether we say nothing else than that space has to be viewed as a carrier of physical qualities.” [34] And space is indeed a carrier of physical qualities: such as the possibility to introduce in every point of space well defined inertial reference systems, or, which is the same, the inertial forces in the accelerated systems. “On the other hand there is a weighty argument to be adduced in favor of the ether hypothesis. To deny the existence of the ether means, in the last analysis, denying all physical properties to empty space. But such a view is inconsistent with the fundamental facts of mechanics.” [35]

Therefore one can say that “physical space” and “ether” are only different terms for indicating the same reality. Furthermore, fields are physical states of space. If no particular state of motion can be attributed to the ether, there does not seem to be any reason for introducing ether as an entity of a special type alongside of space. Naturally it is not forbidden to use the word ether, but only to express the physical properties of space. The word ether changed its meaning many times in the development of science. Around 1920, it no longer stood for a medium built up of particles. Its story, by no means finished, was to be continued by the relativity theory. In conclusion, “Summarizing, we can say that according to the theory of general relativity space is equipped with physical properties; also in this sense an ether exists. According to the general theory of relativity space without ether is unthinkable, as in such a space not only the propagation of light would not take place, but also there

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would be no possibility of existence for clocks and rodes, so that also no spatio-temporal distances "in the sense of physics.” " [36]

Einstein’s new ether was introduced both in connection with the TSR and in connection with the theory of general relativity. The second case is more interesting physically, as: “According to general relativity, the concept of space detached from any physical content does not exist. The physical reality of space is represented by a field whose components are continuous functions of four independent variables - the coordinates of space and time. It is just this particular kind of dependence that expresses the spatial character of physical reality.” [37]

This field whose components are continuous functions of the coordinates of space and time is the gravitational field as described by the potentials. “No space and no portion of space [can be conceived] without gravitational potentials; for these give it its metrical properties without which it is not thinkable at all. The existence of the gravitational field is directly bound up with the existence of space.” [38] Thus depending on the different nearby masses the ether can be found in different states: “The ether of the general theory of relativity therefore differs from that of classical mechanics or the special theory of relativity, in so far as it is not ‘absolute’, but is determined in its locally variable properties by ponderable matter.” [39]

With the mathematics of the general theory of relativity it is possible to implement a continuous transition from the special to the general theory, thus incorporating the laws of nature, already known from special relativity, into the broader framework of general relativity: “The real is conceived as a four-dimensional continuum with a unitary structure of a definite kind (metric and direction). The laws are differential equations, which the structure mentioned satisfies, namely, the fields which appear as gravitation and electromagnetism. The material particles are positions of high density without singularity. We may summarize in symbolical language. Space, brought to light by the corporeal object, made a physical reality by Newton, has in the last few decades swallowed ether and time and seems about to swallow also the field and the corpuscles, so that it remains as the sole medium of reality.” [40]

Thinking one last time of those past events, one realizes that it must have been difficult for Albert Einstein to resist Lorentz’s pressure in favour of ether. One can say that the ether at rest in all inertial frames was his way to concede space to Lorentz while defending relativism. It was not a great idea of the type he had so many times during his life and today it remains half forgotten. It is interesting philosophically as an attempt to put together realism (with ether) and relativism. 5. Simultaneity, the key idea Einstein admitted the conventionality of the velocity of light in his fundamental 1905 paper on relativity, where he wrote: “We have so far defined only an “A time“ and a “B time“. We have not defined a common “time” for A and B , for the latter cannot be defined at all unless we establish by definition that the “time” required by light to

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travel from A to B equals the “time” it requires to travel from B to A .” [41] This statement is remarkable also for showing Einstein’s positivistic inclinations in two different ways. Firstly, he accepts Poincaré’s idea that the speed of light is not measurable and can only be defined in a conventional way; secondly, he writes five times the word time between quotation marks, as if it were a dangerous conception waiting for a replacement. The conventionality of the velocity of light was restated in 1916 when Einstein wrote about the midpoint M of a segment AB whose extreme points are struck “simultaneously“ by two strokes of lightning: “that light requires the same time to traverse the path AM ... as the path BM [M being the midpoint of the line AB ] is in reality neither a supposition nor a hypothesis about the physical nature of light, but a stipulation which I can make of my own free will.” [42]

The relativistic synchronization can be implemented as follows. Suppose that two identical clocks A and B at rest in the same inertial frame are at a distance l from one another. A pulse of light starts from A towards B when the clock in A marks time zero; the clock in B is set at time l / c when the pulse reaches B . From synchronization to relativistic simultaneity the step is short. Two instantaneous point like events in A and B at times t A and t B , as marked by the respective synchronized local clocks, are simultaneous by definition if t A = tB , Naturally a good positivist does not wonder whether the two events are really simultaneous: for him only human manipulations matter and it does not make any sense to think in terms of objectivity of time.

A method coming to everybody’s mind for synchronizing clocks does not work: synchronize them when they are near and carry them in the points where they are needed. It does not work because it is very clear that transport, that is the fact itself of possessing a velocity, modifies the motion of the clock hands as it modifies any periodical motion that one might think to use in order to measure time. The transport of a clock can be executed in a short time at a high velocity, or in a long time at a very low velocity, but there is always an unavoidable delay generated by the clock’s motion. Furthermore such a delay is essentially unknown as it depends on the velocity of the moving clock with respect to the privileged system. This being the case, Poincaré and Einstein decided that the “synchronization“ of clocks could be achieved by ignoring the problem of the objective reality and following criteria of any type, provided only they lead to a non ambiguous identification of events. The simplest choice was made by assuming that the speed of light had the same value in all directions in all inertial frames. In this way the notion of relativistic simultaneity was made to depend on human decisions and not on properties of nature.

We stress that the conventional nature of the relativistic clock synchronization - and then of the relativistic simultaneity of distant events - opens very interesting perspectives. Let us see why. In general time could be different in two different inertial reference systems S0(x0 ,y 0,z 0 , t0 ) and S(x,y,z, t), and the “delay“ t − t0 (positive, null or negative) of S over S0 could depend not only on the time t 0 , but

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also on the geometrical point in which the delay is calculated. This happens in the TSR, as the Lorentz transformation of time contains also a space coordinate. In other words, the time t marked by a clock of S can depend also on the coordinates x, y, z of the point at which the clock is positioned.

In 1925 H. Reichenbach discussed the problem of clock synchronization by examining the following experiment: in the system S a flash of light leaves point A at time t1, is reflected backwards by a mirror placed at point B at time t2 and finally returns at A at time t3 . Of course t1 and t3 are marked by a clock near A , while t2 is marked by a different clock placed near B . The question is how to synchronize the two clocks with one another. In the TSR one assumes that the velocity of light on the one way path A − B is the same as in the two way path A − B − A so that for the time differences one has

t2 − t1 = 12

(t3 − t1) (5.1)

In this way one defines the time t2 of the B clock in terms of the times t1 and t3 of the A clock. One can show that the choice (5.1) fixes the way in which x is present in the (Lorentz) transformation of time. Reichenbach pointed out that Eq. (5.1) is essential in the TSR, but is not epistemologically necessary given its conventional nature. A different rule of the type

t2 − t1 = ε( t3 − t1) (5.2) with any 0 < ε <1 would similarly be adequate, as based on a different convention, and could not be considered false. Reichenbach commented: “If the special theory of relativity prefers the first definition, i.e., sets ε equal to 1/2, it does so on the ground that this definition leads to simpler relations.” [43]

Reichenbach’s coefficient ε was discussed anew in 1979 by Max Jammer who stressed that one of the most fundamental ideas underlying the conceptual edifice of relativity is the conventionality of intrasystemic distant simultaneity. He added: “The “thesis of the conventionality of intrasystemic distant simultaneity ... consists in the statement that the numerical value of ε need not necessarily be 1/2, but may be any number in the open interval between 0 and 1, i.e. 0<ε <1, without ever leading to any conflict with experience.” [44] In a recent book, entirely devoted to the concepts of simultaneity, Jammer stressed that Einstein’s 1905 analysis of the concept of distant simultaneity inaugurated the conceptual revolution of modern physics, and added: “If, as mentioned above, the concept of distant simultaneity is a fundamental ingredient in the logical structure of the theory of relativity but is in reality nothing but a convention, the question naturally arises of whether this does not imply that the whole theory of relativity and with it a major part of modern physics are merely

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fictions devoid of any actual content. A positive answer to this question would have disastrous consequences for the philosophical understanding and epistemological status of physics and with it of the whole of modern science.” [45]

Substitute “disastrous” with “wonderful” and Jammer’s statement expresses also the opinion of the present author. In fact, from the above one can deduce that theories with ε ≠ 1 / 2 should be possible. This obviously means theories different from the TSR only in a conventional matter, probably also in the mathematical structure, but in full agreement with the empirical evidence. I devoted ten years of work to the practical confirmation of this intuition. The confirmation came out ample but with some surprises (see sections 9 - 14). Anyway, there is an important logical space for values ε ≠ 1 / 2 , that is, in the final analysis, for theories alternative to the TSR. This also means that after a century of relativism one can open the doors to a different physics without conflicting with the enormous bulk of experimental results accumulated to date. 6. The basic empirical evidence We move in space at a speed of 2-300 km/sec (about 1‰ of the speed of light) as we take part, with the Sun, to the rotation of our spiral galaxy, the Milky Way. According to the Galilei-Newton physics the velocity of light relative to a terrestrial laboratory should depend on the propagation direction. In fact, let

r c be the velocity of a

punctiform light signal with respect to the privileged system S 0 . If r ′ c is the velocity

of the same signal with respect to a terrestrial laboratory, moving in S 0 with velocity r v , one should have

r ′ c =

r c −

r v . Therefore ′ c should vary from c − v to c + v

when the light propagation direction changes from parallel to antiparallel to r v .

It could be thought that these effects of the first order in v / c are easily observable. One should however recall that even before the birth of the TSR Poincaré had argued the impossibility to measure the velocity of a light signal propagating between two different points. To understand the reasons of this conclusion let us consider a light pulse traveling from A to B . If in B there is a mirror reflecting the signal backwards, it is enough to have a clock near A measuring the times t1 and t3 of start and return. The speed of the pulse is then given by its definition as length over time:

c2 = 2dABt3 − t1 (6.1)

where dAB is the A -B distance that can be measured in the standard way using a rigid rod. However, the so defined c2 is a two way velocity and it is possible that the signal velocity from A to B be different. For measuring the latter, two synchronized

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clocks would be needed, one near A and the other one near B . Unfortunately even today nobody knows how to synchronize two distant clocks.

Years before the formulation of the TSR, Poincaré discussed the independence of the velocity of light of its propagation direction and stated: “That light has a constant velocity and in particular that its velocity is the same in all directions ... is a postulate without which it would be impossible to start any measurement of this velocity. It will always be impossible to verify directly this postulate with experiments.” Agreeing on the impossibility to measure the one way velocities, Einstein decided to solve the problem by decree, assuming the invariance of the velocity of light: the second postulate of the TSR. In fact, he described this hypothesis not as a property of nature.

One of the most precise values of c2 was obtained in 1978:

c2 = (299 792,4588 ± 0,0002) km/sec (6.2) and it was confirmed by subsequent measurements. A precision of 10−9 goes with the known value of c2, a thousand times better than needed for detecting the second order effects due to the Earth motion. Yet, before and after 1978 one always found the same value within errors, and no dependence on the propagation direction was observed, in agreement with the more indirect experiments (such as the Michelson-Morley experiment) that tried to detect the existence of the privileged reference system. Our first fundamental conclusion is: P1. The two way velocity of light is invariant, meaning that it is empirically independent of the propagation direction and of the time at which it is measured. With their 1887 experiment Michelson and Morley concluded that no shifts of the interference figures existed due to the Earth motion. To explain this result Fitzgerald and independently Lorentz supposed that the motion of an object through the ether with velocity v generated its shortening in the direction of velocity by the factor

R = 1 − v 2 / c2 (6.3) The idea of a contraction due to motion was not so strange as it might seem. In fact, using classical physics, Lorentz was able to prove that the motion of an electric charge through ether modifies its electric field by squeezing it towards a plane perpendicular to the direction of motion, and that the degree of squeezing increases with velocity. Thus an electron bound to a moving proton no longer forms a regular atom, but the internal motion takes place on an orbit squeezed similarly to the field. Furthermore, the period of the electronic motion is modified, actually increased for the observer watching the atom move. One must therefore expect that every object (made of atoms)

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be shortened in its dimension parallel to velocity, and that in every moving clock the pace of advancement of the hands be slowed down.

In 1900 Larmor considered a system “composed of two electrons of opposite charge” (one would say today: composed of an electron-positron pair), neglected irradiation, and assumed circular orbits round the common centre of mass of the two particles. Assuming also that the whole system was in motion through ether, he proved that the velocity dependent deformation of the electric fields generated in the bound system exactly the contraction postulated by Fitzgerald and Lorentz. Furthermore Larmor found that the orbital period was necessarily increased by R −1 , where R is given by (6.3). This was the first correct formulation of the idea of a velocity dependent retardation of clocks.

μ

Figure 1. In the CERN storage ring unstable particles (“muons”) circulated with a speed smaller than that of light by only six parts over ten thousand. It was observed that muons disintegrated after a lifetime 29,33 times larger than for muons at rest. The slowing down of moving clocks is nowadays experimentally very well ascertained. An experiment was performed in 1977 [46] when the lifetimes of positive and negative muons were measured at the CERN muon storage ring. Muons with a velocity of 0.9994 c , corresponding to a factor R −1

= 29.33, were circling in a ring with diameter of 14 m, with a centripetal acceleration equal to 1018

g . The lifetime τ was measured and found in excellent agreement with the formula τ = τ0 / R where τ0 is the lifetime of muons at rest.

The lesson learnt from this experiment concerns the transformation of time: the storage ring time interval τ between two events taking place in the same position of the moving system (injection and decay of the muon) is observed to be dilated according to τ = τ0 /R if compared with the corresponding time interval τ0 measured by the laboratory observer on the circling muons.

In the 1972 experiment with macroscopic clocks by Hafele and Keating [47] six accurately synchronized Cesium atomic clocks were used, and: 1) two were carried by ordinary commercial jets in a full eastbound tour around the planet;

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2) other two were carried by ordinary commercial jets in a full westbound tour around the planet; 3) the last two remained on the ground. It was observed that with respect to the latter clocks, those on board the westbound trip had undergone a loss of 59 ±10 ns, while the clocks on the eastbound trip had undergone an advancement of 273± 7 ns. These results were in excellent agreement with the usual formula

τ = τ0 R−1 (6.4) if: a) one used three different factors R −1 for the three pairs of clocks. The largest (smallest) factor was that of clocks that traveled eastward (westward) for which the Earth rotation velocity added to (subtracted from) the jet velocity. That is, it was necessary to refer movements not to the Earth surface, but to a reference frame with origin in the Earth centre and axes oriented toward fixed directions of the sky; b) one kept into account the effect of the Earth gravitational field that varies with altitude and therefore modifies the rates of traveling clocks differently from those left on the ground. The results of the Hafele-Keating experiment have been confirmed by the GPS (Global Positioning System) system of satellites [48]. This system consists of a network of 24 satellites in roughly 12-hour orbits, each carrying atomic clocks on board. The orbital radius of the satellites is about four Earth radii. The orbits are nearly circular, with eccentricities of less than 1%. Orbital inclinations to the Earth equator are about 55°. The satellites have orbital speeds of about 3.9 km / sec in a frame centered on the Earth and not rotating with respect to the stars. Every satellite has on board four atomic clocks marking time with an error of a few ns / day . From every point of the Earth surface at least four satellites are visible at any time. Initially conceived for military aims, the GPS was subsequently used for telecommunications, satellite navigation, meteorology.

The theory of general relativity predicts that clocks in a stronger gravitational field will tick at a slower rate. Thus the atomic clocks on board the satellites at GPS orbital altitudes will tick faster by about 45.900 ns / day because they are in a weaker gravitational field than atomic clocks on the Earth surface. The velocity effect predicts that atomic clocks moving at GPS orbital speeds will tick slower by about 7.200 ns / day than stationary ground clocks. Therefore the global prediction is a gain of about 38.700 ns / day . Rather than having clocks with such large rate differences, the satellite clocks were reset in rate before launch (slowing them down by 38.700 ns / day ) to compensate for the predicted effects. The very rich data show that the on board atomic clock rates do indeed agree with ground clock rates to the predicted

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extent. Thus the theoretical predictions are confirmed, in particular the slowdown of the clock rate due to the orbital velocity. We can then write the second conclusion: P2 Clock retardation takes place according to (6.4) with the factor R given by (6.3) when clocks move with respect to S0 . We left in vagueness the question of the reference frame with respect to which v should be calculated. In the next section we will take P1. and P2. as fundamental empirical facts and eliminate the vagueness by making a new assumption: its validity will be corroborated by the success of the ensuing theory. 7. The new transformations In 1977 Mansouri and Sexl [49] stressed that the Lorentz transformations contain a conventional term, the coefficient of the coordinate x in the transformation of time. Starting in 1994 the present author reconsidered the whole matter and reformulated the transformations of the space and time variables between inertial systems [50] under very general assumptions. The “equivalent transformations” were obtained containing an indeterminate term, e1 , the coefficient of x in the transformation of time: see Eq.s (7.2) below. The reasoning leading to the equivalent transformations is as follows. Given the inertial frames S0 and S one can set up Cartesian coordinates and make the following standard assumptions: (i) Space is homogeneous and isotropic and time homogeneous, at least from the point of view of observers at rest in S0 ; (ii) Relative to the isotropic system S0 the velocity of light is “c ” in all directions, so that clocks can be synchronized in S0 with the Einstein method and the one way velocities relative to S0 can be measured; (iii) The origin of S , observed from S0 , moves with velocity v < c parallel to the +x0 axis, that is according to the equation x 0 = v t0 ; (iv) The axes of S and S0 coincide for t = t0 = 0 . The geometrical configuration is thus the usual one of the Lorentz transformations (see Fig. 2). The assumptions (i) and (ii) are not exposed to objections both from the point of view of the TSR and of any plausible theory based on a privileged system; for the TSR they hold in all inertial systems, in the second case they hold in the privileged system only.

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x

y

z

S

v

z0

y0

x0S0

Figure 2. An inertial system S having coordinates (x, y, z ) moves with velocity v < c with respect to the isotropic inertial system S0 having coordinates (x 0, y0 , z 0 ). The two sets of coordinates overlap perfectly at t 0 = t = 0 . In [51] it was shown that the previous conditions reduce the transformation laws from S0 to S to the form

x = f1 x0 − v t0( )y = g2 y0

z = g2 z 0

t = e1 x0 + e4 t 0

⎧

⎨

⎪ ⎪ ⎪

⎩

⎪ ⎪ ⎪

(7.1)

where the factors f1, g2 , e1, e4 can depend on the velocity v of S measured in S0 . An appropriate name for (7.1) could be “general transformations”. To (i) - (iv) we can add two points discussed in the previous section which, as we saw, are based on solid empirical evidence: (P1) The two way velocity of light is the same in all directions and in all inertial systems: c2 (θ) = c . (P2) Clock retardation takes place with the usual factor R when clocks move with respect to S0 . Notice that we have now eliminated ambiguities by saying that R in the formula τ0 = τ /R has to be calculated relatively to S0 .

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These two conditions were shown [51] to reduce the general transformations of the space and time variables from S0 to S to the form

x = x 0 − v t0

Ry = y0 ; z = z0

t = R t0 + e1 x0 − v t0⎛

⎝ ⎜

⎞

⎠ ⎟

⎧

⎨

⎪ ⎪ ⎪

⎩

⎪ ⎪ ⎪

(7.2)

with R given by (6.3). From (7.1) one can easily see that the “delay” t − t0 of a clock in S , with respect to the clock in S0 which is passing by, in general depends not only on t 0 , but also on the point x of S in which the former clock is placed. Only if e1 = 0 such a complication is absent. The physically free parameter e1 can be fixed conventionally by defining in S the simultaneity of distant events, or, which is the same, by choosing a clock synchronization method in S . Clearly, then, the denomination appropriate for e1 is “synchronization parameter”. The only remaining unknown factor is e1 . This is a conventional term, to be called “clock synchronisation factor”. Length contraction of rods moving with respect to S0 by the usual factor R (independently of e1 ) is also a consequence of (7.2). The velocity of light consequence of (7.2) can be obtained:

c1(θ) = c1 + Γcosθ

(7.3)

with

Γ = v

c + c e1 R (7.4)

The transformations (7.2) represent the complete set of theories “equivalent” to the TSR: if e1 is varied, different elements of this set are obtained, which, according to the conventionality thesis of Reichenbach, should be equivalent for the explanation of experimental results. The Lorentz transformation is recovered as a particular case with

e1 = −v /c2 R, whence Γ = 0 and c1(θ ) = c . Different values of e1 are obtained from different synchronisation conventions. In all cases but the TSR such values imply the existence of a privileged frame.

224

The Lorentz transformations of the TSR introduce a certain symmetry between space variables and time, forcing the latter to a geometrical role in a four dimensional space. With Minkowski’s words: “The views of space and time which I wish to lay before you ... are radical. Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.” [52]

Different values of e1 imply different theories of space and time that are empirically equivalent to a very large extent. One can check with explicit calculations that the empirical data are very often insensitive to the choice of e1 (in experiments by Römer, Bradley, Fizeau, Michelson-Morley, Doppler, etc.). We will do this in the following section. Thus there are infinitely many theories explaining equally well the results of these experiments. It is remarkable that all such theories are based on the existence of a privileged frame, the only exception being the TSR.

The conclusion of physical equivalence of the theories with different e1 would seem to agree with the conventionality idea of clock synchronization. There are however some experimental situations of a different type (Sagnac effect, clock paradox, aberration of starlight, …) allowing one to determine the synchronization which has to be chosen in order to obtain a rational description of natural phenpomena. In sections 9-14 the condition e1 = 0 will be obtained six times, independently. The fact that so many proofs exist is a strong indication of the basic coorrectness in nature of absolute simultaneity. This gives rise to the following transformation of space and time:

x = x0 − v t0R

y = y0 ; z = z0

t = R t0

⎧

⎨

⎪ ⎪ ⎪

⎩

⎪ ⎪ ⎪

(7.5)

As already stressed by Mansouri and Sexl [49], such transformations would have been the logical consequence of the development along the lines of thought of Lorentz-Larmor-Poincaré: they are the very relations one would write down if one had to formulate a theory in which rods shrink and clocks are slow by the usual factor when moving with respect to the ether. That the actual development went along different lines was due to the fact that “local time” was introduced at the early stage in considering the covariance of the Maxwell equations. The one way speed of light predicted by (7.5) can be found by taking e1 = 0 in (7.4):

225

c1(θ) = c1 + β cosθ

(7.6)

“Inertial transformations” is the name proposed for the Eq.s (7.5). They imply a complete liberation of time from the merely geometrical role to which it had been forced in the Minkowski space and predict that the velocity of light relative to an inertial system S moving with respect to the privileged system S0 is not isotropic. A corresponding anisotropy is predicted for Reichenbach’s parameter ε . By studying the multiplication properties of the inertial transformations it has been possible to show that they do not form a group. There are no problems with the existence of the identical and inverse transformations, and also the associative law can be satisfied, but it is not always possible to write a meaningful product of two inertial transformations, due to the presence of two absolute velocities v and ′ v in the transformation. If Ω(v , ′ v ) denotes the transformation (7.5) it is easy to understand that the product Ω(v , ′ v ) Ω( ′ ′ v , ′ ′ ′ v ) is no inertial transformation if ′ ′ v ≠ ′ v . A property implied by (7.5) is absolute simultaneity: two events taking place in different points of S but at the same t are judged to be simultaneous also in S’ (and vice versa). The existence of absolute simultaneity does not imply that time is absolute: on the contrary, the v -dependent factor in the transformation of time gives rise to time-dilatation phenomena similar to those of TSR. A clock at rest in S is seen from S 0 to run slower, but a clock at rest in S 0 is seen from S to run faster so that both observers agree that motion relative to S 0 slows the pace of clocks. The difference with respect to TSR exists because in a meaningful comparison of rates a clock at rest in S 0 must be compared with at least two clocks at rest in different points of S, and the result is therefore dependent on the “convention” adopted for synchronising the latter clocks. Absolute length contraction can also be deduced from (7.5): All observers agree that motion relative to S 0 leads to contraction. The discrepancy with the TSR is due again to the different “conventions” concerning clock synchronisation: the length of a moving rod can only be obtained by marking the simultaneous positions of its end points, and therefore depends on the very definition of simultaneity of distant events. 8. Synchronization independent phenomena In the present section we discuss several famous experiments sharing the property of being explained equally well by all the equivalent transformations (ETs). In other words the theoretical predictions for the measured physical quantities do not depend on the synchronization parameter e1. We start from a generalized version of the Michelson-Morley experiment [53].

226

A laboratory is at rest in the inertial system S moving with absolute velocity r v and in it an interferometric experiment is performed with a beam of light split into two parts in point P by a semitransparent mirror (Fig. 3). The first part propagates along P − A1 − A2...Am − Q , with reflecting mirrors placed at intermediate points, oriented so as to produce the right deviations, the second part along the similar path P − B1 − B2 ... Bn − Q . Finally the two parts superimpose in Q where they interfere, Q being an arbitrary point of an interference figure [54]. On the first path we define the vectors

r l a i (with moduli la i ), i =1,2,.. .m +1, coincident with the

rectilinear segments described by light and all oriented from P toward Q; on the second path we similarly define the vectors

r l b j (with moduli lb j ), j = 1,2,...n +1.

The interference in Q is determined by the time delay ΔT between the two rays. In the TSR light propagates in all directions with the same speed c and:

ΔT = TB − TA =

LB − L Ac

(8.1)

where

L A = la i

i=1

m +1∑ ; L B = lb j

j=1

n +1∑ (8.2)

227

vP

Q

A 1

A 2

A mB 1

B 2

B n

Figure 3. A semitransparent mirror P generates two coherent beams of light, which follow different paths until they meet and interfere in Q. The big grey arrow represents the laboratory absolute velocity. Next we calculate ΔT from the ETs. The inverse velocity of light relative to S is given by Eq. (7.3) and one has:

ΔT = l b j

c1 (θb j )j=1

n+1∑ −

l a ic1 (θa i )i=1

m+1∑ (8.3)

where θa i (θb j ) is the angle between

r l a i and

r v (

r l b j and

r v ). By inserting (7.3) in

(8.3):

ΔT = LB − LAc

+ Γc

l b j cosθb jj=1

n+1∑ − Γ

c l a i cosθa i

i=1

m+1∑

= LB − LA

c + Γ

c

r l b j

j=1

n+1∑ −

r l a i

i=1

m+1∑

⎧

⎨ ⎪ ⎪

⎩ ⎪ ⎪

⎫

⎬ ⎪ ⎪

⎭ ⎪ ⎪

⋅r v v

(8.4)

228

The last term vanishes because the two terms within curly brackets separately equal the vector joining P and Q. Thus (8.1) and (8.4) are the same. Therefore Γ (containing e1 ) disappears from the result and all theories based on the ETs lead to the same predictions for interferometric experiments of the Michelson type.

x

yy0

x0

v

A

B

Cθ

Figure 4. The triangle ABC is at rest in the inertial system S . Mirrors in B and C force a flash of light emitted in A to propagate on the closed path ABC. Along AB light propagates in a moving medium. Next we come to the Fizeau experiment [55]. A simple method exists to obtain, consistently with the ETs, the velocity of a flash of light propagating in a medium in motion with respect to an inertial system S . Consider the triangle ABC of Fig. 4, at rest in the inertial system S , with side lengths lAB , lBC and lCA and with suitably oriented mirrors in B and C. The time t ABC required by a flash of light to propagate on the closed path ABC can be measured with a single clock in A independently of synchronization:

229

t ABC =

lAB

cAB

+ lBC

cBC

+ lCA

cCA

(8.5)

where c AB is the velocity of light from A to B, and so on. The sides BC and CA are in the vacuum, while we assume that the path AB is inside a medium (index of refraction n ) in motion from A to B with velocity u relative to S . In the TSR the velocity of light in such a medium is calculable from the composition of velocities:

cABTSR =

(c / n) + u1 + (u / cn)

(8.6)

In Fig. 4 BC is perpendicular and CA antiparallel to the absolute velocity v of S . Therefore, using eq. (7.3) for the light velocity in the vacuum we have

t ABC =

lAB

cAB

+ lBC

c +

lCA

c(1− Γ) (8.7)

The prediction of the TSR according to (8.6) is instead

tABC = l AB

1 + (u /cn)(c /n) + u

+ l BC

c + lCA

c (8.8)

But t ABC , measurable with a single clock, is independent of synchronization. Therefore (8.7) and (8.8) must be equal. Considering also that lCA = lAB cosθ it follows

1cAB

= n + (u / c)

c + nu +

Γ cosθc

(8.9)

This result shares with eq. (7.3) a property that can be written for any two points X and Y connected by light propagation in the vacuum or in a medium (whether at rest or in motion) as follows:

1cXY

= 1

cXYTSR +

Γ cosθc

(8.10)

where Γ , given by eq. (7.4), is independent of the medium. An equivalent expression for the propagation time t XY over the distance lXY is

230

t XY =

lXY

cXY

= lXY

cXYTSR +

Γc

r l XY ⋅

r v

v (8.11)

where

r l XY is the vector of length lXY oriented from X to Y.

D

A1

Σ

P

T1

T2

A2

A3A4

Figure 5. Scheme of the Michelson-Morley repetition of the Fizeau experiment. In 1851 Fizeau [55] performed an interferometric experiment with light propagating in running water. The experiment was repeated by Michelson and Morley in 1886 [56]. Light from a source at Σ (Fig. 5) fell on a half silvered mirror P , where it divided; one part following the path P A 1 A 2 A 3 A 4 P D and the other P A 4 A 3 A 2 A1 P D . The two parts passed inside two tubes T1 and T2 filled with running water, the first (second) part moving parallel (antiparallel) to the current. The interference between the two parts, observed in D , was found in agreement with Fresnel’s partial drag theory. Actually we will now discuss a generalized Fizeau experiment [57].

231

Consider a laboratory in an inertial reference frame S moving with absolute velocity

r v . A light ray propagates along the closed broken path

P − A1 − A 2 ... A m − P, where suitably oriented reflecting mirrors are placed in the intermediate points (see Fig. 4). The vectors

r l i are defined having moduli li

(i = 1, 2, . .. m +1) coinciding with the rectilinear segments described by light and oriented in the propagation direction. We are interested in the total propagation time on the path, T . The prediction of the TSR is easy to get, since light moves along

r l i

with the velocity

cA i −1A i

TSR = c 1 + ni (ui / c)ni + (ui / c)

where i =1,2,.. .m +1, A 0 and A m+1 coincide with P , ni is the refraction index of the medium present on the i-th segment and ui its fluid velocity relative to S . One has:

T =

l icAi −1A i

TSRi=1

m +1

∑ (8.12)

We calculate next the same quantity T by starting from the ETs, according to which we have, applying (8.10) to the segments of the path of Fig. 6:

t A i −1A i

= li

cA i −1A i

= li

cA i−1A iTSR +

Γc

r l i ⋅

r v

v (8.13)

From (8.13) it follows:

T = tAi−1Ai

i=1

m +1

∑ = l i

cAi−1AiTSR

i=1

m +1

∑ + Γc

r l i ⋅

r v

v

i=1

m +1

∑ (8.14)

Notice that P − A1 − A 2 ... A m − P is a closed line. Therefore the last term in (8.14) vanishes and (8.14) coincides with (8.12). Therefore Γ (containing e1 ) disappears from the result and all theories based on the ETs lead to the same predictions for interferometric experiments of the Fizeau type. We conclude that the ETs predict that a measurement of the time required by light to describe a closed path will necessarily give the same value as predicted by the TSR even if moving fluids are present along the path.

232

P

A1

A2

Am

v

Figure 6. Propagation of light along a broken closed path. The big grey arrow represents the laboratory absolute velocity. When the Ives-Stilwell experiment was published [58] only the TSR proved to be in agreement with the experimental detection of the transverse Doppler effect. Consequently, the ether hypothesis was considered obsolete for one more reason and the absolute motion was discarded. Now it is interesting to see whether it is possible to explain the Doppler effect in the framework of a theory based on the inertial transformations in which the relativity principle does not apply in the usual way. The problem was faced by Puccini and Selleri [59] who concluded that the Doppler effect, as well as the results found by Ives and Stilwell, are perfectly well explained also within the new theoretical framework.

We consider the propagation of a light corpuscle P (a small light pulse propagating in the ray direction) in the privileged frame S0 , relative to which the velocity of light is assumed to be the same in all directions. We describe P with coordinates x 0 and y 0 satisfying, at time t 0 :

x0 = c cosθ0 t0 ; y0 = c sinθ0 t0 (8.15) Our first task is to use the ITs to determine the velocity and the direction of motion of P in S in terms of the S0 quantities c and θ0 , which are considered given. We consider the inverse inertial transformations from a moving inertial system S to the privileged one S0 :

x 0 = Rx + 1

R v t ; y 0 = y ; t 0 = 1

R t (8.16)

2 3 3

Substituting (8.16) into (8.15) we find

x = 1R2 c cosθ0 − β( ) t ; y = 1

R c sinθ0 t (8.17)

Relative to S , which superimposes to S0 at time t 0 = t = 0 , the pulse is assumed to have velocity c1 and propagation angle θ , that is, to satisfy equations similar to (1) but with space-time variables x, y, t and parameters c1 and θ .

Coming to the Doppler effect proper, let us consider a plane electromagnetic wave (in empty space) described in the privileged frame S0 as:

ψ(

r r 0, t0) = ψ0 exp iω0 t0 −

ˆ n 0 ⋅r r 0

c⎛ ⎝ ⎜

⎞ ⎠ ⎟

⎧ ⎨ ⎩

⎫ ⎬ ⎭

(8.18)

where ψ 0 is a constant amplitude, ω0 the angular frequency, ˆ n 0 the unit vector normal to the wave fronts and c the one-way velocity of light. It should be noted that, in the privileged system, ˆ n 0 gives the propagation direction of the wave. In order to study the Doppler effect in a frame S in motion with a velocity

r v relative

to S0 , we must substitute t 0 and r r 0 in the phase φ of the plane wave (8.18) by the

expressions (8.16) of the inertial transformations

φ = ω0

1R

1−vc

n0x⎛ ⎝ ⎜

⎞ ⎠ ⎟ t − 1

cn0xRx + n0y y[ ]

⎧

⎨ ⎪

⎩ ⎪

⎫

⎬ ⎪

⎭ ⎪ (8.19)

This can be written

φ = ω0

1R

1−ˆ n 0 ⋅

r v

c⎛

⎝ ⎜

⎞

⎠ ⎟ t − 1

cˆ n 0 + R −1

v2 ˆ n 0 ⋅r v ( )

r v

⎡ ⎣ ⎢

⎤ ⎦ ⎥ ⋅

r r

⎧

⎨ ⎪

⎩ ⎪

⎫

⎬ ⎪

⎭ ⎪ (8.20)

From (8.20) it is obvious that the angular frequency relative to the S system is

ω = ω0

1R

1 − ˆ n 0 ⋅

r v

c⎛

⎝ ⎜

⎞

⎠ ⎟ (8.21)

which is identical to the relativistic prediction. Therefore the Doppler effect from S0 to S is explained exactly by the ITs, as well as with the TSR.

234

v

v

v

θ1

θ2

A1

A2

B2

B1

0

Figure 7. The rotating Earth seen from the South pole. Two stations A and B at some time of the day occupy the positions A 1 and B1 ; some hours later they have shifted to A 2 and B2 . The propagation direction of a radio signal connecting A and B forms an angle changing from θ1 to θ2 with the Earth absolute velocity v . No need to investigate the Doppler effect with the transformations connecting two moving systems (S and ′ S , say). Once ω0 is given, the frequency relative to S predicted by the inertial transformations from S0 to S is the same as that predicted by the TSR, for all possible S . A transformation from S to ′ S must give the right change of frequency if only the theory is consistent.

Let us come to the discussion of the International Atomic Time system. Atomic clocks distributed around the world communicate with one another by means of radio signals. The synchronization signal sent by a transmitting station always

235

reaches the receiving station ‘on time,’ at any hour of the day despite the motion of the Earth. In Sexl and Schmidt’s opinion [60] the proper functioning of this system demonstrates that light, relative to the Earth surface, has the same speed c in all directions. In fact this may not be so; we will now show that the proper working of the network says nothing about the one way velocity, as it is consistent with another theory, empirically (almost) equivalent to the TSR, in which the one way speed of light has a directional dependence in moving frames.

The 1967 General Conference of Weights and Measures redefined the second as the duration of 9,192,631,770 cycles of the radiation absorbed in the transition between two hyperfine levels of Caesium-133 atom. The International Atomic Time (Temps Atomique Internationale = TAI) is the reference time based on the new definition of the second. For establishing TAI the readings of 250 atomic clocks in 45 institutes of the world are systematically compared.

Two stations A and B at some time of S occupy the positions A 1 and B1 ; some hours later they have shifted to the new positions A 2 and B2 . We show that the theory of the ITs [61] explains the empirical observations just as well as the TSR. According to both theories if a radio signal sent from A at local time ′ t A arrives in B at local time ′ t B , the time difference ′ t B − ′ t A (measured on the Earth) is the same from A 1 to B1 as from A 2 to B2 . These are two generic positions of the stations A and B . The proper working of the TAI, therefore, cannot discriminate the approach based on the ITs from the TSR where the velocity of light relative to the Earth surface is c in all directions.

Consider a rotating circular platform with centre at rest in S . Seen from S the platform remains circular, in spite of its rotation. In fact we can imagine a pen fixed in a point of the platform border drawing on the ground a closed line γ . This line is completely at rest in S and its shape cannot depend on the chosen clock synchronization. Since in the TSR γ is a circle, it must be seen as a circle in all equivalent theories. Seen from a point vertically above the platform centre, γ overlaps exactly with the border of the rotating platform which, therefore, is also circular. We assume the Earth to be circular.

Velocities will instead depend on clock synchronization. In S the rotation velocity generally is not observed as uniform. Let us see why. Let u1 and u2 be the velocities of the Earth surface relative to S on the positions 1 and 2 of Fig. 7. We set, at rest in S near position 1, a suitably oriented segment P − Q with unit length; when the station A passes close to P at velocity u1 , P itself sends out a light signal (at velocity c1 ). In Q we measure the time lag between the arrivals of the signal and of the station A . This delay is the difference between the propagation times: 1 / u1 − 1 / c1. This measurement is made in S with one clock only, so the result must be the same that would be obtained according to the TSR, theory in which

236

rotation velocities are isotropic in S . The same operation can be carried out in position 2. Therefore:

1u1

− 1c1

= 1u

− 1c

and 1u2

− 1c2

= 1u

− 1c

(8.22)

where u = ω r , ω and r being angular velocity and radius of the Earth in the TSR. Clearly Eq.s (8.22) imply the existence of a first dynamical invariant of the rotating Earth:

c1 u1c1 − u1

= c2 u2

c2 − u2 (8.23)

According to the theory of ITs the two synchronization signals of Fig. 7 propagate, in S , respectively, at :

c1 = c

1 + β cosθ1 and c2 = c

1 + β cosθ2 (8.24)

Replacing (8.24) in (8.22) one easily gets

u1 = u c

c + u β cosθ1 and u2 = u c

c + u β cosθ2 (8.25)

Thus the platform rotation velocities in the points A1 and A2 are different. If the segments A1B1 and A2B2 of Fig. 5, judged to be of equal length in the TSR, had equal length also in our approach two observers at rest in S near the points A1 and A2 would see different numbers of such segments pass by in the unit time. This is absurd, as it would imply an accumulation of matter on one side; we write

A1B1 = l1 and A2B2 = l 2, where, in general, l1 ≠ l 2. The station B on the border of the rotating platform passes near a clock U at

rest in S when this clock marks the time t . An observer checks on U the time t + Δt at which also the station A later passes near U . The delay Δt , measured with a single clock, cannot depend on synchronization. Therefore it is the same in all equivalent theories and has the value predicted by the TSR in which the platform rotation is uniform. Thus Δt has a value independent on the position of U in S near the platform border: in this sense rotation is uniform in all the ETs.

In the positions 1 and 2 of Fig. 5 the time interval Δt starts when the station B passes near U and ends when the station A arrives near U . The distances l1 and

237

l2 are traversed by A with velocities u1 and u2 , respectively (all measured in S ). Then Δt = l1 / u1 and Δt = l2 / u2 . This gives a second dynamical invariant:

l1u1

= l2u2

(8.26)

A third invariant can be shown to exist, if u1 and u2 are the rotation velocities of the stations (measured in S ). A short calculation leads to

R1u1

= R 2u2

(8.27)

Three dynamical invariants of the rotating Earth. By multiplying them together one finds

R1 l1u1

2c1 u1

c1 − u1 =

R2 l2u2

2c2 u2

c2 − u2 (8.28)

a result which will be very useful. All the following calculations are performed from the point of view of the inertial system S relative to which the Earth rotates without translating. We want to calculate the times taken by the two synchronization signals to travel from station A to station B in the positions 1 and 2 of Fig. 7. Therefore we are interested in the timing of four event as shown by clocks at rest in S near the points where the events take place: t A1 : the first radio signal leaves A ; t B1 : the first radio signal arrives in B ; t A 2 : the second radio signal leaves A ; t B2 : the second radio signal arrives in B . For t B1 we can write

t B1 = tA1 +

l1c1 − u1

(8.29)

where the last term is the pulse propagation time, justified as follows.

238

The observer in S sees the radio signal and the station B moving in well defined ways, that is according to the equations ξ = c1 t − t A1

⎛

⎝ ⎜

⎞

⎠ ⎟ and

ξ = l1 + u1 t − tA 1⎛

⎝ ⎜

⎞

⎠ ⎟ , respectively, if ξ is a coordinate on the A B line with

origin in A . Clearly the equal position condition (arrival of the signal in B ) will be obtained after a time interval t − tA1 given by l1 / c1 − u1( ).

Let us come to the second position. For t A 2 one can write

t A 2 = tA 1 +

l1u1

+ T12 (8.30)

where l1 /u1 is the station A propagation time from A1 to B1 and T12 is the station A propagation time from B1 to A2 (see Fig. 8). Finally, for tB2 one can write

t B2 = tA 1 + T12 +

l2u2

+ l2

c2 − u2 (8.31)

where T12 is the station B propagation time from B1 to A2 ( T12 is obviously the same for the two stations) and l2 / u2 is the station B propagation time from A2 to B2 . The last term in (8.31) is the pulse propagation time.

Seen from S the paths followed by the stations A and B , due to the rotation of Earth, are not the same (Fig. 8); the symmetry is broken by the presence of a privileged direction, that of the absolute v of translation with which the rotation velocity composes. This point is crucial. The segment B1 − A 2 is common to the two paths and does not introduce any difference. The segment A 1 − B1 is travelled only by the station A at a faster absolute velocity, compared to the segment A 2 − B2 which is travelled only by the station B . The times marked by the two clocks, in these two segments, are different: the clock in A , going faster, develops a delay compared with the one in B .

How will these times be perceived on the Earth? To answer we perform ITs from S to the inertial systems ′ S 1 and ′ S 2 in which the stations A and B can be considered at rest during the short time in which A , seen from S , moves either from A 1 to B1 or from A 2 to B2 .

239

v

A1

A2

B2

B1

Figure 8. While station A moves from A 1 to A 2 , station B moves from B1 to B2 . The path B1 A 2 is common to the two stations and differences in the times marked by the respective clocks can only arise from A 1 B1 and A 2 B2 . According to the golden rule of the time transformation applied separately to the (small) regions 1 and 2 of Fig. 5 the S times will be registered by the stations on the Earth slowed down by R1 / R (if in region 1) and by R 2 / R (if in region 2). For ′ t B1 we obtain

′ t B1 = ′ t A1 + ′ τ +

R1R

l1c1 − u1

(8.32)

where ′ τ is the conventional time actually added to the time marked by the clock of the B station in order to achieve a particular synchronization, e.g. the one ensuring that the Earth based measurement of the velocity of a radio pulse from A to B is c .

Let us come to the second position. For ′ t A 2 we can write

240

′ t A 2 = ′ t A 1 +

R1R

l1u1

+ ′ T 12 (8.33)

because the station A propagation from A 1 to B1 takes place in region 1. In (8.33)

′ T 12 is the station A propagation time from B1 to A 2 as measured on the Earth. Finally, for t B2 we have

′ t B2 = ′ t A1 + ′ τ + ′ T 12 + R2R

l 2u2

+ l 2c2 − u2

⎡

⎣ ⎢

⎤

⎦ ⎥ (8.34)

where ′ T 12 is the station B propagation time from B1 to A 2 as measured on the Earth. Again, it is the same as the station A propagation time from B1 to A 2 . From (8.32)-(8.34) we obtain the time differences

′ t B1 − ′ t A1 = ′ τ +

R1R

l1

c1 − u1 (8.35)

and

′ t B2 − ′ t A2 = ′ τ + R2R

l 2u2

+ l 2c2 − u2

⎡

⎣ ⎢

⎤

⎦ ⎥ − R1

R l1u1

(8.36)

The latter equation is the same as

′ t B2 − ′ t A2 = ′ τ + R2 l 2

R u22

u2c2c2 − u2

− R1R

l1u1

(8.37)

Due to the product of the three invariants the second term in the right hand side can be written with the index everywhere changed from 2 to 1. It is now very easy to check that

′ t B2 − ′ t A 2 = ′ t B1 − ′ t A1 (8.38) Therefore the two stations will not detect any desynchronization between the clocks. On the surface of the Earth it so happens that the delay due to the We applied the ITs to small regions of the rotating Earth. By doing this we used a principle: a small region of an accelerated frame is physically equivalent to the

241

‘comoving’ inertial frame having the same instantaneous velocity. The same principle is applied by people using the relativistic idea of the invariance of the velocity of the electromagnetic radiation. With the Lorentz transformations c is isotropic everywhere and the clocks of two different stations must remain synchronous, as observed. With the inertial transformations there are two phenomena producing a desynchronization: the first one is the anisotropy of the velocity of light, the other one the variable absolute velocity of the clocks generating differences in their pace. The two effects are equal and opposite and cancel, so that the clocks always appear to be synchronous when a signal connects them. The proper functioning of the world time system does not say anything about the one way speed of light and cannot establish which theory is “true”. Thus the theory of the inertial transformations is, in this respect, completely equivalent to the TSR. 9. The Sagnac effect: e1 = 0 It is remarkable that almost a century after the discovery of the Sagnac effect no theoretical justification based on the two relativistic theories has been found. Hasselbach and Nicklaus, describing their own experiment [62], list about twenty different explanations of the Sagnac effect and comment: “This great variety (if not disparity) in the derivation of the Sagnac phase shift constitutes one of the several controversies ... that have been surrounding the Sagnac effect since the earliest days of studying interferences in rotating frames of reference.” Iin this section the reasons for this strange resistance of the relativistic theories at explaining the Sagnac effect are pointed out: A simple explanation is shown to be available, only within a theory based on absolute simultaneity ( e1 = 0). In the Sagnac 1913 experiment a platform was made to rotate uniformly around a vertical axis at a rate of 1-2 full rotations per second. In an interferometer mounted on the platform, two interfering light beams, reflected by four mirrors, propagated in opposite directions along a closed horizontal circuit defining a certain area A . The rotating system included also the luminous source and a photographic plate recording the interference fringes. On the pictures obtained during a clockwise and a counterclockwise rotation with the same frequency, Sagnac observed the interference fringes in different positions and measured the displacement Δz by overlapping the two figures. This Δz is strictly tied to the relative time delay with which the two light beams reach the detector. Sagnac observed a shift of the interference fringes every time the rotation was modified. Considering his experiment conceptually similar to the Michelson-Morley one, he informed the scientific community with two papers (in French) bearing the titles “The existence of the luminiferous ether demonstrated by means of the effect of a relative ether wind in an uniformly rotating interferometer"

242

[63] and “On the proof of reality of the luminiferous ether with the experiment of the rotating interferometer" [64]

S

O

A

B C

D

Figure 9. Simplified Sagnac apparatus. Light from a source S is divided in two parts by the semitransparent mirror A . The first part follows the path ABCDAO concordant with rotation, the second part follows ADCBAO discordant from rotation. Interference fringes observed in O . The experiment was repeated many times in different ways, with the full confirmation of the Sagnac results. Famous is the 1925 repetition by Michelson and Gale [65] for the very large dimensions of the optical interference system (a rectangle about 650m x 360m); in this case the disk was the Earth itself at the latitude concerned. The light propagation times were not the same, as evidenced by the resulting fringe shift. Full consistency was found with the Sagnac formula [see below] if the angular velocity of the Earth rotation was used. Can light propagate with the usual velocity c relatively to the rotating platform? The question was directly faced in the 1942 experiment by Dufour e Prunier [66], in which the mirrors defining the paths of the interfering light beams were partly fixed in the laboratory (directly above the disk) and partly in the spinning disk. The fringe shifts were the same as in a repetition of the test with all

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mirrors fixed on the disk, confirming that the light does not adapt to the movement of the disk, and that it is physically connected with some other reference system, in all probability inertial.

Surprisingly theoreticians were little interested in the Sagnac effect, as if it did not pose a conceptual challenge. For a presentation of Einstein's ideas about the rotating disk see a paper by Stachel [67]. As stated before, the first discussion by Langevin came only 7-8 years later [19] and was as much formally selfassured as substantially weak. One of the opening statements is this: “I will show how the theory of general relativity explains the results of Sagnac’s experiment in a quantitative way.” Langevin argues that Sagnac’s is a first order experiment, on which all theories (relativistic or prerelativistic) must agree qualitatively and quantitatively, given that the experimental precision does not allow one to detect second order effects: therefore it cannot produce evidence for or against any theory. Then he goes on to show that an application of Galilean kinematics explains the empirical observations! In fact his approach is only slightly veiled in relativistic form by some words and symbols, but is essentially 100% Galilean.

The impression that Langevin, beyond words, could not be satisfied with his explanation is reinforced by his second article of 1937 [19] in which two relativistic treatments are presented. The first one is still that of 1921, this time deduced from the strange idea that the time to be adopted everywhere on the disk is that of the rotational centre (motionless in the laboratory). The second one is to define “time” in such a way as to enforce a velocity of light constant and equal to c , falling so flatly in the problem of the discontinuity for a tour around the disk that we will discuss later.

The 1963 review paper by Post [20] seems to agree with the idea that two relativistic proofs of the Sagnac effect are better than one. The first proof (in the main text) uses arbitrarily the laboratory to platform transformation of time ′ t = t R where R is the usual square root factor of relativity, here written with the rotational velocity. The second proof (in an appendix) starts from the Lorentz transformation

′ t = t +r v ⋅

r r / c2( )/ R , but it hastens to make the second term disappear with the

(arbitrary) choice of r r perpendicular to

r v .

The tendency by Langevin and Post to cancel the spatial variable x in the transformation of time shows once more the great difficulty in explaining the physics of the rotating platform with the relativistic theory. Somehow it anticipates the approach based on the inertial transformations, the only ones which can explain the Sagnac effect. Also the space devoted by the Landau and Lifshitz book [21] to the question of clock synchronization on the rotating platform is interesting, especially because the two Russian physicists do not spend a word in criticizing the relativistic theory while they advance in the intricacies of two clocks in fixed positions having different synchronizations relative to one another, depending on the path followed to join them.

Vetharaniam and Stedman [68] developed a theoretical model supposed to describe Lorentz invariance locally, that is for an infinitesimal time interval and in an

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infinitesimal region of space in an accelerated reference frame. They claim that the model can be tested by using a precision ring laser to bound parameters of the theory. I can only remark that the validity of the local Lorentz invariance is immediately in contradiction with the very existence of the Sagnac effect. In fact, Lorentz invariance implies equal local velocities of light along opposite directions and thus - given the circular symmetry of the problem - also equal global velocities of light along opposite directions. After full tours in opposite directions along the rim of the rotating disk, the two pulses would hit the target at the same time of the disk and the Sagnac effect would disappear. Anyway, it remains an important experiment and it is interesting to see if somebody is going to carry it out.

A very interesting modified Sagnac experiment has been carried out recently by Ruyong Wang and collaborators. [69] The instrument was designed to decide whether the travel time difference only appears in rotational motion, or if it also appears in rectilinear uniform motion. The results were unequivocally in favour of the second possibility, in full agreement with our present approach to relativistic physics, which attributes in all cases to a given body the same local velocity relative to an accelerated reference frame and to the inertial frame locally comoving with the latter.

Next we come to the point: the inertial transformations provide a rigorous qualitative and quantitative explanation of the Sagnac effect [70]. In order to simplify calculations, we consider now a monochromatic light source placed on the disk emitting two coherent beams of light in opposite directions. These travel along a circumference concentrical with the disk, until they reunite in a point A and interfere, after a 2π propagation. The circular path can be obtained by forcing the light to propagate tangentially to the internal surface of a cylindrical mirror. The positioning of the interference figure depends on the disk rotational velocity ("Sagnac effect"). Most textbooks deduce the Sagnac formula (our Eq. (9.5) below) in the laboratory, but say nothing about the description of the phenomenon given by an observer placed on the rotating platform: we will see that SRT predicts a null effect on the platform, while our approach based on the inertial transformations gives the right answer. For simplicity we will assume that the laboratory is at rest in the privileged frame. Sagnac effect seen from the laboratory. Light propagating in the rotational direction of the disk must cover a distance larger than the disk circumference length L0 by a quantity l1 = v t01 equaling the shift of A during the time t01 taken by light to reach the interference region. Therefore

L0 + l1 = c t01 ; l1 = v t01 (9.1) From these equations it is easy to get:

245

t01 = L0

c − v (9.2)

Light propagating in the direction opposite to that of rotation must instead cover a distance smaller than the disk circumference length L0 by a quantity l 2 = v t02 equaling the shift of A during the time t02 taken by light to reach the interference region. Therefore

L − l 2 = c t02 ; l 2 = v t02 (9.3) One now gets

t02 = L0

c + v (9.4)

The time diference Δt0 between the two propagations is the parameter fixing the phase difference in the considered interference point. From (9.2) and (9.4) it follows

Δt0 = t01 − t02 = 2L0

c2 v

1− v2 /c2 = 2Lc2

vR

(9.5)

Obviously L0 = L R is the disk circumference length reduced in the laboratory by the usual relativistic factor, if L is the rest length of the same disk. The consistency of Eq. (9.5) with experimental data has been checked in many experiments. Sagnac effect seen from the disk. As a preliminary to the solution of the problem on the disk consider a clock marking the time t fixed in the origin of the moving inertial system S . Seen from S0 it therefore satisfies the equation x0 = v t0. Substituting this equation into the equivalent transformations (7.2) we get x = 0 (the fixed coordinate of the clock in S) and t = R t0. Therefore, all the ETs lead to the same relationship between the times t, t0. For time intervals between two events we write

Δt = R Δt0 (9.6) Eq. (9.6) will be assumed to hold also for a clock on the rim of a disk rotating with speed v . This is consistent with our general philosophy that every small portion of the circumference of the rotating platform can be considered instantaneously at rest in a moving inertial frame of reference locally "tangent" to the disk. Therefore Eq. (7.3) applies for the velocity of light on the disk. Only the cases of light moving parallel

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and antiparallel to the local absolute velocity must be considered. It follows from (7.3) that the inverse velocity of light for these two cases is respectively given by:

1˜ c (0)

= 1 + Γc

; 1˜ c (π )

= 1 − Γc

(9.7)

with Γ given by (7.4). The time difference on the disk is given by

Δt = t1 − t2 = L˜ c (0)

− L

˜ c (π ) = 2L Γ

c (9.8)

Substituting (9.5) in the right hand side of (9.8) we get

Δt = Δt0 R 1 +

c2 e1 Rv

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

(9.9)

where R is the usual square root factor describing the dilation of time intervals in a moving frame. Now, the result aimed at is (9.6): only the inertial transformations, corresponding to e1 = 0 allow us to get (9.6) from (9.9). For all other values of e1 one will get wrong results from (9.9). In particular, the TSR with its e1 = −v /c2R predicts Δt = 0.

We have reached the conclusion that of all theories having different values of e1

only one (e1 = 0 ) gives a rational description of the Sagnac effect on the rotating platform. In the case of e1 ≠ 0 the calculated time difference on the platform disagrees with the prediction (9.5) in the laboratory, prediction which is of course the same for all theories satisfying the equivalent transformations (SRT included), since in the laboratory (assumed to be at rest in the privileged frame) Einstein's synchronization was used.

The Sagnac effect is also important for understanding the nature of the so called Sagnac correction on the Earth surface. As recounted by Kelly [71], in 1980 the CCDS (Comité Consultatif pour la Définition de la Seconde) and in 1990 the CCIR (International Radio Consultative Committee) suggested rules - later universally adopted - for synchronizing clocks in different points of the globe. Two are the methods used to accomplish this task. The first one is to transport a clock from one

247

site to another and to regulate clocks at rest in the second site with the time reading of the transported clock. The second method is to send an electromagnetic signal informing the second site of the time reading in the first site. The rules of the committee establish that three corrections should be applied before comparing clock readings:

(a) the first correction takes into account the velocity effect of the theory of special relativity (TSR). It is proportional to v

2 / 2c2 , where v is the velocity of the airplane, and corresponds to a slower timing of the transported clock;

(b) the second correction takes into account the gravitational effect of the theory of general relativity (TGR). It is proportional to g(φ)h / c2 where g is the total acceleration (gravitational and centrifugal) at sea level at the latitude φ and h is the height over sea level. It corresponds to a faster timing of the transported clock;

(c) the “Sagnac correction” is assumed proportional to 2A Eω / c2 , where A E is the equatorial projection of the area enclosed by the path of travel of the clock (or of the electromagnetic signal) and the lines connecting the two clock sites to the centre of the Earth, and ω is the angular velocity of the Earth.

There are no doubts about nature and need of the first two corrections, but the justification of the third one is unconvincing. I agree completely with Kelly [72] when he says that the only possible reason to include (c) is that the eastward velocity of light relative to the Earth is different from the westward.

In fact one can deduce, for a real experiment, the “Sagnac correction” from Eq.

(9.1) applied to a geostationary satellite, for which the satellite itself and the Earth surface can be thought to be at rest on the same rotating platform.

Saburi et al. carried out their experiment in 1976, before the CCDS and CCIR

deliberations, and made clear that “corrections” were indeed necessary already in the title of their paper [73] (“High-Precision Time Comparison via Satellite and Observed Discrepancy of Synchronization”). They had two atomic clocks, not quite synchronous, one in a first station W (near Washington, USA) the other one in a second station T (near Tokyo, Japan) practically on the same parallel of the two cities. The time difference between the two clocks on August 27, 1976 was measured with two different methods:

(i) by sending an airplane carrying a third clock (initially synchronous with the one in W ) from W to T , via Hawaii (westward);

248

(ii) by sending an electromagnetic signal, via a geostationary satellite, from W to T , again westward.

The uncorrected airplane clock found the T clock 9.42 μs fast with respect to the W clock. The velocity correction and the gravitational correction together were estimated to be about 0.080 μs (to be subtracted to the time shown by the transported clock). By applying such a correction the T -W time difference increased to 9.50 μs .

The electromagnetic signal carried with itself informations about

the time shown by the clock of the transmitting station. Assuming that the signal velocity was c , it was found that the T clock was 9.11 μs fast with respect to the W clock. Thus, the discrepancy between the two measurements was about 0.39 μs .

Let LWS and LST the Washington-satellite and Tokyo-satellite distances,

respectively (see Fig. 10). As most physicists in similar experiments, Saburi and collaborators synchronized clocks by imposing that the velocity of light is c , that is in such a way that tT − tW = LWS + LST( )/c , t W and t T being the times of signal departure from W and arrival in T as marked by the respective clocks. In order to ensure that this formula was correct for their clocks they had to apply the so called “Sagnac correction” to the clock of the receiving station.

Such a correction is given by ΔtT = 2 ω AE /c2 where A E is the area of the

quadrangle OWSTO of 4.

By adopting such an approach Saburi and collaborators made an error because, as we know, the correct velocity of light relative to the rotating Earth is not c, but that given by the inertial transformations, which in the appropriate directions is

1. c WS = c

1 + β cosα WS ; cST =

c1 + β cosαST

(9.10)

where β = ω r / c (r is the radius of the W -T parallel and ω is the Earth angular velocity), α WS is the angle between the line W S and the local velocity (normal to the radius OW ), αST is the angle between the line S T and the normal to the radius OT in Fig. 10. Therefore αWS = θW − π/2 and αST = θT − π/2 where θW and θT are the angles O ˆ W S and O ˆ T S of Fig. 10, respectively.

249

Figure 10. An electromagnetic signal travels between two points W and T on the Earth via a geostationary satellite S (seen from the North pole). The equations (9.10), however, do not give the velocities adopted in this experiment in which the choice was always in favour of c . Having imposed the impossible condition that the velocity of light is c relatively to the rotating earth the quoted authors had now to apply the mysterious “Sagnac correction” ΔtT on the time of arrival in Tokyo. Such a correction, from our point of view, is best calculated by replacing c with

c WS and cST as follows

ΔtT =LWScWS

−LWS

c+

LSTcST

−LST

c (9,11)

which is positive as c > cWS , cST . We now can easily show that the mystery of the “Sagnac correction” of Earth physics disappears completely if one adopts the inertial transformations.

TWθ r

O

S

250

Using the definition β = ω r / c we can write

ΔtT = ω r LWS cosαWS + LST cosαST( )/c2 but

r LWS sinθW + r LST sinθT = 2 AE

where A E is the area of the quadrangle OWSTO of Fig. 10. We have thus provided a full physical justification of the Sagnac correction 2A Eω / c2 . Our results confirm the qualitative observation of Hayden [74]: electromagnetic signals need more time for a full tour around our planet toward east than toward west and this can only mean that relatively to the Earth the velocity of light in the two senses is not the same. 10. The rotating platform: e1 = 0 Next we review earlier results showing that the comparison between the relativistic descriptions of rotating platforms and inertial reference systems points out to another fundamental difficulty. Furthermore we show that the difficulty can be overcome only by substituting the Lorentz transformations between inertial systems with the “inertial” transformations based on e1 = 0 [75]. The problem is related to the Sagnac effect. It is well known that no perfectly inertial frame exists in practice because of Earth rotation, of orbital motion around the Sun, of Galactic rotation. All knowledge about inertial systems has therefore been obtained in frames having small but non zero acceleration a . For this reason the mathematical limit a → 0 taken in the theoretical schemes should be smooth and no discontinuities should arise between systems with small acceleration and inertial systems. This requirement will be shown not to be satisfied by the existing relativistic theory. Consider an inertial reference system S0 and assume it is isotropic so that the one-way velocity of light relative to S0 has the usual value c in all directions. In relativity the latter assumption is true in all inertial frames, while in other theories only one frame satisfying it exists. In a laboratory there is a circular platform (radius r and centre constantly at rest in S0) which rotates uniformly around its axis with angular velocity ω and peripheral velocity v = ω r . On its rim, consider a single clock CΣ (marking the time t ) and assume it to be set as follows: When a clock of the laboratory momentarily very near CΣ shows time t 0 = 0 then also CΣ is set at time t = 0.

251

When the platform is not rotating, CΣ constantly shows the same time as the laboratory clocks. When it rotates, however, motion modifies the pace of CΣ and the relationship between the times t and t0 is taken to have the general form

t 0 = t F v , . ..( ) (10.1) where F is a function of velocity v and eventually acceleration a = v 2 r and higher derivatives of position (not shown). Eq. (10.1) is a consequence of the isotropy of S0. Its validity can be shown in three elementary steps: 1. In the inertial system S0 all directions are physically equivalent. If a clock is moving on a straight line l with a certain speed v relative to S0 , the change in rate of advancement of its hands cannot depend on the orientation of l . 2. Similar is the case of the clock CΣ at rest on the rim of a uniformly rotating platform, with centre at rest in S0 . If S0 is isotropical the rate of advancement of its hands cannot depend on the angle between the clock instantaneous velocity vector and any given direction in S0 but only on speed v . 3. This conclusion, clearly correct by symmetry reasons, was confirmed experimentally by the 1977 CERN measurements of the anomalous magnetic moment of the muon [76]. The decay of muons was followed closely in different parts of the storage ring and the results showed a decay rate constant in the different points of the trajectory. Thus we have every reason to believe (10.1) to be correct. We are of course far from ignorant about the function F . There are strong experimental indications that the dependence on a is totally absent and that F v, ...( ) = 1/R . This is however irrelevant for our present needs as the results obtained below hold for all possible factors F . On the rim of the platform besides the clock there is a light source Σ placed in a fixed position near CΣ . Two light flashes leave Σ at time t1 of CΣ and are forced to move on a circumference, by “sliding” on the internal surface of a cylindrical mirror placed at rest on the platform, all around it very near its border. Mirror apart, the light flashes propagate in the vacuum. The mirror behaves like a source (“virtual”) whose motion never changes the velocity of emitted light signals. Therefore the motion of the mirror cannot modify the fact that relative to the laboratory, the light flashes propagate with the usual velocity c .

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The description of light propagation given by the laboratory observers is the following: two light flashes leave Σ at time t 01. The first one propagates on a circumference, in the sense discordant from the platform rotation, and comes back to Σ at time t 02 after a full circle around the platform. The second flash propagates on the same circumference, in the sense concordant with the platform rotation, and comes back to Σ at time t 03 after a full circle around the platform. These laboratory times, all relative to events taking place in a fixed point of the platform very near CΣ , are related to the corresponding platform times via (10.1): t0i = ti F v, ...( ) (i =1,2,3) . The circumference length is assumed to be L0 and L , measured in the laboratory S0 and on the platform, respectively. If c 0( ) and ˜ c π( ) are the light velocities, relative to the disk, for the flash propagating in the direction of the disk rotation and in the opposite direction, respectively, one can show with a few elementary steps using the very definition of velocity

1˜ c π( )

= t2 − t1L

= L0 /LF c (1+ β)

; 1˜ c 0( )

= t3 − t1L

= L0 /LF c (1− β)

(10.2)

From (10.2) one has

˜ c (π )˜ c (0)

= 1 + β1 − β

(10.3)

Notice that the function F has disappeared in the ratio (10.3). Therefore the light instantaneous velocities relative to the disk will also coincide with the average velocities ˜ c (0) and c (π ), and Eq. (10.3) will apply also to the ratio of the instantaneous velocities [thus we do not need a different symbol for the instantaneous velocities].The result (10.3) holds with the same numerical value for platforms having different radius, but the same peripheral velocity v . Let a set of circular platforms be given with centres at rest in S0 . Let their radii be

r1, r2 , ... ri, .. . , with r1 < r2 < . .. < ri < .. . , and suppose they are made to spin with angular velocities ω1, ω2, ... ω i, .. . such that

ω1r1 = ω2r2 = .. . = ω iri = . .. = v , where v is constant. Obviously, then, (10.3) applies to all such platforms with the same β β = v /c( ). The centripetal accelerations decrease regularly with increasing ri . Therefore, a small part AB of the rim of a platform, having peripheral velocity v and large radius, for a short time is completely equivalent to a small part of a "comoving" inertial reference frame (endowed with the same velocity). For all practical purposes the segment AB will belong to that inertial reference frame. But the velocities of light in the two directions AB and BA have to satisfy (10.3).

253

BA

O

rφ

Figure 11. By symmetry reasons, the velocity of light relative to the rotating disk between two nearby points A and B does not depend on the angle φ fixing the position of the segment AB on the rim of the disk. It follows that the one way velocity of light relative to the comoving inertial frame cannot be c and must instead satisfy

c1(π )c1(0)

= 1 + β1 − β

(10.4)

The equivalent transformations (of which the inertial transformations are a particular case) predict the inverse one way velocity of light relative to the comoving system S :

1c1(θ )

= 1c

+ βc

+ e1 R⎡ ⎣ ⎢

⎤ ⎦ ⎥ cosθ (10.5)

where θ is the angle between the light propagation direction and the absolute velocity r v of S . Eq. (10.5) applied to the cases θ = 0 and θ = π easily gives

c1(π)c1(0)

= 1 + β + c e1R1 − β − c e1R

(10.6)

254

Clearly Eq. (10.6) is compatible with (10.4) only if e1 = 0 . We thus see that also our fundamental result (10.4) is consistent with the physics of the inertial systems only if absolute simultaneity is adopted. For a better understanding of the reasons why the TSR does not work consider again the ratio

ρ ≡ ˜ c (π )˜ c (0)

(10.7)

which, owing to (10.4), is larger than unity. Therefore the light velocities parallel and antiparallel to the disk peripheral velocity are different. For the TSR this conclusion is unacceptable, because a set of platforms, all endowed with the same peripheral velocity locally approximates an inertial system better and better with increasing radius. The logical situation is shown in Fig. 12. Figure 12. The ratio ρ = ˜ c (π ) / ˜ c (0) plotted as a function of acceleration for rotating platforms of constant peripheral velocity and decreasing radius (increasing acceleration). The prediction of the TSR is 1 (black dot on the ρ axis) and is not continuous with the ρ value of the rotating platforms. Thus the TSR predicts for ρ a discontinuity at zero acceleration. While all the experiments are performed in the real physical world [where of course a ≠ 0, ρ = (1 + β) /(1 − β) ], the theory has gone out of the world ( a = 0, ρ = 1)!

acceleration

ρ

1

(c+v)/(c−v )

255

Notice that the velocity of light given by Eq. (10.5) with e1 = 0 is required for all inertial systems but one, the isotropical system S0 . In fact, for every small region AB of every such system it is possible to imagine a large rotating platform with center at rest in S0 and rim locally comoving with AB and the result (10.5) can be applied. Therefore the velocity of light depends on direction in all inertial systems with the sole exception of the privileged one S0 . 11. Linear accelerations: e1 = 0 The hypothetical indifference of physical reality with respect to clock synchronization exists only insofar as one neglects accelerations. In fact, when a body is accelerating, one can consider it at rest in different inertial systems during infinitesimally small time intervals, and it is therefore impossible to adopt in those systems a procedure, such as Einstein’s, requiring a finite time to synchronize clocks placed in different points. Nevertheless physical events take place and synchronization must somehow be fixed by nature itself. Essentially, this is what we saw in the two previous sections with rotating disks. We will now see how this happens with linear accelerations and we will discover that also in this case nature, left alone, gives rise to the so called absolute synchronization [77]. With our notation this corresponds to the choice e1 = 0 . Two identical spaceships A and B are initially at rest on the x 0 axis of the (privileged) inertial system S0 at a distance d0 from one another. Their clocks are synchronous with those of S0 . At time t 0 = 0 they start accelerating in the +x0 direction, and they do so in the same identical way, in such a way as to have the same velocity v (t0 ) at every time t 0 of S0 , until, at a common time t 0 = t 0 of S0 , they reach a preassigned velocity v = v (t 0 ) parallel to +x0 ; For all t 0 ≥ t 0 the spaceships remain at rest in a different inertial system S , which they concretely constitute, moving with velocity v . With respect to S0 the positions of A and B at any time t 0 ≥ t 0 are given by:

x0A t0( )= x

0A0 + d ′ t 0

0

t 0∫ v ′ t 0( )+ t0 − t 0( )v

x0B t0( )= x

0B0 + d ′ t 0

0

t 0∫ v ′ t 0( )+ t0 − t 0( )v

(11.1)

so that

x 0B t0( )− x0A t0( ) = x 0B 0( )− x0A 0( ) = d0 (11.2)

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Eq. (11.2) implies that the motion of A and B does not modify the distance d0 between the spaceships as seen from S0 . The same distance seen from S (call it d ) instead increases during acceleration, as the unit-rod measuring it undergoes a contraction. One has:

d =

d0

1 − v 2 / c2 (11.3)

In fact the observer in S0 will check: (i) That the distance d0 between A and B remains the same after the acceleration, as shown by (11.2); (ii) That the unit rod in S is Lorentz contracted if compared with a similar rod at rest in S0 ; (iii) That the observer in S using his shortened rod to measure the A -B distance finds a value larger by a factor 1 / 1− v 2 / c2 than found before departure when the spaceships were at rest in S0 . This measurement is an objective procedure and its result (= number of times the rod of S fits into the A -B distance) cannot depend on the subjective point of view. Therefore the observer in S finds indeed what the observer in S0 sees him finding, namely a distance between A and B larger by a factor

1 / 1− v 2 / c2 , as given by (11.3). It will next be shown that the transformation relating S0 and S is necessarily the inertial one, if no final clock re-synchronization is applied correcting what nature itself generated during the acceleration of the two spaceships. Since A and B accelerate exactly in the same way, their clocks will accumulate exactly the same delay with respect to those at rest in S0 . Motion is the same for A and B and all effects of motion will necessarily coincide, in particular time delay. Therefore two events simultaneous in S0 will be such also in S , even if they take place in different points of space. Clearly we have a case of absolute simultaneity and the condition e1 = 0 must hold in (7.2), reducing these transformations to their inertial form (7.5). In order to make the point as clear as possible we check next that the velocity in S of a light pulse traveling from A to B when the two spaceships are at rest in S (while, of course, they move with velocity v with respect to S0 ) is consistent with the inertial velocity of light formula (7.6). Let a light signal leave A at time t A and reach B at time t B , both times being measured in S . Its velocity ˜ c in S is by definition

˜ c = d

t B − t A (11.4)

257

where d is given by (11.3). Now define:

t 0A : time of emission of light signal from A as seen in S0;t 0B: time of its arrival at B as seen in S0 ( t0B > t 0A );τ: delay generated by velocity up to time t 0 at which

acceleration stops (t0B > t0 A > t 0 ).

Since for all t 0 ≥ t 0 time dilation in S is due to the constant velocity v, one has:

tA = t 0 − τ + (t0A − t 0) 1− v2 /c2

tB = t 0 − τ + (t0B − t 0) 1− v2 /c2 (11.5)

By subtracting the first equation from the second, one gets

t B − tA = t0B − t0A( ) 1 − v 2 / c2 (11.6) Eq. (11.6) is the clock retardation formula for the travel time of the light pulse. The point ˜ x 0B of S0 where the light is absorbed by B must satisfy:

˜ x 0B = x0A + c(t0B − t0A )

˜ x 0B = x0B + v(t0B − t0A ) (11.7)

if x 0A and x 0B are the positions of the spaceships A and B respectively at time t 0A of emission of the light signal. In fact, while the light signal goes from x 0A to ˜ x 0B with velocity c, spaceship B moves from x 0B to ˜ x 0B with velocity v. From (11.7) it follows:

x0B − x0A = c(1− v c)(t0B − t0A ) (11.8) Remembering that (11.2) holds for all times t 0 and using (11.3) and (11.6), Eq. (11.8) gives

tB − tA

d = 1+ v c

c (11.9)

258

which compared with (11.4) gives finally:

˜ c = c

1+ v c (11.10)

Therefore the velocity of light in S satisfies (7.6) with θ = 0 . This is what one expects from the inertial transformation since the straight line connecting the spaceships A and B has been assumed parallel to their velocity.

Before:

After:

B

B

A

A

Figure 13. Two identical spaceships A and B are initially ar rest on the x 0 axis of the inertial system S0 . After having accelerated in exactly the same way A and B are at rest in a different inertial system S which they concretely constitute. These considerations show with clarity the arrival point, a new theory in which the slowing down of clocks is no longer relative, but only dependent on velocity with respect to the privileged frame. The existence of a vacuum endowed with concrete physical properties becomes acceptable. Naturally, one needs also to verify the new theory experimentally, but in a certain sense this has already been done, at least in the case of the Sagnac effect [78]. Not only the absolute simultaneity is concretely realised in the moving frame of the two spaceships, but one can find other convincing arguments showing that it

259

gives the most natural description of the physical reality. We will suppose that our spaceships have passengers PA and PB , who are homozygous twins. Of course in principle nothing can stop them from re-synchronizing their clocks once they have finished accelerating and the two spaceships are at rest in S . If they do so, however, they find in general to have different biological ages at the same (re-synchronized) S time, even if they started the space trip at exactly the same S0 time and with the same velocity, as stipulated above. Everything is regular, instead, if they do not operate any asymmetrical modification of the time shown by their clocks. In fact we already concluded that clocks in A and B are retarded in the same way, and that the transformations S − S0 must be the inertial ones. Also the ageing of the twins must have been the same, since at every time before, during and after the acceleration they were in identical physical conditions. Therefore the twins have the same age when the times shown by their clocks are the same if they have been synchronised in S0 before departure and never modified after. Naturally PA and PB can inform one another of their biological ages (e.g., via telefax) by exchanging pictures in which the times they were taken is marked: the twin receiving a picture can check in his archives that at the time shown on his brother’s picture he had exactly the same look, and therefore the same age. Naturally the twins PA and PB can use a different synchronization of clocks, if they wish, e.g. Einstein's synchronization leading to the validity of the Lorentz transformations between S and S0 . To do so they must send a light signal, e.g. from A to B , and they must reset at least one clock. We can even suppose that every twin has two clocks and keeps the first one set on absolute time, while regulating the second to show the Lorentz time. More exactly we assume that:

PA has a first clock TA marking the natural time tA

PA has a second clock ˆ T A marking Einstein' s time ˆ t APB has a first clock TB marking the natural time tB

PB has a second clock ˆ T B marking Einstein' s time ˆ t B

After a certain initial time interval during which t A = ˆ t A = tB = ˆ t B only ˆ T B is resynchronised in the following way. At a certain preestablished time a light signal is sent from A to B. The convention that the one-way velocity of light in S is c forces the observer in B to rotate the hands of his clock ˆ T B in such a way that the time necessary for the signal to cover the distance d from A to B be measured to be d / c. Clearly, after resynchronising, at a given time t 0 of S0 one will have t B = tA , but ˆ t B ≠ ˆ t A . The simultaneity of ˆ T A and ˆ T B is now different from that of

260

TA and TB! If PA and PB exchange pictures of themselvs in which also the times marked by the clocks T and ˆ T are shown, they discover having had the same age at the same time t, but different ages at the same time ˆ t . This provides a strong argument in favour of the inertial transformations, because not all natural “clocks” can be synchronised: irreversible processes exist such as the ageing of PA and PB ! Nature itself favours the inertial transformations from S0 to S for describing the time of inertial systems concretely produced. Bibliography: general [1] [LK, p. 21] [2] [LK, p. 104] [3] A. Einstein, Nature, 112, 253 (1913). [4] A. Einstein, Remarks to the Essays Appearing in this Collective Volume, in the book [PS]. [5] Letter to M. Solovine of 1938 [6] [SD, pp. 35-36] [7] [SD, p. 36] [8] [UQ, pp. 152-153] [9] [UQ, pp. 96-97] [10] [PS, pp. 683-684] [11] [HR , p. 96] [12] [SP, p. 35] [13] [DB, p. 7] [14] [EB, p. 116] [15] [EB, p. 149] [16] [EB, p. 82] [17] Letter to M. Solovine [18] [EF, p. xx] [19] M. Langevin, Comptes Rendus 173, 831 (1921); Ibid. 205, 304 (1937). [20] E. Post, Rev. Mod. Phys. 39, 475 (1967). [21] [LL, §§ 82-87] [22] H. Dingle, Nature, 179, 866 and 1242 (1957); H. Dingle, Introduction,

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AUTOBIOGRAPHY, Fontana/Collins, Glasgow (1978)

266

Photon-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

Abstract

Novel photon-like solutions of Maxwell’s equations in free-space are constructed where transverse fields, propagating at frequency ω with phase (group) velocities vp (vg), possess local helical rotations at a frequency Ω over the whole cross-section. These are referred to as distributed spin rotations. The frequencies Ω and ω are independent with the helical modulation propagating at vg, unlike single frequency classical solutions with helical phase fronts. These novel solutions are accessible only with vector formalisms although the axial fields satisfy the standard scalar wave-equation. The theory is outlined using the compact Riemann Silberstein formulation of Maxwell’s equations with a field vector F = E + icB. Light-cone coordinates facilitate a manifestly Lorentz invariant theory. Appropriately chosen distributed spin rotations provide a wide variety of Lorentz invariant packets that envelope the classical fields and contain energy that is proportional to the total helical rotation over the length of the packet. The requirement that both transverse and axial fields are enveloped together leads to quantisation of the rotational energy in integer units, N. Solutions with different N are orthogonal. Operators can be formed, which increase (decrease) the rate of helical rotation and hence increase (decrease) the energy, and behave as promotion and demotion operators of standard quantum theory supporting a view that these new solutions form a photon-analogue. The paper concludes with a review of single-photon experiments that are in keeping with this model. Appendices contain detailed mathematics, speculative material and theorise on quantum-like features of the photon-analogue with regard to interference, polarisation and entanglement.

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1. Introduction

In this book series on ether space-time and cosmology it is appropriate to acknowledge Einstein's concluding observations in his 1920 Leyden lecture that general relativity requires space to have physical qualities. This is Einstein’s ether which he saw as essential for the propagation of light and the existence of standards of space and time. Moreover, our knowledge of the universe arises mainly through studies of the electromagnetic spectrum: e.g. extragalactic background light [1]. Yet despite such reliance on optical phenomena, the photon itself continues to be an enigma [2-4]. On the one hand its wave description is beautifully and practically described by Maxwell’s equations [5-8]. On the other hand, links from wave equations to particle descriptions rely heavily on analogues between electromagnetic resonators and quantised harmonic oscillators [9-12]. A variety of approaches can create closer links between classical electromagnetic waves and quantum phenomena [13-19] ranging from Cook’s sophisticated photon dynamics [13] to the approximate paraxial approach of Nienhuis and Allen [19]. Significant research recognises similarities between Maxwell and Dirac equations [20-24], but similarities cannot imply equivalences between boson and fermion equations [25] and do not help to visualise a photon’s physical structure that might enrich our understanding of this particle.

A valuable mathematical tool for exploring quantum-classical connections is the compact Riemann-Silberstein (RS) formulation of Maxwell’s equations with the RS vector F = E + icB. The RS vector has been considered as a photon wave function [26, 27] furthering a classical-quantum correspondence for the photon [28], describing photon polarization dynamics [29, 30] and calculating angular momentum [31]. This present work also uses the RS formulation in a manifestly Lorentz invariant form.

Although some theories suggest that a photon may not be localised [32-34], there are other theories in favour of localisation [35-38]. Single photon interference [38], measurements of group and phase velocity [40-42] and experiments such as that of Hong and Mandel [43] support a practical view that photons can be localised. There is then a question of whether Maxwell’s classical equations allow fully confined packets of energy to be formed and propagate in a way that might provide helpful photon-analogues. Focussed Wave Modes (FWM) and Bessel beams [44-47] provide successful attempts at localisation of classical energy propagating along an axis. However, there are difficulties to be overcome to obtain a bounded energy when integrating over an unbounded cross sectional area [28, 45]. Because there is no experimental evidence that photons exist only in specific modal forms it is unattractive to have packets that require specific profiles such as Bessel, Gaussian or other.

The motivation here is directly related to this previous work and, in particular, expands on original speculative work reported at PIRT 2006 [48]. The aim is to show that Lorentz invariant packets of electromagnetic energy with photon-like properties

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can be formed directly from almost arbitrary classical solutions to Maxwell’s equations for transverse modes propagating along an axis (Oz). These packets are defined through counter-rotating helical modulations that propagate with the group velocity of the wave. Associated with these local helical rotations is an energy that is proportional to the rotation which is quantised because of a defined packet length. The associated energy is then also quantised. It is possible to construct operators that increase or decrease the rate of helical rotation by integer units. These operators mimic the promotion and demotion operators of quantum theory. This mainly theoretical chapter, formulating a photon-analogue from Maxwell’s equations, ends by examining old and new experimental evidence that is consistent with our analogue.

A useful starting point recognises that general electromagnetic modes have to be Transverse Electric (TE) or Transverse Magnetic (TM) modes or a mixture of these modes [7, 8, 49]. The TE and TM modes can be determined by their axial B-fields or axial E-fields respectively. The TEM (transverse electric and magnetic) mode it is claimed can be considered correctly only as a mixture of TE and TM modes and then taking a limit where the axial fields tend to zero. The fact that there are non-zero axial fields does not mean that this work is connected to the theories of Evans, Vigier or others who extend Maxwell’s equations through additional fields or currents [4, 50, 51]. Maxwell’s classic equations are used making full use of their vector properties and Lorentz invariance. For TE and TM modes, the group velocity is always less than c enabling phase and group velocity to be distinguished and allowing non-singular Lorentz transformations to a frame of reference moving with the group velocity.

Figure 1

In general waves are mixtures of TE (Ez = 0) and/or TM (Bz = 0) waves.

(vgroup < c)

Figure 1 sketches some essential TE/TM features. The transverse vector fields ET, cBT, and n (the unit vector in the direction of propagation) always form a right handed set with |ET|/|cBT| a constant over the whole cross section. These waves always have a non-zero transverse propagation constant of magnitude κ that gives a measure of the divergence/convergence/diffraction of the wave. While there can be a superposition of waves with different values of κ, here we consider a single value and explore the range of solutions that this allows.

ET

cBT

cBz / Ez n

Diverging Electromagnetic TE / TM Wave

O z

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The next point is to recognise that the TE/TM Maxwell’s vector wave-equations have a surprising set of recently recognised solutions [48,52] referred to here as Distributed Spin Rotation solutions (DSRs) because they are associated with local helical rotations of the transverse fields that are distributed over the whole cross section. Just as two frequencies can form a wave-packet enveloping the axial fields while travelling at the group velocity, counter-rotating DSRs are able to create an envelope of the transverse fields that also moves with the group velocity. These envelopes are structurally invariant under Lorentz transformations along the direction of propagation. The duration of the envelope is not measured by length or time, but by a Lorentz invariant phase with shortest phase ~ π. Another key property of DSRs is that their rotation or spin, distributed over the whole packet, integrates to give a Poynting-like theorem with a new measure of energy. When this energy is integrated over the packet’s total length it is found to be proportional to (2N+1)ω where ω is the classical frequency and N is a Lorentz invariant integer that quantises the helical rotations. These helical rotations, travelling with the wave’s group velocity, do not represent those classical modes with a single frequency and helical phase fronts, such as Laguerre-Gaussian (LG) modes [53, 54] or helical Focus Wave Modes [55]. Indeed such modes themselves could, in principle, be enveloped in these new wave-packets.

One difficulty, as for Focussed Wave Modes [45], is that energy integrals may diverge. DSRs are also able to create a lateral envelope of the transverse fields that ensures convergence. The proposed new packets then confine transversely, longitudinally and in a Lorentz invariant manner, arbitrary Maxwellian modes. The number of such packets in a closed system is then also Lorentz invariant [56].

The proposed new packets are extremely flexible with an ability to be extended or confined both axially and transversely. The packets have a definite group velocity and can be assigned a relative phase compared to other packets and have a definite phase velocity. It will be possible to envisage packets that exhibit a long or short coherence length as may be desired. Their lateral extent is also flexible enabling one to envisage how such packets might interact with compact sources or detectors.

Fourier synthesis with ideal plane waves allows virtually any classical Maxwellian mode to be formed. The virtue of starting with TE and TM modes is that one is now able to naturally and correctly take the limit, as the axial fields tend to zero, to find a TEM wave. This it is claimed gives the correct properties of an almost ideal plane wave along with its wave-packet formation and without the problems of infinite energy and lack of angular momentum over unbounded areas [57].

The structure of this chapter is as follows. Section 2 considers a Riemann-Silberstein representation of Maxwell’s equations where transverse fields are set apart from axial fields. A matrix representation permits all transverse components to be readily rotated using a rotation matrix. This is an important foundation. Section 3 then

270

uses ‘Dirac light-cone coordinates’ [58, 59] that restructure Maxwell’s equations into a manifestly Lorentz invariant form with three equations.

Section 4 introduces the important algebra of DSRs. Section 5 shows how to envelope axial fields in a Lorentz invariant way, using different frequencies while Section 6 shows how to envelope transverse fields with different DSRs. ‘Energy’ of the DSR fields is considered in section 7 and convergence of this energy integral is considered in section 8 where different DSRs provide ways of transversely confining the packets so as to ensure that energy integrals can be normalised.

Section 9 will consider the relationship of this present work to previous ideas about Packets of Retarded and Advanced Helically Modulated (PRAHM) modes reported at PIRT 2006. Concepts of retarded and advanced fields used previously are renamed ‘reference’ and ‘adjoint’ fields and are discussed throughout sections 4-9. These will be seen in appendices C and K to be essential to obtain photon-like behaviour with polarisation and entanglement. Section 10 considers promotion and demotion operator analogues for these wave-packets giving the standard rules of commutation.

Section 11 shows that there is considerable experimental evidence about photons that is in keeping with the features of the photon-analogue wave-packets proposed here. Section 12 summarises the available evidence that provides good reasons to consider these new wave-packets as providing the potential for a better understanding of the enigmatic and elusive photons.

Several appendices consider detailed mathematics and more speculative material such as an estimate of Planck’s constant, problems of ‘which path’ in interference experiments, uncertainty in polarisation and how even and odd symmetries might explain entanglement with this photon-analogue.

2. Matrix representation of Riemann-Silberstein Maxwell fields

This section considers an essential matrix representation of Maxwell’s equations applicable for waves propagating along an axis, Oz. An important axial rotation matrix ϕ is recognised, similar to work by Allen et al. [60] but does not lead to their massive matrices. This matrix representation allows for the RS formulation but, separating transverse elements from axial elements and keeping to free space, leads to a more straightforward analysis than the full matrix route followed by Khan [24]. TE and TM modes can be isolated even though they are handled simultaneously.

Because the fields are in free space, it is convenient to use E and cB as the electric and magnetic vector fields (normalised with c) with identical dimensions. The RS formulation defines a complex field F = (E + i c B) giving Maxwell’s equations as:

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curl F = i ∂ctF ; div F = 0 2.1

The physical significance of i in this work is considered in the context of the TE and TM modal structure of Maxwell’s equations. The TM mode is ‘driven’ by Ez while the TE mode is ‘driven’ by cBz. The two modal types are independent so that ( Ez + i cBz) can represent a complex field in an Argand diagram that gives a measure of the strengths of the independent TE and TM fields. The operator i, interchanges the fields so that i can be said to rotate TE modes into TM modes and vice-versa. Once it is accepted that Fz = (Ez + i cBz) is a useful way to describe the relative magnitudes of TM/TE fields then it follows naturally that one describes the total fields using the widely used RS formulation F = ( E + i c B).

All transverse fields FT are now written in a matrix vector form:

FT = ⎥⎦

⎤⎢⎣

⎡

y

xFF

= ⎥⎦

⎤⎢⎣

⎡++

yy

xx

cBFcBE

ii

2.2

The matrix transverse gradient vector is defined from:

∇T = ⎥⎦

⎤⎢⎣

⎡∂∂

y

x 2.3

where ∂x = ∂/∂x and similarly for y and z. The short-hand ∂ct = ∂/∂(ct) will also be used. A rotation about the Oz axis is defined by a matrix operator ϕ:

ϕ = ⎥⎦

⎤⎢⎣

⎡ −0110

; ϕϕ = −1 2.4

The vector notation n × FT is then replaced by ϕ FT so that the operator ϕ rotates transverse vectors by 90°. Given an exponential matrix operator:

Θ = exp(ϕ θ) = cos(θ) + ϕ sin(θ) 2.5

then Θ FT (and Θ∇T) causes FT (and ∇T) to be rotated through an angle θ (see end of appendix A). Notice that although Θ rotates FT (and ∇T), there is no rotation of coordinates (x, y) so that Fz is unaffected by Θ (see also equations C.23-24). Transposition of rows and columns is denoted by a raised bold solidus / to avoid confusion with ' which will denote a change of frame of reference. Conventional conjugation is denoted by *. For example FT* FT =FT

/ FT* is the scalar product of FT

and its conjugate FT*. From Appendix A, Maxwell’s divergence equations become:

∇T/ FT = − ∂zFz 2.6

Maxwell’s curl equations become:

272

∇T / (iϕ FT) = ∂ct Fz 2.7

∇TFz = ∂ct (iϕ FT) + ∂z FT 2.8

To derive the wave-equation, the first step is to use a standard method of separation of variables where a wave-number κ gives:

∇T/ ∇T Fz = (∂x

2 +∂y2) Fz = −κ2Fz

2.9

Physically, κ is the transverse propagation constant that gives a measure of the divergence, convergence or diffraction at right angles to the propagation along Oz. Given a wavelength λ and |κλ| ∼ 0 then the solution approximates to a plane wave. Now, οperate on equation 2.8 by ∇T

/ and use equations 2.7 and 2.6 to obtain:

∇T/ ∇T Fz = −κ2Fz

= [(∂ct)2 − ∂z

2 ]Fz 2.10

A typical solution used here with frequency ω and wave-vector k is:

Fz = Fzo exp[i(kz − ωt)] 2.11

(ω/c)2 − k2 = κ2 2.12

The phase and group velocity are connected independently of the value of κ:

(vp/c) = (ω/c)/k ; (vg/c) = (dω/dk)/c = (c/vp) 2.13

By selecting terms in the real and imaginary parts of Fz appropriately, it is possible to isolate the TM and TE modes, given from:

FTm = ETm + ic BTm = [∇T∂zEz − iϕ ∇T∂ct Ez]/(κ2) 2.14

FTe = ETe + ic BTe = i [∇T∂z cBz − iϕ ∇T∂ct cBz]/(κ2) 2.15

From the fact that ϕ rotates the fields by 90o it is possible to see that ET and cBT are always at right angles in each of TE and TM modes with ET and ϕ cBT parallel and proportional:

|ETe|/|c BTe| = |ϕ ∇T∂ct cBz| /| ∇T∂z cBz | = ω/kc > 1 2.16

|ETm|/|c BTm| = |∇T∂zEz| / |ϕ ∇T∂ct Ez | = kc/ω < 1 2.17

Circularly polarised waves require ET and cBT to be in phase quadrature but also to have exactly the same magnitude. Equations 2.16 and 2.17 therefore show that if κ is never precisely zero, as will be argued here, ideal circularly polarised waves have to be a mixture of TE and TM waves.

273

3. Light-cone coordinates and Maxwell’s equations

This section forms further essential background looking at the relativistic invariance of Maxwell’s equations using the RS formulation developed in section 2. It is believed that the role of relativity is most easily recognised by using light-cone coordinates [58, 59] defined here from:

(1/√2)(∂z −∂ct) = ∂τf ; (1/√2)(∂z +∂ct) = ∂τr 3.1

(1/√2)(z − ct) = τ f ; (1/√2)(z + ct) = τr 3.2

For variations as in equation 2.11, define light-cone propagation constants from:

kf = (1/√2)[k + (ω/c)] ; kr = (1/√2)[k − (ω/c)] 3.3

Waves moving with a phase velocity vp = ω/k vary as:

Fz = Fzo exp[i(kz − ωt)] = Fzo exp[i(kf τ f + kr τr )] 3.4

Envelopes moving with the group velocity vg = c2 k/ω vary as:

exp[i(ω/c)(z −vgt)] = expi[(ω/c) z − k ct)] = exp[i(kf τ f − krτr )] 3.5

With a primed frame moving at a velocity v (where v = c tanh α) , then light-cone coordinates transform in particularly simple ways:

τ f ' = τ f exp(α) ; τr' = τr exp(−α) 3.6a

kr ' = krexp(α) ; k f '= kf exp(−α) 3.6b

Any term with a relevant superscript is said to lie on the forward branch of the light-cone and to transform as τ f. Any term with a relevant subscript is said to lie on the reverse branch of the light-cone and to transform as τr. Products with relevant super- and sub-scripts are then automatically Lorentz frame invariant e.g.

∂τrτr = 1 = ∂τf τ f ; (kf τ f + kr τr ) = (kz − ωt) ;

(kf τ f − kr τr ) = (ω/c)(z − vgt) 3.7

The transverse gradient operator ∇T, and axial fields Fz remain, as usual, frame invariant for changes of frame moving along Oz. Using light-cone coordinates, Maxwell’s equations will next be written in a manifestly invariant form for such changes of frame moving along Oz. To further this task, ‘projection operators’, with roles that become clear later, are defined with their properties stated as:

f = ½(1 − iϕ) ; r = ½(1 + iϕ) 3.8

ff = f ; rr = r; fr = rf = 0 ; r + f = 1; r − f = iϕ 3.9

274

Appendix B shows that Maxwell’s equations 2.6 and 2.7 combine into equations 3.10 and 3.11 below while equation 2.8 is re-written as equation 3.12:

∇T/ ( r FT) = − ∂τf [(1/√2) Fz] 3.10

∇T/ ( f FT) = − ∂τr

[(1/√2) Fz] 3.11

∇T(1/√2)Fz = ∂τr (r FT) + ∂τf (f FT) 3.12

In the same way as the wave-equation was derived previously, operate on equation 3.12 with ∇T

/ :

∇T/ ∇T(1/√2)Fz = −κ2 (1/√2)Fz = ∂τr ∇T

/ (r FT) + ∂τf ∇T/ (f FT) 3.13

Eliminate ∇T/(r FT) and ∇T

/(f FT) using equations 3.10-11:

κ2 Fz = 2∂τr∂τf Fz 3.14

For variations as in equation 3.4 one requires:

κ2 = − 2 k f kr = −k2 + (ω/c)2 3.15

Equations 3.10-12 are able to show explicitly how Maxwell’s equations are invariant under axial Lorentz transformations with both sides of each equation transforming in the same way. In equations 3.10-11, ∇T and Fz are frame invariant and do not change with axial Lorentz transformations. The differentials ∂τf and ∂τr respectively transform as τr or τ f as do the projected vectors rFT and f FT; this latter feature is checked in a different way at the end of Appendix B. The projected fields rFT and f FT can then be said to lie on the reverse and forward branches of the light-cone respectively. This is consistent with equation 3.12 which exhibits frame invariance with invariant products on the right hand side and invariant terms on the left hand side. The solution for FT in terms of Fz is given from a mixture of fields on the forward and reverse branches of the light-cone:

FT = (r ∇T∂τf + f ∇T ∂τr ) (1/√2)Fz /(κ2) 3.16

Besides projecting the fields into components on the forward or reverse branches of the light-cone, the operators f and r reveal another important effect:

r FT = ⎥⎦

⎤⎢⎣

⎡+−

xy

yxiFFiFF

; f FT = ⎥⎦

⎤⎢⎣

⎡−+

xy

yxiFFiFF

3.17

(rFT)x = −i(rFT)y (f FT)x = +i(f FT)y 3.18

275

It can be seen that the x- and y- components of each of the projected fields are equal in magnitude and are also in ‘+’ or ‘−’ phase quadrature. This is precisely the requirement for what can be called ‘+’ or ‘−’ circular polarisation. Hence f FT represents ‘+’ circular polarisation while r FT represents ‘−’ circular polarisation demonstrating a clear link between circular polarisation and relativity. Equation 3.16 hides considerable detail about circular polarisation that is placed for reference in Appendix C. It might now be argued that f and r are circular polarisation projection operators. This is only partially correct. While f and r always project the forward and reverse components of the field along the light-cone, f and r project out the ‘+’ or ‘−’ circular polarisation respectively only if Fz varies as Fzp = Fzpo exp(ikz − iωt). Now there is nothing to prevent Fz varying as Fzq = Fzqo exp(−ikz + iωt). This change of the sign of i is simply changing the phase of the TE waves with respect to the TM waves and is not necessarily conjugating all complex fields. In this case, f and r project out the ‘−’ and ‘+’ circular polarisations respectively. Equation 3.19 then shows that the total field is the sum of the two opposing circular polarisations as well as being the sum of fields along the two branches of the light-cone:

FT =(f ∇T ∂τr Fzp + r ∇T∂τf Fzq) + (r ∇T∂τf Fzp + f ∇T ∂τr Fzq) /(√2κ2) 3.19

Finally in this section, it is of interest to evaluate light-cone wave-vectors in the frame travelling at the group velocity vg of the waves where (vg/c) = (ck/ω) = tanh(α):

k r ' = kr exp(α) ; kf '= kf exp(−α) 3.20

kf ' = (1/√2)(ω/c)[tanh(α) + 1]exp(−α) = (√2)(ω/c)/cosh(α) 3.21

kr ' =(1/√2)(ω/c)[tanh(α) − 1]exp(α) = −(√2)(ω/c)/cosh(α) 3.22

The two light-cone wave-vectors are then equal and opposite in the frame moving with the electromagnetic energy. This is true no matter how close the group velocity vg is to the velocity of light c so long as vg < c.

4. Distributed Spin Rotations of Maxwell’s equations

This section discusses in more detail the major topic of this work, showing how novel solutions of Maxwell’s vector equations arise that cannot be observed with a purely scalar field formulation and yet these solutions still lead to the classic scalar wave-equation. This solution has been referred to as a ‘spin rotation’ [52] but is referred to here as a Distributed Spin Rotation (DSR) to emphasise the fact that the rotations are distributed over the whole cross section. It must be noted that one is not rotating the frame of reference as discussed by other authors [61, 62]. Here the frame of reference is fixed but the DSR solutions consider the transverse fields to be rotating or spinning about every Oz axis that goes through each point (x, y). This unusual

276

solution may appear to mix the transverse fields inextricably but by superimposing a second ‘adjoint’ field with a counter-rotating DSR, order can be restored as discussed in section 6 (see Figure 3).

The concept of the DSR may be explained by starting with a local rotation through a given angle θ with a rotation operator given by (equations 2.5, A.12-13):

Θ = exp(ϕ θ) = cos(θ) + ϕ sin(θ) ; Θ/ Θ = 1 4.1

This rotation is applied to every transverse vector and transverse gradient operator about each Oz axis passing through every point (x, y), though the frame of reference defining (x, y) is not rotated hence Fz(x, y) is unchanged (see equations C.24-24). Equations 3.10-11 can then be re-written by substituting Θ∇T for ∇T and ΘFT for FT:

(Θ∇T) / ( f ΘFT) = − ∂τr [(1/√2) Fz] 4.2

(Θ∇T) / ( r ΘFT) = − ∂τf [(1/√2) Fz] 4.3

Because (Θ∇T) / = ∇T

/ Θ/ and Θ/ Θ = 1 equations 4.2 and 4.3 are always true

even if θ varies with space and time. It is also clear that the presence of Θ does not alter the value of Fz in these equations. Now spin-rotate, by Θ, equation 3.12 giving:

Θ∇T(1/√2)Fz = Θ ∂τr(r FT) + Θ ∂τf (f FT) 4.4

= ∂τr (r ΘFT) + ∂τf (f ΘFT) − R 4.5

where

R = (∂τr Θ) (r FT ) + (∂τf Θ) (f FT ) 4.6

Now follow the previous method for finding the scalar wave-equation from equation 3.12 but with the requirement that ∇T, FT are replaced with Θ∇T, ΘFT. First note that operating on equation 4.4 with (Θ∇T)/

:

(Θ∇T) Θ∇T(1/√2)Fz = ∇T/ Θ/ Θ ∇T(1/√2)Fz = ∇T

/ ∇T(1/√2)Fz

= (Θ∇T) / [Θ∂τr(r FT) + Θ ∂τf (f FT)] = ∂τr ∇T/ (r FT) + ∂τf ∇T

/ (f FT) 4.7

Indeed, just because Θ/ Θ = 1 for all rotations, equation 4.7 is trivially the same equation as equation 3.13 which leads to the wave-equation 3.14-15. However, if the premise of replacing ∇T, FT with Θ∇T, ΘFT is correct then one should be able to write the second line of equation 4.7 as:

(Θ∇T) / [∂τr (rΘ FT) + ∂τf (f ΘFT)] = ∂τr ∇T

/ (r FT) + ∂τf ∇T/ (f FT) 4.8

Comparing equations 4.4, 4.5 and 4.7, one can see that equation 4.8 is valid, provided:

277

(Θ∇T) /R = 0 4.9

Provided also that Θ/(∂τf Θ) and Θ/(∂τr Θ) are scalar numbers then, taking R from

equation 4.6, one can re-arrange equation 4.9 to give:

Θ/(∂τr Θ) ∇T/ (r FT ) + Θ/(∂τf

Θ) ∇T/ (f FT )

= − Θ/(∂τr Θ) ∂τf [(1/√2) Fz] − Θ /(∂τf Θ) ∂τr

[(1/√2) Fz] = 0 4.10

Equation 4.10 shows, from equation 3.4 with Fz = Fzo exp[i(kf τ f + kr τr )] , that any distributed spin rotation Θ associated with the fields FT, Fz requires:

Θ/(∂τr Θ) kf + Θ/(∂τf Θ) kr = 0 4.11 Θ = exp[W ϕ(kf τ f − kr τr)] = exp[ϕ W (ω/c)(z −vgt)] 4.12

Notice how, in the light-cone formulation, P(kf τ f + krτ r ) represents a function P moving at the phase velocity of the wave while Q(kf τ f − krτ r ) represents a function Q moving at the group velocity: both (kf τ f +/− krτ r ) are Lorentz invariants. Consistent with references 47 and 51, the distributed spin rotation Θ imparts a helical motion travelling with the group velocity of the underlying classical waves. To maintain Lorentz invariance, W must be a Lorentz invariant number but is otherwise arbitrary. The frequency of helical rotation Ω is then given by Ω= W(ω vg/c).

In summary, the DSR concept is that for every classical Maxwellian vector field represented by ∇TFz , FT there is a DSR solution given from Θ∇TFz , ΘFT . Here Θ is an arbitrary distributed spin rotation that can be a constant rotation or, from equation 4.10, can give a helical motion travelling with the group velocity of the underlying fields ∇TFz , FT.

Although these additional solutions are available within the vector formalism of Maxwell’s equations, they may not be used in exactly the same way as standard Maxwellian solutions because a DSR solution cannot satisfy the rule of superposition. For example, given Θm∇T

Fz , ΘmFT and Θn∇T Fz , ΘnFT then these two solutions

cannot be superimposed to find a rotation Θo where one might try to demonstrate:

Θm∇T Fz , ΘmFT + Θn∇T

Fz , ΘnFT = Θo∇T Fz , ΘoFT 4.13

The incorrectness of equation 4.13 means that if power or energy is to be discussed in a meaningful way then some averaging over whole periods of helical rotation (denoted below by ⟨ ⟩ ) gives:

⟨ Θm/ Θn⟩ = 0 4.14

In other words, distinct DSR solutions of the same classical mode can be regarded as orthogonal, analogous to quantum theory where eigen solutions are orthogonal.

278

Figure 2 below, indicates the difference between a standard rotation and a DSR. In a DSR, the transverse field vectors and gradient operators are rotating but the axes are fixed. A further distinctive feature is that a DSR is a helical rotation with angular frequency Ω moving with the group velocity. At present, Ω has no necessary link with the angular frequency ω of the wave: i.e. W in equation 4.12 is arbitrary – though a link between Ω and ω will be revisited later.

Standardrotations

Distributedspin rotations

Ox

EcB

1

2

1

2

3

4

3

4

OyOx

Figure 2 Standard Rotations and Distributed Spin Rotations Standard rotations rotate the axes and the pattern as a whole; distributed spin rotations take every local vector at x,y and rotate that vector about the axis through x,y.

5. Axial-field envelopes

This section investigates a Lorentz invariant wave-packet that envelopes the axial field Fz and considers if one can simultaneously envelope the transverse fields FT. To this end, solutions for equations 3.10-12 are now chosen allowing Fz to vary in a Lorentz invariant manner using light-cone coordinates:

Fz+ = Fzo expi[kf exp(δ) τ f + k rexp(−δ)τr ] 5.1

The wave-vectors appear as if they have changed to a frame of reference moving with a velocity (v/c) = tanh(δ) , although the z-t coordinates remain unaltered. The Lorentz invariant number δ (typically |δ|<< 1) will be called the phase length parameter. For any value of δ one still has kf and kr defining frequencies ω and wave-vectors k as in equations 3.3-4 with equation 3.14 still requiring that:

2 kf exp(δ) kr exp(−δ) = 2 k r k f = −κ2 5.2

See also appendix D, in particular equations D.2 and D.14.

279

The axial field of equation 5.1 can be re-written symbolically as:

Fz+ = Fzo exp( i φ) exp( i Δ) 5.3

where Fzo is some function of (x, y) dependant on the classical mode and:

φ = (kfτ f + k rτ r ) cosh(δ) = (kd z − ωd t) 5.4

where kd = k cosh(δ) , ωd = ω cosh(δ) [k and ω as in equations 3.3-4] and:

Δ = cosh(δ)(kf τ f − krτ r ) tanh (δ) = [(ωd/c) z − kd ct] tanh(δ)

= (ωd/c)(z − vgt) tanh(δ) 5.5

Here vg is the group velocity at the angular frequency ωd that is observed inside the packet (equation 5.4). While φ is commonly known as the phase and is a Lorentz invariant, Δ can be called a group-phase and is similarly Lorentz invariant.

Now change the sign of δ and call this solution Fz− :

Fz− = Fzo exp( i φ) exp(− i Δ) 5.6

Consequently the superposition of the two axial fields Fz+ and Fz− gives:

Fztot = Fz+ + Fz− = 2Fzo exp( i φ) cos(Δ) 5.7

Here cos Δ defines a wave-packet propagating at the group velocity vg . The packet edges are determined from Fztot = 0 giving a minimum group-phase value:

Δ = +/−π/2 5.8

[(ωd /c)(z − vg t)]Δ = +/−π/[2 tanh(δ)] 5.9

Notice that the packet length is determined in terms of group-phase and phase-length parameter δ ensuring that the packet is Lorentz invariant.

This appears to define a wave-packet with a single frequency ωd and a finite duration thus defying the uncertainty principle. However inspection of the coefficient of t within equation 5.3 shows two frequencies: [ωd +/− (kd c) tanh(δ)] giving an uncertainty in angular frequency of Dωd ∼ +/−(kd c) tanh(δ). The temporal uncertainty is Dt ~ +/− π/[2 kd c tanh(δ)] giving a frequency-time uncertainty product of:

Dωd Dt ~ π/2. 5.10

Although a Lorentz invariant packet for Fz can be created by this method with Fz going to zero at the packet’s edges, the derivatives of Fz do not also go to zero at the edges. Consequently further investigation is required to see if and how FT can be enclosed within similar envelopes with the same net group-phase duration as that for Fz. The substantial algebra is relegated to appendix D and discussed in the next section.

280

6. Transverse-field envelopes

In the last section, it was shown that the selection of two pairs of axial fields Fz+ and Fz− , with corresponding light-cone wave-vectors kf exp(δ), krexp(−δ) and kf exp(−δ), krexp(δ), could beat together so as to create a Lorentz invariant wave-packet that trapped the net axial field Fz within a finite phase interval. Now a related analysis is needed to show how the transverse fields can be enveloped. This is accomplished by considering appropriately counter-rotating DSRs.

Consider for the moment a single field solution Fz , FT with light-cone wave-vectors kf , kr corresponding to a conventional wave-vector k and angular frequency ω (equation 3.3). We shall call this field a reference wave axial field Fz with a DSR transverse field Fdsr where:

Fdsr = Θ FT ; Θ = exp [W cosh(δ) ϕ (kf τ f − krτ r)] 6.1

where W is at present arbitrary and cosh(δ) has been inserted to make the notation easier later. Now consider an adjoint wave with axial and transverse fields FzA and FTA where although FzA = Fz and FTA= FT the adjoint field has a DSR transverse field, reversing the sense of rotation (i.e. reversing ϕ):

FdsrA = Θ−1FT 6.2

Looking carefully at equations 2.6-2.8 it can be seen that if ϕ is reversed but Fz and FT are to remain a solution of exactly the same equations, then c has to be reversed. [N.B. The normalising c in cB (cB having the same dimensions as E) never changes sign in this c reversal process]. One concludes that it is possible to have adjoint and reference fields appearing identical to an observer looking simply at amplitudes, phase and group velocity etc. FzA, FTA= Fz, FT. Nevertheless the associated light-cones for the two fields are interchanged. In other words, looking at equations 3.10-3.12, reversing both c and ϕ interchanges r and f and interchanges ∂τf and ∂τ r as can be seen from equation 3.1. This leaves the equations 3.10-3.12 unaltered overall. Reversing the sign of c is analogous to the concept of Cramer’s transactional analysis [63] where a wave function Ψ has a return handshake Ψ*. Here instead of i changing sign, it is ϕ that changes sign. The deeper physical significance of reversing c is discussed in more detail for example by Kotel’nikov [64]. This reversal will be discussed further in considering the relationship of the present work to previously published work about PRAHM modes [48]. For the present, the philosophical implications can be ignored in favour of concentrating on the mechanics of creating an envelope. Consider the fields [Fz + FzA] = 2 Fz with the total DSR transverse fields:

FTot = (Fdsr + FdsrA) = (Θ + Θ−1)FT = 2cos [W cosh(δ) (kfτ f − krτ r)] FT

= 2 cos W[(ωd/c) z − kd ct] FT 6.3

281

FTot

FTotmax

FTot

FTot = 0

FTot = 0−FTot

−FTotmax

−FTot

FTot = 0

1

2

3

4

5

6

7

8

9

ETcBT

Figure 3 Counter-rotating distributed spin rotations of linearly polarised fields remain linearly polarised fields. Solid (dotted) arrows illustrate ET (cBT) field. Phase increases going from picture 1 to 9 (= 1 and repeat). Note that periodically (at 1, 5, 9) the total transverse fields go to zero: these are the phases that define the envelope.

Figure 3 illustrates how the counter-rotating fields of equation 6.3 produce a modulated field where linearly polarised fields FTot remain linearly polarised. Circularly polarised projections such as f FT

or rFT also remain circularly polarised with value 2f FT

or 2rFT at the peak value of the envelope. Notice though, the total fields go to zero periodically along the envelope ‘length’ because of the beating of the counter-rotating helical fields. These zeros are the reason for being able to envelope the transverse fields along the Oz axis and define appropriate envelope phase lengths.

Of course for a satisfactory envelope propagating at the group velocity, it is necessary that each of FT+ and FT− (associated with Fz+ and Fz−) as well as the net Fztot

= Fz+ + Fz− all have envelopes going to zero at the same ‘group-phase points’. However, because FT+ and FT− have different values for their light-cone wave-vectors [ i.e. kf exp(δ), krexp(−δ) and kf exp(−δ), krexp(+δ)] it might be thought that there cannot be just one value of Θ that is a DSR for both. Fortunately, as shown in Appendix D, the particular light-cone wave-vectors that have been chosen for FT+ and FT− allow the same DSR where, choosing W in equation 6.3 appropriately, one may write: Θ = exp[ M sinh(δ) ϕ(kf τ f − krτ r )] (equation D.21) with M arbitrary, regardless of whether δ is positive or negative.

282

∝ cos[(ωd/c)(z – vg t ) tanh(δ)] Fzo

π

∝ cos[(2N+1) (ωd /c)(z − vg t ) tanh(δ) ] FT

FT enveloped by counter-rotating fields

Fz enveloped by two frequencies

Envelopes may now be formed as follows. Take fields Fz+, FT+ andFz-, FT-, and add the ‘adjoint’ fields FzA+, FTA+FzA-, FTA- where both ϕ and c have been reversed and consider a DSR given by Θ for the reference fields and a DSR given by Θ−1

for the adjoint fields. The net result, following equation 6.3, of all the combined fields is shown schematically in Figure 4 and summarised in equations 6.4-6 as:

F+dsr +F+dsrA = 2 cos[M cosh(δ)] (kf τ f − krτ r) tanh(δ) FT+

= 2 cos M [(ωd/c) z − kd ct] tanh(δ) FT+ 6.4

F−dsr +F−dsrA = 2 cos M [(ωd/c) z − kd ct] tanh(δ) FT− ; 6.5

Compare equations 6.4 and 6.5 with the envelope for the combined reference and adjoint Fz:

Fz+ + FzA+ + Fz− + FzA − =

4Fzo exp[ i (kf τ f + krτr) cosh(δ) ] cos [ cosh(δ) (kfτ f − krτ r) tanh (δ)]

= 4Fzo exp[ i (kd z− ωd t)] cos [ (ωd /c) z − kd ct] 6.6

The total FT and the total Fz (equations 6.4-6) can be seen to become zero at the edges of a minimum group-phase length packet (equation 5.8 rewritten for convenience):

[(ωd/c) z − kd ct] tanh (δ)Δ = +/− π/2 6.7

provided that:

cos (M π/2) = 0 6.8

or equivalently with an integer N:

M = (2N+1) 6.9

Figure 4 Wave-Packets

Schematic for envelopes confining Fz and FT. The different mechanisms for producing each wave-packet allows ‘FT’ packets to have multiple periods to a single period of ‘Fz’ packets

283

The fact that there are different mechanisms for producing the envelopes to contain the axial fields and the transverse fields along the Oz axis, allows the transverse field mechanisms to have a multiple period of the axial envelope. Figure 4 sketches a concept of compatible phase lengths so that transverse and axial fields vanish at the same group-phase points, confining all the electromagnetic energy. The compatible group-phase lengths lead to ‘quantisation’ of the helical DSR frequencies.

The result of compatible group-phase lengths for the envelope of Fz and FT is still required even as Fz → 0, as will be assumed in part of section 7. It appears that the ideal plane wave does not really exist because there is always an effect of the axial envelope even for otherwise negligibly small values of Fz. Practically of course there is always some diffraction (convergence/divergence) when one interacts with a real environment and this can be said to mean that Fz is never actually zero. In summary, a circularly polarised plane wave has to be taken as a limiting case of an appropriate mix of TE+TM modes within the RS formulation as Fz → 0 but Fz ≠ 0.

Having introduced adjoint waves in this section, it will be helpful to end with a short discussion about ‘reference’ and ‘adjoint’ waves. The reference waves are the standard causal solutions of Maxwell’s equations. Although the key difference for ‘adjoint’ waves is that both c and ϕ are reversed, they also satisfy the same Maxwellian equations and are able to match the reference waves in phase, frequency, velocity, field profile and boundary conditions. However the c-reversal means that such adjoint waves cannot be causal and can only participate or be partners in interactions with the causal reference waves. It is imagined that there is a sea of adjoint waves that can interact with any and every reference wave provided that phases, rotations etc. match appropriately. This is discussed again in section 9.

7. ‘Energy’ associated with DSR solutions

The classical electromagnetic energy density for fields varying as exp[i(kz − ωt)] is given from:

½εο F*.F = ½εο ET*/ET + cBT*/cBT +Ez*Ez+cBz*cBz. 7.1

A classical Poynting theorem is obtained from the RS formulation (appendix F) to give in our notation with ⟨⟨ ⟩⟩ implying integration over the whole cross section:

P = ½ εο⟨⟨ ∂ct (FT*/ FT + Fz *Fz) + ∂z (FT*/ iϕ FT) ⟩⟩ = 0 7.2

It is helpful to also understand this notation in terms of conventional fields:

P = ½εο ⟨⟨ ∂ct (ET*. ET + cBT*. cBT + Ez *Ez + c Bz *cBz)

− ∂z (ET* ×cBT + ET ×cBT*) ⟩⟩ 7.3

284

With the matrix notation, it is important to recognise that (FT*/ iϕ FT) is proportional to the Poynting vector giving the power density flowing along the Oz axis while (FT*/

FT) is proportional to the transverse energy density. Later, power and energy contributed by Fz will be considered negligible as the plane wave limit is approached.

Because the product exp(−iωt)* exp(−iωt) is time-independent, it can be seen how conjugation ensures that energy integrals do not average to zero over time. From equations 2.6-8 it can be seen that changing the sign of i also requires that the sign of c changes if the conjugate solutions are still to satisfy exactly the same Maxwellian equations. As previously remarked, reversing the sign of c appears analogous to the concept by Cramer [62] where a wave function Ψ has a return handshake Ψ*.

Now DSR solutions slightly change this perspective on conjugation. Consider a DSR transverse field and its adjoint DSR field that is rotating in the opposite direction as in equations 6.1-3:

Fdsr = expϕ W[(ωd/c) z − kd ct ] FTo exp[i(kd z − ωd t)] 7.4

FdsrA = exp−ϕ W[(ωd/c) z − kd ct ] FToA exp[i(kd z − ωd t)] 7.5

Although adjoint fields FTA will always, in this work, be equal to FT it is helpful to keep track of where the reference field contributes and where the adjoint field contributes by adding a subscript A for the adjoint field. In equation 7.5 the adjoint field FzA satisfies exactly the same equation as the reference field Fz only if both ϕ and c have been reversed, which means that W has changed sign in equation 7.5. Note that i is not reversed in equation 7.5 nor in its associated FzA. It is now required to produce a z-t invariant product of the reference and adjoint fields inside the envelope (sections 5 and 6). This requires a new operation C to be defined in the following way:

FdsrAC / Fdsr

= exp−ϕ Ω[(ωd/c) z − kd ct ] FToA exp[i(kd z − ωd t)] C / expϕ Ω[(ωd/c) z − kd ct ] FTo exp[i(kd z − ωd t)] = FToA*

/ FTo 7.6

Analysis of the requirement for equation 7.6 shows that the operation C needs to reverse both i and ϕ . This is referred to here as ϕ−conjugation. If FdsrA is a DSR solution for Maxwell’s vector equations (albeit with c reversed), it can be seen from equations 2.6-8 that FdsrA

C is also a DSR solution of exactly the same equations. This ϕ−conjugate product FdsrA

C/ Fdsr in equation 7.6 then suggests that FdsrA and Fdsr can

interact wherever they overlap and this overlap of these two fields will be determined by (or equally will determine) the length of the wave-packet that they jointly create.

285

The changes that should be made to the Poynting analysis that led to equations 7.2 and 7.3 are considered in Appendix F. The result is two equations (F.11 and F.13) with some normalising quantity N that has yet to be discussed give ‘Poynting’ flows:

Pc = N ⟨⟨ (FTAC /

)(i∂ctFT) − FTAC /

(iϕ)(−i∂z FT) − i∂ct (FzA

C) Fz ⟩⟩ 7.7

PcA = N ⟨⟨ −i ∂ct (FTAC /) FT + (i∂z FTA

C /)(iϕ FT) + i FzAC∂ct (Fz)⟩⟩ 7.8

with both Pc and PcA being zero. These terms are evaluated first with both fields varying as exp[i(kz − ωt)] when both give the same result:

Pc = PcA = N ⟨⟨ ω [(FTAC /)(FT) +(FzA

C) Fz] − k FTAC / (iϕ)FT) ⟩⟩ 7.9

Comparing the term (FTAC /)(FT) in equation 7.9 and FT*/

FT in equation 7.2 and 7.3 one concludes that the DSR energy density is proportional to the conventional energy density proportional to the angular frequency ω. The final term gives the momentum flow for DSRs proportional to the wave vector k, consistent with special relativity.

Now consider DSRs where Θ = exp ϕ M tanh(δ) [(ωd /c)z − kd ct] for FT and Θ−1 for FTA respectively but now ignore the variations in exp[i(kd z − ωd t)]. Ignoring the axial fields as we approach the plane wave limit, equation 7.7 gives:

Pc = N M tanh(δ) ⟨⟨ − kd (FTAC /)(iϕ FT) + (ωd/c)FTA

C / FT) ⟩⟩ = 0 7.10

The constant values within ⟨⟨ ⟩⟩ are of course only over the length of the wave-packet where FT and FTA could be said to overlap for the phase interval [(ωd/c) z − kd ct]Δ = π/[sinh(δ)]. Denoting this extra integration by an additional bracket ⟨ ⟩ one finds:

Pc packet = (2πN /c) ⟨⟨⟨ −(½Mckd)(FTA C /)(iϕ FT) +(½M ωd)(FTA

C / FT) ⟩⟩⟩ 7.11

Comparing terms in equation 7.11 with similar terms in equations 7.2 and 7.3 along with their traditional interpretation, one concludes that (½Mωd)(FTA

C / FT) is a measure of the energy within the packet for an excitation where ½M= (N + ½) with N integer; equivalent to the quantum number in the Schrödinger representation. If this were to be accepted then the normalisation pararameter (2πN /c) becomes Planck’s constant with ⟨⟨⟨ (FTA

C / FT)⟩⟩⟩ normalised to unity. Section 7 will show that this last term is

indeed normalisable. Similarly, the term ⟨⟨⟨ (½Mkd c) (FTAC /)(iϕ FT)⟩⟩⟩ is the measure of

the momentum within the packet. Because Pc and PcA are both zero (equation E.13-15), Pc packet = 0. This is to be interpreted that the net power flow along the packet is equal to the stored energy.

It is of interest to note that if one starts with equation 7.8 (instead of 7.7) with the same DSRs then the equivalent of equation 7.11 is given by:

Pc packet = (2πN /c)⟨⟨⟨ +(½Mckd) (FTAC /)(iϕ FT) +(½M ωd)(FTA

C / FT) ⟩⟩⟩ 7.12

286

The difference of sign in equations 7.11 and 7.12 is taken to mean that FT interacting with FTA gives power flow in one direction but FTA interacting with FT gives the power flow in the opposite direction. This is consistent with the concept that there is a resonant process where power flows both ‘forward’ and ‘backward’ along the resonant system with the net power flow as the difference of these two. Using that concept, Appendix I gives a heuristic estimate of Planck’s constant ~ 8×10-34 Joule sec. As further support for the moving resonance interpretation, one notes that, on changing to a frame of reference moving with the group velocity, the light-cone wave-vectors are equal and opposite (equations 3.20-21). Fabry Perot resonators for example always have equal and opposite wave-vectors.

Of course the alleged existence of DSR solutions for Maxwell’s equations poses a problem as to why such spin-rotations of the transverse fields are not observed. It is argued here that phase and amplitude are fundamentally measured by power measurements. Engen and Hoer first noted this feature for microwaves [65, 66]. Walker and Carroll were amongst the first to use this multiport method for optical measurements [67] and the method is capable of giving a full characterization of optical fields [68]. Power does not fluctuate with the rotational frequency. It fluctuates only with the granularity caused by finite packets of energy. Therefore it is argued that these fundamental ways of measuring phase and amplitude can explain why the helical rotations would not be observed directly but have to be inferred by counting packets of energy.

There may also be a perceived problem as to how planar boundary conditions can be met with helically rotating fields. This problem is answered by considering Figure 3 with the counter-rotating fields. As can be seen from Figure 3, if FT is plane polarised then net fields Fdsr + FdsrA are also plane polarised and so can match planar boundary conditions.

8. Transverse field convergence

From section 7 it follows that, for a finite energy, it is essential to show how to create a wave-packet where the integral J = ⟨⟨ FTA

C / FT⟩⟩ is bounded even though the whole of transverse space is included. Obtaining a finite energy is a problem also encountered with wave-packets of Bessel beams [41, 42]. The problem arises because one can envisage Bessel beam wave-packets with a transverse propagation constant κ and with field amplitudes decaying as 1/(κR)½ at large enough radii R from the main Oz axis . The energy density then reduces only as 1/(κR) for large enough R so that on integrating over an area πR2 there is a prima facie problem of convergence. The packets proposed, at present, neatly envelope the electromagnetic fields along the Oz axis but suffer the same problem transversely as Bessel beams. This section shows how a new type of DSR solution achieves convergence for the electromagnetic energy.

287

Start as for section 7 with the two fields given in 7.4 and 7.5. Then using the projection operators f and r, the integral J can be split into contributions from the two branches of the light-cone:

J = ⟨⟨ FTaC /

FT⟩⟩ = ⟨⟨ (rFTaC)/ (

f FT)⟩⟩ + ⟨⟨ (f FTaC)/ (

r FT)⟩⟩ 8.1

Here use is made of the relationships f C = f ; r C = r ; f /= r ; r / = f . Appendix E gives more detail of how to use imaginary distributed spin rotations in the form:

Φ = exp[iϕ ½ξ2 x/ x ] [ x = (x, y)/ ; x/ x = x2+ y2] 8.2

where ξ is an arbitrary reciprocal distance; Φ/ = exp[−iϕ ½ξ2 x/ x ] ; Φ/Φ = ΦΦ/

= 1. The remarkable properties of the projection operators show that:

f Φ = f G [G = exp(−½ξ2 x/ x)] 8.3

r Φ/ = r G 8.4

The DSR operator Φ is applied to f FT and the operator Φ / is applied to r FT. The integral J then has its |fields|2 enveloped within a transverse Gaussian envelope G2:

J → ⟨⟨ (Φ/ rFTA

C)/ ( Φf FT)⟩⟩ + ⟨⟨ (Φ fFTA

C)/ ( Φ/ r FT)⟩⟩ →

⟨⟨ (G rFTAC)/ (

Gf FT)⟩⟩ + ⟨⟨ (G f FTaC)/ ( G r FT)⟩⟩ = ⟨⟨ FTA

C / G2 FT⟩⟩ 8.5

Now reconsider the example where the asymptotic field amplitudes, appropriate for Bessel functions where the transverse propagation constant is κ, diminish in amplitude as 1/(κR)½ for sufficiently large radii R (R2 = x2 + y2). From equation 3.16, seek some value Fb that is sufficiently large so that |FT | < Fb

/ (κR)½ then |FT|2 <

|Fb|2 /(κR). The singularity at R=0 integrates out with an elemental area (2π RdR) so that the integration can be bounded for any field pattern with a finite κ giving:

⟨⟨ GFTsaC / GFTs⟩⟩ ≤ ⟨⟨ G2 |FT|2⟩⟩ ≤ ∫

∞0

|Fb |2 [exp(−ξ2 R2)/(κR)] (2π RdR)

< |Fb |2 (2π/ξκ) 8.6

By choosing ξ, it can be seen that the lateral confinement of this photon-analogue is not fixed and in principle the lateral packet can be closely confined to the axis or spread out over a large area. Similarly, because of the flexibility of the phase length parameter δ in section 5, the longitudinal confinement of this photon-analogue is also very flexible. The photon-analogue packet is typically more than half a wavelength and may extend over many wavelengths if required.

As a final comment to this section notice that the integral J is a Lorentz invariant. This is seen immediately from equation 8.1. Here |FT| = |FTa| and one takes products for example of f FT and rFTa which lie on opposing branches of the light-cone

288

and just as τ f τr is invariant so (r FTaC) / ( f FT) and (f FTa

C) / ( r FT) are also invariant. The number of such packets then is expected to be a Lorentz invariant – as indeed would be desirable for any classical photon-analogue [56].

9. PRAHM modes

One of the authors (JEC) put forward the concept of counter-rotating propagating helical motions as additional solutions to Maxwell’s vector equations in a speculative paper given at PIRT 2006 [48]. Similar to the concepts in this present paper, one of these helical rotating wave- packets was based on a conventional causal wave (previously referred to as a retarded wave) and a counter-rotating helical wave where c was reversed (previously referred to as an advanced wave). This then led to a concept of Packets of Retarded and Advanced Helically Modulated modes (PRAHM modes): der Prahm is also German for a flat bottomed boat used for ferrying goods. PRAHM is then seen as a vehicle for carrying ‘energy’. Regrettably, insufficient attention was paid to how the limiting case of the ideal plane wave should be reached and the work failed to ensure that both transverse and axial fields had compatible envelopes moving at the same group velocity.

A feature recognised in the present work is that properties of ideal plane waves cannot be found simply by setting the transverse propagation constant to zero. One has to take the limit as the propagation-constant-wavelength-product tends to zero: |κλ| → 0. The present RS analysis has shown that TE or TM waves cannot by themselves be perfectly circularly polarised because the ratios of |ET|/|cBT| in either TE or TM modes are not unity. The plane wave considered in a PRAHM mode is therefore best considered as the limiting case of a mixture of TE and TM modes where the energy in the axial fields becomes negligible. Then as in the PRAHM mode, it is hypothesized that all the energy and momentum in the plane wave arises from the DSRs. This is the speculative proposition of this work.

The terms advanced and retarded related to the use made by Wheeler and Feynman of these terms in a classic discussion [69]. In reference 48 they were used essentially to mean a c reversal while keeping the phase velocity the same, similar to c reversal here. Unlike Wheeler and Feynman’s advanced and retarded fields or Jackson’s advanced and retarded potentials [5], the fields Fz, FT and FzA, FTA, appear identical to any observer in amplitude, group and phase-velocity even though they have an internal difference of c-reversal leading to an interchange of the forward and reverse branches of the light-cone. Consequently it is suggested that the retarded waves are now called reference waves Fz, FT and remain as the causal waves considered by all classical field calculations. The advanced waves are now called adjoint waves, FzA, FTA, and have both c and ϕ reversed. Although adjoint waves have light-cone branches that are interchanged with respect to the reference fields,

289

they still satisfy exactly the same Maxwell equations in free space and so to a classical observer appear identical to the reference fields. In this work then, PRAHM modes will now mean Packets of Reference and Adjoint Helically Modulated modes.

Quantisation of the helical rotations now occurs through matching the net phase length of the axial envelope and transverse envelope with both envelopes moving together with the group velocity and representing a Lorentz invariant packet trapping all fields. This is always held to be required even as the energy in the axial fields appears to become negligible. A new feature of the wave-packets in the present work is that lateral or transverse confinement is also maintained by appropriate DSRs (section 8) so that fields can always be normalised. The considerations of confinement now show significant flexibility in both length and breadth for what might constitute a Maxwellian analogue of a photon; a flexibility that is in keeping with the difficulty of assigning a precise length or breadth to a photon. It is still recognised, as in the earlier paper, that localised helical rotations add energy to the system (section 7).

Time

Distance

FUTURE

PASTReference waveground state

Adjoint waveground stateexcited helical

modulations forphoton- analogue

A key point in the PRAHM mode model is that there is always a ‘ground

state’ formed by the reference wave with c travelling ‘forward’ and an adjoint wave with c ‘travelling’ in the ‘reverse’ direction and with ϕ reversed. The value of Fz it is claimed is never completely zero because of diffraction. Fz and FzA then form the basis of the ground state. The reference and adjoint solutions are indistinguishable as far as field patterns, phase velocity and group velocity but will have their DSRs rotating in opposite directions. The change of the arrow direction on the dotted lines in Figure 5 should remind one that c and ϕ have been reversed. The single-photon analogue then is an excited state where the helical rotations defining a wave-packet have been increased by one unit relative to the helical rotation appearing in the ground state and consequently the single photon-analogue carries one unit of available energy over and above the ground state which has no available energy. The excited packet, travels with the group velocity of the underlying classical wave, sketched schematically in Figure 5.

Figure 5 Schematic formation of PRAHM mode The reference and adjoint waves, define a ground state path along which one may form packets of excited Reference and Adjoint Helically Modulated modes to create a ‘resonance’ propagating at the group velocity: a photon-analogue.

290

Because both the reference and adjoint fields satisfy the same equations and boundary conditions it is not possible to say from a conventional Maxwellian theory where this excited packet lies along the permitted path. One finds the reference and adjoint fields from a single classical calculation that fails to distinguish between these two fields. The photon-analogue can then lie anywhere along this path. This, it is claimed, is why it is so difficult to locate the photon in any classical theory, and why there are difficulties of knowing which way a packet travels at a beam splitter (see Appendix J).

By having possible ground state paths defined by adjoint waves it may be thought that this resurrects the concept of de Broglie waves or Bohm’s pilot wave that are ‘empty’ of photons [70], a theory that has been found wanting for experimental evidence [71]. The de Broglie waves [72] are not the same as the adjoint waves. We repeat a point made earlier at the end of section 6. The reversal of c in every adjoint wave implies that such a wave can only be an adjunct to action that is initiated by the causal reference wave: the adjoint wave can never of itself excite or initiate action without violating causality. Adjoint waves are envisaged as forming a background field that is essential if energy is to propagate in packets, but such a background field cannot initiate energy transfer, energy propagation or control the phase: these are functions of the reference wave only. The absence of an appropriate adjoint wave simply means that there can be no emission, spontaneous or other wise, because the packet initiated by the reference wave cannot be completed. However, because both waves are solutions of the same classical equations, absence of all reference waves (e.g. in a stop band of a filter) also means absence of adjoint waves and vice-versa.

The PRAHM model envisages that the two counter-rotating fields form a free space resonance. The development of the Poynting theorem considers spin rotations as the essential source of energy. Equations 7.11 and 7.12 suggest that this is a viable supposition where FT interacting with FTA gives power flow in one direction but FTA interacting with FT gives power flow in the opposite direction. It is also noted that, within the frame of reference moving with a wave-packet, the light-cone propagators kr and kf are equal and opposite – another indication of a resonance.

10. Promotion and Demotion

The analysis of promotion and demotion operations for the approximate paraxial Gaussian modes by Nienhuis and Allen [19] makes exciting classical and quantum connections. However to be totally satisfactory it would need to be independent of the field profiles, so that it could apply to any mode. In our analysis, promotion operators will increase the rate of the distributed spin while demotion operators will decrease the rate of the distributed spin. This statement is indeed independent of the transverse field profile.

291

For the sake of the discussion, accept that energy is created by the distributed spin rotations and consider plane electromagnetic waves as the limit where the RS axial fields tend to zero. Arbitrary electromagnetic fields are then considered, through Fourier analysis, to be a superposition of plane waves. For each plane wave of frequency ω, one forms a reference RS transverse field FT with its DSR giving a column matrix now written in a suggestive form where ΘΝ+½ gives the distributed spin:

ΨN = AN ΘΝ+½ Ψ0 10.1

Here AN is some complex amplitude where AN*AN = 1 while (ΘΝ+½)/ΘΝ+½ = 1 as usual. The adjoint field is identical to the reference field except that now the DSR rotates in the contrary direction and is given a subscript A. In section 7 it was seen in equations 7.7 and 7.8 that the energy associated with DSRs required calculation of terms of the form FTA

C / FT where ϕ−conjugation denoted by C changes the sign of both ϕ and i. Applying a similar calculation here:

ΨNA = AN Θ−Ν−½ Ψ0 10.2

ΨNAC / = Ψ0

C / (ΘΝ+½) / AN* 10.3

ΨNAC/ ΨN = Ψ0

C / Ψ0 10.4

Section 7 showed how the photon-analogue fields could always be normalized. Equations 10.2-4 allow a formal mechanism whereby the field amplitudes are then normalised to the value Ψ0

C / Ψ0 taken conveniently as unity. Because the energy is identified with both ω and the helical rotation integrated over the whole phase length ωτ of the wave-packet, one writes symbolically for as phase length ωτ:

ΘN = exp N ϕ ω τ ; 10.5

Averaging any non-zero multiple of exp(ϕ ω τ) over the phase length ωτ = π for the packet will give an average of zero so that there is a sense in which these DSR solutions are orthogonal ;

⟨⟨⟨ ΨNAC / ΘN Θ−MΨN

⟩⟩⟩ = 0 (M ≠ N , M & N integer) 10.6

Now it is permitted to generalise differential expressions of exponentials so as to include square roots of differentials [73]. Equation 10.7 below differentiates an exponential M times and it is then straightforward to allow M →½ :

(∂/ ϕ τ)Μ Θ = (N ω) Μ Θ → (∂/ ϕ τ)½ Θ = (N ω) ½ Θ 10.7

It is then possible to define two operators. The first operator, A+, increases the rate of helical rotation and a second, A− , decreases the rate of helical rotation. These operators can be defined as:

A+ = Θ½ [∂/ ∂(ϕ τ)½] Θ½ ; A− = Θ−½ [∂/(ϕ τ)½] Θ−½ 10.8

292

In particular taking the conjugate products: A+

C /A− = Θ½ (∂/∂ ϕ τ)½ Θ½ Θ−½ (∂/∂ ϕ τ)½ Θ−½

= Θ½ (∂/∂ ϕ τ) Θ−½ 10.9

A−C /A+ = Θ−½ (∂/ ∂ϕ τ)½ Θ−½ Θ+½ (∂/ ∂ϕ τ)½ Θ+½

= Θ−½ (∂/ ∂ϕ τ) Θ½ 10.10

Now examine the action of these operators on the fields ΨN = AN ΘΝ+½ Ψ0

ΨNAC / A+

C /A− ΨN = ΨNAC Nω ΨN Nω = Nω 10.11

ΨNAC / A−

C /A+ ΨN = ΨNAC (N +1)ω ΨN = (N +1)ω 10.12

Here one recognises that formal promotion and demotion operations can be formed where one achieves the conventional operational values after normalisation:

½ ΨNAC / (A−

C /A+ + A+C /A− ) ΨN = (N +½)ω 10.13

ΨNAC / (A−

C /A+ − A+C /A− ) ΨN = ω 10.14

A − ΨN = N½ ΨN-1 ; A+ ΨN = (N+1)½ ΨN+1 10.15

These results are the direct analogues of standard equations for demotion and promotion operators but now with a physical interpretation of increasing or decreasing the helical rotations in the DSRs. There is now no mystery as to the meaning of the Schrödinger frequencies (N +½)ω; these are not real frequencies but related to the net helical rotation over the length of the wave-packet and determine the number of DSR energy quanta within a single packet.

These quantum-like operators that demote or promote the helical rotation support the concept that the helical DSR rotations could indeed be linked with photons. The section also supports the concept proposed by others [21] that the RS formulation is a link between quantum mechanics and wave-mechanics.

11. Experimental Tests

A feature of the photon-analogue wave-packet is that it has a group velocity for the packet and the internal waves have a phase velocity. As a consequence it is expected that one should be able to measure a phase velocity and a group velocity for photons if such packets were indeed photon-analogues. Simultaneous measurements have been made of group and phase velocity [42] and also measurements have been made of the group velocity of photons in bulk glass and along fibres [40, 41]. The photon-analogue that has been described is in keeping with these features.

293

Experiments that examine the interference of a photon with itself or the interference between pairs of correlated photons clearly suggest that one can have photons that are highly localised [40,74,75] perhaps over a few microns distance. To offset such experimental results that support a highly localised photon, one need only look at diffraction gratings and how exactly the same diffraction pattern is given by building the pattern up for single photons from the wide classical beam. The beam may illuminate a few cms2 of grating and, because the single photon gives the same interference pattern, the implication is that the single photon must also cover the same area. Similar arguments may be put forward looking at modern versions of the Hanbury Brown-Twiss experiment [76] where second order interference from thermal light sources show correlations can extend over hundreds of microseconds suggesting that photons need not be well confined spatially [77]. Experiments therefore suggest that the granularity (quantisation) in photonic descriptions of microwave/optical fields take on a wide variety of forms. The DSR (PRAHM) model has such flexibility.

|a(t)|2

64 fs

time

13 ns

Periodic Optical Pulse

2.06 2.00 1.94 1.88

|A(f)|2

×1014 Hz

60 80 100 120 140 160Relative time of photon arrival (ns)

Dispersed Pulse

|b(t)|2

Spectrum

Frequency and arrival timecorrelated by the group velocity at the frequency f

a

b

c

The authors have been interested in the problems of characterising single

photons that emerge from short (< 100 femtoseconds) pulses into a dispersive fibre. The questions that were of interest were whether the photons were characterised by a temporal duration related to the pulse width or related to the pulse period (e.g. mode locking) or perhaps related to the photon’s coherence length. In short, what happens when photons pass through a dispersive fibre? Figure 6 shows schematically an experimental ‘short’ pulse and its associated ‘broad’ spectrum. The standard interpretation is that the power against time shown as |a(t)|2 in Figure 6a represents, schematically, the probability of a photon being emitted at a time t. We inferred the pulse width from auto-correlation measurements as the pulse is too short to measure conventionally. However, the key feature required was that the emission time of the

Figure 6 Expected relationship between pulse spectrum and photon arrival times.

294

photon was well defined compared with other time scales measured later. Figure 6b shows the measured classical spectral power |A( f ) | 2 which again is to be interpreted as the probability of a photon being emitted with a frequency f. Now if we were able to select a photon with a given frequency f then, putting aside the problem of locating the position of the photon in time, the energy in the photon and hence the photon itself would have to propagate with the group velocity vg( f ) appropriate to f. So if we have a single photon detector situated at some position L along a sufficiently dispersive fibre, that detector will be activated by single photons, characterised by a frequency f, arriving at different times [L/vg( f )] after their emission time. With a dispersion characteristic for the fibre giving an increase in the group velocity with increasing frequency, the higher photon frequencies should mean an earlier arrival time. Now because the initial pulse is so much shorter than the dispersed pulse we can assign an emission time to+/−δτ and neglect the less than 1 picosecond error δτ in relation to the spread of arrival times [L/vg(fmin)] − [L/vg(fmax)] ~ 80 ns. If one assumed the photon’s energy could be located, for example, to within a nanosecond, then Figure 6c shows how the photon emission at frequency f is expected to correlate with the arrival time. On the other hand, one might argue that because the photon was assigned a particular frequency f, then the uncertainty in its temporal position could be sufficiently large that there was little or no correlation between the spectrum and the arrival times in the dispersed pulse.

F

F

Figure 7 shows the experimental equipment schematically with the details placed in Appendix H. The optical gate A after the pulsed laser source allows the pulse rate to be a sub-multiple of the pulsed laser source (a mode-locked laser). The optical gate B in front of the photon detector isolates the time of arrival of a photon to the interval where the gate is open.

Optical path

Electrical path

Single Photons (average 0.1 photons

per pulse)

Pulse generator

Delay &width control

Display

Dispersive fibre

Pulsed Laser source

Optical Gate A

Single Photon Detector

Optical spectrumanalyzer

Optical Gate B

NeutralDensityFilter(attenuator)

Delay &width control

Figure 7 Simplified Experimental Set-up

295

With gate B ‘wide open’ (i.e. for a time longer than the dispersed pulse duration) there can be no restriction on the passage of any photons. With such a setting, Figure 8 shows good correlation between photon arrival times and the original spectrum; as discussed in Figure 6 b and c. Suppose that the wave-packet is located to within Δτ ~ 100 ps, then the photon-analogue model would have a spectral width of order Δf ~ 10 GHz. These uncertainties Δf and Δτ are unfortunately too small to detect when plotting the frequency ‘f’ associated with the packet and its arrival time ‘to + L/vg( f )’ where the approximate emission time is to. With the equipment available, it was not possible to make sufficiently accurate measurements of both the frequency and the arrival time so as to verify the uncertainty product predicted by equation 5.10.

60 80 100 120 1400

6000

0

3.01450 1500 1550 1600

phot

on c

ount

s

relative photon arrival time (ns)

Pow

er (μ

W)

spectral wavelength (nm)

photon countsspectrum

60 80 100 120 140 160

Relative time of arrival

Dispersed PulsePhoton count/ns

Gated measurements

Figure 9 shows an idealised result if the temporal width of gate B could be

reduced in length (~ 6 ns for the sketch here) and its open time delayed at regular intervals across the width of the dispersed pulse. In this experiment, one examines the photon count within the interval of the reduced width gate and provided that all the photons get through the gate, the photon count over this interval should take the same value as when the gate is ‘wide open’. If however the photon count is reduced, then

Figure 8 Correlation between relative photon arrival times and pulse spectrum as expected from the photon-analogue model.

Figure 9 Ideal gated measurements of arrival times, measuring photon lengths’

296

the some of the photons must have a ‘length’ that is longer than the gate opening time preventing these ‘long’ photons getting through gate B.

Figure 10 looks at an experimental example of gating the photons with 10 ns gates in comparison to gate B being ‘wide-open’. It might be thought that, with a period of 13 ns for the mode locked laser pulses, the photon length could be tied to this time scale, or perhaps tied to the longer time scale of the pulse period (determined by gate A) ~ 1.7 microseconds. Unfortunately the lack of a sharp enough switching time in our optical gates prevented us from secure measurements with gating time scales less than 10 ns. The particular photon detector that was used, when combined with its display, has a limiting resolution of 1 ns for single photon arrival times (see

60 80 100 120 1400

6000

Phot

on c

ount

s


10 ns gatewidths

photon counts (+5000)wide-open gate

Appendix H where more detail is given and additional sets of experimental results demonstrate the important attribute of repeatability).

This work can be compared to work by Brendel et al. who selected frequencies from a short pulse from a light emitting diode and then measured the time of flight of single photons in order to measure the dispersion properties of a fibre [78]. Our experiment differs from their experiment in two respects. First our optical pulse is much shorter and second we did not pre-select any frequency before sending the single photon into the fibre. Our measurement can also estimate the dispersion giving an averaged value (see Figure 8) over 1450-1600 nm of 17.9 ps/nm/km which is in accord with conventional measurements for the type of fibre that was used.

Figure 10 Gated photon arrival rates compared with arrival rates where the gate was longer than the dispersed pulse width. No significant reduction in photon arrival rates for 10 ns gates indicating photons are shorter than 10 ns.

297

12. Discussion

The DSR solutions are essential steps towards forming the packet referred to as a PRAHM mode (section 9). This work suggests that the PRAHM mode model has the potential to be a helpful analogue for a photon where the photon number is increased or decreased by increasing or decreasing the number of helical rotations over the length of the packet. Many points in favour of this model can be given.

(i) The photon-analogue or PRAHM model is in keeping with the experimental tests outlined in section 11 (and appendix H).

(ii) The flexibility of the photon-analogue model explains why it is so difficult to specify the photon’s length or breadth because a wide variety of envelopes all with the same energy are permitted when normalisation is taken into account (section 5, 6, 8).

(iii) The helical rotation rate and hence the associated energy can be increased or decreased by operators which are entirely in keeping with the quantum theory, obeying the standard commutation rules (section 10).

(iv) The photon-analogue wave-packet satisfies the uncertainty principle (section 5).

(v) The number of wave-packets in any closed system is a Lorentz invariant, as one would expect for the number of photons in a closed system where they were all trapped without interaction (end of section 8).

(vi) Appendix C explains how this model supports the concept that the polarisation of a photon is not necessarily fixed until it is measured. This is an important point for the physics of entanglement and a speculative explanation for entanglement using this photon-analogue model is given in appendix K.

(vii) A quasi-classical estimate of Planck’s constant gives the correct order of magnitude (appendix I). More work is clearly needed to explain how electrons and fields interact in this model.

At present a rigorous extension of this Lorentz invariant system into dispersive dielectric media looks as difficult a task as the rigorous extension of quantum optics into similar media [79]. However there may be some utility when considering an isotropic material with refractive index n in reforming Maxwell’s equations (as in equation 2.1) with c replaced by c' = c/n and B replaced by B' = B/n. The equations then allow a formal analysis that is similar to the present analysis except that it will not be fully Lorentz invariant. Further changes would be required to allow for weak material dispersion such as using the methods of Haus [80].

The concept of probability can also be considered within the context of the PRAHM model. The model suggests that the photon resides only where the helical excitations of reference waves and adjoint wave overlap. This overlap may occur

298

anywhere along the identical permitted paths of the reference and adjoint waves that are solutions of Maxwell’s equations. The density of the classical energy is then proportional to the average number of photons found in that region which in turn is proportional to the probability of finding an overlap of helically excited reference and adjoint fields: i.e. proportional to the square of the fields: F*.F.

At a beam splitter, both the reference and adjoint waves have permitted solutions that split either way. In order for a wave-packet to form with the reference wave there has to be an adjoint wave. Because the adjoint waves have c reversed there is knowledge in the reference wave of the future permitted path ahead of the current overlap position. When there are two (or more) equally permitted paths ahead, then which path is taken (Welcher Weg: Appendix J) is a random process that, a priori in a balanced geometry, gives equal probability for either path. One cannot know which way the PRAHM packet has actually travelled except by interacting with both the reference and adjoint waves and so destroying the interference. This seems to be entirely in line with clever delayed choice interference experiments such as devised by Hellmuth et al. [81]. It is also believed that the feature of two waves being required to define the position of the photon will be able to explain experiments on entangled photons such as Shih and Alley [82] (see also appendix K outlining a possible classical explanation of entanglement using reference and adjoint waves).

This new theory is believed to be the first model, based on the solid foundations of Maxwell’s equations that offers such a good potential for resolving so many of the dilemmas posed by wave-particle duality within a relativistic context afforded by Maxwell’s equations. It gives a more intuitive and visual account for a photon model by explaining the significance of the Schrödinger frequencies as normalised helical rotational rates. Although the helical rotations determine the energy, this energy is not modulated except by the granularity of the wave-packet. This explains why such Schrödinger frequencies are not observed directly. All this suggests that the theory may be more than just another classical skirmish into quantum territory.

13. Acknowledgements

The authors are indebted to the continued support of Ian White with many suggestions on what to do and also to their colleagues Jonathan Ingham and Jose Rosas-Fernandez for practical advice. One of the authors (JEC) is particularly indebted to Ken Sander who taught him some fifty years ago so many of the techniques used in this analysis.

299

Appendix A. Matrix form for RS Maxwell propagating along Oz

Writing F = [E + i cB], Maxwell’s equations 2.1 can be written:

∂yFz − ∂zFy = (i/c) ∂t (Fx) A.1

∂zFx − ∂xFz = (i/c) ∂t (Fy) A.2

∂xFy − ∂yFx = (i/c) ∂t (Fz) A.3

∂xFx + ∂yFy + ∂zFz = 0 A.4

For greater transparency over the operations, rearrange A.1 and A.2 as:

∂xFz − ∂zFx = −(i/c) ∂t (Fy) A.5

∂yFz − ∂zFy = (i/c) ∂t (Fx) A.6

This is recognised as:

∇TFz − ∂z FT = (1/c) ∂t iϕ FT A.7

This is equation 2.8. Rearrange A.3 and A.4:

−∂xiFy + ∂y iFx = (1/c) ∂t (Fz) A.8

∂xFx + ∂yFy = − ∂zFz A.9

These are recognised as:

∇T / ( iϕ FT) = (1/c) ∂t (Fz) A.10

∇T / FT = − ∂zFz A.11

These are equations 2.6 and 2.7.

To rotate FT = ⎥⎦

⎤⎢⎣

⎡

y

xFF

through an angle θ we need to arrive at:

FTθ = ⎥⎦

⎤⎢⎣

⎡+−

θθθθ

sincossincos

xy

yxFFFF

= [cos θ + ϕ sin θ] FT A.12

where ϕ is the matrix given in equation 2.4. Now because ϕ2 = −1 it is possible to expand exp(ϕθ) using a power series to arrive at:

Φθ = exp ϕθ = [cos θ + ϕ sin θ] ; FTθ = Φθ FT A.13

300

Appendix B. Relativistic form of Appendix A

With r − f = iϕ and r + f = 1, equations 2.6-8 are rewritten as:

∇T/ [( r + f ) FT] = − ∂zFz B.1

∇T/ [( r − f ) FT] = ∂ct Fz B.2

∇TFz = ∂ct [( r − f ) FT] + ∂z [( r + f ) FT] B.3

Use equation 3.1 where (1/√2)[∂z −∂ct] = ∂τf ; (1/√2)[∂z +∂ct] = ∂τr

Add equations B.1 and B.2 to find:

∇T/ ( r FT) = − ∂τf [(1/√2) Fz] B.4

Subtract equations B.2 from B.1 to find:

∇T/ ( f FT) = − ∂τr [(1/√2) Fz] B.5

Re-arrange equation B.3:

∇T(1/√2)Fz = ∂τr (r FT) + ∂τf (f FT) B.6

B.4-6 becomes 3.10-12 in the main text.

The solution for FT in terms of Fz is found from B.4 and B.5 noting that r f =0 so that by substitution one can see that:

FT = r∇T∂τf [(1/√2) Fz] + f ∇T∂τr [(1/√2) Fz] /κ2

B.7

where ∇T/ ∇T Fz = − κ2 Fz B.8

The last section of this appendix looks at an independent derivation that r FT and f FT transform in the same way as τr and τf respectively. First evaluate:

r FT = ⎥⎦

⎤⎢⎣

⎡+−

xy

yx

FFFFii

; f FT = ⎥⎦

⎤⎢⎣

⎡−+

xy

yx

FFFFii

B.9

noting that:

(r FT)x = −i(r FT)y (f FT)x = +i(f FT)y B.10

Consequently we need consider only the ‘x’ term in each of r FT and f FT. A standard Lorentz transformation gives relationships for transverse fields p.79 of [6] as:

301

⎥⎦

⎤⎢⎣

⎡−+⎥

⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡−−⎥

⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡

x

y

y

x

y

x

x

y

y

x

y

x

cBcB

EE

EE

EE

cBcB

cBcB

)sinh()cosh(''

;)sinh()cosh(''

αααα

B.11

Taking the appropriate terms from B.11:

(Fx− iFy)' = (Ex + cBy)' − i (Ey − cBx)'

= (Ex + cBy)[ cosh(α) − sinh(α)] − i(Ey− cBx) [cosh(α) − sinh(α)]

= (Fx− i Fy) exp(−α) B.12

(Fx + iFy)' = (Ex − cBy)' + i (Ey + cBx)'

= (Ex − cBy)[cosh α + sinh(α)] + i(Ey+ cBx) [cosh(α) + sinh(α)

= (Fx + i Fy) exp(+α) B.13

Hence r FT and f FT do indeed transform in the same way as τr and τ f .

Appendix C. Polarisation

Equation 3.19, repeated below as C.1, relates transverse and axial fields:

FT =(r ∇T∂τf Fzp + f ∇T ∂τr Fzq) + (f ∇T ∂τr

Fzp + r ∇T∂τf Fzq) /(√2κ2) C.1

where:

Fzp = Fzpo exp[+ i(k z − ω t)] or as Fzq = Fzqo exp[− i(k z − ω t)]. C.2

As pointed out in equations 3.17 and 3.18:

(rFT)x = −i(rFT)y (f FT)x = +i(f FT)y C.3

The sign of i in equation C.3 is controlled by the decision to define F = E + i c B. Equation C.2 is independent of this decision made in C.3. Polarisation physics can be formalised as follows. Left-handed circular polarisation is defined [6, p.12] if:

Ex ∝ cos(ω t − k z ) = cos(k z − ω t) = real part exp[+ i(k z − ω t)]= real part exp[− i(k z − ω t)] C.4

Ey ∝ − sin(ω t − k z) = sin(k z − ω t) = real part−i exp[+ i(k z − ω t)]= real part+i exp[+ i(k z − ω t)] C.5

It follows from the above definitions and equation C.3 that r ∇T Fzp and f ∇T Fzq project out left handed circular polarisations (denoted previously as ‘−’ circular polarisation).

302

Similarly f ∇T Fzp and r ∇T Fzq project out right-handed circular polarisations (denoted previously as ‘+’ circular polarisation).

The magnitudes of the contributions to each polarisation are determined from:

∂τf Fzp → i (1/√2)[k + (ω/c)] Fzp ; ∂τr Fzp → i (1/√2)[k − (ω/c)] Fzp C.6

∂τf Fzq → −i (1/√2)[k + (ω/c)] Fzp ; ∂τr Fzq → −i (1/√2)[k − (ω/c)] Fzp C.7

Consequently as k → (ω/c), the ideal plane wave value, the two polarisations tend towards the contributions:

FT → r i (ω/c)∇T√2Fz p (LH polarisation) − r i (ω/c)∇T√2Fz q (RH polarisation) C.8

However it is wrong to argue that because ∂τr Fzp ~ ∂τr

Fzq ~ 0 they should be completely neglected. For example, in the frame of reference moving at the group velocity one has ∂τr

Fzp/q = − ∂τr Fzp/q : all these are equally ‘small’ in magnitude but

they cannot all be neglected!

The second part of this appendix will not be followed by a reader until after reading sections 4-7 where the concepts and use of adjoint fields are discussed. Nevertheless, it is useful to show here how the approach of our photon-analogue model allows polarisation to be indeterminate until a measurement is made.

Figure C.1

(A) Generation of reference fields with both circular polarisations denoted by circular arrows and same phase of field denoted by ⇑: polarisation undetermined at point of generation denoted by star. (B), (C) Possible adjoint fields. Note that B & C fields are in anti-phase, ⇑ and ⇓, in each polarisation so that total adjoint B + C fields add to zero.

Figure C.1 (A) illustrates schematically the output from a source with the reference fields having both circular polarisations (as in equation C.9). Polarisations are indicated by ‘circular’ arrows and + and − signs. These reference fields propagate in the direction of solid black arrows. For simplicity, both polarisations are

⇓

+

⇑−(A)

(B)

(C)

−

−

+

+ ⇑

⇑

⇑

⇑

reference field

adjoint field

adjoint field

303

represented as having the same phase, indicated by the vertical arrows ⇑ and ⇑. Because both circular polarisation fields exist together, one forms an output with, as yet, an undetermined state of circular polarisation. Now every reference field has adjoint fields associated with them. The concept of the PRAHM mode model is that only pairs of reference fields and adjoint fields can carry net energy in localised regions. The adjoint fields propagate with exactly the same phase and group velocity as the reference fields except that c is reversed (along with ϕ). Τhis c- reversal is indicated by a broken line with reverse arrow direction. However one recalls that c- reversal means that the adjoint fields can neither initiate energy transfer nor carry energy by themselves without violating causality. Consequently it is only at the measurement stage that the adjoint fields play their unique role.

Two pairs of possible adjoint fields are shown here with + and − circular polarisations. One pair of fields is indicated in Figure C.1 (B) and a similar pair, but in antiphase with those in (B), is shown in Figure C.1 (C). With the two polarisations of the reference fields in (A) shown as being symmetrical in phase (⇑ and ⇑) then the two polarisations in the adjoint fields of (B) are shown as having asymmetrical phases (⇑ and ⇓). In the photon-analogue model energy is available only at appropriate locations of overlap of a pair of appropriately phased reference and adjoint fields. Reference fields alone cannot transfer energy but can initiate an interaction with the adjoint fields as discussed in sections 4-7 provided the requisite adjoint fields are available. If the fields (A) + (B) are captured together (along with their DSR fields discussed in sections 4-7 and appendix D) in a measurement, they form a ‘+’ polarised state with the ‘−’ polarised state cancelled. Alternatively one could have the adjoint fields as in (C) now in anti-phase (⇓ and ⇑) to those (⇑ and ⇓) in (B). If the fields (A) + (C) are captured together in a measurement, then they form a ‘−’ polarised state with the ‘+’ polarised state cancelling. However the adjoint fields (B) + (C) together cancel one another so that (A) + (B) + (C) add up to (A) as these fields propagate. This means that there is no decision as to which definite circular polarisation is actually present until the detection attempts to measure a ‘+’ or ‘−’ circular polarisation. The success of any detection stage is determined by which adjoint field is available with the right phase at the right time. This availability is considered here to be a random process. The model then states that polarisation need not be fixed at the time of emission but is fixed only at a time of measurement. Of course, with the right source, it is possible to fix the polarisation at the time of emission but in that case only one reference field along with its appropriate adjoint field propagate together.

304

Appendix D. DSRs to form envelopes of transverse fields

Consider an axial field Fz as in 5.1:

Fz = Fzo expi[kf τ f exp(δ) + krτr exp(−δ) = Fzo exp( i φ) exp( i Δ) D.1

where φ = (kf τ f + krτr) cosh(δ) ; Δ = (kf τ f − krτr) sinh(δ)

It is immediately seen from the construction of kf exp(δ) and kr exp(−δ) that equation 3.15 still holds

−κ2 = 2 kf exp(δ) exp(−δ) kr = 2 kf kr D.2

The aim of this section is to show that a distributed spin rotation Θ = exp(MΔ ϕ Δ) can hold for either +/− δ where MΔ is a Lorentz invariant number. In other words the value of δ need have no effect on the wave-relation of equation D.2 or on the value of the distributed spin rotation that depends on the invariant phase factor (kf τ f − krτr). This result is demonstrated by going back to section 3.

Take equations 3.10 -11 repeated here for convenience

∇T/ ( r FT) = − ∂τf [(1/√2) Fz] D.3

∇T/ ( f FT) = − ∂τr

[(1/√2) Fz] D.4

Then feed in D.1 and, with a little re-arrangement, obtain

∇T/ [ exp(−δ)r FT] = − ikf [(1/√2) Fz] D.5

∇T/ [ exp(+δ) f FT] = − ikr

[(1/√2) Fz] D.6

Projection operators have a remarkable effect as seen by evaluation below:

r exp(−iϕ δ) = r[cosh(δ) − iϕ sinh(δ)] = ½(1+ iϕ )[cosh(δ) − iϕ sinh(δ)

= ½(1+ iϕ )[cosh(δ) − sinh(δ)] = r exp(−δ) D.7

Similarly one can show that

f exp(−iϕ δ) = f exp(δ) ; D.8

r exp(iϕ δ) = r exp(δ) ; f exp(iϕδ) = f exp(−δ) D.9

In other words, a simultaneous rotation ϕ in the x,y plane along with an ‘interchange’ of E and cB caused by the operator i combine to be equivalent to a magnification exp(δ) or demagnification exp(−δ) dependent on the particular light-cone branch.

Consequently in D.5 and D.6 one can define

FTp = exp(−iϕδ) FT D.10

305

and replace r exp(−δ)FT and f exp(δ)FT so that D.5 and D.6 can be re-arranged into:

kr∇T/ [ r FTp] = kf ∇T

/ [ f FTp] = − ikf kr [(1/√2) Fz] D.11

Review equation 3.12 again repeated below for convenience

∇T(1/√2)Fz = ∂τr (r FT) + ∂τf (f FT) D.12

Multiply right through D.12 by the imaginary rotation exp(−iϕδ) and use D.10

exp(−iϕδ) ∇T(1/√2)Fz = ∂τr (r FTp) + ∂τf (f FTp) D.13

Now operate on both sides by ∇T/ noting that ∇T

/ ϕ ∇T = 0 so that:

∇T/ exp(−iϕδ) ∇T(1/√2)Fz = cosh(δ) ∇T

/ ∇T(1/√2)Fz = − κ2 cosh(δ) (1/√2)Fz

= ∂τr ∇T/ (r FTp) + ∂τf ∇T

/ (f FTp)

= − ikf ∂τr [(1/√2) Fz] − ikr

∂τf

[(1/√2) Fz]

= 2 cosh(δ) kf kr [(1/√2) Fz] D.14

Hence the wave-equation relationship of D.2 is confirmed irrespective of δ

Consider equation D.12 multiplied by Θ exp(−iϕδ) in order to find the permitted distributed spin rotations Θ giving:

Θ exp(−iϕδ)∇T(1/√2)Fz = Θ ∂τr (r FTp) + Θ ∂τf (f FTp) D.15

= ∂τr (r ΘFTp) + ∂τf (f ΘFTp) − R D.16

− R = (∂τr Θ)(r FTp) + (∂τf Θ)(f FTp) D.17

We know that operating on D.15 with (Θ ∇T)/ we shall be able to recover the wave equation D.2 as we did in equation D.14, because Θ/ Θ = 1. To reach the same result using equation D.16, where the rotation Θ now operates directly on FT in the way that the DSR hypothesis demands, requires that:

(Θ ∇T)/ R = 0 D.18

Assuming that Θ/ (∂τr Θ) and Θ/(∂τf Θ) are constants, then equation D.18 can be re-arranged to give:

Θ/ (∂τr Θ)∇T/(r FTp) + Θ/(∂τf Θ)∇T

/(f FTp) = 0 D.19

Θ/(∂τr Θ) kf + Θ/(∂τf Θ) kr = 0 D.20

However Θ = exp(ϕΔ) satisfies equation D.20 where, as in equation 5.5, Δ = sinh(δ)(kf τ f −kr τr). Consequently a more general DSR is given from:

Θ = exp(M ϕΔ) D.21

3 0 6

where M is any Lorentz invariant number. From equation 5.5 one notes that:

Δ = sinh(δ)(kf τ f −kr τr) = (ωd/c)(z − vgt) tanh(δ) D.22

It can be observed that the whole analysis can be repeated with δ having changed sign but the results of D.21-22 still hold.

It was noted in the main text that any DSR Θ did not ‘rotate’ Fz. This important point can be seen in another way. It is shown below that Fz(x+dx, y+dy) remains unaltered with or without distributed spin-rotations:

Fz(x+dx, y+dy) = Fz(x, y) + (∇T Fz)/dr ; dr/ = [dx, dy] D.23

Under a distributed spin rotation where Θ /Θ ≡ 1 :

Fz(x+dx, y+dy) = Fz(x, y) + (Θ∇T Fz)/ Θdr

= Fz(x, y) + (∇T Fz)/ dr D.24

Appendix E. A classical ‘Poynting’ theorem

From A.7 and A.10 repeated here for convenience:

∇TFz = ∂ct( iϕ FT) + ∂z FT ; E.1

∇T/ ( iϕ FT) = ∂ct (Fz) E.2

Hence with standard conjugation:

∇TFz* = ∂ct( −iϕ FT*) + ∂z FT* ; E.3

∇T/ (− iϕ FT*) = ∂ct (Fz*) E.4

Then transposing E.1 and E.3:

(∇T/Fz*) = ∂ct (FT*/

iϕ ) + (∂z FT*/) E.5

(∇T/Fz ) = ∂ct(−FT

/ iϕ) + (∂z FT

/) E.6

Post multiply E.5 by iϕ FT and premultiply E.2 by Fz*:

(∇TFz*)/( iϕ FT) = ∂ct (FT*/ iϕ )( iϕ FT) + (∂z FT* /)( iϕ FT) E.7

Fz*∇T/ ( iϕ FT) = Fz*∂ct (Fz) E.8

Combine E.7 and E.8:

∇T/(Fz* iϕ FT)

= [∂ct (FT*/ iϕ )]( iϕ FT) + (∂z FT*/)( iϕ FT) + Fz*∂ct (Fz) E.9

Take the conjugate equations and form the conjugate result to E.9:

307

∇T/ (− Fz iϕ FT*)

= [∂ct (−FT/ iϕ )]( −iϕ FT*) + (∂z FT

/)(−iϕ FT*) + Fz ∂ct (Fz*) E.10

Integrate E.9 and E.10 over the whole cross sectional area (denoted by ⟨⟨ ⟩⟩ ) with the fields vanishing at the boundaries of that cross sectional area. This integration removes the terms in ∇T setting the l.h.s of the equations zero:

0 = ⟨⟨ ∂ct (FT*/) FT + (∂z FT*/)( iϕ FT) + Fz*∂ct (Fz)⟩⟩ E.11

0 = ⟨⟨ ∂ct (FT/ ) FT* + (∂z FT

/)(−iϕ FT*) + Fz ∂ct (Fz*)⟩⟩ E.12

Adding together E.11 and E.12 along with the usual normalising factor of ½ :

0 = ½ ⟨⟨ ∂ct (FT*/ FT) + ∂ct (Fz *Fz) + ∂z (FT*/ iϕ FT) ⟩⟩ E.13

This recovers then the classic Poynting equation as in the text.

Appendix F. ‘Poynting’-like theorem for DSR solutions

Here C implies changing the sign of both i and ϕ and is referred to as ϕ−conjugation. The raised solidus / as usual implies transposition. Again starting with equations A.7 and A.10 repeated here for convenience:

∇TFz = ∂ct( iϕ FT) + ∂z FT F.1 ∇T

/ (iϕ FT) = ∂ct (Fz) F.2 Consider then an adjoint field FzA ,FTA:

∇TFzAC

= ∂ct( iϕFTAC) + ∂z FTA

C F.3

∇T/ (iϕ FTA

C) = ∂ct (FzAC) F.4

Transpose F.1 and F.3: (∇T

/Fz ) = ∂ct(−FT/ iϕ) + (∂z FT

/) F.5 (∇T

/FzAC) = ∂ct (−FTA

C / iϕ ) + (∂z FTA

C / ) F.6

Post multiply F.6 by (ϕ FT): (∇T

/FzAC)( ϕ FT) = [∂ct (−FTA

C / iϕ )]( ϕ FT) + (∂z FTA

C /)( ϕ FT) F.7

Premultiply F.2 by iFzAC:

− FzAC ∇T

/ ( ϕ FT) = iFzAC∂ct (Fz) F.8

Subtract F.7 from F.8: −∇T

/(FzAC ϕ FT) = [∂ct (FTA

C / iϕ )]( ϕ FT) −(∂z FTAC /)( ϕ FT) + iFzA

C ∂ct (Fz) = (− i ∂ct FTA

C /) FT + (i∂z FTAC /)(iϕ FT) + i FzA

C∂ct (Fz) F.9

308

The equivalent operations with F.5 and F.4 yield the ϕ-conjugate equations: ∇T

/ ( Fz ϕ FTAC) = i (∂ct FT

/) FTAC

+ (−i∂z FT/)(iϕ FTA

C) − i Fz ∂ct (FzAC)

F.10 Integrate, as in Appendix D, over the whole cross sectional area (denoted by ⟨⟨ ⟩⟩ ) with the fields vanishing at the boundaries of that cross sectional area so that the integrated terms in ∇T on the l.h.s of equations F.9 and F.10 go to zero:

0 = ⟨⟨ −i (∂ct FTAC /

) FT + (i∂z FTAC /)(iϕ FT) + i FzA

C∂ct (Fz)⟩⟩ F.11

0 = ⟨⟨ i (∂ct FT/) FTA

C + (−i∂z FT

/)(iϕ FTAC) − i Fz ∂ct (FzA

C)⟩⟩ F.12 Transpose F.14:

0 = ⟨⟨ (FTAC /)(i∂ctFT) − FTA

C / (iϕ )(−i∂z FT) − i∂ct (FzAC) Fz ⟩⟩ F.13

F.11 and F.13 are then used in the main text (equations 7.7-8).

Appendix G. DSRs to ensure cross sectional convergence

Given: Φ = exp[iϕ ½κ x/ x ] [ x = (x, y)/ ; x/ x = x2+ y2] G.1

The features of projection operators used in equation C.8-9 yield:

f Φ = f G [G = exp(−½κ2 x/ x)] G.2

r Φ/ = r G G.3

Because ∂τr Φ , ∂τr Φ/ , ∂τf Φ and ∂τf

Φ/ are all zero, then irrespective of using either Φ or Φ/ one finds R = 0 in equation 4.5 so that either Φ or Φ / are satisfactory DSR operators that can replace Θ in equation 4.4

Now project both sides of equation 4.4 using f and r so that the f FT fields and the r FT fields can be allowed to have the different distributed spin rotations Φ and Φ / respectively. Equations 4.2 and 4.3 along with 4.5, split into its f and r branches, yield

(f Φ / ∇T)

/ (r Φ/ FT) = − ∂τf [(1/√2) Fz] ; r Φ / ∇T(1/√2) Fz = ∂τr (r Φ

/ FT) G.4 (r Φ ∇T)

/ ( f Φ FT) = − ∂τr [(1/√2) Fz] ; f Φ∇T(1/√2) Fz = ∂τf (f Φ FT)

G.5

From the respective equations G.4 or G.5, it can be checked by eliminating rFT or f FT and noting ∇T

/ ϕ ∇T = 0, that the wave-equation for Fz still holds:

(f Φ / ∇T)

/ (∂τrr Φ/ FT) =(f Φ / ∇T)

/ r Φ / ∇T(1/√2) Fz

= ∇T / r ∇T(1/√2) Fz = ½ ∇T

/ ∇T(1/√2) Fz = ∂τr∂τf [(1/√2) Fz] G.6

309

Hence the different DSRs for f FT and r FT are still valid solutions. Now use equation 4.5 (with R = 0) along with the appropriate DSRs for f FT and r FT and equations 6.2-3:

[ ∂τr (r Φ / FT) + ∂τf (f ΦFT)] = G[ ∂τr (r FT) + ∂τf (f FT)]

= [ r +f ] G ∇T(1/√2)Fz = G ∇T(1/√2)Fz G.7

It follows with this special case, that rFT and f FT may be replaced with

r Φ /FT = r GFT and f ΦFT = f GFT respectively. Similar changes can be made for the

adjoint fields rFTA and f FTA. Because f + r = 1, the addition of these imaginary DSRs yields:

⟨⟨ FTAC/ FT⟩⟩ → ⟨⟨ (rΦ/

FTA) C /(fΦ FT)⟩⟩ + ⟨⟨ (f Φ FTA)

C / (rΦ/ FT)⟩⟩ G.8

⟨⟨ FTAC/ FT⟩⟩ → ⟨⟨ (GFTA)

C / f (GFT)⟩⟩ + ⟨⟨ (GFTA) C /r (GFT)⟩⟩ G.9

⟨⟨ FTAC/ FT⟩⟩ → ⟨⟨ (GFTA)

C /(GFT)⟩⟩ G.10

Hence equation 8.5 is obtained.

Appendix H. Experimental Equipment and Methods for section 11

In this appendix a detailed description is given of the experimental method and equipment used to perform the experimental tests described in section 11. In addition further sets of results are presented to demonstrate the repeatability of the measurements.

The pulse source used in the experimental results presented in section 11 is a mode-locked fibre ring laser, the Toptica FemtoFiber Scientific FFS Ultrafast Erbium Fiber Laser System. A core-pumped fiber, doped with Erbium ions, acts as the active laser medium with mode-locking achieved by a polarization sensitive crystal. The laser produces pulses of less than 100 fs duration with spectral width of 55 nm, centred at ~ 1571 nm and a repetition rate of 76.9 MHz.

Single photons are created by strongly attenuating the pulses using variable neutral density filters. The use of strong attenuation is standard practice in today’s practical quantum cryptographic systems [83]. The average photon number per pulse is set to be less than 0.1 so that following Poisson statistics, less than 5 % of non empty pulses contain more than one photon. At the point at which the laser pulse is attenuated to single photon level the pulse width is ~ 8 ps; it is broadened from the initial sub picosecond width by the immediate insertion of fiber at the pulse exit point from the laser. This short pulse width allows multi-nanosecond time dispersion over 26 km of Single Mode Fibre (SMF).

310

The experimental set-up is shown in Figure H.1 below; this is a detailed version of the schematic shown in Section 11, Figure 7. The laser emits pulses into free space, which are then immediately coupled into single mode fibre. The laser pulses are initially passed through a ‘JDSU’ Lithium Niobate Mach-Zehnder modulator (MZ 1) acting as a gate that reduces the pulse repetition rate from 76.9 MHz, directly from the ring laser, to close to 600 kHz. Dispersion over many nanoseconds can then be introduced without adjacent pulses overlapping. Following the reduction in pulse repetition rate, JDSU fiber miniature variable neutral density attenuators are used to change the power levels between classical pulse regime and ‘single photon’ level.

PulseGenerator 1

SMFMZ 1

PulseGenerator 2

MZ 2

trigger

DCA

SPAD

OSA

Variable Attenuator

Frequency Divider

Fiber ring laser

MSA

polarizationcontrol

DelayGenerator

polarizationcontrol

Figure H.1 Experimental set-up for single photon dispersion measurement in single-

mode fiber. The optical path is shown in bold. DCA = Digital Communications Analyser; OSA = Optical Spectrum Analyser;

SPAD = Single Photon Avalanche Photodiode; MSA = Multiscalar

The optical gates shown in Figure 7 are both Mach-Zehnders with MZ1 shown in Figure H.1 controlled by a pulse generator that is in turn controlled from an electronic trigger signal originating from a sampling photo-diode internal to the laser. This trigger signal has its repetition rate reduced to 600.5 kHz by a custom made electronic frequency divider circuit, the output of which triggers a Stanford Research Systems DG535 four channel digital delay generator which in turn drives two pulse generators. The first pulse generator, Agilent 33250A, controls MZ1 to gate the optical pulses with 600.5 kHz. Dispersion is introduced after MZ 1 by 26.4 km of Corning SMF, which gives dispersion of 18 ps/nm/km at 1550 nm. The second pulse generator, Hewlett Packard 8082A, allows a possible second JDSU Lithium Niobate Mach-Zehnder (MZ2) to be inserted after the 26.4 km of SMF.

In the classical regime the pulse can be measured with a Hewlett Packard 83480A digital communications analyser (DCA) or Hewlett Packard 70950B optical spectrum

311

analyser (OSA). At the ‘single photon’ level an id Quantique id200 Single Photon Avalanche Photodiode (SPAD) module detects the photon counts, which are input into a photon-counting PCI card, a Becker and Hickl Multiscaler, MSA-1000. The accompanying software allows a histogram of photon arrival counts in 1ns time intervals over 200ns to be accumulated. The photo-detection, while allowing arrival rates to be measured over 1ns intervals, cannot make any measurement about the duration of a single photon.

Initial measurements were made with the second gate (MZ2 in Figure H.1 and optical gate B in Figure 7) ‘wide-open’ with a width of 500 ns, considerably greater than the dispersed pulsed width. Subsequently the width of the second gate was reduced and it was delayed at intervals relative to the dispersed pulse. As discussed in section 11, photons can arrive only within the interval that the gate is open and only photons ‘shorter’ than the gate length will be counted.

In Section 11, Figure 8, a good correlation is shown between photon arrival times and the original pulse spectrum, as expected from the photon-analogue model. Two further examples are shown in Figure H.2 below. These also demonstrate a good correlation between photon arrival times and pulse spectral shape. The measurements were performed on different occasions and the difference in spectral shape between measurements can be accounted for by different operating conditions of the laser and ambient temperature.

80 100 120 140 1600

60001500 1550 1600

0

4

phot

on c

ount

s

relative photon arrival times (ns)


spectrum

pow

er (u

W)

100 120 140 1600

6000

1450 1500 1550 1600

0

15

phot

on c

ount

s


pow

er (u

W)


spectrum photon countsphoton counts

Figure H.2 Examples of correlation between relative photon arrival times and pulse spectrum (the spectra have been offset by the pedestal of dark counts to best demonstrate the correlation between photon counts and spectra).

Repeatability of the ‘gated’ photon measurements has also been demonstrated as shown by the two further examples given in Figure H.3. In these measurements the width of the second gate, MZ2, after the dispersion, is reduced from 500 ns (wide-open) to 10 ns (gated). These ‘gated’ photon counts are then compared to the ‘wide -

312

open’ measurements and no significant reduction in the photon arrival rates across the dispersed pulse width, is observed with the 10 ns gate. As discussed in section 11, this suggests that the photon ‘length’ is less than 10 ns.

60 80 100 120 140 1600

10000

phot

on c

ount

s



60 80 100 120 140 1600

10000

phot

on c

ount

srelative photon arrival times (ns)


Figure H.3 Repeatability of the comparison of ‘gated’ photon arrival rates with ‘wide-open’ gate photon arrival times. No significant change in photon arrival rates is observed.

Appendix I. A heuristic estimate of Planck’s constant

This present Maxwellian theory of photon-like wave-packets does not give any method of calculating the coupling of the electron and the fields. In a plane wave limit, the classical energy over the wave-packet is given from a three dimensional integral over the packet expressed symbolically as:

U classical = ⟨⟨⟨ ½ εο ( FTc*/ FTc)⟩⟩⟩ I.1 On the other hand the DSR energy integrated over the smallest packet (equation 7.11 last term with M=1) requires knowledge of a normalisation constant N :

(2πN/c) ⟨⟨⟨ (½ωd)(FTAC /

FT) ⟩⟩⟩ = (2πN/c) ⟨⟨⟨ (½ωd)( FT*/ FT ⟩⟩⟩ I.2 The classical values of the RS vector are seen to be normalised differently from the DSR values which could then give a correspondence with quantum mechanics.

Suppose we have a classical circularly polarised magnetic field HT on one side of a circularly polarised current IT distributed over an area A. This current launches the time varying electromagnetic fields into the wave-packet. From dimensional reasoning, the average H-field is related to the average current from:

⟨⟨|HT|2⟩⟩ = ⟨⟨|IT2|/A⟩⟩ I.3

Because this is a circularly polarised current IT the average current and average charge should be related by the frequency associated with the polarised field

313

⟨⟨|IT2|/A⟩⟩ = ⟨⟨ ω2 e2 /A⟩⟩ I.4

The power flow integrated for the time of flight over the packet length gives the energy stored as:

⟨⟨⟨ ET HT τ ⟩⟩⟩ = ⟨⟨⟨ Ζο ω2 e2 τ /A⟩⟩⟩

A hypothesis consistent with section 6 is that reference waves with frequencies ω = ωd +/− ckd tanh(δ) have associated adjoint waves with frequencies ωΑ = ωd −/+ ckd tanh(δ). The power flow of a reference waves and that of its adjoint are in opposing directions giving a net power flow determined from:

Upacket ~ ⟨⟨⟨ ET HT τ ⟩⟩⟩ net ∼ ⟨⟨⟨ Ζο e2 [4 ωd ckd tanh(δ)]τ /A⟩⟩⟩ I.6

Integration over the cross sectional area removes the area A, while the integration over the duration τ of the packet makes ckd tanh(δ) τ = π so that writing ωd = 2πν one arrives at an estimate for the packet energy:

Upacket ~ 8 π 2 Ζο e2 ν ∼ 7.8 ×10−34 ν joule I.7

The point of including this heuristic estimate is because it gives the right order of magnitude for Planck’s constant (6.626 ×10−34 Joule sec) supporting the proposition that this speculative theory is worth pursuing further.

Appendix J. Welcher Weg

It is argued that the PRAHM mode model with its reference and adjoint ground states and the excited photon-analogue gives a straightforward model that explains why one cannot tell which way a photon travels within an interferometer. A perfect Mach-Zehnder with equal arms is shown schematically in Figure J.1 below. The path lengths are all adjusted so that the interference pattern ensures that any photon entering at P is detected only at D1. The reference and adjoint waves have ground states that are identical in the path they take. The reference wave is marked as a solid line and the adjoint wave is marked as a dotted line with the arrow reversed to remind one that both c and ϕ have been reversed.

The photon-analogue or PRAHM is then a packet where these two waves each have extra counter-rotating helical rotations over a limited interval giving the additional available energy. This packet can in principle lie anywhere along the identical paths travelled by the reference and adjoint ground states, but once excited the packet must travel with its group velocity following any of the permitted classical paths. The interference pattern, adjusted by precise path lengths, ensures that any photon entering at P is detected only at D1. The path can be either A, AB, ABC or A, AD, ADE. There is no way that one can tell which way the packet went without destroying the interference. Even with a non-demolition measurement [84] [indicated by a shaded area in Figure J.1(F)] there is still a phase change between the two

314

previously balanced paths (paths ADF and ADE) so that there is a non-zero probability that the packet will emerge at D2 rather than at D1, demonstrating that the perfect interference has been destroyed. The PRAHM should behave analogously to a photon.

Figure J.1 The paths of reference and adjoint waves inside an interferometer are identical; the photon analogue simply excites these two waves with additional counter-rotating helical rotations giving one unit of available energy. Remember that the reversal of the arrow on the dashed (adjoint) paths is simply to remind one that both c and ϕ have been reversed.

Appendix K. Classical Entanglement

This appendix builds on the model of polarisation given at the end of Appendix C using the concepts of reference and adjoint fields. This model suggests that entanglement of two photons arises through even and odd symmetries of emission. The importance of even and odd symmetries in understanding space-time geometry is well illustrated by the work of Hestenes and also Doran and Lasenby [85, 86] taken as exemplars in an extensive field. Carroll has also considered how correlations of even and odd signals can create a four dimensional space-time with relativistic properties [87].

Reference waveground state

Adjoint waveground state

Beam splitter Mirror Photon packet

A ABABC

AD ADE ADF

D1D2

D1D2

D1D2

P

315

One might start with a microwave generator in the middle of a resonant cavity as in Figure K.1. No matter how long the length of the cavity, there are always even modes and odd modes, though of course the actual resonant frequency may change with length. The polarisations at the cavity ends A and B are always correlated. However entanglement is not so straightforward because we do not know until we measure whether the polarisation will be horizontal or vertical. Nevertheless, the concept of symmetrical and asymmetrical fields appears to be a key.

Figure K.1

Even and Odd Modes in a cavity (schematic).

Now in the light of Appendix C on polarisation, consider even and odd symmetries of emission with the reference fields giving an even symmetry with the two photon fields flying off equally and symmetrically from the source as sketched in Figure K.2 (A). Here, for greater simplicity and comparison with quoted experiments [88], we allow for the possibility of either vertical polarization (denoted by either ⇑ or ⇓) or horizontal polarisation (denoted by either ⇐ or ⇒) so that there are two fields with ‘opposite’ polarisations simultaneously present as there were for circular polarisation in appendix C. Given that the reference fields are symmetric, the adjoint fields are required to be asymmetric about the source for reasons that become apparent. As in Appendix C there are two sets of adjoint fields (B) and (C) which, because of their phasing, add up to zero so that (A) + (B) +(C) add to give (A). Remember that the adjoint fields cannot carry energy or initiate energy flow. Energy can be carried only along the paths where adjoint and reference fields overlap. Now Bob sets out to detect say horizontal polarisation (denoted by ⇒) and provided that state (C) is present in an appropriate phase then he can capture (A) + (C) in his measurement and discover horizontal polarisation. His vertical polarisation fields all cancel but (A) + (C) leaves Alice able to measure only vertical polarisation (⇑ ) with certainty. Equally if Bob sets out to measure vertical polarisation and is able to capture (A) + (B), then again Alice will be able to measure only horizontal polarisation with certainty. There is no predetermination that either vertical or horizontal polarization will be measured. Nevertheless Bob and Alice will measure correlated polarisations. Because symmetry is independent of length, provided that the possibility for even and odd symmetry is not destroyed, entanglement should be regardless of path length. Even and odd symmetries are invariant with special relativity and so this argument based on symmetry suggests that entanglement is also a Lorentz invariant property.

EVEN MODE

ODD MODE

A

A

B

B

316

Figure K.2 Quasi-classical Entanglement.

(A) gives V (⇑ ) and H(⇒) polarisations on emission with symmetrical emission on either side.

(B) and (C) give adjoint asymmetrical emissions. Note: (B) +(C) = 0.

Consider (A)+(C) or (A)+(B).

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322

Cosmological Coincidence and Dark MassProblems in Einstein Universe and Friedman

Dust Universe with Einstein’s Lambda

James G. Gilson [email protected] of Mathematical Sciences

Queen Mary University of LondonMile End Road London E14NS

September 27th 2007

Abstract

In this paper, it is shown that the cosmological model that was intro-duced in a sequence of three earlier papers under the title A Dust UniverseSolution to the Dark Energy Problem can be used to analyse and solve theCosmological Coincidence Problem. The generic coincidence problem thatappears in the original Einstein universe model is shown to arise from a mis-understanding about the magnitude of dark energy density and the epochtime governing the appearance of the integer relation between dark energyand normal energy density. The solution to the generic case then clearlypoints to the source of the time coincidence integer problem in the Fried-man dust universe model. It is then possible to eliminate this coincidenceby removing a degeneracy between different measurement epoch times. Inthis paper’s first appendix, a fundamental time dependent relation betweendark mass and dark energy is derived with suggestions how this relationcould explain cosmological voids and the clumping of dark mass to becomevisible matter. In this paper’s second appendix, it is shown that that darkenergy is a conserved with time substance that is everywhere and for alltime permeable to the dark mass and visible mass of which the contractingor expanding universe is composed. There are more detailed abstracts givenwith both appendices.

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1 Introduction

The work to be described in this paper is an application of the cosmologi-cal model introduced in the papers A Dust Universe Solution to the DarkEnergy Problem [23], Existence of Negative Gravity Material. Identificationof Dark Energy [24] and Thermodynamics of a Dust Universe [32]. Theconclusions arrived at in those papers was that the dark energy substanceis physical material with a positive density, as is usual, but with a neg-ative gravity, -G, characteristic and is twice as abundant as has usuallybeen considered to be the case. References to equations in those paperswill be prefaced with the letter A, B and C respectively. The work in A,B and C, and the application here have origins in the studies of Einstein’sgeneral relativity in the Friedman equations context to be found in ref-erences ([16],[22],[21],[20],[19],[18],[4],[23]) and similarly motivated work inreferences ([10],[9],[8],[7],[5]) and ([12],[13],[14],[15],[7],[25],[3]). Other usefulsources of information are ([17],[3],[30],[27],[29],[28]) with the measurementessentials coming from references ([1],[2],[11]). Further references will bementioned as necessary. The application of the cosmological model intro-duced in the papers A [23], B,[24] and C [32] is to the extensively discussedand analysed Cosmological Coincidence Problem. This problem arose fromEinstein’s time static cosmology model derived from his theory of generalrelativity. The Einstein first model is easily obtained from the Friedmanequations (1.1) and (1.2) with the positively valued Λ > 0 term that heintroduced to prevent his theoretical universe from collapsing under thegravitational pull of its material contents,

8πGρr2/3 = r2 + (k − Λr2/3)c2 (1.1)

−8πGPr/c2 = 2r + r2/r + (k/r − Λr)c2. (1.2)

Einstein’s preferential universe was of the closed variety which involves thecurvature parameter being unity, k = 1 and with a positively valued Λ, wehave,

8πGρr2/3 = r2 + (1− Λr2/3)c2 (1.3)

−8πGPr/c2 = 2r + r2/r + (1/r − Λr)c2. (1.4)

To get a static universe from these equations that holds for some finite timeinterval we have to impose the non expansion condition, v = r = 0, togetherwith the none acceleration condition a = v = r = 0 and if, additionally, we

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choose the dust universe condition, P = 0, we get (1.5) and (1.6).

8πGρr2/3 = (1− Λr2/3)c2 (1.5)

0 = (1/r − Λr)c2. (1.6)

Einstein identified his cosmological constant Λ as arising from a density ofdark energy in the vacuum, ρΛ = Λc2/(8πG), so that equations (1.5) and(1.6) could be put into the forms (1.7) and (1.8) with the radius of theEinstein universe given by (1.9),

8πG(ρ + ρΛ) = 3c2/r2E (1.7)

8πG(ρΛ) = c2/r2E (1.8)

rE = Λ−1/2. (1.9)

From (1.7) and (1.8) it follows that

8πGρ = 2c2/r2E (1.10)

ρΛ = ρ/2 (1.11)

ρ = 2ρΛ = ρ†Λ. (1.12)

Equation (1.12) is the generic version of the so called cosmological coinci-dence problem. I think that Einstein would not have recognised the rela-tionship between ρ and ρΛ at (1.12) as a problem in the early years afterdiscovering it. He probably thought that the 2 factor was interesting andneeded explaining but did not see it as a problem. In those early yearshe was convinced the universe was a time static entity and had no visionof the possibility that the relation might have a different coefficient fromthe integer 2 which could come about by the now recognised and acceptedexpansion process. Only after expansion was accepted does the questionfollowing arise. If at time now equation (1.12) holds in an expanding uni-verse of decreasing density ρ with time and with ρΛ an absolute constant,is it not an extraordinary coincidence that at time now the coefficient in(1.12) is exactly the integer 2? Clearly the significance of the factor 2 mustbe seen against the likely possible values of ρ which probably varies from ∞to 0 with ρΛ remaining constant over the whole positive life time history ofthe universe. Einstein’s generic cosmological coincidence problem is com-pletely resolved by the cosmological model introduced in references A [23],B,[24] and C [32] as I shall next explain. However, there is one importantreservation about this claim that will be discussed in the next section. I

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call this first cosmological coincidence critical because it involves the inte-ger point value number, 2, which would have zero probability of occurringin any finite time ranged variable quantity. Such coincidences need to beexplained in any structure. The model introduced in those papers revealsthe true nature of dark energy material and that is the clue to resolvingthe generic coincidence problem. One conclusion from those papers wasthat the dark energy density, contrary to Einstein’s identification, shouldbe theoretically and physically measured as ρ†Λ (1.12) rather than as ρΛ.The second conclusion from those papers was that dark energy has posi-tive mass density but is characterised by carrying a negative gravitationalvalue of the gravitational constant, −|G|. Thus equation (1.12) achievesEinstein’s purpose of stopping the gravitation collapse of the universe bychoosing conditions such that the positive mass material, ρ+ρ†Λ, within the

universe is gravitationally neutral, Gρ + (−G)ρ†Λ = 0. Thus although thatcould have happened some time or other it would not necessarily hold forever as in a constant universe or indeed occur at the time now. The modelI am suggesting is a flat universe with, k = 0, and the actual time whensuch conditions apply is denoted by tc and can be calculated. At that timev(tc) 6= 0 contrary to the what is implied in the Einstein universe wherev = 0 given above. The time tc is the important time greatly in the pastand recognised recently by astronomers when the acceleration of the uni-verse changes through zero from negative to positive or when dark energytakes over from normal mass energy. The critical coincidence in the genericEinstein universe is completely resolved by the conceptual aspects of theFriedman dust universe that I have been proposing. This reinterpreted oldand modified model which is closely related to an early Lemaıtre model hasa structure that has identified the cause of the Einstein critical coincidence.The nature of this coincidence can be described as, mistaking the Einsteinradius for a possible constant present time radius . This mistake is com-pletely excusable on the grounds that Einstein did not recognise that theuniverse radius was in truth a variable with time quantity and he was com-pletely unaware that at some time in the past the dark energy density ashe defined it was exactly half the normal mass density. The explanation ofthe root cause of the critical Einstein coincidence can be used to identifiedthe cause of another critical time coincidence between the present time t†

and time tc, t† = 2tc, in the Friedman dust universe. This will be explainedin the next section.

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2 Coincidence in Friedman Dust Universe

The coincidence in the Friedman dust universe model involves, t†, the timenow and, tc, the time when the universe changed from deceleration to ac-celeration.

t† = 2tc. (2.1)

This equation involves again the exact numerical integer value, 2. This isclearly critical because if two events over time are so related, then theremust be some physical explanation because the probability of two suchtime-point events on any finite time line range is zero. The generic Einsteincoincidence was critical in the same sense. This coincidence seems obviouslyrelated to the generic Einstein coincidence which suggests it is also totallyexplainable. The reservation I mentioned earlier is that you might see itas ironic that a model with a coincidence can completely solve the coinci-dence in an earlier model. This can be explained by the fact that theoreticalstructures involve patterns of abstract symbols as one aspect and numericalconstants as another aspect when they are applied to physical situations.The new model is correct in the first aspect but in the second aspect, thenumerical values have not all been associated with the measurement time,t†, but rather some with a conceptual time, t0, the time that would beassociated with the centre of the values given by the astronomical measure-ments. There is some subtlety in this situation because in this model, itseemed that t† should be equal to t0. However, this equality created thedegeneracy that led to the coincidence. It can all be resolved by using theformula for Hubble’s constant, the formula for the radius and the formulafor the constant C,

H(t) = (c/RΛ) coth(3ct/(2RΛ)) (2.2)

r(t) = b sinh2/3(3ct/(2RΛ)) (2.3)

C = ΩM,0H2(t0)r

3(t0). (2.4)

These expressions involve the numerical parameter, RΛ. It is necessary tofind the correct value for this parameter that is to be associated with theseformulae. To make this step, we need the astronomical measurements of theΩs. The accelerating universe astronomical observational workers [1] give

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measured values of the three Ωs, and wΛ to be

ΩM,0 = 8πGρ0/(3H20 ) = 0.25+0.07

−0.06 (2.5)

ΩΛ,0 = Λc2/(3H20 ) = 0.75+0.06

−0.07 (2.6)

Ωk,0 = −kc2/(r20H

20 ) = 0, ⇒ k = 0 (2.7)

ωΛ = PΛ/(c2ρΛ) = −1± ≈ 0.3. (2.8)

From these equations assumed to hold at a conceptual time, t0, when theuniverse passes through the centre value of the measurement ranges, we getthe formulae,

t0 = (2RΛ/(3c)) cosh−1(2) (2.9)

RΛ = 3ct0/(2 cosh−1(2)) (2.10)

tc = (2RΛ/(3c) coth−1(31/2) (2.11)

t0/tc = cosh−1(2)/ coth−1(31/2) = 2. (2.12)

Having found RΛ in terms of t0 this value of RΛ can be substituted into theformula for Hubble’s constant, (2.2), to find the value of the time now , t†.

H(t†) = (c/RΛ) coth(3ct†/(2RΛ)) (2.13)

t† = (2RΛ/(3c)) coth−1(RΛH†/c) (2.14)

=

(t0

cosh−1(2)

) (coth−1

(3t0H

†

2 cosh−1(2)

))(2.15)

=

(2tc

cosh−1(2)

) (coth−1

(6tcH

†

2 cosh−1(2)

)), (2.16)

where H† = H(t†) is the present day measured value of Hubble’s constant.Equations (2.15) or (2.16) is essentially the solution to the coincidence prob-lem. If we write (2.16) in the form

t†/tc =

(2

cosh−1(2)

) (coth−1

(6tcH

†

2 cosh−1(2)

))(2.17)

t†/tc = 2f(2tc), (2.18)

where f(2tc) gives the deviation of the ratio t†/t0 from the value unity andremoves the degeneracy. Expressed in another way it is the multiplicativefunction that breaks the coincidence at (2.12) and converts the integer 2 toa much less notable non integral value. However, we can give the formulae

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(2.17) and (2.18) together an interpretation in terms of the uncertaintiesof the measurement process. This is achieved by defining the measurementdeviation function dmeas(t0) as follows,

dmeas(t0) = t†/t0 − f(t0) (2.19)

f(t0) =

(1

cosh−1(2)

) (coth−1

(3t0H

†

2 cosh−1(2)

)). (2.20)

The function (2.19) is a dimensionless measure of how much the centralΩ values from astronomy assumed to have occurred at t0 differ from thetime now measurement from the Hubble variable quantity H(t†) taken attime now, t†. It is sufficient to assume that the event at t0 is still yet tooccur, t0 > t†, then we see that the function dmeas passes through zerowhen the full degeneracy holds at t0 = t† and it has a maximum at t0 ≈0.643× 1018s when t† and t0 assume the approximate maximum deviation,0.17. When t0 = 0.643×1018, t† can be assumed constant at the coincidencevalue 4.34467× 1018 so that the maximum deviation times ratio is t†/t0 ≈0.43467/0.643 ≈ 0.6757 or

t† = 0.6757t0. (2.21)

It follows that t†, the time now value, can vary from t0 down to a value oft† ≈ 0.6757t0 = 1.3514tc. Thus the coincidence is decisively removed witht† 6= t0 = 2tc.

3 Conclusions

It has been shown that the generic Einstein coincidence problem can beresolved in terms of a correction in the value of the density he associatedwith his cosmological constant Λ and a rethink about the significance ofthe radius of his model. This solution then points clearly to resolution ofthe coincidence in the recent dust universe model as essentially the sameconcepts are involved. The conceptual centre Ω value measurements fromthe astronomers can not necessarily be assumed to occur at exactly the sameepoch time t0 as the measurement of the value of the Hubble constant atepoch time now, t†. The usually assumed degeneracy t0 = t† can be removedto find the true range of values within which t† has to reside so that theinteger 2 aspect of the same degeneracy t† = 2tc sees the 2 replaced with a

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less mysterious non integer. The time tc is when the expansion accelerationchanges from negative to positive.

Acknowledgements

I am greatly indebted to Professors Clive Kilmister and Wolfgang Rindlerfor help, encouragement and inspiration over many years.

4 Appendix 1

Fundamental Dark Mass, Dark Energy TimeRelation in a Friedman Dust Universe

and in a Newtonian Universewith Einstein’s Lambda

Abstract

In this appendix, it is shown that the cosmological model that was intro-duced in a sequence of three earlier papers under the title A Dust UniverseSolution to the Dark Energy Problem can be used to recognise a fundamen-tal time dependent relational process between dark energy and dark mass.It is shown that the formalism for this process can also be obtained fromNewtonian gravitational theory with only the additional assumption thatNewtonian space contains a constant universal dark energy density distri-bution dependant on Einstein’s Lambda, Λ. It thus seems that the processis independent of general relativity and applies in more contexts than justthe expansion of the entire universe. It is suggested that the process can bethought of as a local space and time packaging for dark mass going throughpart transmutations into locally condensed visible material. The processinvolves a contracting and then expanding sphere of conserved dark matter.At two stages in the process at special times before and after a singularityat time zero, the spherical package goes through a condition of gravitationalneutrality of very low mass density which could be identified as cosmologicalvoids. The process is an embodiment of the principle of equivalence.

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5 Dark Mass, Dark Energy Relation

The work to be described in this appendix is an application of the cosmolog-ical model introduced in the papers A Dust Universe Solution to the DarkEnergy Problem [23], Existence of Negative Gravity Material. Identificationof Dark Energy [24] and Thermodynamics of a Dust Universe [32]. Furtherreferences will be mentioned as necessary. Application of the cosmologicalmodel introduced in the papers A [23], B,[24] and C, [32], is to be found inthe paper D, ([34]), to the extensively discussed and analysed CosmologicalConstant Problem. In the next section a relation between Dark Mass andDark Energy over epoch tine is deduced and analysed.

6 Cosmological Vacuum Polarisation

Consider the result for gravitational vacuum polarisation derived in paper(D)

GρΛ = G−ΓB(t) + G+∆B(t) (6.1)

0 = G−ΓZ(t) + G+∆Z(t), (6.2)

where G− = −G and G+ = G. The upper case Greek functions ΓB(t),∆B(t), ΓZ(t) and ∆Z(t) are defined from the equations of state for ∆ andΓ substances which together are assumed to form all the time conservedmaterial of the universe,

P∆B/c2 = ρ∆B,νc(t)ω∆(t) = ∆B(t) (6.3)

PΓB/c2 = ρΓB,νc(t)ωΓ(t) = ΓB(t) (6.4)

P∆Z/c2 = ρ∆Z,νc(t)ω∆(t) = ∆Z(t) (6.5)

PΓZ/c2 = ρΓZ,νc(t)ωΓ(t) = ΓZ(t). (6.6)

The Z subscript above denotes zero-point values. Let us now consider theEinstein cosmological constant, Λ, in relation to the Friedman equations,

8πGρr2/3 = r2 + (k − Λr2/3)c2 (6.7)

−8πGPr/c2 = 2r + r2/r + (k/r − Λr)c2. (6.8)

Einstein introduced a physical explanation for his Λ term by associatingit with a density of what is nowadays called dark energy in the form of an

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additional mass density, ρΛ, where ρΛ = Λc2/(8πG). Thus with this densitythe Friedman equations can be written with the Hubble function of epochtime H(t) as,

8πGρr2/3 = r2 + (k − 8πGρΛr2/3)c2 (6.9)

−8πGPr/c2 = 2r + r2/r + (kc2/r − 8πGρΛr) (6.10)

H(t) = r(t)/r(t) = (c/(RΛ)) coth(3ct/(2Rλ)). (6.11)

Thus the first friedman equation can be expressed as

8πG(ρ + ρΛ)/3 = H2(t) + (kc2/r2) (6.12)

8πGρTE = 3(H2(t) + kc2/r2) (6.13)

ρTE = ρ + ρΛ, (6.14)

where ρTE is the total density for mass at points within the boundary of the

universe as perceived by Einstein. Rearranging the first Friedman equation,we have

8πG(ρ + ρΛ)− 3(kc2/r2) = 3H2(t) (6.15)

8πGρ

3H2(t)+

8πGρΛ

3H2(t)− kc2

r2H2(t)= 1. (6.16)

The three Omegas which the astronomers use to display their measurementsare defined using the three terms on the left hand side of (6.16) accordingto which they have to add up to unity ,

ΩM(t) = 8πGρ/(3H2(t)) (6.17)

ΩΛ(t) = 8πGρΛ/(3H2(t)) (6.18)

Ωk(t) = −kc2/(r2H2(t)) (6.19)

ΩM(t) + ΩΛ(t) + Ωk(t) = 1. (6.20)

There is a very strong case (A,B,C,D,E) for identifying the dark energymass density that should account for Einstein’s constant Λ term as givenby twice the density introduced by Einstein,

ρ†Λ = 2ρΛ (6.21)

ρT † = ρ + ρ†Λ (6.22)

and this implies the formula (6.22) for the total amount of physical massdensity within the boundaries of the spherical universe in contrast with

332

(6.14). Thus equation (6.15) should be replaced by

8πG(ρ + ρ†Λ)− 3(kc2/r2) = 3H2(t) + 8πGρΛ (6.23)

8πGρ

3H2(t) + c2Λ+

8πGρ†Λ3H2(t) + c2Λ

− 3kc2

r2(3H2(t) + c2Λ)= 1. (6.24)

Thus we now have three new Omegas

Ω†M(t) = 8πGρ/(3H2(t) + c2Λ) (6.25)

Ω†Λ(t) = 8πGρ†Λ/(3H2(t) + c2Λ) (6.26)

Ω†k(t) = −k3c2/(r2(3H2(t) + c2Λ)) (6.27)

Ω†M(t) + Ω†Λ(t) + Ω†k(t) = 1. (6.28)

Here I shall be mostly concerned with the flat space case k = 0 so that thetwo possible and equivalent sets of Omegas satisfy the relations

ΩM(t) + ΩΛ(t) = 1 (6.29)

Ω†M(t) + Ω†Λ(t) = 1. (6.30)

Inspection of the formulae for H(t), ΩM(t) and ΩΛ(t) shows that ΩΛ(t)varies between 0 and 1 as t varies between 0 and ∞ and consequently from(6.29), ΩM(t) varies between 1 and 0. It follows that there will be a timewhen

ΩM(t0) = 1/4 (6.31)

ΩΛ(t0) = 3/4 (6.32)

and this event will happen regardless of any measurements. I have assumedthat the epoch time of this event in the history of the universe is givenby t0. Thus the usual use of the subscript 0 to denote time now has beenabandoned and time now will in future be denoted by t†. The correspondingand more realistic time t0 relation between non-dark energy materials anddark energy will with a simple calculation be represented in terms of thedagger Omegas by

Ω†M(t0) = 1/7 (6.33)

Ω†Λ(t0) = 6/7. (6.34)

This implies that about 85.7% of the universe mass is dark energy ratherthan the usually assumed 75%, a substantially changed assessment. If this

333

assessment of the percentage of dark energy to conserved mass is accepted,it will also have some effect on the amount of visible mass assumed tobe present within the total mass of the universe. The ratio dark mass tovisible mass is often taken to be 4 to 1. Thus the percentage of dark massin the universe according to (6.33) and (6.34) would become reduced to20× (4/7)% ≈ 11.44%. The total non-visible mass would then be 85.7% +11.44% ≈ 97.14% leaving us with being able to see just about 2.86% of thetotal mass. If it is taken that we know nothing about the dark elements, asis often suggested, then our actual knowledge of the universe is mass wiseabysmal. However, fortunately it is not true that we have no knowledgeof the dark elements. We do have indirect knowledge of these aspects.The theory associated with this model give a definite relation between darkenergy and dark mass this relation can be read off from the gravitationpolarisation equations (6.1, 6.2) repeated next

GρΛ = G−ΓB(t) + G+∆B(t) (6.35)

0 = G−ΓZ(t) + G+∆Z(t) (6.36)

ρ(t) = ρ∆,νc + ρΓ,νc . (6.37)

The third equation above expresses the total time conserved density ρ(t)in terms of the CMB mass density, ρΓ,νc , and the rest of the universemass density ρ∆,νc . The νc subscript indicates that zero point energies areincluded in these terms. The second equation above defines the zero-pointenergy of the dark energy as being zero, effectively defining energy zero forthis cosmology theory. The total energy density for this model equation(6.22) can thus be written as (6.41)

ρ†Λ = 2ρΛ (6.38)

ρT †(t) = ρ(t) + 2ρΛ (6.39)

ρT †(t) = ρ∆,νc + ρΓ,νc + 2(∆B(t)− ΓB(t)) (6.40)

ρT †(t) = ρ∆,νc + 2∆B(t) + ρΓ,νc − 2ΓB(t) (6.41)

ρT †(t) = ρ∆,νc + ρΓ,νc (6.42)

ρ∆,νc = ρ∆,νc + 2∆B(t) (6.43)

ρΓ,νc = ρΓ,νc − 2ΓB(t). (6.44)

The tilde versions of the basic two densities are the resultants of a gravi-tational vacuum polarisation process in which the basic Γ and ∆ densities

334

induce, via their pressures and coexistence, the two polarisation densities2ΓB(t) and 2∆B(t) which together represent the dark energy density ρΛ,equation (6.35). This process takes place through the equations of motionof the two components. Thus from this point of view dark energy withinthe universe boundary is a vacuum polarisation consequence of the of theexistence of the basic Γ and ∆ fields in interaction under general relativ-ity. The dark energy density also exists outside the universe boundary butin an un-polarised condition. Thus the polarisation within the universe isconstrained by the constant value that exists everywhere. To examine theweight of this gravitational vacuum polarisation on the none polarised fieldsseparately at time t† using the numerical results from (A,B,C)

2ω∆(t†) ≈ 6 (6.45)

2ωΓ(t†) = 2/3 (6.46)

they must be expressed in terms off the none polarised fields as in (6.47)and (6.48)

2∆B(t†) ≈ 6ρ∆B,νc(t†) (6.47)

2ΓB(t†) = (2/3)ρΓB,νc(t†) (6.48)

ρ∆B,νc(t†) ≈ (104/1.9)ρΓB,νc(t

†) (6.49)

ρΓB,νc(t†) ≈ 1.9× 10−4ρ∆B,νc(t

†) (6.50)

ρΛ = Λc2/(8πG) ≈ 7.3× 10−27 (6.51)

ρΓB,νc(t†) = aT 4(t†) ≈ 4.66× 10−31. (6.52)

The relation (6.49) also comes from (A,B,C). Thus we can express the pos-itively weighted ∆B and negative weighted ΓB vacuum polarisation densitypoles as

2∆B(t†) ≈ (6× 104/1.9)ρΓB,νc(t†) (6.53)

2ΓB(t†) ≈ (2/3)1.9× 10−4ρ∆B,νc(t†) (6.54)

2∆B(t†) ≈ 3× 104ρΓB,νc(t†) (6.55)

2ΓB(t†) ≈ 1.26× 10−4ρ∆B,νc(t†). (6.56)

Returning to the gravitational vacuum polarisation equation (6.1) repeatedhere for convenience,

GρΛ = G−ΓB(t) + G+∆B(t) (6.57)

0 = G−ΓZ(t) + G+∆Z(t), (6.58)

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we can do a spot numerical check using the values above and without theG factor as follows

7.3× 10−27 ≈ ρΛ = ∆B(t†)− ΓB(t†) (6.59)

= ρ∆ω∆ − ρΓωΓ (6.60)

≈ (3× 104 − (1/3))ρΓ (6.61)

≈ (3× 104)ρΓ (6.62)

≈ (3× 104)× 4.66× 10−31 (6.63)

≈ (13.98/1, 9)× 10−27 (6.64)

≈ 7.3× 10−27. (6.65)

This is just a rough check that does give a good though approximate resultwhile showing that the induced ∆ and induced Γ fields in the form of adifference are the source of the dark energy density within the universesboundaries. At step (6.61), the −1/3 term from the Γ field is abandonedbecause it contributes negligibly in relation to the 104 from the ∆ term.However, at step (6.62) the Γ field only appears to be a main contributorbecause it occurs as multiplicatively weighted by the ∆ factor, 104. As the∆ field is all the conserved universe field density less the CMB the induceddelta field ∆ is all the induced conserved density universe field less the in-duced CMB field. The ∆ field includes the so called dark matter as itsmajor contributor of about 80% with normal visible mass making a smallerpercentage of about a 20% contribution. Thus the important conclusion isthat dark energy value within the universe is a direct consequence of theinduced mass from the ∆ field which itself is largely dark mass . Briefly,dark energy within the universe is numerically very close in value to thevacuum polarised dark mass and if the Γ field is also classified as dark thecloseness becomes coincidence. From the preceding discussion and equation(6.57) it should not be inferred that dark mass is a primary source of darkenergy. I think the reverse is nearer to the truth and equation (6.57) isthe direct result of a mechanical equilibrium between pressure equivalentinduced density from the CMB and the sum of the pressure induced densi-ties from the ∆ and Λ field at the boundary and within the universe. Thusthis mechanical equilibrium effectively transfers the dark energy pressurefrom outside the universe to its boundary and hence by homogeneity to in-side the universe. The PEID concept will be explained in the next sectionon pressure equivalent induced densities.

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7 Pressure Equivalent Induced Density, PEID

It turns out to be very useful to introduce the concept of Pressure EquivalentInduced Density, PEID , in relations to the equations of state associatedwith specific subsystems of the total system. For example, suppose onesubsystem is called the ∆ system with the equation of state,

P∆(t) = c2ρ∆ω∆(t) (7.1)

∆(t) = ρ∆(t)ω∆(t) (7.2)

= P∆(t)/c2, (7.3)

then I take the definition for the PEID, ∆(t), to be given by equation (7.2).Thus ∆(t) has the same dimensions as density because in common with allthe omegas, ω∆(t), is dimensionless and it is derived from ρ∆(t) throughthe multiplicative action of the inducing function, ω∆(t). From (7.3) it isclearly essentially a pressure with the dimensions of density. It representsthis pressure in the form of the mass density , ∆(t). I am not aware thatthe PEID slant on equations of state has any important part elsewherein physics but it seems that it does play an essential role in cosmology inrelation to the understanding of dark energy and its connection to otherkey densities. This is clear from inspection of equation (6.1) again with andwithout the G weightings,

GρΛ = G−ΓB(t) + G+∆B(t) (7.4)

ρΛ = ∆B(t)− ΓB(t). (7.5)

Thus from equation (7.5) the source of dark energy density within the uni-verse is just the difference of the PEIDs for the ∆ and Γ fields which to-gether constitute all the conserved mass of the universe. Thus the mysteryof the origin of the dark energy density, ρΛ = Λc2/(8πG) in Einstein’s form

or in my revised form ρ†Λ = 2ρΛ, within the universe is completely resolvedby this theory. Possibly this is the reason that dark energy is not visible.It could be because pressures are not usually visible and the pressure sta-tus of the dark energy density is its dominant characteristic. However, itseems to me that dark energy with approximately an equivalent density of5 hydrogen atoms per cubic meter would not be visible anyway. The for-mula (7.5) can also be used to show a simple relation between dark massand dark energy but before discussing that aspect it is useful to considerin the next paragraph the way this theory structure has developed and can

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continue developing. In the first two papers, A and B of the four A,B,C,D ,I found the dust universe model from scratch by just integrating the Fried-man equations. The result subsequently turned out to be a reincarnation ofthe first model introduced by Lema ıtre [25] but with substantially differentinterpretations and additional details. The version of the model in A andB , like most cosmological models, involved the assumption that the massdensity of the universe only depended on time and so was space-wise homo-geneous. However, the structure unearthed in that version of the model wascompletely adequate to describe cosmological expansion and its change fromdeceleration to acceleration at some time tc in the past and various othernew understandings of the cosmological process, all in complete agreementwith up to date measurement. Thus this basic structure did not dependon differentiating the mass density into separate components to representvarious contributory fields such as the electromagnetic or heavy particlecontributions. The dark energy contribution was involved in that version ofthe theory but not included as part of the conserved mass of the universe,it was rather treated as a permanent constant density resident of the hyper-space into which the universe expands. I shall here denote that model byU0 = UΛ(DM), meaning that it can be assumed to only contain an energyconserved over all time quantity of dark mass, MU , while, as we have seen,it swims in and is permeated with the dark energy content of an enveloping3D-hyperspace. The conserved mass density, ρ(t) ∼ ΩM(t), in this modelmust represent all the dark mass, if we assume that none of this dark masshas converted into visible mass and further because it satisfies the equation(6.29) which has to add up to unity to ensure that fact. Thus the modelUΛ(DM), can be conceived as not containing any visible hadronic matter,which as we know can only be present in a very small proportion anywayand it would also likely be none uniformly distributed. It follows that themodel U0 = UΛ(DM) can be regarded as a very bland, over all time, ap-proximation to the actual universe and which can be built up in stages torepresent the universe with increasing accuracy. I emphasise the usual cos-mological basic assumption that the model’s density function is space-wisehomogeneous means that if the model contains any dark mass within itsboundaries then it contains only uniform dark mass and together with theuniformly distributed dark energy background. The next stage in the buildup process in which the cosmic microwave back ground was added was pub-lished in C and will be denoted by U1 = UΛ(DM = ∆(t) ∪ Γ(t)). This

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means that the fixed amount of dark mass in the first version is now ableto transform into time dependent components ∆(t) for one part and Γ(t)for the complementary part, the CMB, with the same total mass quantityas the original dark mass. The next stage of complexity is the introductionof the possibility that part of the ∆ mass, MU can transform into visi-ble mass, often called hadronic mass. This universe can be represented byU2 = UΛ(DM = (∆(t) = ∆D(t) ∪∆V (t)) ∪ Γ(t)) with now the quantity of∆ mass being shared between the dark and visible versions as denoted bythe D and V subscripts. Clearly the increasing complexity procedure cancontinue to produce universes with lower homogeneity described by U3 andso on. Let us now return to discussing the relation between dark mass anddark energy.

8 Dark Mass, Dark Energy Ratio

Consider firstly the basic universe type Friedman dust universe, U0. Themodel in this basic case is an excellent representation of the modern as-tronomical measurements. However the basic density function is assumedto be rigorously homogeneous and contains only conserved with time darkmass and the hyperspace permeating dark energy. The density functionsfor the dark mass, dark energy and the ratio, rΛ,DM(t), of dark energy todark mass as functions of time are respectively represented by

ρ(t) = (3/(8πG))(c/RΛ)2 sinh−2(3ct/(2RΛ)) (8.1)

ρ†Λ = (3/(4πG))(c/RΛ)2 (8.2)

rΛ,DM(t) = ρ†Λ/ρ(t) = 2 sinh2(3ct/(2RΛ)) (8.3)

rΛ,DM(±tc) = 2 sinh2(±3ctc/(2RΛ)) = 1. (8.4)

Equation (8.3) is a general result but in the case of a U0 universe it can beexpressed differently by using equation (7.5) with the Γ term taken zero as

ρΛ = ∆B,0(t) (8.5)

= ρ(t)ω∆,0(t), (8.6)

the zero subscripts having been added to differentiate the functions con-cerned from those in the U1 version. From paper C, we know that

ω∆(t) =

(MΓ

3MU

+3(c/RΛ)2ρ−1(t)

8πG

)/(1−MΓ/MU). (8.7)

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Thus the zero Γ version for U0 is given by

ω∆,0(t) =

(3(c/RΛ)2ρ−1(t)

8πG

). (8.8)

Substituting this into equation (8.6) confirms the validity of (8.6). Thus therather trivial equation (8.6) gives the all time dependent relation betweendark energy and dark mass for the nontrivial model U0. However, trivialor not, the dark energy and dark mass densities are strongly numericallyrelated through the function ω∆,0(t) and this applies for all time, (−∞ <t < +∞). Let us now consider the ratio, rΛ,DM(t), of dark energy to darkmass in the case of a universe in which the homogeneity has been brokenby the addition of the cosmic microwave background, replacing some by theCMB. From (8.3), we have generally,

rΛ,DM(t) = ρ†Λ/ρ(t) = 2 sinh2(3ct/(2RΛ)). (8.9)

However, with the addition of the Γ field

ρ(t) = ρ∆(t) + ρΓ(t) (8.10)

so that the dark energy dark mass ratio of U0 at (8.9) becomes in U1

rΛ,DM,1(t) =ρ†Λ

ρ∆(t) + ρΓ(t)= 2 sinh2(3ct/(2RΛ)). (8.11)

The denominator of the ratio remains unchanged as also does the secondequality because the numerical values are unchanged. It might be thoughtthat the left and right sides of the first equality do not now agree becauseonly the ∆ part contains dark mass, that which is left from the U0 universecase after some has converted to CMB. Numerically there is no problemas the quantity of dark mass is presumably shared between the ∆ and Γfields. However, the terminology might be questioned. Arguably , the CMBis composed of photons which are not visible and therefore the CMB canbe classified as dark mass equivalent material. Of course photons conveyinformation about other visible materials to the eye but photons themselvesare not seen in the usual meaning of the word. I have added the extra sub-script 1 in the U1 ratio so that no confusion can arise if the case I have justmade is not accepted. The dark energy dark mass ratio in either form aboverepresents a fundamental time conditioned relation between dark mass and

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dark energy. This result and the formula (7.5) both of which hold insideand on the boundary of the universe show how totally interdependent arethe two dark facets. The ratio rΛ,DM(t) is of great generality and couldplay an important part in helping to understand cosmological voids , a re-cent astronomical discovery. This ratio has come out of general relativitybut it can be shown that it is independent of general relativity and itsexistence only depends on some simple assumptions added to Newtoniangravitational theory. The very basic and major significance of this ratio willbe discussed and demonstrated in the next section by showing that it isdirectly derivable from Newtonian gravitational theory. It will be indicatedhow this implies a context for its significance within smaller regions of spacewithin the universe’s boundary.

9 Newtonian Dark Mass and Dark Energy

Consider an infinitely extended 3-dimensional Euclidean space such as thatin which Newtonian gravity is usually considered to act between objectshaving the physical characteristic called mass. I shall make the usual as-sumption that Newtonian gravity acts between enclosed regions of spaceof spherical shape that enclose a uniform density distribution of mass thatcan change with time but retaining an overall fixed quantity with respect totime of the usual positive gravitational mass within it boundary, an amountM , say. Usually there will be some moving gravitational centroid at whichthe gravitation force between objects will be thought to be acting. I alsoonly use configurations in which this centroid is the centre of a sphere. Thedifference from Newtonian theory that I am about to introduce is the as-sumption that this Euclidean space is filled uniformly throughout all itsextent by a positively mass density field of negatively characterised grav-itational material such as the dark energy found to exist in the cosmos.This negative gravity material will be denoted by the constant density,ρ†Λ = c2Λ/(4πG) just as in my double version of the Einstein theory quan-tity, ρΛ = c2Λ/(8πG). Consider now a spherical region of this space of radiusr about the origin of this space as centre. Suppose this sphere contains atotal amount of dark mass , M , with its positive gravitation characteristic,G. The sphere will also contain an amount of negative gravity, −G, dark

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energy given by

MΛ = ρ†ΛV (t) (9.1)

V (t) = 4πr3(t)/3. (9.2)

Thus the total gravitational acceleration caused by the sphere’s contents atits surface will be given by the Newtonian gravitational formula,

r(t) = M †ΛG/r2(t)−MG/r2(t) (9.3)

= 4πr3ρ†ΛG/(3r2)− C/(2r2) (9.4)

= 4πrρ†ΛG/3− C/(2r2) (9.5)

= rc2Λ/3− C/(2r2). (9.6)

If we multiply equation (9.5) through by r, we obtain

rr = 4πrrρ†ΛG/3− Cr/(2r2) (9.7)

d

dtr2/2 =

d

dtr2Λc2/6 + C

d

dtr−1/2 (9.8)

r2 = (rc)2Λ/3 + Cr−1 (9.9)

C = 2MG. (9.10)

The constant of integration that could occur in integrating (9.8) can betaken to be zero under the conditions that r(t) is taken to be infinite withr(t) = 0 at t = 0. Thus the spherical region expands with high speed fromthe origin, r = 0 at time t = 0. The solution to equation (9.9) was obtainedin paper A in the form

r(t) = b sinh2/3(3ct/(2RΛ)) (9.11)

RΛ = (3/Λ)1/2 (9.12)

b = (RΛ/c)2/3C1/3 (9.13)

C = 2MG (9.14)

where M here is any dark mass value. It follows that the dark mass densityof the spherical region containing total dark mass, M , is as in (8.1) givenby

ρ(t) = M/(4πr3(t)/3) = M sinh−2(3ct/(2RΛ))/b3 (9.15)

= (3/(8πG))(c/RΛ)2 sinh−2(3ct/(2RΛ)). (9.16)

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Thus the ratio of dark energy mass density to dark mass density within thisregion over time is

rΛ,DM(t) = ρ†Λ/ρ(t) = 2 sinh2(3ct/(2RΛ)) (9.17)

which again is the same as (8.3). The formula for the ratio of dark energyto dark mass, rΛ,DM(t), depends only on the dark energy mass densitythrough t and RΛ. The time variable origin t = 0 depends only on wherethe sphere expansion is assumed to have started from with radius zero, anarbitrarily chosen space origin r(t) = 0 at time t = 0, in Euclidean threespace. Thus it seems that this is a fundamental formula governing a timeevolutionary process relating dark energy and dark mass. The consequenceof this situation is that we can visualise, quite independently of relativity,such mixed mass region expansions. They can take place over time fromanywhere in astro-space and apparently originate from a point quantity ofdark mass , M , with infinite density. Further, the formula is time reversibleso that it suggests that spherical contractions of spherical dark mass regionscan also be visualised as a possible cosmological sequence of events resultingin the appearance of a point dark mass, M, with infinite density locally. Assuch an expansion proceeds the spherical region picks up dark energy massfrom the enveloping Newtonian space, the expansion continuing with theexpanding region having then a mixture of the two gravitational types ofmass, ±G. An important event in the history of such an expansion is whenthere are equal quantities of the two mass types within the sphere. At thisevent occurring, the sphere will be gravitationally neutral. The sphere willat that time exert no gravitational force on material outside its boundary,it will be gravitationally isolated from any material exterior to itself. If wedenote the time when the sphere is so isolated by tc this time can be foundfrom the formula of dark mass and dark energy mass equivalent equality,either equation (9.18) or equation (9.19)

rΛ,DM(tc) = ρ†Λ/ρ(tc) = 1 (9.18)

ρ†Λ = ρ(tc) (9.19)

sinh2(3ctc/(2RΛ)) = 1/2 (9.20)

⇒ tc = ±(2RΛ/(3c)) sinh−1(1/21/2) (9.21)

and, curiously, the times ±tc do not depend on the amount of dark masswithin the expanding sphere but only depends on the value of the cosmo-logical constant, Λ. It follows that the time tc has exactly the same value

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as the relativistic epoch time when the universe changes from decelerationto acceleration. The time tc is a fundamental universal time interval in thecosmological context. It is important to note that, as the process is timereversal invariant, the contraction sequence, in negative time, with mass Mcan be immediately followed by an expansion sequence with the same massM , in positive time, so that conservation of mass is assured and mass isneither created from nothing nor is it destroyed at the singular event whent = 0. The non dependence of the process on the amount of dark masswithin the boundary of the contracting or expanding sphere of dark masshas a surprising explanation. The process conforms exactly to the principleof equivalence. Just as the acceleration of a falling mass in a gravitationalfield does not depend on the value of the falling mass so the accelerationrΛ,DM(t) of the collapsing sphere process does not depend on its mass. Thecollapsing sphere in its own gravitational field conforms exactly too and isa manifestation of the principle of equivalence. It can occur locally andis a basic part of the description of the whole universe motion with epochtime. Recognition of this fundamental process in relation to other physicalprocesses in cosmology will be discussed in the final section.

10 Appendix 1 Conclusions

The cosmological model introduced in references A, B, C and applied tothe finding of solutions to the cosmological constant problem in D has herebeen applied to unravelling the dark mass problem. Here it has been shownthat a fundamental time moving relation holds between dark energy anddark mass . This relation was first shown to hold at the scale of the wholeuniverse by using the Friedman equations from Einstein’s general relativityand involving his positively valued cosmological constant Λ. Here it has beenshown that the same relation can be derived from Newtonian gravitationtheory with only the addition of a constant and universally distributeddensity of dark energy, ρ†Λ = 2ρΛ, twice the Einstein value ρΛ, in Newtonianspace and only subject to Newtonian gravity theory. This result implies thatthe formula relating dark mass and dark energy is independent of generalrelativity and the way it is derived also show that it can have applicationsat a much smaller scale than that of the entire universe. It can describelocal space and time small scale movements of dark mass in relation todark energy. Thus I suggest the formula could play an important role in

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explaining the way that dark mass, if taken to be primary positive gravity,+|G|, mass, can condense, precipitate or clump to become galaxies or justempty voids [35] in the cosmological fabric. As we have seen, there are fivemain events in the time sequences evolution of this dark energy dark massprocess, E0, E±1, E±∞, say. They involve E0 when some definite randomquantity of dark mass M is located at some definite point in three spaceat some definite time labelled as t = 0 for the process. At that time thedark mass is by itself because a point cannot contain any of the uniformand finite constant density of dark energy mass. Thus in space around thepoint mass it will own a Newtonian gravitational potential field −MG/r. Atboth the events E±1 at times ±tc because of the time reversal invariance thecontracting or expanding sphere will contain equal quantities of the darkmass and dark energy so that the sphere will be gravitationally neutral.It will thus be isolated gravitationally and so not own any gravitationalpotential. However the total mass density within the spheres boundarieswill be ρ(tc) + ρ†Λ, a numerically very small value ≈ 9 proton masses percubic meter. I think that such a sphere being gravitationally isolated and ofsuch low density could qualify for the title cosmological void . At the eventsE±∞, the sphere will own a gravitation potential at points within its surfaceinvolving both the dark energy and dark mass within concentric spheres ofradius r < ∞ but dominated by the repulsive dark mass for relatively largevalues of r. The contraction phase between E−∞ and E0 might representa moving platform for an original dark mass concentration to convert frompure dark mass to becoming dark mass contaminated with visible mass whileits volume descends to occupying some relatively small region containing agroup of visible galaxies or, a single galaxy or even a single particle. In otherwords, the descending spherical volume could represent a time dependantpackaging process for cosmological clumping. A final remark about therelation of this theory structure to aether theory is appropriate. It is darkenergy rather than dark mass that seems to play a role much like the allpervading aether which has been used to give a physical explanation forelectromagnetic wave motion in so called empty space. The dark energydensity is certainly an all-pervading effect in this cosmological theory ashas been shown in this article and as it is also perceived in the present dayarena of astronomical observations. It seems to be an everywhere presentbackground reference level against which many astrophysical and quantumproblems can be understood and measured. The dark mass or positive

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gravity element appears to represent a measure of a soliton like wave effecteither universally or locally of a boundary motion at an interface betweendark mass and dark energy described by the inverse of the ratio, rΛ,DM(t).

11 Appendix 2

Expanding Boundary Pressure ProcessAll Pervading Dark Energy Aether

in a Friedman Dust Universewith Einstein’s Lambda

Abstract

In this appendix, the cosmological model that was introduced in a sequenceof three earlier papers under the title A Dust Universe Solution to the DarkEnergy Problem is used make a more detailed study of the role of dark energymass as a conserved with time substance that permeates the expandinguniverse. It shown that if dark energy is to be conserved over all time ithas to satisfy the cosmological vacuum polarisation equation over the pre-singularity range of contraction and the post-singularity range of expansionin order for it to remain in a self mechanical equilibrium inside and outsidethe boundary of the expanding universe and so be able to be always andeverywhere permeable to the positive gravity dark mass and visible massmaterial within the universe.

12 Effect of Boundary Pressures

In the paper D, [34], it was shown that the quantum vacuum polarisationidea can be seen to play a central role in the Friedman dust universe modelintroduced by the author. An essential part of that role involves the rela-tions between three pressures at the boundary of the expanding universe.In particular of fundamental importance is a relation between pressure fromthe CMB, PΓ, pressure from all the rest of the universe which is not CMBand not dark energy, P∆, and pressure from dark energy itself, PΛ. This

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relation takes the form

PΓ = P∆ + PΛ (12.1)

PΛ = c2ρΛωΛ = −c2ρΛ (12.2)

ωΛ = −1. (12.3)

Equation (12.2) together with equation (12.3) is the well known specifica-tion implying negative pressure, PΛ, from the dark energy density in theEinstein form ρΛ = Λc2/(8πG). In earlier work, I have referred to the equa-tion (12.1) as representing a mechanical equilibrium between the Γ field andthe ∆ and Λ fields combined. I now think that designation while remainingformally correct should be presented with a changed interpretation becauseof the negative pressure associated with the dark energy field, Λ. In theusual specification of a mechanical equilibrium two pressures P1 = P2 aresaid to be equal where there is no complication of possible negative parts foreither of them. Pressures on either side of a boundary between non-miscibleliquids for example are said to be in mechanical equilibrium if the bound-ary is not accelerating. In such a case, although the pressures act at theboundary in opposite directions they are both taken as positive. Mechani-cal equilibrium in thermodynamics is a very contentious area of research sothat my explanation in this context is very minimal. As a result of thesecomplications it is desirable to express equation (12.1) in the alternativeform using a modulus sign, | |. The term boundary of the universe at timet refers to a conceptual sphere of radius given by the function r(t) definedearlier (A).

PΓ = P∆ − |PΛ| (12.4)

P∆ = PΓ + |PΛ|. (12.5)

In this form, all the pressures are expressed as positive quantities and themechanical equilibrium between these three field can now be more safelyreinterpreted as a mechanical equilibrium between the ∆ field and the Λand Γ fields combined. This version of the equilibrium condition at theboundary of the expanding universe makes good sense physically for atleast two reasons. The first reason is that dark energy material exists onboth sides of the expanding boundary of the universe so the ±|PΛ| versionsrefer to the pressure direction on the boundary from dark energy materialon one side or the other, whilst the gravitational pressure, P∆, is directedtowards the material within the universe and so, on the boundary, is only

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effectively equivalent to a positive pressure towards the centre of expansion.This last property is a well-known result originating from Newtonian gravi-tational theory. Taking the total pressure, P , in the Friedman equations aspositive is a rather anomalous convention which has caused much confusionwhich I attempt here to unravel. The second reason is that the form (12.4)rearranged as in (12.6)

PΛ = −|PΛ| = PΓ − P∆ (12.6)

P (t) = +PG + PΛ = P∆(t)− PΓ(t) + PΛ ≡ 0 ∀ t. (12.7)

clearly expresses the physics of the equilibrium condition. It is that thenegative outward pressure PΛ that would be exerted on the boundary fromthe dark energy inside the universe is equal to the difference of the outwardCMB pressure, PΓ, less the inward pressure P∆ exerted on the universeboundary from within by the positive G or normal gravitating materialwithin the universe. The equations (12.6) and (12.2) firmly identify boththe pressure PΛ and the mass density ρΛ of the dark energy as coming fromthe quantities PΓ and P∆ both defined with meanings within the universe.Elsewhere, I have expressed the equation (12.6) using the PEID form whichexplains it in terms of the mass densities, Γ(t), ∆(t) rather than the equiv-alent pressures,

ρΛ = ∆(t)− Γ(t). (12.8)

GρΛ = G+∆(t) + G−Γ(t) (12.9)

G+ = +G (12.10)

G− = −G. (12.11)

The equation (12.7) uses (12.6) to bring us back to the total pressure P (t)which as indicated is identically zero for all t and so indicates that the wholehistory of the universe in this model is that of a dust universe. Clearly thenthe total pressure P (t) cannot be responsible for the acceleration. Thisconclusion agrees with what was noted earlier that the acceleration is accu-rately determined by a generalisation of the Newtonian gravitation theoryonly involving adding to the inverses square law a linear law term involvingEinstein’s cosmological constant Λ. The realisation that the various pres-sures that we have been discussing earlier do not determine the dynamicalbehaviour of the system generates the question, what is this complicatedpressure structure all about?

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The immediate answer to the question at the end of the last paragraphis that the pressure structure that derives from the equilibrium relation be-tween the three pressures, PΛ, P∆ and PΓ generates the important relation,(12.8) or (12.9). This relation shows that within the spherical volume ofthe universe the constant valued dark energy density, ρΛ, is determined bythe dark mass dominated quantity pair ∆(t) and Γ(t). I emphasise withinbecause both these quantities are part of the constant space and time con-served energy of the universe, MU . However, the space-time constant darkenergy mass density ρΛ, by initial assumption, exists everywhere in the uni-verse’s enveloping space with the same definite numerical value outside asinside. Thus the following subsidiary question presents itself: If the darkenergy density inside the universe is given by (12.8) in terms of the inter-nal constituents, ∆ and Γ, how is it that outside the universe involvingregions which will not have been reached by the internal constituents ofthe expanding universe, the dark energy density ρΛ exist in its own right byassumption, with the same constant value as inside and apparently not gen-erated by any internal influence? The unique character of this model doesallow a satisfactory answer to this question which depends on the model’sstrict conformance to the principle of conservation of energy in contrastwith the standard big bang model . This model involves two basic positivetypes of mass defined by their gravitational character, which is determinedby whether the mass appears in the theory multiplied with G+ = +G orG− = −G, where the gravitational constant G is always constant, G > 0.Dark mass and normal mass belongs to the G+ category and dark energybelongs to the G− category. An important way in which this model differsfrom the big bang type universe is that the beginning of time in this theory,rather than occurring at time t = 0, occurs at time t = −∞ and the end oftime occurs at t = +∞. This can be interpreted as this universe lasts forever . The reader may prefer to regard this infinite time scale as just oneout of a possible infinite number of infinite contiguous periodic time scalesand so reinforcing the lasting for ever concept. This last extension can beusefully incorporated in the theory, see paper (C). Let us now consider thesituation at and after the start of time taken as t = −∞ + tε, tε ≈ +|0|at this stage the radius of the universe is infinitely large, r(−∞ + tε), andwill decrease with advancing time. In other words, near the beginning oftime the universe is a sphere occupying almost the whole of hyperspace andso the internal generating dark energy process (12.8) is operative almost

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everywhere in hyper space. It follows that the universe is full of the darkenergy density ρΛ, though this density is itself small it adds up in total overthe whole universe volume to a very large amount of dark energy mass. Theuniverse also contains a much smaller density value, ρ(−∞+tε), of conservedpositive gravitational mass, MU , so that ρ(−∞ + tε)VU(−∞ + tε) = MU .This conserved mass contains the mass of the universe that we see. Thebasic assumption in this model is that dark energy density, ρΛ(t), existseverywhere and at all time so that if the radius of the universe at t = −∞is infinite and if the space is flat Euclidean then the universe has no outsideand takes in all the hyper-universe so that all the dark energy is enclosed.After the small time elapse, tε, the universe will have acquired a small out-side volume and a slightly smaller inside volume than it had initially. Itfollows that in principle there are two types of simple likely possibilities.Firstly, the contracting universe leaves no dark energy density outside asit evolves in time and keeps the original value inside at the same valueas given by formula (12.8). Secondly, as it evolves in time it leaves out-side sufficient dark energy density to keep to the uniform constant densitycondition everywhere and so keeping the dark energy density inside at thesame value given by formula (12.8) as outside. The first option means thatthe dark energy within the universe would decrease with decreasing volumeconsequently losing density to no recognisable sink and so implying darkenergy is not conserved. This would also violate the assumption that darkenergy density is constant everywhere and at all time. Thus we are left withonly the second possibility and consequently the actual scenario has to bethat as the universe contracts the internal pressure process described by theinternal ∆ and Γ fields precipitates the right amount of dark energy mate-rial outside in the space produce by the contracting universe. This processwill continue for all time and, in particular, past the singularity at t = 0when the volume is zero. Thus after the singularity, when the universe is inan expanding mode, it will encounter the pre-singularity dark energy den-sity outside its boundaries precipitated in its contracting mode. Thus themain role of formula (12.9) is to keep the conserved and bounded dark masswithin the universe freely permeable to or non interacting with the darkenergy in which it swims by maintaining the self mechanical equilibrium ofthe dark energy mass density in the form,

PΛ,in = PΛ,out (12.12)

ρΛ,in = ρΛ,out, (12.13)

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where PΛ,in is the dark energy pressure just inside the boundary and PΛ,out

is the dark energy pressure just outside the boundary of the universe. Equa-tion (12.13) gives the same condition in terms of densities.

13 Appendix 2, Conclusions

All this suggests that the dark energy density, although pressure identified,has also to be taken seriously as a genuine mass density. It also shows thatthe pre-singularity negative time phase is a necessary adjunct to makingsense of this theory. The conclusion associated with this section is that theformula (12.9) together with the full time history of this model assures thatdark energy and dark mass are both conserved over all time. The above ar-gument is not meant to be mathematically rigorous but rather a plausibilityconstruction. No doubt the reader can think of various improvements.

References

[1] R. A. Knop et al. arxiv.org/abs/astro-ph/0309368New Constraints on ΩM , ΩΛ and ω froman independent Set (Hubble) of Eleven High-RedshiftSupernovae, Observed with HST

[2] Adam G. Riess et al xxx.lanl.gov/abs/astro-ph/0402512Type 1a Supernovae Discoveries at z > 1From The Hubble Space Telescope: Evidence for PastDeceleration and constraints on Dark energy Evolution

[3] Berry 1978, Principles of cosmology and gravitation, CUP

[4] Gilson, J.G. 1991, Oscillations of a Polarizable Vacuum,Journal of Applied Mathematics and Stochastic Analysis,4, 11, 95–110.

[5] Gilson, J.G. 1994, Vacuum Polarisation andThe Fine Structure Constant, Speculations in Scienceand Technology , 17, 3 , 201-204.

[6] Gilson, J.G. 1996, Calculating the fine structure constant,Physics Essays, 9 , 2 June, 342-353.

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http://arxiv.org/abs/astro-ph/0309368

http://xxx.lanl.gov/abs/astro-ph/0402512

[7] Eddington, A.S. 1946, Fundamental Theory, Cambridge UP

[8] Kilmister, C.W. 1992, Philosophica, 50, 55.

[9] Bastin, T., Kilmister, C. W. 1995, Combinatorial PhysicsWorld Scientific Ltd.

[10] Kilmister, C. W. 1994 , Eddington’s search for a FundamentalTheory, CUP.

[11] Peter, J. Mohr, Barry, N. Taylor, 1998,Recommended Values of the fundamental Physical Constants,Journal of Physical and Chemical Reference Data, AIP

[12] Gilson, J. G. 1997, Relativistic Wave Packingand Quantization, Speculations in Science and Technology,20 Number 1, March, 21-31

[13] Dirac, P. A. M. 1931, Proc. R. Soc. London, A133, 60.

[14] Gilson, J.G. 2007, www.fine-structure-constant.orgThe fine structure constant

[15] McPherson R., Stoney Scale and Large NumberCo-incidences, to be published in Apeiron, 2007

[16] Rindler, W. 2006, Relativity: Special, Generaland Cosmological, Second Edition, Oxford University Press

[17] Misner, C. W.; Thorne, K. S.; and Wheeler, J. A. 1973,Gravitation, Boston, San Francisco, CA: W. H. Freeman

[18] J. G. Gilson, 2004, Mach’s Principle II

[19] J. G. Gilson, A Sketch for a Quantum Theory of Gravity:Vol. 17, No. 3, Galilean Electrodynamics

[20] J. G. Gilson, arxiv.org/PS cache/physics/pdf/0411/0411085v2.pdfA Sketch for a Quantum Theory of Gravity:

[21] J. G. Gilson, arxiv.org/PS cache/physics/pdf/0504/0504106v1.pdfDirac’s Large Number Hypothesisand Quantized Friedman Cosmologies

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http://www.fine-structure-constant.org/

http://arxiv.org/PS_cache/physics/pdf/0409/0409010.pdf

http://arxiv.org/PS_cache/physics/pdf/0411/0411085v2.pdf


[22] Narlikar, J. V., 1993, Introduction to Cosmology, CUP

[23] Gilson, J.G. 2005, A Dust Universe Solution to the Dark EnergyProblem, To be published in Ether, Spacetime and Cosmology ,PIRT publications, 2007,arxiv.org/PS cache/physics/pdf/0512/0512166v2.pdf

[24] Gilson, PIRT Conference 2006, Existence of Negative GravityMaterial, Identification of Dark Energy,arxiv.org/abs/physics/0603226

[25] G. Lemaıtre, Ann. Soc. Sci. de Bruxelles Vol. A47, 49, 1927

[26] Ronald J. Adler, James D. Bjorken and James M. Overduin 2005,Finite cosmology and a CMB cold spot, SLAC-PUB-11778

[27] Mandl, F., 1980, Statistical Physics, John Wiley

[28] Rizvi 2005, Lecture 25, PHY-302,http://hepwww.ph.qmw.ac.uk/∼rizvi/npa/NPA-25.pdf

[29] Nicolay J. Hammer, 2006http://www.mpa-garching.mpg.de/lectures/ADSEM/SS06 Hammer.pdf

[30] E. M. Purcell, R. V. Pound, 1951, Phys. Rev., 81, 279

[31] Gilson J. G., 2006, www.maths.qmul.ac.uk/∼ jgg/darkenergy.pdfPresentation to PIRT Conference 2006

[32] Gilson J. G., arxiv.org/abs/physics/0701286

[33] Beck, C., Mackey, M. C. http://xxx.arxiv.org/abs/astro-ph/0406504

[34] Gilson J. G., 2007, Reconciliation of Zero-Point and Dark Energies ina Friedman Dust Universe with Einstein’s Lambda

[35] Rudnick L. et al, 2007, WMP Cold Spot, Apj in press

[36] Gilson J. G., 2007, Cosmological Coincidence Problem in an EinsteinUniverse and in a Friedman Dust Universe with Einstein’s Lambda

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http://arxiv.org/abs/physics/0603226

http://www.arxiv.org/abs/gr-qc/0602102

http://hepwww.ph.qmw.ac.uk/~rizvi/npa/NPA-25.pdf

http://www.mpa-garching.mpg.de/lectures/ADSEM/SS06_Hammer.pdf

http://www.maths.qmul.ac.uk/~jgg/darkenergy.pdf

http://arxiv.org/abs/physics/0701286

http://xxx.arxiv.org/abs/astro-ph/0406504

http://arxiv.org/abs/0704.2998


http://arxiv.org/abs/0704.0908v2



354

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

by James E. Beichler

I. Introduction

Nearly a half century before Einstein developed his general theory of relativity, the Cambridge geometer William Kingdon Clifford announced that matter might be nothing more than small hills of space curvature and matter in motion no more than variations in that curvature. Clifford assumed the reality of a fourth dimension of space according to the new non-Euclidean geometries. In this respect, Clifford merely followed the common assumption that geometry modeled physical reality, so the new non-Euclidean geometries represented real possibilities that space could be curved rather than Euclidean flat. These ideas were further elaborated in Clifford's Common Sense of the Exact Sciences of 1885,1 partially written and edited by Karl Pearson six years after Clifford's unfortunate death from consumption.

The short abstract of 1870 in which Clifford explained his model of space, "On the Space-Theory of Matter," 2 has long been recognized in studies on general relativity and its history, but Clifford's concepts of space and their relationship to physics have been limited to the role of "anticipation" 3 of Einstein's theory. Within this context, Clifford's model has been branded a "speculation" 4 that was "untenable" 5 during his brief professional career. E.T. Bell has gone so far as to liken Clifford's "brief prophecy" 6 to hitting "the side of a barn at forty yards with a charge of buckshot." 7 Yet these opinions of Clifford’s contributions are completely inaccurate within the context of Clifford’s time period era as well as when more recent trends in physics are taken into account. Clifford’s work should now be regarded as the first significant step toward a unification theory in physics, rather than a simple ‘precursor’ to general relativity.

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II. Clifford’s work in a modern context

Recently, there has been some renewed interest in the relationship between Clifford's theory and its role in the development of modern physics. Ruth Farwell and Christopher Knee have looked at Clifford's work as a "nineteenth century contribution to general relativity." 8 Joan Richards has written on the development of non-Euclidean geometry in Victorian England, a movement in which Clifford played a significant role,9 while Howard E. Smokler, a philosopher, has taken a new look at Clifford's concepts within a philosophical context.10

This apparent renewal of interest is not a totally new phenomenon, but has been occurring in regular cycles for some time. Nearly forty years ago, James R. Newman noted that the "neglect of Clifford [was] difficult to explain," 11 yet nothing strikingly new has been published on Clifford. The newest revelations offered by these scholars in recent years have not yet shed the preconceived historical outlook on Clifford that can be found in older post-relativity publications. They offer more of the same old inaccuracies tempered by a few bright spots of original historical research.

The renewed interest extends beyond the historical significance of Clifford's work to his mathematical system of biquaternions, first developed in 1873. A.E. Power, a mathematician at the University College in London, has published articles comparing Clifford's mathematical system to modern work in both quantum mechanics and relativity.12 Feza Gurney has written of the "hope and disappointment connected with the role of quaternions in physics," 13 another episode in which Clifford's mathematical system can be found to play an important role. A conference dedicated to Clifford's mathematical system and its application to modern physical theory was held at Canterbury, England, in 1985.14 The interest of physicists in Clifford's mathematical work received a large boost some years ago when John Archibald Wheeler developed some of Clifford's ideas in his "Geometrodynamics," 15 but that approach to relativity theory seems to have been abandoned. Something of a philosophical obituary has been written for it by Adolph Grunbaum.16

All of these new inquiries into Clifford's work are inadequate from the historical perspective. In many respects, they merely perpetuate the small myths concerning Clifford's role in the development and popularization of non-Euclidean geometry and physical space. In turn, these myths form part of a

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legend that has grown up around Einstein's discovery of general relativity and its acceptance by the scientific community. Einstein's discovery has been presented as the first successful attempt to describe physical space as non-Euclidean.17 This statement is beyond debate, but past historical studies on the subject either imply or state that attempts to associate the new forms of geometry with the physical world before Einstein were either non-existent or quite rare and isolated, i.e., "untenable." This attitude forms the general historical context against which Clifford's accomplishments are normally evaluated.

Regardless of the recent interest in Clifford's work, no one has noted the logical paradoxes that arise from these commonly accepted historical views. While it is generally accepted that Clifford's ideas were "untenable" in the 1870's, no one has yet addressed the complementary issue of how scientific attitudes had changed so much in the intervening years that nearly the same concept was "tenable" during Einstein's early career. In fact, general relativity was accepted quite rapidly by scientists, philosophers, mathematicians and scholars, as well as educated laymen, in spite of the radical notion of representing matter by space curvature.

On the other hand, if it could be demonstrated that the non-Euclidean geometries were popular enough that Einstein's version of curved space was not as radical in 1915 as Clifford's in 1870, then it would become necessary to find the event or factor between the two periods when the turning point in attitude occurred. The explanation of the rapid acceptance of relativity would be made easier, but at the expense of a cherished legend. The point at which the shift from a purely mathematical non-Euclidean geometry to the belief in a possible physical interpretation of such geometries cannot be so easily pinpointed. The obvious question would then become, what was Clifford's connection to this evolutionary process of accepting a physical non-Euclidean space? Historians could take the Whiggish way out of the dilemma and state that Einstein's concept was "tenable" because it was correct and/or accepted, but history cannot be served by having it both ways by denying the affect of Clifford's work while accepting Einstein's theory as "tenable" in the presence of a void.

Those authors who characterize Clifford's work as "speculation," support their conclusions by further stating that he had no followers who continued his work after his death,18 or he never published his theory.19 Since Clifford's was the strongest Victorian voice supporting a physical interpretation of non-Euclidean

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geometry, these accusations and misrepresentations imply that the physical interpretations of the non-Euclidean geometries were disregarded by the vast majority of scholars before Einstein.

In this regard, Poincaré's conventionalist attitude, that it is preferable to change the laws of physics (or optics) and retain Euclidean space rather than consider the possibility that space might be non-Euclidean,20 is usually cited as representing the prevalent view of the scientific and academic communities immediately prior to relativity. By accepting Poincaré's view as scientific doctrine during that period, historians and others have neglected the fact that Poincaré’s conventionalism was a reaction to the growing tendency to associate the non-Euclidean geometries with physical space well before the end of the nineteenth century. It was geometric escapism. The story that has evolved around these misinterpretations of the historical record forms one of the basic foundations upon which Einstein's theory is considered a solid break with past theoretical work in physics and attitudes on the non-Euclidean geometries.

Several years ago, Arthur I. Miller attempted to destroy another of the pillars of mathematical history that might challenge Einstein's absolute originality in relating a non-Euclidean geometry to physical space. He argued that J.K.F. Gauss' survey measurements from three mountaintops in Hanover in the 1820's originally had nothing to do with space curvature as many authors have indicated. According to Miller, that particular interpretation of Gauss' survey was made only after the development of general relativity.21 However, there is strong evidence that Gauss' survey measurements were interpreted as a measure of space curvature long before relativity theory was first introduced.22 This evidence also emphasizes the fact that the physical consequences of non-Euclidean geometry were a popular subject for discussion and debate within the scientific and academic communities during the late nineteenth century. The popularity of the non-Euclidean geometries and the possibility that they rendered physical consequences that could be investigated implies that there is no real basis for accepting the conclusion that Clifford's ideas were "untenable." Further investigation of Clifford's work would seem warranted, but has not yet been carried out. While admitting the necessity of investigating Clifford's work, recent authors have perpetuated the mistaken images of the past.23

These apparent paradoxes can be dispensed with quite readily by taking a fresh look at both Clifford's work and its reception, but this must be done within the

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greater historical context of the attitudes toward the non-Euclidean geometries during the period before relativity theory. The story that emerges from such a study demonstrates that Clifford's ideas were both "tenable" and popular, and raised profound questions within the academic community on the role of non-Euclidean geometry in physics. Clifford was working on a specific theory that was partially completed before his untimely death. Therefore, his ideas were not "speculations," but a serious effort to "solve the universe," 24 as Clifford would say. He also had followers who attempted to extend his work after his death.

Enough of Clifford's theory can be reconstructed from his various papers to indicate the general principles upon which he based his theory. In essence, Clifford's theory cannot be evaluated from just a cursory reading of his "Space-Theory" and the Common Sense, as past writers and investigators have done. All of his published papers must be investigated to understand the depth and breadth of his theoretical outlook. It was in this manner that Clifford's colleagues and peers interpreted his concepts. However, Clifford's mathematical theory was so abstract and so intimately bound to quaternions that it had minimal affect on the later development of relativity and perhaps disguised Clifford's work from the scrutiny of later scholars and historians. The purely mathematical portion of Clifford's theory was continued by Sir Robert S. Ball and Arthur Buchheim, among others, and primarily involved the mechanics of motion in an elliptical space.

The fundamental element of space curvature in Clifford's mathematical model was the twist, which he hoped to use to describe electromagnetic and atomic phenomena. Karl Pearson continued Clifford's development of the twist without reference to its relation to space curvature in his own development of the "ether-twist." Both Pearson's and these other extensions of Clifford's work were all bound to the Victorian principles and attitudes toward science that were already in decline before relativity struck. Correctly or incorrectly, they suffered from either an association with quaternion algebras or ether-vortex theories, or both, at the time when these physical concepts lost favor within the scientific community.

Of greater historical importance was the more general, philosophical concept of space as expressed by Clifford and its relation to the mathematical studies of non-Euclidean geometry. Included within this perspective would be the general model of non-Euclidean space presented in Clifford's "Space-Theory" abstract,

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but not exclusively by that presentation. In this regard, Karl Pearson, Frederick W. Frankland and Charles H. Hinton spread Clifford’s ideas. Clifford's purely mathematical studies were not immune from involvement in this aspect of the non-Euclidean debate, but the complexity of the mathematics involved allowed only the best and brightest mathematicians to draw conclusions regarding the physics of space and time directly from Clifford's mathematics. On the other hand, the philosophical contributions allowed anyone who could read and use their imagination to draw conclusions from Clifford's more general concepts.

Two generations of thinkers who expressed an interest in the non-Euclidean geometries and hyperspaces separated Clifford's original work from general relativity. During that time, the popularity of physical interpretations of the non-Euclidean geometries grew. When Einstein first completed and published his theory, it found an eager audience which already accepted the possibility that physical phenomena could be affected by space curvature. Although many scholars and laymen added their own thoughts to this general attitude during the decades before the adoption of relativity theory, Clifford's contributions went beyond all others in both content and breadth of view. His "Space-Theory" set the limits to which all others attained in their belief of a physical non-Euclidean space before Einstein institutionalized the concept that matter could be represented by space curvature

II.Clifford's Theory and the Issue of "Tenability"

The first public announcement that Clifford had been working on a new concept of space and matter came in J.J. Sylvester's presidential address before the British Association in 1869. The speech was subsequently published with footnotes in Nature, where it became available to a much larger audience with a more varied background. What they first learned of Clifford's work was that our space of three dimensions might be "undergoing in a space of four dimensions ... a distortion analogous to the rumpling" of a piece of paper.25 Sylvester also committed himself to a belief in a fourth dimension and mentioned that Immanuel Kant thought of space as a "Form of Intuition." Sylvester's interpretation of Kant's doctrine on space immediately triggered a debate over the Kantian meaning of the phrase "Form of Intuition" and Kant's notion of space as "a priori." 26 The ensuing debate over Kant's concept of space thus became the first line of defense for those who accepted the absolute truth of a three-dimensional Euclidean space.

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As the initial stages of this debate began to subside, C.M. Ingleby, a contributor to the Kant debate, took up the crusade against Clifford's concepts. Both the early Kant debate and the row over Clifford's statements took place in the "Letters to the Editor" column of Nature, where all could follow. Ingleby was known as an expert on Shakespeare, but he was also well acquainted with other areas of philosophy, especially Kant's work. Ingleby first criticized Clifford's characterization of Kant's concepts as expressed in Clifford's address, "On the Aims and Instruments of Science," 27 presented before the British Association in 1872.

The infraction against Kant was small and after the initial round of charge and countercharge,28 Ingleby progressed to the real point of contention. Clifford had presented his first statements on the general concept of space and non-Euclidean geometry in this speech and had put a scare into an unnamed friend of Ingleby's.29 It had not been Clifford's inconsequential comments on Kant that had raised Ingleby's ire, but the challenge to Kant's notions of space as implied by Clifford's references to the non-Euclidean geometries.

For historical purposes, this minor debate between Clifford and Ingleby might seem of little significance, but it was just the tip of the iceberg in a larger debate proceeding out of public view within the scientific community. Ingleby was an intimate friend of Sylvester and they discussed the physical consequences of a four-dimensional non-Euclidean space in private. Evidence of these discussions comes from yet another source, the surviving portions of an ongoing correspondence between Ingleby and C.J. Monro.

Ingleby and Monro were debating the concept of a possible higher dimension to space as early as August of 1870.30 In October of 1871 Ingleby informed Monro that Sylvester maintained that "there is nothing in geometry that is not wholly based on order in time." 31 The following month, Ingleby stated that Sylvester had reached a "new position, that the mathematicians are now in possession of evidence that space is curved. This is now what he says and sticks to." 32 In the next letter, Monro was once again updated on Sylvester's opinions of curved space and informed that Sylvester had "assured [Ingleby], ..., that he had reason to believe that our space is curved. But [Sylvester] did not base this on Celestial Observations." 33 Unfortunately, Sylvester would not tell Ingleby why he thought space was curved. If he had, historians would have a record of Sylvester's thoughts on the subject today.

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Thus, before the late winter or early spring of 1872, Ingleby had only related what he thought were Sylvester's private thoughts and concerns of a curved space, as best he could, to Monro. Although the thoughts were attributed to Sylvester, Sylvester had informed Ingleby that the "mathematicians" accepted these notions. Then, in a letter written on 24 May 1872, Ingleby spoke of "the speculations of Riemann, Helmholtz and Clifford." 34 From that day forward, whenever he spoke of a curved space, the reference was to Clifford, not Sylvester. Clifford's opinion on the subject had been elevated to a position above that of Sylvester's, and it was implied that Sylvester was following Clifford's lead on the subject of curved space. When Sylvester referred to "the mathematicians" he had really been referring to those who followed Clifford's ideas. Clifford's time had come, and the issue of curved space became an open argument between Clifford and Ingleby in the pages of Nature.

Monro was also a friend and correspondent of James Clerk Maxwell, who obtained for Monro a membership in the London Mathematical Society,35 as well as a friend of Arthur Cayley. In March of 1871, Maxwell wrote to Monro and enumerated his arguments against a fourth dimension.36 Another letter, this time from Monro to Maxwell in September of 1871, indicates that these thoughts were part of an ongoing discussion between Maxwell and Monro.37

Maxwell may not have been completely convinced by his own arguments against the non-Euclidean geometries and hyperspaces. He was still somewhat perplexed on the issue and not totally sure of his own concepts on 11 November 1874 when he wrote to his friend and fellow physicist Peter G. Tait, once again expressing his opinions.

The Riemannshe Idee is not mine. But the aim of the space-crumplers is to make its curvature uniform everywhere, that is over the whole of space whether that whole is more or less than infinity. The direction of the curvature is not related to one of the x y z more than another or to -x -y -z so that as far as I understand we are once more on a pathless sea, starless, windless and poleless totus feres abque rotundus.38

Maxwell was somewhat incredulous and ambivalent toward the new concept of a curved space, but none-the-less concerned. His reference to the "space-crumplers" indicates his disagreement with them, but also indicates that he

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could not ignore their arguments. Nor could he ignore the possibility that the non-Euclidean hypothesis might bear some relevance to physics.

The depth of the debate within the scientific community is demonstrated by the fact that Maxwell's opinion in this instance came in answer to a question posed by Tait in a letter to Maxwell just two days earlier. Both men were concerned with the new mathematical hypotheses, but Tait seemed slightly more willing to accept their possibility. Tait exhorted Maxwell to "Xplane why it is bosh to say that the Riemannsche Idee may, if it is found to be true, give us absolute determinations of position." 39 It is obvious that Tait attributed some special knowledge of Riemann's geometry to Maxwell or that he knew Maxwell was in contact with those who did have such knowledge. Otherwise, he would not have made this inquiry. It is equally obvious that both Maxwell and Tait were searching for counter-arguments to the "space-crumplers" continuing onslaught.

The term "space-crumplers" referred directly to Clifford with regard to his stated opinions on the feasibility of using curved space to develop a physical theory of "solving the universe." The idea that absolute position could be found via the use of space curvature was a basic tenet of Clifford's geometrical model of space, as later expressed in Common Sense.40 Maxwell did not agree with these concepts, but he did know of them and still had a great deal of admiration for Clifford and Clifford's abilities. He offered a glowing letter of recommendation when Clifford applied for a professorship at University College in London.41

Ingleby also admired Clifford even while he debated with him. His venomous attack on Clifford in Nature was not as serious as it would seem. Ingleby wrote to Monro that Monro would not agree with some of his criticisms of Clifford. Indeed, Ingleby himself saw "things in it to be excepted to. But [he had] an object to serve thereby." Ingleby could not "seem ignorant of such a speculation" as Clifford announced in his open portrayal of non-Euclidean space,42 and he felt obliged to overreact in public to the implications of Clifford's geometrical concepts of space. He only wished to "open the oyster" and air his arguments against the new ideas espoused by Clifford in a public forum. Thus, in private Monro tempered his attack on Clifford, but did not alter his position.

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Clifford refused to further answer Ingleby's charges within the pages of Nature. Instead, he promised Ingleby that his answers would be forthcoming in a series of lectures at the Royal Institution.43 The lectures to which Clifford referred were given in March of 1873. The three lectures constitute Clifford's "Philosophy of the Pure Sciences." They answered all of Ingleby's charges within a far more comprehensive philosophy of science. The second lecture of the series, "The Postulates of the Science of Space," dealt specifically with Clifford's concept of space. This lecture became one of Clifford's most popular expositions of the non-Euclidean geometries as well as his general concept of space. Clifford ended the lecture with a statement that he often found relief from the boredom of our homaloidal space by picturing an elliptic space which he hoped would someday explain physical phenomena.44

The year of 1873 marked a distinct turning point in Clifford's quest for a space-theory of matter. In the early summer, Clifford published the essay "Preliminary Sketch on Biquaternions," 45 which described a new calculus of twists and screws. This calculus was the three-dimensional counterpart of an elliptic space. Clifford accomplished this feat by combining both William Rowan Hamilton's quaternions and some of the features of Hermann Grassmann's "Ausdehnungslehre." It seemed that Clifford's work was reaching a point of climax, given his recent public lectures and his new mathematical system. Then, in the British Association meeting of 1873, Clifford offered a paper entitled "On some Curves of Zero Curvature and Finite Extent." 46 In this paper, Clifford presented a new non-Euclidean geometry which exhibited Euclidean flatness over large distances, but Riemannian characteristics in the infinitesimal connections between consecutive points of space. This geometry was an extension of his algebra of biquaternions.

Both systems, the new geometry and biquaternions, made use of Clifford's concept of geometric parallelism whereby parallel lines need not exist in the same plane. It is curious that Clifford never published an explanation of this geometry while his mathematical theory of space and matter suffered the same fate. It is quite possible, given Clifford's basic tenet that geometry is a physical science, that this geometric model in fact represented his spatial model of matter and that he so stated in his presentation. However, as far as history is concerned, that conjecture will probably never be proven. What history has recorded is that Clifford's new geometry would have died away had not Felix Klein and W. Killing revived and begun to develop Clifford's new geometry about 1890.47

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At this meeting of the British Association, Clifford also met with Robert Stawell Ball and Klein. In long, all night discussions, he converted Ball to the non-Euclidean point of view and traded ideas on the non-Euclidean geometries with Klein.48 Clifford had known of Ball's work on screws before this meeting and had adopted the screw system for his own use in the system of biquaternions. In so doing, he explored the geometry of motion to a far greater extent than Ball would during his own lifetime.

From this time forward, Clifford's emphasis changed from his general model of space and matter to the dynamical study of matter in motion in an elliptical space via his use of biquaterions, screws and twists. It can be assumed that the scientific community did not receive Clifford's new geometry very well. Otherwise, there would have been more development of it before Klein revived it in 1890 and Clifford would have continued to develop it before his unfortunate death. Perhaps Clifford was discouraged by this lack of faith in his geometry. Clifford could not have known that his new geometry was far too advanced for a scientific community that was just beginning to cope with the repercussions pursuant to the discovery of ordinary non-Euclidean geometries. The more immediate concern of science was the development and understanding of Maxwell's theory of electromagnetism, and it was to that end that Clifford applied his biquaternions and twists.

It is not that Clifford ignored the more general problems of space. He published papers in 1878 related to general spaces, "On the Classification of Loci" and "Applications of Grassmann's Extensive Algebra." 49 In both cases, Clifford dealt with the problem whereby properties in a flat space of lesser dimensions were analogous to properties in an elliptic space of a higher dimension by one. The connection of this research to his physical theory should be obvious. He was looking for the mathematical terminology to convert physical properties in a three- dimensional space to a curved four-dimensional space.

Clifford also penned several related articles on philosophical subjects during this period. In the essay "On the Nature of Things-in-Themselves," 50 Clifford introduced the concept of "mind-stuff" which offered a philosophical method to deal with the problem that the human mind could not perceive curved space. He also wrote a critique of the Unseen Universe 51 in which he implied that the ether was secondary to space curvature.52 Clifford was clearly trying to establish a complete philosophical picture of the universe, rather than just a space-theory of matter.

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At this point, Clifford's thoughts on "solving the universe" evolved in several directions at once, but his true passion was a description of matter in motion. To that end, he published what would prove to be his magnum opus, the Elements of Dynamic, in 1878. The Elements was both simple enough for anyone with a mathematical education through trigonometry to understand as well as deep enough for anyone with experience in advanced mathematical physics to find enlightening. Clifford was trying to rebuild the mathematical universe on the basis of non-Euclidean principles without appearing to do so. In this first volume of the Elements, Clifford combined quaternion algebra, projective geometry and vector techniques to describe kinematical motion, but he was insidiously preparing his readers for the acceptance of the biquaternions that would turn his work into the mathematical equivalent of a non-Euclidean space. This fact can only be understood by a comparison to Clifford's other published work from the same period of time. It was within this context that his colleagues understood his work.

Although the title to the book promised a study of dynamics, Clifford only delivered kinematics. In his grand scheme of an ultimate reality there was no such thing as a force, therefore a study of the dynamics of motion was unnecessary. Clifford only believed in energy of motion, or kinetic, and energy of relative position, or potential, but not in force. He had explained this in a lecture in 1872, but the lecture was never published in its entirety. An abridged version of "Energy and Force" was published in Nature from Frederick Pollock's notes of the lecture after Clifford's death.53 This reduction of force to energy as a property of space fits Clifford's overall scheme of reducing three-dimensional dynamics to the four-dimensional kinematics in an elliptic space and conforms with his "Space-Theory" model.

In the Elements, Clifford refused to mention quaternions, projective geometry and such advanced terminology and symbols as found in those studies. In their place he introduced many visually descriptive terms such as twists, squirts, sinks, shells, vortices and so on. Tait praised Clifford's use of quaternion methods in his review of the book for Nature, but condemned Clifford's introduction of such terms as confounding the issue.54 On the other hand, Clifford was praised by others for using such terms so that even the most modestly educated person could understand his explanations.55

The concluding statement of the book more accurately described what Clifford was trying to accomplish. Clifford refrained from use of the term ether

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throughout the book. In only one case of an example did he stray from this pattern. Otherwise, all references to an elastic medium, or in this one particular case an infinite body, could just as easily be interpreted as representing the whole of curved space.

Thus we have shown that if the expansion and the spin are known at every point, the whole motion can be determined, and the result is, that every continuous motion of an infinite body can be built up of squirts and vortices.56

Since the "squirts" and "vortices" were in essence composed of twists of "stuff" within an infinitely extended elastic "stuff," Clifford's system of algebra could be of instrumental use in describing such motions. On one level the "stuff" could be interpreted as the ether, on another level space curvature, and on yet another level Clifford's "mind-stuff." Thus, the system of kinematics which Clifford proposed in the Elements was a thinly disguised application of his "Space-Theory of Matter," and the Elements was a literary vehicle for the development and application of his biquaternion system of algebra.

The relation between the Elements and Clifford's "Space-Theory" is not at all obvious to anyone unfamiliar with Clifford's concept of space and complete mathematical researches. It would appear to be just another Victorian attempt to explain ether vortices, but it was much more than that in actuality. For this reason, historians, philosophers, scientists and other scholars who later studied Clifford's published writings have utterly failed to recognize the import and extent of Clifford's theoretical researches.

The second volume of the Elements was unfinished at Clifford's death. The fragments that remained were collected and published by Robert Tucker,57 but they did not measure up to the promise offered by the first volume. Some hints were given in this reconstruction of Clifford's book of the direction in which Clifford hoped to take physical theory. For example, he related gravitation to a strain in space,58 a fact confirmed by an earlier publication,59 but he did not delve into this matter any further than a few simple statements. He also planned to give a more precise definition of matter than just "stuff" whose mass was determined by comparison to some agreed upon standard, but this definition was never completed. Clifford stressed the mathematics of screws which would imply that he planned to use screws and biquaternions for most of his mathematical analysis of physics.

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The model of space upon which Clifford settled could be briefly described as a four-dimensional elliptic space in the large. The constant of curvature was too small for detection through astronomical observations, but that fact did not negate the possibility that space could be other than Euclidean. Our three-dimensional space was probably considered a boundary between two four-dimensional sections of space. It may not be proper to use the term space in the sense of four dimensions, since there is evidence that Clifford considered the fourth dimension to be time. However, it is more likely that Clifford believed in a purely four-dimensional space with time as a separate but connected quality or quantity. A four-dimensional kinematics would be adequate to describe all physical phenomena, but at the same time it would be analogous to a three dimensional dynamics. Clifford's own geometry could circumvent the problem of non-observation of space curvature on the astronomical scale since his geometry approximated Euclidean space in the large.

The infinitesimal scale of nature presented other problems. On this scale the connections of contiguous points of space exhibited curvature in the fourth dimension. The three-dimensional analogue of this curvature was an elastic medium in which twists were the most fundamental element. The twists, in turn, were composed of vortices and squirts, which supplied the strains in the elastic medium that gave rise to electromagnetic and gravitational forces. This particular reconstruction of a model of space conforms well to all aspects of Clifford's work, but it must be remembered that it is a reconstruction. Several parts of this model are confirmed by the later work and comments of Clifford's students and followers. It is far too tempting to attempt such a reconstruction even though its historical accuracy is debatable. What is assured is that some model, at least similar to this one, represented the goal toward which Clifford was moving.

The chief mathematical obstacle to Clifford's theory was the projective interpretation of space. Clifford freely used projective methods in his mathematical researches and his work has been categorized as projective.60 However, the projective view of space implies an intrinsic alteration in the definition of distance while strictly adhering to a three-dimensional Euclidean space of experience rather than adopting an extrinsic curvature of space. Although Clifford used projective methods, he was not a dedicated projective geometer in the same sense as Arthur Cayley. On the whole, Clifford's work was not projective.

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Cayley argued for the projective view at least as late as 1883 in his address before the British Association.61 Modern historians62 and Victorian scholars63 alike have accepted this view as an accurate description of Cayley's attitudes toward physical space. Indeed, this was the face he put on for the public, but in private he was unsure of his own position in the face of Clifford's arguments.

In 1889, ten years after Clifford's death, a group of Lord Kelvin's popular lectures were published.64 Upon receipt of his copy of the book, Cayley questioned Kelvin's use of vortices of ether to describe physical phenomena. In a letter to Kelvin, Cayley wrote that

In the lecture on the wave theory, you parenthetically ignore the notion of the curvature of space - Clifford would say that, going far enough, you might come - not to an end - but to the point at which you started. I have never been able to see whether this does or does not assume a four- dimensional space as the locus-in-quo of your [vortical] & therefore finite space.65

Cayley, possibly the staunchest advocate of Euclidean three-dimensionality, even while a friend, teacher and colleague of Clifford, had been swayed by Clifford's arguments. This admission demonstrated a crack in the thick veneer shrouding both Cayley's and the Victorian dedication to Euclideanism since Cayley was the inventor of a mathematical system, the projective geometry, which offered the only logical alternative to the radical concept of a curved space. Here we see the steadfast pillar of Victorian geometry with cracks heretofore unnoticed by historians.

Under these circumstances, the use of words such as "speculation," "prophecy" and "untenable" to describe Clifford's work and its reception among his peers is, to say the least, historically inaccurate as well as unfounded. Clifford had begun to publish his theory while the extent of his research far exceeds anything implied by either of the terms "speculation" or "prophecy." The term "untenable" implies nearly absolute rejection of Clifford's theory and concepts, the case of which has been demonstrated as an historical inaccuracy. The new historical issue thus becomes a question of whether Clifford's research died with Clifford. If Clifford's ideas and their influence on the study of non-Euclidean geometry ended with his death, as other authors have contended, then Clifford's work could have had no influence on general relativity.

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However, Clifford had followers who continued different facets of his research and further popularized his concepts.

III. The Followers

Clifford must have felt a great deal of gratification in 1877 when Frederick W. Frankland's essay on non-Euclidean space appeared in Nature. Before moving to New Zealand for reasons of health, Frankland had been a student of Clifford. The paper was an effort to study the characteristics of a special type of Riemannian or elliptic geometry, but only for the case of two dimensions. Frankland had originally presented the essay before the Wellington Philosophical Society in November of 1876. It was subsequently read before the London Mathematical Society before publication in Nature in April of 1877.66 A similar geometry was investigated by the American astronomer, Simon Newcomb, with the results published in the German journal Crelle's in 1877.67 While Frankland's presentation was more philosophical, tracing the logical development of a curved two-dimensional surface, Newcomb developed the purely mathematical characteristics of a similar three-dimensional curved surface. This type of surface, which later came to be known as the single elliptic or polar form of Riemannian geometry, had been discovered by Klein.68 Newcomb's discovery was independent of Klein's and Newcomb has been given credit as co-discoverer of this geometric system.69

Given the date of Newcomb's publication, it is possible that Clifford's work influenced Newcomb's research. Newcomb had traveled to England before the publication and it is quite possible that he met and spoke with Clifford, the "Lion of the season" 70 on his visits to London. Otherwise, there are enough references to Clifford in Newcomb's later publications to conclude that it would be wrong to think that Newcomb had never been influenced by Clifford's thoughts. After the turn of the century he referred to Clifford as the only person who had ever truly understood gravitation,71 implying that he had a more intimate knowledge of Clifford's thoughts than could be gleaned from Clifford's publications.

When Frankland's paper was published it initiated some small controversies. In New Zealand the whole concept was attacked72 and in England Monro noted a few small points of difficulty in the pages of Nature.73 After reading Newcomb's paper and checking Clifford's Elements, the difficulties experienced by Monro were resolved.74 Monro raised an even greater

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question75 over Newcomb's second paper on the non-Euclidean geometry, published in the American Journal of Mathematics in 1878.76 In this paper, Newcomb proved that a hollow sphere could be turned inside-out without tearing or rupturing its surface by transiting a single elliptic space. Monro published, under Cayley's sponsorship, a paper on this phenomenon in the Proceedings of the London Mathematical Society.77 This paper constituted Monro's only mathematical publication on the non-Euclidean geometries. Newcomb published no other papers on the mathematical aspects of the non-Euclidean geometries, but returned on several occasions to popular expositions of them as well as commenting on them from time to time in other publications and presentations.

However, Frankland's paper inaugurated a more lengthy study of the possibility of explaining physical phenomena by space curvature. Frankland's researches were based, by admission, on Clifford's concept of the connection between contiguous points of space.78 Frankland moved to America in 1892, settling in New York. Living in the United States where he found a more open and receptive audience offered Frankland a greater opportunity to discuss his theories with mathematicians and scholars. He presented his "Theory of Discrete Manifolds" at the summer meeting of the American Mathematical Society in 1897 79 Newcomb presided over this meeting. Except for a short description of his presentation in the Society's Bulletin,80 Frankland's theory was not published. Commentators complained that his theory suffered from obscurity from the failure to publish it. They could not evaluate the theory since they could not obtain copies of it. However, a collection of the dozen or so separate papers which constituted his theory were finally published in New Zealand in 1906.81 In spite of this publication, the theory still remained obscure and did not greatly influence the development of the non-Euclidean geometry.

Perhaps the greatest influence on the development of the non-Euclidean geometries in America was the arrival of Sylvester in 1877 as the professor of mathematics at Johns Hopkins University. He taught alongside Newcomb and Charles S. Peirce. His first student was George Bruce Halsted who became world renowned for both his contributions to the history of non-Euclidean geometry and his mathematical publications on geometry. Halsted privately believed that physical space was hyperbolic or Lobachewskian,82 but publicly he only admitted that it was impossible to distinguish which type of geometry was the true geometry of space.83 Peirce actually developed a theory of

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hyperbolic space in the early 1890's. He thought that he had detected a discrepancy in parallax measurements between stars which could only be accounted for by assuming a Lobachewskian type of space.84 Unfortunately, the lack of support for his ideas forced him to abandon the effort and his theory was discarded, never having been published nor committed to paper.85

Sylvester also taught W.I. Stringham who continued Clifford's work on "Loci" and conducted a mathematical investigation of rotations in spaces of four dimensions.86 It is no coincidence that the work of these men closely reflected the ideas of Clifford. Clifford's own essay on "Grassmann's Extensive Algebra" was published in the first volume of the American Journal of Mathematics, as was Newcomb's paper on the transformations of surfaces in spaces of four dimensions. The journal was founded by Sylvester and carried the stamp of his influence just as his influence generated the early American interest in the non-Euclidean geometries. It would be erroneous to think that Sylvester did not inform his American colleagues and students of Clifford's theory as well as his own concept of geometrical reality while he was in America. Each of these American scholars had been influenced by Sylvester's tenure at Johns Hopkins, before Sylvester returned to a professorship at Oxford in 1885.

Of far greater importance were Karl Pearson's extensions of Clifford's work. In 1885, Pearson published the Common Sense of the Exact Sciences, which had been left partially completed by Clifford at his death.87 When Clifford died the English academic community deeply felt the pain and loss of his passing. Even his detractors found kind words for him and expressed the great loss for England and society as a whole by his death. Ingleby wrote Monro that he took Clifford's death "to heart" and wished that he "had the brain of Clifford." He thought that the "death of Clifford might well throw all our churches into deepest mourning." 88 These private comments concerning Clifford's death are all the more important when it is considered that Ingleby had been Clifford's most vocal detractor in the earlier part of the decade.

The loss felt by scholars in England was exacerbated by the fact that Clifford failed to write down many of his lectures. Friends and scholars alike feared that his ideas might be lost to posterity. Hence, there developed a movement to publish anything and everything of Clifford's as quickly as possible after his death. This was a frantic effort to save the work that was considered too valuable to be lost to the world. Clifford's friends Frederick Pollock and Leslie Stephen collected and published Clifford's Lectures and Essays,89 the papers

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which were to constitute the Common Sense went to Professor Rowe at Oxford while Robert Tucker collected Clifford's mathematical papers90 and gathered together the fragments which were to become the second volume of the Elements.91 The extent of these endeavors was unprecedented and represented a tribute to the friendships that Clifford built during his life as well as the deep respect that his peers and colleagues had for his work.

Rowe died a few years after Clifford and the manuscripts for Common Sense then passed to Pearson. Pearson published the book early in 1885. Common Sense summarized Clifford's concept of space and time and offered a unique view of Clifford's method of mathematics as well as the best exposition of his concepts of curved space. Large parts of the Common Sense, including the section which described Clifford's concept of space curvature, were written by Pearson.92 But Pearson only meant to describe Clifford's concepts, not his own, and the work was accepted as an accurate portrayal of Clifford's ideas.

It is difficult to understand why all those authors who have studied Clifford's work have insisted upon the fact that Clifford had no followers while quoting passages written by Pearson in Clifford's Common Sense. A closer study of Pearson's scientific researches during the decade of the 1880's shows that Pearson was developing a theory of electromagnetism and atomism based directly upon Clifford's twists. He combined two strands of theoretical work, one on pulsating spheres of ether93 and the other on twists,94 to develop his final theory of "ether-squirts," published in 1891.95 His theory was published in the American Journal of Mathematics, once again demonstrating the greater American tolerance for such ideas. The English mathematical community was already beginning to slip into the doldrums of philosophical introspection, a movement that was largely a stepchild of the philosophical crises brought on during the earlier debates on the non-Euclidean geometries.

Pearson's ether-squirts were sources and sinks where ether flowed into and out of our space from a fourth dimension. The theory was a purely mechanical theory of the ether, rather than a mathematical theory of space curvature as Clifford had intended. In the publication, Pearson refused to speculate on the source of the ether in the fourth dimension, leaving that task for the transcendentalists.96 He also made no public comments on the relation of his theory to space curvature, but privately he acknowledged that space curvature was the bottom line in the human perception of reality.

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In a letter to his friend Robert J. Parker, written in 1885, Pearson commented that Kelvin's attempts to weigh the ether were conceptually erroneous, "as if empty space could weigh anything! I am going to weigh a twist!" 97 In this one private statement, passed between intimate friends, Pearson confirmed that Clifford's twist, which had been associated with the ether, was no more than an element of space curvature. Pearson had also commented on the Clifford's twist in a footnote in the Common Sense. In this case, he likened the twist to magnetic induction.98 Although this suggestion was made in an editor's footnote, which would seem to suggest Pearson's personal opinion rather than Clifford's thought, the fact that Clifford considered this possibility was later confirmed by Charles T. Whitmell,99 another student of Clifford's, and Frankland.100

Pearson's development of a strictly mechanical theory of ether-twists over the period of a decade was accompanied by an evolution in his own philosophy and methodology of science. In December of 1885 he presented a talk on "Matter and Soul" before the Sunday Lecture Society. In this lecture, he described and evaluated the prevalent theories of matter: The Boscovichean atom, Kelvin's vortex theory of the atom and Clifford's space-theory of matter. Boscovich's atom represented no more than "non-matter in motion," 101 an absurdity, and was therefore rejected. Kelvin's vortex atom was "very like non-matter in motion" since stopping the motion would create a massless void.102 The possibility was not rejected outright, but severely questioned. On the other hand, Clifford's space-theory was non-mechanical. Matter was something in motion, but the something was geometric, the changing shape of space.103 Boscovich's and Kelvin's theories were examples of how matter could be explained as a product of motion, while Clifford's theory sought to explain motion itself. Pearson concluded that matter could never be explained by a mechanical theory and Clifford's was the only non-mechanical theory available. This conclusion implied the ultimate superiority of Clifford's point of view.

Pearson further criticized the definition of mass. Since matter could not be explained by a mechanical theory, then mass could not be defined as the "quantity of matter," as it had been by Newton. Mass was merely the ratio of a force to the acceleration resulting from that force. Here, Pearson differed from Clifford who had defined mass as "stuff," 104 but perhaps he was being too hard on Clifford. Clifford had stated his intention to redefine mass more precisely, but died before he could do so. Even so, differences of opinion were beginning to show between Clifford and Pearson's concepts.

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The evolution of Pearson's own ideas on the philosophical and methodological aspects of science ended with the publication of the Grammar of Science in 1892. The ideas that he expressed in this book were as similar to Ernst Mach's as they were to Clifford's. Space and time were reduced to "modes under which we perceive things apart," rather than realities in the world of phenomena.105 Scientific concepts became limits extrapolated by our perceptions of the phenomenal world.106 Mach's influence was evident in these attitudes rather than Clifford's. It seemed that Pearson had abandoned Clifford's notion that space curvature was the underlying reality, yet his new ideas still reflected the influence of Clifford's "mind-stuff" with a Machian turn (or twist).

Pearson had become disillusioned with the reception of his work on an ether theory of matter prior to his publication of the paper on ether-squirts. He had also been involved in a sometimes frustrating debate with Kelvin over the ultimate existence of the ether.107 In a letter to Kelvin, which was never finished or mailed, Pearson stated his final opinion on the matter of space curvature in a clear and concise manner, the like of which cannot be found in any of his published materials. He claimed that space curvature did not represent ultimate reality.108 In this way he moved beyond his earlier opinions on the subject and decided that space curvature was only the final step toward a reality of which we could not have any physical knowledge. In terms of his statements on space in the Grammar, the representation of matter by space curvature was not reality, but the limit to which the human mind extrapolated its best and most precise perceptions of reality. In this sense, he had not so much withdrawn his earlier conviction to the reality of space curvature as he had decided that it was not possible for the human mind to have knowledge of ultimate reality.

When coupled with other events in his life, Pearson's disillusionment with the reception of his theory of matter gave him the opportunity to abandon any further attempts to realize Clifford's goal and relate his ether-squirts to Clifford's mathematical twists and space curvature. Pearson turned away from his past theoretical work and began working in the new field of the statistics of heredity which was far more rewarding. He never returned to his work on Clifford's theory, nor is he remembered for that work. Pearson's Grammar is still used to portray his philosophical ideals as well as the overall philosophical temper of science in the late 1890's while Pearson is quite well known for his work in statistics. In fact, Pearson is regarded as the father of modern statistics.

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While Pearson worked toward his own theory of twists in the form of ether-squirts, Ball was developing a purely mathematical and analytical theory of screws in a non-Euclidean space. Each researcher attempted to continue Clifford's work in his own manner, but there was apparently no collusion between the two men. Their theories were characteristically different. Unlike Pearson, Ball made very few statements regarding the physical applications of his theory of screws, steered clear of the philosophical aspects of his work and never attempted to relate his mathematical researches to the ever-popular ether theories. Each of these men could look to different aspects of Clifford's Elements for inspiration, but their paths of research were divergent. Unlike Pearson's work, Ball's research was well accepted within the scientific community.

Ball's theory of screws began with a purely physical assessment of a simple mechanical motion.109 The problem of describing this motion mathematically intrigued Ball, but as he developed the mathematical theory of the motion of a screw the theory took on a life of its own and captivated his imagination. Ball never meant to describe all mechanical motions with his theory, but limited his research to those small oscillations or vibrations that could be described by the generalized screw-like motion. Over the course of years, the theory evolved from the description of a simple motion to the study of a system of screws and the mathematical study of the motion of the system.

Ball's theory was quite well known in England as well as internationally. Many mathematicians contributed to its development in small ways, but it was primarily Ball's theory. In the early years of the 1880's, Arthur Buchheim made some important contributions to the theory.110 He was interested in generalized algebraic and geometric systems and worked directly from Clifford's published work and unpublished fragments. He also corresponded with Ball and Sylvester. It seemed as if Buchheim might be a worthy mathematical successor to Clifford, but, like Clifford, consumption claimed his life before he could reach his full mathematical potential.

In 1897, Ball completed the task that he had originally set for his research.111 At first, the simple screws gave way to instantaneous, reciprocal and impulsive screws until Ball could completely describe a system by a screw-chain. His theory was complete when he found the method of finding the instantaneous screws given the corresponding impulsive screws. Ball had taken his theory to its logical limits within the normal Euclidean context.

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However, Ball's 1873 discussions with Clifford had convinced him that the theory of screws was "obsolete; it is all going over into non-Euclidean space." 112 This early assessment was not completely true, there was still work to be done in completing the Euclidean portion of the theory. Ball did not attempt to study the non-Euclidean aspects of his theory during Clifford's lifetime. It was only after Clifford died that Ball 113 began to develop a non-Euclidean mechanics of vibrational motion.

Ball was not a mathematician, but an astronomer. His intrigue with the non-Euclidean geometry was twofold and extended beyond just the mechanical theory of screws. In his professional duties as the Astronomer Royal of Ireland, Ball made many parallax observations. In 1881, he announced the results of some of these observations before a group at the Royal Institution in London. He concluded his presentation by stating that his observations were not accurate enough to determine between the Euclidean and non-Euclidean nature of space.114 This one statement clearly demonstrates that Ball believed in the reality of a non-Euclidean phsyical space. In 1885, when Ball wrote the article on "Measurement" for the ninth edition of the Encyclopaedia Britannica, he summarized his findings on parallax measurements, but did not commit himself to any particular geometry of space.115 The article was actually an exposition of the latest advances in the non-Euclidean geometries, further enhancing the recognition of his expertise in this area of study.

During the early 1880's, Ball wrote several articles and memoires on the non-Euclidean aspects of his theory of screws, but he came to an impasse during the latter part of the decade and returned to the completion of his original Euclidean study of screws. The deeper he journeyed into the non-Euclidean aspects of mechanics, the more difficult it became for him to philosophically justify his work. The problem revolved about the intrinsic projective interpretation of distance, which limited geometry to the Euclidean space. All the terms traditionally associated with geometry carried an Euclidean bias. In 1887, Ball penned his essay "On the Theory of Content" 116 in an attempt to come to terms with the resulting philosophical discrepancies in the non-Euclidean geometries. Ball developed a complete new terminology for the study of geometry that he thought devoid of any bias or preconceived notions of space. For example, he no longer referred to the distance between points of space, but the interval between elements in a content. His hope was to dissociate geometric terms in Euclidean geometry with similar concepts in the non-Euclidean geometry. It is worthwhile to note that he dealt with a "content"

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of four elements in his derivations, or rather, a four-dimensional space in the biased language of Euclidean geometry. The "content" or space that he dealt with was elliptic,117 just like space in Clifford's theory.

Ball rarely commented on the physical phenomena to which his theory might be applied. However, in an 1885 review entitled "The Theory of Screws," Olaus Henrici implied that Ball's theory would eventually be used to describe the vibrations of molecules and the transmission of vibrations between molecules.118 In other words, the implied physical applications of the theory were the absorption, emission and transmission of electromagnetic vibrations. Ball confirmed this interpretation in his 1887 presidential address before the British Association, "A Dynamical Parable." In this allegorical dialogue between scholars, Ball stated "all instantaneous motions of every molecule in the universe were only a twist about one screw-chain while all other forces of the universe were but a wrench upon another." 119 In this statement and following comments, Ball confirmed Clifford's belief and his own opinion that all motion could be reduced to the geometry of position in an elliptical space. In Ball's perspective, the three-dimensional Euclidean analogue to that description would be exhibited by his theory of screws.

After he completed his theory, Ball collected and summarized all of his work in A Treatise on the Theory of Screws.120 The last chapter of the Treatise was an exposition of the non-Euclidean aspects of his theory while his "Dynamical Parable" was included as an appendix to the volume. This book should have been the final chapter of the story on screws, but the final chapter on non-Euclidean geometry was incomplete and opened new vistas for the expansion of Ball's theory. For the past few years, Ball had been working with Charles Jasper Joly in the hope that his screw system could be expanded through the quaternion algebra. Ball was now too old to carry on the task alone and he found in Joly both a willing and able collaborator. But Joly died in 1905 and Ball could do little more to further his theory in the direction of non-Euclidean spaces. He continued work on the expanded theory nearly until his death in 1913, but the attempt was futile. The association of screws with quaternions in conjunction with advances in physics in unexpected new directions after the turn of the century doomed the theory of screws to an undeserved respite in historical oblivion even though it was popular among mathematicians and scientists at least until Ball's death.

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Ball continued Clifford's work along the lines of the dynamics of non-Euclidean space, but Pearson considered the ethereal mechanics that corresponded to the twist as well as the most general philosophical and methodological aspects of science implied by Clifford's ideas on science as an academic discipline. In the view of later scholars, Ball's theory of screws, Pearson's ether squirts and Clifford's Elements would all suffer from too close an association with their Victorian counterparts. As science changed, their ideas fell by the wayside. Yet, in each case these scientists were forging new ground in breaking away from the Victorian attitudes with which their work was associated. In the case of Charles H. Hinton, the association with Victorian attitudes neither harmed nor helped. Hinton's early work on hyperspaces and non-Euclidean geometries played to the more spectacular interests of the common public.

Hinton became a student at Oxford in 1871 and was associated with the University until receiving his Masters degree in 1886. Hinton may have had no direct contact with Clifford, but there was ample opportunity for him to come into contact with Clifford's ideas. He studied geometry at Oxford while H.J.S. Smith, who wrote the introduction to Clifford's Mathematical Papers, held the chair of Savilian professor of geometry. After Smith's death in 1883, Sylvester was elected to the chair and returned to England from America. This was two years before Hinton finished his work at Oxford. Given the fact that Hinton studied and taught geometry, he would have undoubtedly come into contact with these two men while at Oxford.

However, Hinton's first publication, "What is the Fourth Dimension?" came in 1880,121 before Sylvester came to Oxford. It was republished with other popular essays and pamphlets that Hinton had written in a book under the title of Scientific Romances between 1884 and 1886.122 Hinton developed a rough model of a four-dimensional space in his essays, but the greater part of his writing was devoted to the visualization of the fourth dimension by the human mind as well as ethical and metaphysical aspects of the fourth dimension. His writing was aimed at the general reading public rather than scientists and mathematicians. This early work included no mathematical development other than a crude verbal model of space.

The model of space first proposed by Hinton was a three-dimensional sheet of ether in which atoms were embedded. The complete structure was curved within a fourth dimension. The material atoms were likened to threads passing

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through the sheet from outside the three dimensions of the sheet, the points of intersection representing the individual atoms.123 In this model, Hinton could only account for some of the fundamental properties of matter. In another of his essays, he introduced "twists" as mechanical models of electrical activity.124

Although he used terms similar to those used by Clifford and his twists were very like Clifford's, Hinton's twists were non-mathematical visual gimmicks. If Hinton had not been aware of Clifford's work before these essays were published, it would be nearly impossible that no one would have pointed out the similarities between his concepts and Clifford's. Unfortunately, Hinton gave no one credit for his ideas, nor did he make any references or citations of previous work in these early essays. His model of space became more elaborate in succeeding essays. The model evolved into a sheet of ether, curved in the fourth dimension like a sphere upon which material particles followed grooves on their courses through time,125 like the needle of a phonograph following the grooves in a record.

Hinton eventually moved to America where he taught at Princeton and other schools before settling in Washington, working at the Naval Observatory. He began this job shortly after Newcomb retired from the post of director of the Observatory and there has been some speculation that Newcomb, knowing of Hinton through their mutual interest in hyperspace theories, secured the job for him.126 In 1891, Newcomb offered a brief model of the physical world as a four-dimensional ether. Our three-dimensional space was sandwiched between layers of the ether. The primary purpose of this model was to account for the negative results of the Michelson-Morley experiments in detecting the ether.127

In 1891, W.W. Rouse Ball also published a theory which assumed a four-dimensional curved ether.128 Rouse Ball's purpose was to explain gravitation and other physical phenomena. While Newcomb's model was only presented before the Washington Philosophical Society, Rouse Ball's theory was published in the Mathematical Messenger as well as several editions of his Mathematical Recreations that were published before 1915. In both cases, the models of curved space were quite similar to Hinton's. Rouse Ball discovered Hinton's work after publishing his own theory and pointed out that his theory represented a case of independent discovery.129

It cannot be accepted as purely coincidental that those men who developed such models were all associated with Clifford in some manner. Rouse Ball had taken

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over Clifford's duties at University College when Clifford took his first leave to deal with his illness. It would not be unfair to conclude that these models of space were in some part derivative, either knowingly or unknowingly, from Clifford's theoretical work. In the case of Hinton, especially when considering his use of twists to explain electrical phenomena, the similarities are too remarkable to assume that they were developed in a vacuum.

Hinton's early development was purely philosophical, if not metaphysical, and aimed toward a popular audience. After settling in America, he turned to developing the mathematical aspects of his model. In 1902, he presented a paper before the Washington Philosophical Society, "The Recognition of the Fourth Dimension," in which he finally presented a theory of non-Euclidean space.130 The philosophical and explanatory portions of the presentation were republished in The Fourth Dimension of 1904, but the mathematical portion of the theory was deleted. Hinton explained that the positive and negative aspects of electricity depended upon the anti-symmetric parts of a Hamiltonian quaternion.131 He published a short paper in a mathematical journal indicating the relation of his theory of quaternions to Cayley's work in algebra,132 but the complete mathematical theory was never published.

Of all these theories and publications, Ball's was the most popular among serious mathematicians. Pearson's Grammar was quite popular, but some of his ideas were met with skepticism when he first published the book in 1892. The third edition of 1911 contained a chapter on the new advances in physics including a summary of recent research on both the electrical theory of matter and relativity.133 In spite of the new advances in science, Pearson chose to leave most of his Grammar unaltered, including his long explanation of ether-squirts and the sections on matter and space-curvature. Hinton's earlier books became very popular. They seemed to attract everyone from serious mathematicians to spiritualists and mystics. Hinton did not support spiritualism, nor did he approach the subject in his publications, but some of his speculations could be termed mystical, allowing quite a wide interpretation of his work. In any event, Hinton's 1902 theory remained obscure and apart from his earlier popular publications.

Even while these scientists were working on theoretical models, the popularity of the non-Euclidean geometries and hyperspaces grew rapidly. Ball's and Buchheim's work did little to popularize the concepts, but Pearson, Hinton, Frankland, Halsted and others wrote enough to pique the interest of scientists,

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scholars and laymen alike who read their publications. During the early 1880's, mathematical expansions of non-Euclidean dynamics were in order. Not only did Ball and Buchheim work in this area, but major contributions were made by R.S. Heath134 and Homersham Cox.135 Clifford had founded the study of non-Euclidean dynamics and these men carried on that work after his death. But this early expansion of the study of dynamics was accompanied by a slight lull in philosophical discussion of the concepts as Clifford's ideas were assimilated and evaluated. That evaluation was not all positive. Attacks on the non-Euclidean concepts of space were mounted by Cayley,136 Samuel Roberts137 and J.B. Stallo.138 In spite of this negative reaction to the non-Euclidean geometries, there was an explosion of interest and popular articles on the subject in the latter part of the 1880's and the 1890's.

The geometry which Clifford developed in 1873 was rediscovered and redeveloped by Klein and W. Killing about 1890, inspiring a new look at the physical possibilities of non-Euclidean spaces.139 Philosophical discussions of the physical reality of higher dimensions and non-Euclidean spaces became quite common in popular journals as well as professional publications. Perhaps by this time the initial shock of the possibility of non-Euclidean geometries had worn off and scholars became used to the idea, but the discovery by Michelson and Morley that the ether was undetectable cannot be discounted as an incentive to find alternate hypotheses. Newcomb's 1891 suggestion of a non-Euclidean ether model was a direct result of that failure to detect the ether.

In 1892, when Poincaré's conventionalist philosophy was first presented to an English audience in a translation of his "Non-Euclidean Geometry," 140 it was a reaction against both the rising tide of popular non-Euclidean heresies and original scientific work on the subject. The scientific community was beginning a period of radical change that affected both its methods and attitudes. The change was as much a product of the non-Euclidean geometries as it was of recent scientific discoveries and experimental results. A vast amount of popular literature on the non-Euclidean geometries and hyperspaces was published throughout this period. There are enough direct references to Clifford as well as allusions to his original ideas, that it can be said with certainty that the seeds he had sown did not lay fallow on the ground. His program was beginning to grow and mature, but not in the way that he had planned. His mathematical work had been severed from its philosophical basis. The mathematical system that he was trying to develop withered on the vine with the failure of Ball and Pearson's theoretical work, but the philosophical concepts were carried forward. The

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imagination of the educated public caught fire on the subject, as had the more disciplined imaginations of some scholars and scientists. This development was not to abate throughout the period up to and including the rise of relativity theory. Any claims that Clifford's ideas died with him or that he had no followers to continue his work are not supported by the historical evidence and thus unfounded.

IV. The New Century and a New Physics

The turn of the century brought no magical changes in the world of non-Euclidean geometries. If anything, it offered a unique opportunity for everyone to reminisce on the changes that had taken place within geometry and mathematics in the previous century. At this time, scholars documented the fact that mathematicians were leaning toward giving credence to the possibility of a physical interpretation of the non-Euclidean geometries. Edward Kasner wrote about the changes from "attempts to discover universal methods" and develop an "ultimate geometric analysis," such as the quaternion analysis, to a more modest search for different theories of geometry.141 He further confessed that it was the duty of mathematicians to study the "geometric foundations of the various branches of mechanics and physics." 142 It is obvious that he was not speaking of the strictly Euclidean basis of science. Corrado Segre, an Italian mathematician well known for his work in non- Euclidean geometries, expressed similar sentiments.143

Mathematicians were seriously considering their scholarly right to use the non-Euclidean geometries to represent physical phenomena. The trend was toward the acceptance of a physical connection between the mathematical and physical studies of non- Euclidean space, but the scope and method of application were ill defined. Federigo Enriques went still further in his 1906 book on the Problems of Science. He explained the connection between physical space and geometry, both the Euclidean and non-Euclidean varieties, and then stated that geometry was the basis of mechanics.144 He did not distinguish between which geometry formed the basis of physical reality, but left the clear impression that he was fully willing to accept that physical space was non-Euclidean if that hypothesis was found necessary.

In the present state of our knowledge, physical space must be positively regarded as Euclidean. But this does not justify the assertion that matter could not be otherwise. And it is unjust to

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accuse the non-Euclidean geometers of having raised a doubt, which is only removed for the present, and perhaps postponed to a distant future.145

Enriques would have had no reason to believe that the future date of which he speculated was only one decade away. Although he was a mathematician, his book was quite explicit in the explanation of physical theories.

A Treatise on Electrical Theory and the Problem of the Universe was still more explicit on current physical theories. Although G.W. de Tunzelmann of England published it in 1910, it provides a unique window on the physical attitudes of British science immediately prior to the development of special relativity. Relativity in a broader sense than expressed by Einstein was discussed, but only with regard to the theories of Henri Poincaré and H.A. Lorentz. De Tunzelmann made no references to either Einstein or Minkowski in spite of the fact that he made copious use of recent publications and scrupulously documented his references.

De Tunzelmann also offered a unique suggestion that time could be represented as a fourth dimension and explained the fundamental aspects of such a physical model.146 After a discussion on absolute space, he also professed that an elliptic geometry fulfilled the necessary conditions for experiential space.147 Although common three-dimensional space was completely relative, absolute space could be determined relative to an ether associated with a fourth dimension.

When we think of space as filled with something, such as the ether, it seems to be much easier to think of position or direction relatively to it, even if we think of the ether only as a perfectly uniform continuous medium; and it becomes easier still when we think of space as full of ether whirls or spins which have to be traversed in moving from point to point.148

This model of an elliptical space was quite crude, but the source of de Tunzelmann's thought is not difficult to locate. The terms "whirls and spins" are reminiscent of Clifford's Elements. This fact should come as no surprise, since de Tunzelmann had been a student of Clifford four decades earlier.

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It may not be fair to draw the conclusion that de Tunzelmann's thoughts on this matter reflected a general sentiment among scientists. But then, it is not necessary that scientists and scholars completely rejected Clifford's ideas at this date to demonstrate the influence of Clifford's work on the acceptance of general relativity. Just the fact that many were already familiar with Clifford's concepts of space immediately prior to 1915, disregarding their denial or acceptance, is adequate to indicate the influence of Clifford's work. It would be an unexpected bonus to prove that scientists fully believed Clifford's model of elliptic space represented reality, but that cannot be accomplished. Einstein presented a theory that Clifford reputedly was unable to develop and Einstein derived physical consequences of that theory which could be experimentally verified. Their methods were clearly different as were their immediate goals. Clifford was trying to explain electromagnetic phenomena with gravitation a secondary consideration while Einstein explained gravitation.

The fact that the academic community in its larger sense was already familiar with the notion that matter might be expressed as space curvature introduced a palatability factor that was missing when Clifford introduced his "Space-Theory of Matter" in 1870. Yet the historical consequences go deeper than just the question whether curved space was more palatable in 1915 due to Clifford's "Space-Theory." Clifford's actual theory was only a small, albeit extremely important part of a larger trend in accepting the possibility of a physical non-Euclidean space. In many cases, Clifford's direct influence cannot be discerned and that is justly so. However, at the very least an indirect influence can be assumed since Clifford was the founding father of the English concept of a physically curved space.

In 1908, Hermann Minkowski presented his space-time model of Einstein's theory of special relativity. Until that time, Einstein's theory of relativity was just one among many in which the Lorentz-Fitzgerald formulas could be justified. Minkowski presented a model by which the world was non-Euclidean, not just four-dimensional. In a community where the possibility of a non-Euclidean space was already being considered, where many scholars would not admit that our experiential space was Euclidean simply because astronomical observations could not prove otherwise, Minkowski's model of space-time was not just four-dimensional, but implied a non-Euclidean four-dimensional structure for space-time.

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Halsted wrote of the space-time model "the theory of relativity has made non-Euclidean geometry a powerful machine for advance in physics." 149 He specifically singled out the work of a Croatian mathematician, Vladimir Varicak, who was able to derive the equations of special relativity directly from his studies of Lobatchewskian geometry.150

Henry P. Manning of Columbia University also confirmed the non-Euclidean interpretation of space-time. He characterized space-time as a "system [which] may be regarded as a non-Euclidean geometry in which the conical hypersurface plays the part of absolute angles, while distances along lines of the two classes are independent and cannot be compared." 151 Like these other men, Manning had a long association with the non- Euclidean geometries before the development of relativity theory. Manning's association with the purely mathematical studies of geometry did not overshadow his willingness to look at the physical interpretations of geometry.

In 1910, an anonymous donor gave Scientific American magazine five hundred dollars to hold an essay contest on the fourth dimension. The competition proved so popular that two- hundred and forty five essays were submitted from nearly every civilized country in the world.152 The contest was judged by Manning and S.A. Mitchell of Columbia University. Manning published a group of the better essays in 1914 under the collective title The Fourth Dimension Simply Explained. He wanted to save these essays for posterity. Within the published essays, there was absolutely no mention of Einstein's relativity theory or Minkowski's space-time model, but there was ample evidence of the seriousness with which the non-Euclidean geometries and their physical counterparts were taken by the educated populace during the period of time immediately prior to Einstein's discovery of general relativity.

In the book in which he first mentioned relativity, a purely mathematical study of Non-Euclidean Geometry, Manning also mentioned the work of Gilbert N. Lewis and Edwin Bidwell Wilson. In 1912, these two men collaborated on a non-Euclidean theory of relativity based upon the Minkowski model of space-time. They felt that "any line in our four-dimensional manifold which represents motion with velocity of light must bear the same relation to every set of axes" was "sufficient to determine the properties of" our non-Euclidean space.153 Both men had some previous experience with non-Euclidean geometries, but Wilson's experience was quite extensive. As early as 1904, he had criticized the overly philosophic trends that were exhibited by many

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geometers. He thought that mathematicians had been displaying "a mania for logic" which was wholly unjustified and that there was nothing of reality behind this logic.154 Something more was needed in geometry beyond the logic of axioms, something intuitive, perhaps a "postulate of reality." 155 From these observations, it is obvious that Wilson did not accept a complete distinction between abstract geometry and the real world.

Harry Bateman also developed an essentially non-Euclidean theory of space and time. In this case, the theory preceded even Minkowski's space-time by a short time.156 Bateman worked on expressing electromagnetic waves by a geometry of spheres in his four-dimensional space with time as a fourth dimension, a situation analogous in many ways to a non-Euclidean geometry. Bateman, who had some previous experience studying pure rotations in a four-dimensional space, developed the mathematics of general covariance by 1910,157 a feat not accomplished by Einstein using Christoffel tensors until several years later. Bateman did not claim that his geometry was non-Euclidean, but implied this description.158

Not only were non-Euclidean versions of Minkowski's space-time model being developed before general relativity, but Hans Kleinpeter remarked on the similarity between Clifford's concepts of space and time and Minkowski's space-time in his 1913 German translation of Clifford's Common Sense.159 Kleinpeter's note to this effect appeared on the page preceding Pearson's original editor's note relating space curvature to physical phenomena and the twist to magnetic induction. It is unlikely that Kleinpeter, a German, was the only person with knowledge of Clifford's most popular published work to draw this analogy.

Perhaps the earliest public mention of Clifford's work in conjunction with general relativity came at the hands of Ludwik Silberstein in 1918. Silberstein did not fully accept general relativity as written, but investigated its tenets and consequences. In particular, he considered the theory without the principle of equivalence. In the course of this study, he noted that Clifford had already equated curvature with matter.160 The fact that he mentioned this is not so important as the context. His attitude was that equating curvature to matter should not be regarded as a new accomplishment. Clearly, he would not have given Einstein credit for this particular advance in science, but would have awarded Clifford the honor.

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Silberstein compared general relativity to the "Space-Theory" and Common Sense, but other writers made early comparisons with Clifford's other publications. Henry L. Brose recommended that readers of his translation of Erwin Freundlich's Foundations of Einstein's Theory of Gravitation refer to Clifford's article on "Loci" and H.J.S. Smith's introduction to Clifford's Mathematical Papers.161 Sir Oliver Lodge, by no means a supporter of general relativity, attempted to explain away the positive results of the light bending measurements by arguing that either the ether near the sun changed the refractive index of space or the ether composing the light beam reacted to the gravitation of the sun. Only if these hypotheses could be decisively refuted, could Einstein's theory be considered. He then referred to Clifford's "Philosophy of the Pure Sciences." 162 Even then, general relativity was only a mathematical gimmick to give the correct experimental results, and was only palatable since Clifford had already shown the comparison of ether and curvature, or so Lodge implied by his reference to Clifford's work. But only those scientists, who were familiar with Clifford's work, as were the British scientists of that era, would have recognized the implication. So the implication is lost to anyone reading Lodge's paper today.

Neither de Tunzelmann 163 nor Bateman referred to Clifford in their limited adoptions of relativity, but neither left any doubt that their acceptance of general relativity was limited by their own preconceived notions of space curvature. In de Tunzelmann's case this proceeded directly from Clifford. On the other hand, Bateman's references to general relativity were especially significant. Bateman thought that he had discovered the same theory several years before when he discovered the "general principle of relativity," the general covariance under all transformations.164 There might be some small amount of legitimacy to this claim. Some scientists who first adopted relativity considered the "general principle of relativity" as the more important aspect of Einstein's theory rather than the expression of space curvature as matter. This aspect of the development of general relativity would explain why Silberstein gave Clifford rather than Einstein credit for equating space curvature to matter. Willem de Sitter had noted this very fact in his 1916 article on "Space, Time, and Gravitation" in The Observatory.165 If the "general principle of relativity" were considered the more significant part of Einstein's theory at this early date, then Clifford's priority for equating matter to curvature would be preserved and the early references to Clifford's other works explained.

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But it was the work of Sir Arthur S. Eddington, who led the expedition to confirm Einstein's light bending prediction, which so clearly demonstrates the greatest influence of Clifford. Eddington became intrigued with general relativity after reading de Sitter's 1916 accounts of the astronomical consequences of the theory.166 In his earlier publications on the theory, Eddington indicated that he did not fully believe in the literal truth of space curvature.167 His early interpretations of the theory were decidedly Victorian with talk of strains in the ether, but Eddington's ability to handle the different non-Euclidean concepts as well as his perspective on the theory developed very rapidly and continuously. He admitted that he originally knew little of the non-Euclidean geometries,168 so it can be concluded that he made a study of the non-Euclidean geometries to fill in the gaps in his own knowledge of the subject.

It is quite likely that his basic concepts on the non- Euclidean geometries came from Clifford. If he didn't already know of Clifford, he must have become very interested in Clifford's work because he was able to show a great familiarity with Clifford's work in just a few years. In his 1921 popular exposition of the theory, Space, Time, and Gravitation, Eddington introduced one chapter by a quote from Common Sense169 while he began the chapter on "Kinds of Space" with a quotation from Clifford's "Postulates." 170 The quote from Common Sense was the same paragraph that ended Clifford's chapter on "Position," and the very words to which Pearson added the note that twists may well represent magnetic induction.

However, Eddington also quoted a passage from the "Unseen Universe" in which Clifford expressed his desire that physical reality would one day be expressed as the geometry of position. "Out of these two relations [nextness or contiguity of space and succession in time] the future theorist has to build up the world as best he may." What might help the scientist in this endeavor, suggested Clifford, was the description of distance as an expression of position as in the mathematics of 'analysis situs' and the fact that space curvature could be used to describe matter in motion.171 It was implicit in Clifford's original context of this statement that the ether could be replaced by space curvature for a total theory of the physical world of matter. Eddington's first work on general relativity clearly displays his Victorian heritage and education. But as his ideas about general relativity evolved beyond Eddington's Victorian bias, Clifford's words and influence seemed to exert an ever-stronger presence in Eddington's own work.

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Two and a half decades later, E.T. Whittaker wrote a history of scientific conceptions of the external world, From Euclid to Eddington. The book ended with a statement that Eddington was attempting to reduce all of physics to "one kind of ultimate particle, of which [the known elementary particles] are, so to speak, disguised manifestations." 172 A comparison of this with Clifford's goal, as expressed in the closing remarks of the Elements, indicates that Clifford's and Eddington's goals were essentially the same, the physical expression of the universe based upon the various manifestations of a single particle. But their methods of achieving that goal were quite different. Eddington did not use Clifford's twists, but did adopt Clifford's basic philosophy as well as borrow some of Clifford's mathematics. Regarding the similarity between their philosophies, Smokler even suggests that Eddington's book The Nature of the Physical World be referred to for an explanation of Clifford's philosophy.173

The theory to which Whittaker referred was Eddington's "fundamental theory." Eddington had already presented various papers and articles on the theory and these were collected, edited and published by Whittaker in 1946, after Eddington's death. The fundamental theory was meant to be the pinnacle of Eddington's considerable work and long association with the theories of relativity, the quantum theory and cosmology. The theory was based upon the mathematics of E-numbers, which represented the elements of an E-frame that Eddington associated with our physical space-time. This E-frame, in conjunction with an F-frame to which it was related, then allowed a new interpretation of the Christoffel tensors from which Einstein had constructed his own mathematical model of space-time curvature.

The E-numbers were quaternions and shared many characteristics with both Clifford's biquaternions and Ball's screws. But Eddington's application of quaternions was different because the essential problem of finding a mathematical model was different for Eddington than it had been for Clifford. It had become necessary for Eddington to account for all of the physical concepts and phenomena that had been discovered since Clifford's death: quantum theory, the Bohr atom, radioactivity, the atomic nucleus, electrons, protons, neutrons, the theories of relativity and others. So Eddington's theory was different from Clifford's even though they were philosophically similar and could not be considered a simple continuation of Clifford's work.

In 1944, Eddington published a paper entitled "The Evaluation of the Cosmical Number." He had intended that the ideas presented in this paper be included as

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the epistemological basis of his theory in its final version. So Whittaker added the paper as an appendix to the posthumous publication of the Fundamental Theory. In his epistemological explanation, Eddington stated that the central problem that he had addressed was "to discover a structure of measures and measurables which is such that this promise [of distinguishing between measures and measurables] can be fulfilled." 174

Measures, which are strictly geometrical in nature, formed the basis of Eddington's model of space-time, while measurables could be interpreted as purely physical particles. It was necessary that both contribute to the structure of space-time even though they had to be distinguished one from the other at the same time. The problem for Eddington was that measures and measurables were both the same and different. The science of space-time was thus reduced to a question of distinction between the two.

The data of physics are measures; but we can make nothing of a mere collection of measures without any note of the objects and circumstances to which they refer. The crux of the problem is to supply 'connectivity' to the measures; so that in the theoretical treatment there may be an equivalent for that part of the procedure of measurement which consists in noting the objects and circumstances to which the measures relate.175

From this statement alone, Eddington's philosophical debt to Clifford is clearly evident. Eddington's measurables were in a very broad sense the same as Clifford's twists. The problems faced by Clifford in discovering the mathematics of ‘connectivity’ between the individual contiguous points of space were the same as those described by Eddington. This problem lead Clifford to the development of that particular non-Euclidean geometry which he had hoped to use to describe the 'structure' of space in his own space-theory of matter, just as it lead Eddington to the development of his own fundamental theory. If Eddington had used any other word than 'connectivity,' which he himself had emphasized, the case for Eddington's debt to Clifford would have been harder to make, but not impossible to make. But the idea of 'connectivity' was so essential and unique to Clifford's mathematical development that this word alone proves Eddington's debt to Clifford. This single statement reflected Clifford's concepts as much as Eddington's.

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Eddington made no reference to Clifford's earlier work nor would he have been compelled to cite Clifford as the source of his ideas since his own theory was far more comprehensive than Clifford's and thus quite different in application. Also, Clifford's concepts had long been accepted as part of the public domain of science so there was no need to cite Clifford directly. What can be determined from these examples with historical accuracy is that Eddington's views on science and the physical world, from the very beginning of his association with general relativity until his death, if not before 1916, were profoundly influenced by Clifford's earlier researches and conceptual developments. This influence could not have been unique in Eddington's experience alone, but would have been true for many others.

For his own part, Whittaker made no reference to Clifford in his book From Euclid to Eddington and only briefly mentioned Clifford in his History of the Theories of Aether and Electricity,176 but this oversight is insignificant. By the time that these books were written, Clifford had already received adequate recognition by many scientists as the originator of the concept of matter as space curvature as well as inaccurate recognition as the "anticipator" of general relativity. Actually, what Clifford had anticipated went well beyond just the use of space curvature as matter as described in general relativity.

Nor did Thomas Greenwood directly mention Clifford in his 1922 essay "Geometry and Reality," even though Greenwood did relate other interesting facts regarding the general attitude toward space curvature. After explaining that astronomers had been searching for space curvature for some time by careful observation of stellar parallax, Greenwood continued to describe another aspect of non-Euclidean science that was common knowledge before relativity.

But all these [parallax] observations proved negative: space presented itself as Euclidean. Nevertheless there was an idea amongst men of science, that more accurate observations and the development of mechanical consequences of non-Euclidean geometry with regard to astronomical problems, would certainly favour the legitimacy of non-Euclidean postulates as physical hypotheses.177

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These simple historical facts, as explained by Greenwood, seem all but forgotten by modern historians and scholars who study the genesis of general relativity.

Clifford had translated Riemann's work into English. He was the first scientist in the English speaking world to describe the problem of parallax measurements with respect to space curvature before the public and the first to popularize the concept of non-Euclidean geometry in his presentation of "The Aims and Instruments of Science" and the "Philosophy of the Pure Sciences." His "Postulates" was considered a classic of science by the turn of the century, as was his Common Sense. Clifford was the founder of mathematical studies on the dynamics of non-Euclidean space and discovered a whole class of non-Euclidean geometries. These were no mean accomplishments and Greenwood did not need to mention Clifford's name within the context of the pre-relativistic search for a physical non-Euclidean space. When he referred to the mechanics of motion in a non-Euclidean space, he could have been speaking of no one but Clifford.

By the date of general relativity's initial development, Clifford's ideas had been disseminated throughout science and culture and in many cases were no longer associated with Clifford's name. Although the suggestion that space curvature could have physical consequences can be attributed to Riemann alone, only Clifford had gone so far as to link small variations of curvature with the concept of matter itself and begin the task of redefining the very concept of force in terms of such a space curvature. It is also quite evident that a continuous historical line can be found flowing from Clifford's initial ideas through the work of Hinton, Robert Ball, Pearson and others, to the more generally held belief by many scientists and the common educated populace that the real physical space of human perception could possibly be non-Euclidean. This was an accepted fact on the eve of the discovery of relativity theory, as Greenwood implied.

On the other hand, there are no causal links between the mathematical theory of relativity and Clifford's mathematical researches because the mathematization of space curvature did not follow the path originally explored by Clifford. Under Einstein's direction, the mathematical model of space-time curvature was to be based upon tensors rather than quaternions. So whenever historians and scholars have seen fit to trace the historical roots of general relativity into the past, they have generally followed the concrete examples of the

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mathematical lineage rather than the more spurious development of concepts and attitudes deriving primarily from Clifford's work. Such a "quick fix" of history does not tell the whole story. The development of tensor calculus was a purely mathematical exercise, devoid of physical content. So it would seem to anyone tracing the mathematical development of tensors that the space-time curvature represented by tensors was devoid of physical interpretation before Einstein's work was completed and thus accept the fact that Einstein was the first to give a new "physical meaning" to the purely mathematical model of curved space based on tensors. This seems to be true at least for the case of Einstein himself, for whom no evidence exists of a previous knowledge of either Clifford's "Space Theory of Matter" or Clifford's other physical interpretations of space curvature. But it must be remembered, and rightly so, that the new theory of general relativity was grafted on to an already considerable and growing recognition of the fact that space could well be and probably was non-Euclidean.

For many of those who were interested in the scientific problem of space curvature before general relativity, this attitude was supplemented by a previous knowledge of Clifford's physical concepts of mass and force. So we have such nonchalant statements as that made by Frank Kassel in his 1926 doctoral thesis that the "principle [which demonstrates that Euclidean geometry should be abandoned with general relativity] is an outcome of a thought emphasized by Clifford: that, namely, the metrical properties of space are wholly determined by the masses of bodies." 178 Kassel's statement came a decade after the inception of general relativity and he drew no historical connections between Clifford and Einstein, but neither did he hesitate to associate their ideas on a purely philosophical level. Even then, neither the physicist E.H. Kennard nor the philosopher Edgar H. Singer of the University of Pennsylvania objected to this association of ideas or Kassel's statement would not have accepted in the published text of his thesis. Many such statements can be found in the decade following 1916 which would lend further support to the conclusion that many of the scientists and scholars who first accepted Einstein's theories had a previous knowledge of Clifford's work.

There is no way to absolutely "prove" that these scientists and scholars accepted general relativity "because" they had a previous knowledge of Clifford's concepts of matter and space, and it may well be inaccurate to even voice such an opinion. But there is certainly a preponderance of evidence indicating that Clifford's work influenced the following generations in such a

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way that, in essence, he laid the foundations for the positive attitudes toward physical interpretations of curved space upon which general relativity later built its own following and found a comfortable home. Therefore, the acceptance of general relativity by the scientific community was enhanced and accelerated by the previous knowledge of Clifford's work. There was no longer a need for the numerous philosophical arguments against space-curvature that had plagued Clifford's original ideas, so such arguments did not develop after the advent of general relativity. This is especially true in England and America where Clifford's concepts remained popular throughout the years between his death and Einstein's success.

Clifford was a major player in opening a whole new field of scientific inquiry in which our basic notions of space, time and force and their relationships to electromagnetism and gravitation were challenged, even unto this day. Even the recent theory of "twistors" in which Roger Penrose attempts a grand unification of the natural forces is based upon Clifford's earlier work. By introducing the concept of a "twist" as an element of space curvature, Clifford began an intellectual movement to tear down the house that forms our preconceived prejudice toward a physics based solely upon Euclidean space and replace it with a more general concept of space curvature which could account for both gravitation and electromagnetism.

ENDNOTES

1. William Kingdon Clifford, The Common Sense of the Exact Sciences, edited by Karl Pearson, newly edited with an introduction by James R. Newman and preface by Bertrand Russell, (New York: Dover, 1955; Reprint of the 1946 Knopf edition; Unaltered reprint of the third edition of 1899; First English edition, London: Macmillan, 1885; First American edition, New York: Appleton, 1885). See Chapter on "Position," pp.184-204.

2. William K. Clifford, "On the Space-Theory of Matter," presented 2 February 1870, Transactions of the Cambridge Philosophical Society, 1866/1876, 2: 157-158; Reprinted in William K. Clifford, Mathematical Papers, edited by Robert Tucker, with an introduction by H.J.S. Smith, (New York: Chelsea, 1968; An unaltered reproduction of the 1882 original): 21-22.

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3. Sir Arthur S. Eddington, Space, Time, and Gravitation, (Cambridge: at the University Press, 1921), on p.192; E.T. Bell, The Development of Mathematics, second edition, (New York: McGraw-Hill, 1945; Reprint of the 1940 original), on p.360; Lloyd S. Swenson, The Genesis of Relativity, (New York: Burt Franklin, 1979), on p.36. At least in Eddington's case, the term "anticipated" was followed by the qualifiers "with marvelous foresight." Those who borrowed the term "anticipation" from Eddington dropped the qualifiers.

4. For example, see Max Jammer, Concepts of Space, (Cambridge: Harvard University Press, 1954), on p.160; A. d'Abro, The Evolution of Scientific Thought from Newton to Einstein, (New York: Dover, 1950; Second and enlarged edition of the 1927 original), on p.58; C.W. Kilmister, General Theory of Relativity, (New York: Pergamon Press, 1973), on p.124. Jammer used the term "suggestion" rather than speculation while D'Abro described the concepts as "exceedingly speculative."

5. Jammer, Concepts, pp.160-161; D'Abro, Scientific Thought, p.58; Banesh Hoffman, Albert Einstein, Creator and Rebel, with the collaboration of Helen Dukas, (New York: Viking Press, 1972), on p.176. Jammer used the descriptive terms "fantastic in [Clifford's] own day" rather than "untenable."

6. E.T. Bell, Men of Mathematics, (New York: Simon and Schuster, 1965; Reprint of 1937), on p.503.

7. E.T. Bell, The Development of Mathematics, (New York: McGraw-Hill, 1945; Second edition of the 1940 original), on p.360.

8. Ruth Farwell and Christopher Knee, "The End of the Absolute: A Nineteenth-Century Contribution to General Relativity," Studies in the History and Philosophy of Science, March 1990, 21: 91-121.

9. Joan L. Richards, Mathematical Visions: The Pursuit of Geometry in Victorian England, (Boston: Harcourt, Brace, Jovanovitch, 1988).

10. Howard E. Smokler, "W.K. Clifford's Conception of Geometry," Philosophical Quarterly, 1966, 16: 249-257; "William Kingdon Clifford," The Encyclopedia of Philosophy, edited by Paul Edwards, eight volumes, (New York: Macmillan, 1967), Volume 2: 123-125.

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11. James R. Newman, "William Kingdon Clifford," Scientific American, February 1953, 188: 78-84, on p.78.

12. E.A. Power, "Exeter's Mathematician - W.K. Clifford, F.R.S. 1845-79," Advancement of Science, 1970, 26: 318-328; "The Application of Quantum Electrodynamics to Molecular Forces," a copy was received directly from E.A. Power without references.

13. Gurney, Feza, "Quaternionic and Octonionic Structures in Physics: Episodes in the relation between physics and mathematics," in Symmetries in Physics, Proceedings of the First International Meeting on the History of Scientific Ideas held at Sant Feliu de Guixols, Catalonia, Spain, September 20-26, 1983, edited by Manuel G. Doncel, Armin Hermann, Louis Michel and Abraham Pais, (Bellaterra, Spain: Seminari d'Historia de les Cienceis, Universitat Autonoma de Barcelona, 1987): 557-592.

14. J.S.R. Chisholm and A.K. Common, editors, Clifford Algebras and their Applications in Mathematical Physics, Proceedings of the Nato and SERC Workshop on Clifford Algebras, Canterbury, UK, September 1985, NATO ASI Series, Series C, Mathematics and Physical Sciences, 183, (Dordrecht: Reidel, 1986).

15. John Archibald Wheeler, Geometrodynamics, (New York: Academic Press, 1962). References to Clifford on pp.8, 123, and 129.

16. Adolph Grunbaum, "The Ontology of the Curvature of Empty Space in the Geometrodynamics of Clifford and Wheeler," in Patrick Suppes, editor, Space, Time and Geometry, Synthese Library, (Dordrecht: Reidel, 1973): 268-295, on pp.292-293.

17. Lewis S. Feuer, Einstein and the Generations of Science, (New York: Basic Books, 1974), on p.64.

18. Bell, Men, p.490.

19. This idea actually goes back to H.J.S. Smith in his introduction to Clifford, Mathematical Papers, p.xliii.

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20. Henri Poincaré, "Non-Euclidean Geometry," translated by W.J.L., Nature, 25 February 1892, 45: 404-407, on p.407; The passage on conventionalism was repeated in Science and Hypothesis, translated by W.J. Greenstreet with an introduction by J. Larmor, (New York: Dover, 1952; Reprint of 1905 edition), on p.50.

21. Arthur I. Miller, "The Myth of Gauss' Experiment on the Euclidean Nature of Physical Space," Isis, 1972, 63: 345-348, on p.348.

22. For example, see George Chrystal, "Theory of Parallels," the ninth edition, Encyclopaedia Britannica, (Edinburgh: Adam & Charles Black, 1885), Volume 18: 254-255; Also see Federigo Enriques, Problems of Science, translated by Katherine Royce with a note by Josiah Royce, (London: Open Court, 1914; Original Italian published in 1906), on pp.192-193.

23. Richards, Visions, pp. 93-94, 113. Richards uses such terms as "not widely accepted" and "exuberantly speculated" to describe Clifford's physical concepts.

24. Letter from William K. Clifford to Frederick Pollock, partially reprinted in Lectures and Essays, edited by Frederick Pollock and Leslie Stephen, two volumes, (London: Macmillan, 1879), Volume I, on p.30. "Solving the Universe" was a phrase often used by Clifford to describe his private thoughts and published work.

25. J.J. Sylvester, "A Plea for the Mathematician," Nature, 30 December 1869, 1: 237-239, on p.238; Sylvester's "Plea" was an abridgement of his British Association address, the complete text was printed in the Reports of the British Association, (Exeter), 1969: 1-9; The complete text was reprinted with further footnotes in an appendix in J.J. Sylvester, The Laws of Verse, (London: Longmans, 1870), and thus reprinted in Collected Papers, (Cambridge: at the University Press, 1904): 650-719.

26. The various commentators on "Kant's View of Space" in the pages of the first volume of Nature were George Henry Lewes, on pp.289, 334, 386; T.H. Huxley, on p.314; C.M. Ingleby, on pp.314, 361; J.J. Sylvester, on pp.314, 360; G. Croom Robertson, on p.334; and W.H. Stanley Monck, pp.335, 387.

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27. William K. Clifford, "On the Aims and Instruments of Scientific Thought," a lecture delivered before members of the British Association at Brighton on 19 August 1872, Macmillan's Magazine, October 1872: 499-512; Reprinted in Clifford, Lectures and Essays, Volume 1: 124-157.

28. C.M. Ingleby, "The Antinomies of Kant," Nature, 6 February 1873, 7: 262; William K. Clifford, "The unreasonable," Nature, 13 February 1873, 7: 282; C.M. Ingleby, "The Unreasonable," Nature, 20 February 1873, 7: 302-303.

29. C.M. Ingleby, "Prof. Clifford on Curved Space," Nature, 13 February 1873, 7: 282-283, on p.282. Ingleby had already made a general criticism of hyperspace theories in "Transcendent Space," Nature, 13 January 1870, 1: 289; and 17 February 1870: 407.

30. Letter from C.M. Ingleby to C.J. Monro, 30 August 1870, #2438, Acc.1063, Monro Collection, Greater London Record Office, London, England. All of the following letters from Ingleby to Monro are from this same collection.

31. Ingleby to Monro, 16 October 1871, #2454.

32. Ingleby to Monro, 29 November 1871, #2469.

33. Ingleby to Monro, 3 February 1872, #2491.

34. Ingleby to Monro, 24 May 1872, #2502B.

35. Letter from C.J. Monro to James Clerk Maxwell, 10 September 1871, Acc.#1063, #2109, the Monro Collection, Greater London Record Office, London, England.

36. James Clerk Maxwell to C.J. Monro, 15 March 1871, partially reprinted in Lewis Campbell and William Garnett, The Life of James Clerk Maxwell, with selections from his correspondence and occasional writings, (London: Macmillan, 1884), on p.290.

37. Monro to Maxwell, 10 September 1871, #2109. Campbell and Garnett only offered a few paragraphs in their publication of this letter and ignored the much larger portion of the letter dealing with the question of a fourth dimension.

399

38. Note card from James Clerk Maxwell to Peter G. Tait, 11 November 1874, Add. 7655/Ig/72, Maxwell Papers, University Library, Cambridge, England.

39. Letter from James Clerk Maxwell to Peter G. Tait, 9 November 1876, Add.7655/#38, Maxwell Papers, University Library, Cambridge, England.

40. Clifford, Common Sense, pp.193-197.

41. Frederick Pollock quoting James C. Maxwell in Clifford, Lecture and Essays, p.14.

42. Ingleby to Monro, 9 February 1873, #2540.

43.Clifford, "The unreasonable," p.282.

44. W.K. Clifford, "The Philosophy of the Pure Sciences. II. The Postulates of the Science of Space," Contemporary Review, 1874, 25: 360-376, on p.376; Reprinted in Lectures and Essays, Volume I, pp.295-323. The lectures on the "Philosophy of Pure Science" were part of an afternoon lecture series at the Royal Institution on 1, 8, and 15 of March, 1873. They were subsequently reprinted in Contemporary Review and The Nineteenth Century, in October 1874, February 1875, and March 1879, as well as being included in Clifford's Lectures and Essays. The lectures were attended by 165, 212 and 143 people, respectively, which were substantial crowds for that type of lecture.

45. William K. Clifford, "Preliminary Sketch of Biquaternions," presented on 12 June 1873, Proceedings of the London Mathematical Society, 1873: 381-395; Reprinted in Mathematical Papers: 181-200.

46. William K. Clifford, "On some Curves of the Fifth Class," and "On a Surface of Zero Curvature and Finite Extent." Only the titles were listed in the Reports of the British Association, (Bradford), 1873, 43. No mention is made of these references in any of standard bibliographies on the non-Euclidean geometries. It is nearly as if the lectures were never given.

47. Felix Klein, "Zur nichteuklidische Geometrie," Mathematische Annalen, 1890, 37: 544-572. Klein also presented this topic before an American audience in 1893 and published as The Evanston Colloquium, Lectures on Mathematics, delivered 28 August to 9 September 1893 before members of the Congress of

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Mathematics held in connection with the World's Fair in Chicago, at Northwestern University, Evanston, Illinois, and reported by Alexander Ziwet, (New York: Macmillan, 1894), on pp.89-90; Wilhelm Karl Joseph Killing, "Clifford-Klein'sche Raumformen," Clebsch, Mathematische Annalen, 1891, 39: 257-278.

48. Sir Robert S. Ball quoted by Sir Joseph Larmor in Ball, Reminiscences and Letters of Sir Robert Ball, edited by W. Valentine Ball, (London: Cassells, 1915), on p.155; Ball repeats the story in "Non-Euclidean Geometry," Hermathena, 1879, 3: 500-541, on p.537.

49. W.K. Clifford, "On the Classification of Loci," read 8 April 1878, Philosophical Transactions of the Royal Society, Part II, 1878: 663-681: Reprinted in Mathematical Papers: 305-331; "Applications of Grassmann's Extensive Algebra," American Journal of Mathematics, 1878, 1: 350-358; Reprinted in Clifford, Mathematical Papers: 266-276, on p.271.

50. W.K. Clifford, "On the Nature of Things-in-Themselves," Mind, 1878, 3: 57-67; Reprinted in Lectures and Essays, Volume II, pp.71-88.

51. Peter G. Tait and Balfour Stewart, The Unseen Universe, or Physical Speculation on a Future State, (London: Macmillan, 1875). The book was published anonymously at first, but became such an instant success that the authors revealed themselves in later publications. At the time of Clifford's review, the authors were unknown although there was much rumor and speculation as to their identities.

52. William K. Clifford, "The Unseen Universe," Fortnightly Review, January-June 1875, 17: 776-793, on p.788; Reprinted in Lectures and Essays, Volume I, pp.228-253.

53. William K. Clifford, "Energy and Force," edited by Frederick Pollock and J.F. Moulton, Nature, 10 June 1880, 22: 122-124.

54. Peter G. Tait, "Clifford's Dynamic," Nature, 23 May 1878, 18: 89-91, on p.91.

55. Unnamed reviewer, "Professor Clifford's Elements of Dynamic," The Saturday Review, 22 June 1878, 45: 792-794, on p.793.

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56. William K. Clifford, Elements of Dynamic: An Introduction to the Study of Motion and Rest in Solid and Fluid Bodies, Part I. Kinematics, (London: Macmillan, 1878), on p.221.

57. H.J.S. Smith in W.K. Clifford, Mathematical Papers, edited by Robert Tucker, with an introduction by H.J.S. Smith, (New York: Chelsea, 1968; Reproduction of the first edition of 1882).

58. William K. Clifford, Elements of Dynamic, Book IV and Appendix, edited by Robert Tucker, (London: Macmillan, 1878), on pp. 59-62.

59. William K. Clifford, "Instruments Illustrating Kinematics, Statics, and Dynamics," in Mathematical Papers: 424-440; on p.437. This paper was originally included in the South Kensington Handbook to the Special Loan Collection of Scientific Apparatus, 1876, and was also reprinted in Clifford, Lectures and Essays, Volume II: 9-30.

60. Richards, Visions, pp.143-148, 154; Joan Richards, "Projective Geometry and Mathematical Progress in Mid-Victorian Britain," Studies in History and Philosophy of Science, 1986, 17: 297-325, on pp.320-321.

61. Arthur Cayley, Presidential Address to the British Association, Report of the British Association, (Southport), 1883: 3-37; Reprinted in The Collected Mathematical Papers, thirteen volumes, with volumes I-VII edited by Arthur Cayley, volumes VIII-XIII edited by A.R. Forsyth, (Cambridge: At the University Press, 1889): Volume XI: 429-459, on p.436.

62. Richards, Pursuit, pp.90, 138-140.

63. John William Withers, Euclid's Parallel Postulate: Its Nature, Validity, and Place In Geometrical Systems, (Chicago: Open Court, 1905), on pp.49-50. This book represented a publication of Wither's doctoral dissertation.

64. William Thomson, Lord Kelvin, "The Wave Theory of Light," a lecture delivered before the Academy of Music, Philadelphia, under the auspices of the Franklin Institute, 29 September 1884, printed in the Journal of the Franklin Institute, November 1884, 118: 321-341; Reprinted in Sir William Thomson, Popular Lectures, (London: Macmillan, 1888).

402

65. Letter from Arthur Cayley to Lord Kelvin, 25 March 1889, Add.7342/C63, Kelvin Papers, University Library, Cambridge.

66. Frederick W. Frankland, "On the Simplest Continuous Manifoldness of two Dimensions and Finite Extent," Transactions of the New Zealand Institute, 1876, 9: 272-279; And in Proceedings of the London Mathematical Society, 1876, 8: 57-64; Also reprinted in Nature, April 1877, 12: 515-517.

67. Simon Newcomb, "Elementary theorems relating to the geometry of a space of three dimensions and of uniform positive curvature in the fourth dimension," Crelle's Journal fur die reine und angewandte Mathematik, 1877, 83: 293-299.

68.Felix Klein, "Ueber die sogennante Nicht-Euklidische Geometrie," Mathematische Annalen, 1871, 4: 573-611.

69. Robert Stawell Ball, "Measurement," ninth edition, Encyclopaedia Britannica, (London: Adam & Charles Black, 1885), Volume 15: 659-668, on pp.664-665; George Chrystal, "Non-Euclidean Geometry," Proceedings of the Royal Society of Edinburgh, 1878-1879, 10: 638-664, on p.644; Bertrand Russell, "Non-Euclidean Geometry," in tenth edition, Encyclopaedia Britannica, Volume 18: 664-674, on p.668; Bertrand Russell and A.N. Whitehead, "Non-Euclidean Geometry," in the eleventh edition, Encyclopaedia Britannica, Volume 11: 724-730, on p.729.

70. Thomas Archer Hirst in William H. Brock and Roy M. MacLeod, editors, Natural Knowledge in Social Contexts: The Journals of Thomas Archer Hirst, (London: Mansell, 1980), on p.1828.

71. Simon Newcomb, "Is the Airship Coming?" McClure's Magazine, September 1901, 17: 432-435, on pp.432-433.

72. William Skey, "Notes upon Mr. Frankland's paper 'On the Simplest Continuous Manifoldness of Two Dimensions and of Finite Extent'," Transactions and Proceedings of the New Zealand Institute, 1880, 13: 100-109.

73. C.J. Monro, "On the Simplest Continuous Manifoldness of Two Dimensions and Finite Extent," Nature, 26 April 1877, 17: 547.

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74. C.J. Monro, "On the Simplest Continuous Manifold of Two Dimensions and of Finite Extent," Nature, 8 July 1880: 218. This letter should not be confused with Monro's earlier note of the same title.

75. C.J. Monro, "Inside Out," Nature, 30 May 1878, 18: 115.

76. Simon Newcomb, "Note on a Class of Transformations Which Surfaces May Undergo in Space of More than Three Dimensions," American Journal of Mathematics, 1878, 1: 1-4.

77. C.J. Monro, "On Flexure of Spaces," read 13 January 1878, with a comment by Arthur Cayley, Proceedings of the London Mathematical Society, November 1877 to November 1878, 9: 171-176.

78. Frederick W. Frankland, Thoughts on Ultimate Problems: Being a Series of Short Studies on Theological and Metaphysical Subjects, fifth and revised edition, (London: David Nutt, 1912), on p.13.

79. This meeting was held in Toronto, Canada, on 17 August 1897. Although Frankland's "Theory of Discrete Manifolds" was obscure and copies were difficult to obtain, it was published as Appendix C of Thoughts on Ultimate Problems: 37-42.

80. F.N. Cole, "Fourth Summer Meeting of the American Mathematical Society," Bulletin of the New York Mathematical Society, October 1897, 4: 1-11, on p.10.

81. Frederick W. Frankland, Collected Essays and Citations, Volume I, Theology and Metaphysics, 1872-1906, (Foxton, New Zealand: G.T. Beale, 1906).

82. Charles H. Chandler, "Transcendental Space," Transactions of the Wisconsin Academy of Sciences, Arts and Letters, 1896-1897, No.11: 237-248, on p.243.

83. George Bruce Halsted, "The Old and the New Geometry," Educational Review, 1893, 6: 144-157, on p.150; "Non-Euclidean Geometry," Popular Astronomy, 1900, 8: 189-202, on p.189. These are only two of the many examples available.

404

84. C.S. Peirce in Carolyn Eisele, ed., "The Charles Peirce-Simon Newcomb Correspondence," Proceedings of the American Philosophical Society, October 1957, 101: 409-433, on pp.420-423.

85. C.S. Peirce, "The Architecture of Theories," The Monist, January 1894, 1: 161-176, on pp.173-174, 176.

86. W.I. Stringham, "Rotation in Four-Dimensional Space," Johns Hopkins University Circular, March 1880, 1: 49; "On the Rotation of a Rigid System in Space of Four Dimensions," Proceedings of the American Association, 1884, 33: 55-56.

87. Pearson in Clifford, Common Sense, pp.lxiii-lxiv.

88. Ingleby to Monro, 15 March 1879, #2649. Ingleby praised Clifford again in another letter to Monro, 19 March 1879, #2650.

89. Frederick Pollock and Leslie Stephen, editors, William K. Clifford, Lectures and Essays. These men were also biographers of Clifford. Pollock's touching biography first appeared in The Fortnightly Review, 1879, 25: 667-687, and was then revised and published as the introduction to Lectures and Essays, Volume 1: 1-43. Leslie Stephen went on to write a short biography of Clifford for the Dictionary of National Biography, from early times to 1900, edited by Sir Leslie Stephen and Sir Sydney Lee, (London: Geoffrey Cumberlege, Publisher to the University, Oxford University Press, 1917), Volume 4: 538-541.

90. Before editing Clifford's Mathematical Papers, Robert Tucker had the task of collecting them. He searched far and wide, placing a call for any of Clifford's papers in Nature: "Professor Clifford's Mathematical Papers," Nature, 26 June 1879, 20: 195.

91. William K. Clifford, Elements of Dynamic: An Introduction to the Study of Motion and Rest in Solid and Fluid Bodies, Part 1. Kinematic, Book IV. And Appendix, edited by Robert Tucker, (London: Macmillan, 1887).

92. Pearson later repeated these comments in a letter to Ernst Mach, although at that time he claimed to have written them in 1883, which was impossible. The letter has been published by Joachim Thiele, "Karl Pearson, Ernst Mach, John

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B. Stallo: Briefe aus den Jahren 1897 bis 1904," Isis, 1978, 69: 535-542, on p.538.

93. Karl Pearson, "On the Distortion of a Solid Elastic Sphere," Quarterly Journal of Pure and Applied Mathematics, 1879, 10: 375- 381; "On the Motion of Spherical and Ellipsoidal Bodies in Fluid Media," Quarterly Journal of Pure and Applied Mathematics, 1883, 20: 60-80; Part II: 184-211; "On a certain Atomic Hypothesis," read 2 February 1885, Transactions of the Cambridge Philosophical Society, 1883-1889, 14: 71-120, on p.120. The date appearing at the end of the paper was 11 March 1883, the actual date that it had been written; "On a certain Atomic Hypothesis," Proceedings of the London Mathematical Society, 8 November 1888, 20: 38-63.

94. Karl Pearson, "Note on Twists in an Infinite Elastic Solid," Messenger of Mathematics, 1883, 13: 79-95; "On the Generalised Equations of Elasticity, and their Application to the Wave Theory of Light," Proceedings of the London Mathematical Society, 1889, 20: 297-350.

95. Karl Pearson, "Ether Squirts. Being an attempt to specialize the form of ether Motion which forms an Atom in a Theory propounded in former papers," American Journal of Mathematics, 1891, 13: 309-362.

96. Pearson, "Ether Squirts," p.309. However, Pearson gave a slightly more extensive explanation on pp.312-313.

97. Letter from Karl Pearson to Robert J. Parker, dated 6-4-85, file #922, Pearson Papers, University College, London.

Cambridge very quiet, did nothing but look over proofs & talk mathematics, metaphysics with Macauley. He does not see how I can create the universe out of empty space, by twisting it, and is a perfect slave to the matter superstition. So Sir William Thomson, who has been writing about the weight of the ether, as if empty space could weigh anything! I am going to weigh a twist! ... That might mean something."

98. Pearson in Clifford, Common Sense, p.203.

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99. Charles T. Whitmell, Space and Its Dimensions, (Cardiff: South Wales Printing, 1892), on p.24.

100. Frankland, Thoughts, pp.12-13.

101. Karl Pearson, "Matter and Soul," read before the Sunday Lecture Society on 6 December 1885, and printed as a pamphlet by the Society; Reprinted in The Ethic of Freethought and other Addresses and Essays, second edition, (London: Adam and Charles Black, 1901; Originally printed in 1887): 21-44, on p.30-32.

102. Pearson, "Matter," pp.28-29.

103. Pearson, "Matter," pp.29-30.

104. Karl Pearson, Review of Elements of Dynamic - Part 1. Kinematic, Book IV, and Appendix, The Athenaeum, 16 July 1887, No.3116: 86-87, on p.86.

105. Karl Pearson, The Grammar of Science, (London: Walter Scott, 1892; Third edition, London: Adam and Charles Black, 1911; New York: Meridian Books, 1961; Reprint of the third edition of 1911), on p.229.

106. Pearson, Grammar, p.229.

107. Letters from Lord Kelvin to Karl Pearson, 10 January 1893, 16 January 1893, 2 March 1893, 11 March 1893, file #871/1, Pearson Papers, University College, London.

108. Letter from Karl Pearson to Lord Kelvin, undated, #104, Pearson Papers, University College, London. This letter is listed as an essay on "The Nature of Physical Space" in the catalogue to the Pearson Papers, rather than a letter to Kelvin. It is probably from the period about 1892-1893, the time during which Pearson was arguing with Kelvin over the nature of the hypothetical ether.

109. Robert S. Ball, "On the Small Oscillations of a Rigid Body about a Fixed Point under the Action of any Forces, and, more particularly, when Gravity is the only Force acting," read 24 January 1870, Transactions of the Royal Irish

407

Academy, 1860-1870, 24: 593-627; A summary was published in Proceedings of the Royal Irish Academy, 1870-1874, 5, second series: 11-14.

110. Arthur Buchheim, "A Memoir on Biquaternions," American Journal of Mathematics, 1884, 7: 293-326; "On Clifford's Theory of Graphs," Proceedings of the London Mathematical Society, 12 November 1885, 17: 80-106; "On the Theory of Screws in Elliptic Space," Proceedings of the London Mathematical Society, 10 January 1884, 15: 83-98; 13 November 1884, 16: 15-27; 10 June 1886, 17: 240-254; 11 November 1886, 18: 88-96.

111. Sir Robert S. Ball, "The Twelfth and Concluding Memoir on the 'Theory of Screws,' with a Summary of the Twelve Memoirs," read 8 November 1897, Transactions of the Royal Irish Academy, 1896-1902, 31: 145-196, on p.536.

112. Joseph Larmor in Ball, Reminiscences, pp.154-155.

113. Robert Stawell Ball, "Non-Euclidean Geometry," Hermathena, 1879, 3: 500-541; "Notes on Non-Euclidean Geometry," Reports of the British Association, 1880, 50: 476-477; "On the Elucidation of a Question in Kinematics by the Aid of Non-Euclidean Space," Reports of the British Association, (York), 1881, 51: 535-536; "Certain Problems in the Dynamics of a Rigid System Moving in Elliptic Space," read 14 November 1881, Transactions of the Royal Irish Academy, 1880-1886, 28: 159-184; "Notes on the Kinematics and Dynamics of a Rigid System in Elliptic Space," read 9 June 1884, Proceedings of the Royal Irish Academy, 1884-1888, 4: 252- 258; "Note on the Character of the Linear Transformation which Corresponds to the Displacement of a Rigid System in Elliptical Space," read 9 November 1885, Proceedings of the Royal Irish Academy, 1884-1888, 4: 532-537; Dynamics and the Modern Geometry: A New Chapter in the Theory of Screws, the Cunningham Memoir, No.IV., (Dublin: Published by the Academy at the Academy House, 1887).

114. Robert Stawell Ball, "The Distance of Stars," read 11 February 1881, Proceedings of the Royal Institution of Great Britain, 1879-1882, 9: 514-519, on p.519.

115.Ball,"Measurement," p.664.

408

116. Robert Stawell Ball, "On the Theory of Content," read 12 December 1887, Transactions of the Royal Irish Academy, 1889, 29: 123-182.

117. Ball, "Content," p.151.

118. Olaus Henrici, "The Theory of Screws," Nature, 5 January 1890, 42: 127-132, on p.131.

119. Robert Stawell Ball, "A Dynamical Parable," Nature, 1 September 1887, 36: 424-429; Originally presented as the presidential address to the Physical Section of the British Association, and published in the Reports of the British Association, (Manchester), 1887, 57; Reprinted in Ball, Treatise: 496-509, on pp.508-509.

120. Robert Stawell Ball, A Treatise on the Theory of Screws, (Cambridge: Cambridge University Press), on p.519.

121. Charles H. Hinton, "What is the Fourth Dimension?" originally published in the Dublin University Magazine, 1880, and then republished in the Cheltenham Ladies' College Magazine. It was printed as a separate pamphlet in 1884 with the subtitle "Ghosts Explained" added by the publisher.

122. Hinton's "What is the Fourth Dimension?" was first published as a pamphlet before it was added to a collection his other essays and published in Scientific Romances, two volumes, (London: Swann & Sonnenschein, 1884-1886), Volume 1: 1-32; Reprinted in Speculations of the Fourth Dimension: Selected Writings of Charles H. Hinton, edited by Rudolf v.B. Rucker, (New York: Dover, 1980): 1-22.

123. Hinton, "What?" in Speculations, pp.16-20.

124. Hinton, "A Plane World," in Speculations: 23-40, on pp.36- 37. Hinton also developed the idea of "twists" as a physical concept, independent of electrical phenomena, in "Many Dimensions," in Speculations: 67-79, on pp.74-75. Both of these essays were published in Scientific Romances.

125. Hinton, "A Picture of Our Universe," in Speculations, on pp.52-55. Originally published in Scientific Romances.

409

126. Rucker in Hinton, Speculations, p.v.

127. Simon Newcomb, "On the Fundamental Concepts of Physics," presented before the Washington Philosophical Society, roughly typed copy in the Simon Newcomb Papers, Box #94, Library of Congress; Abstract printed in Bulletin of the Washington Philosophical Society, 1888/1891, 11: 514-515.

128. William Walter Rouse Ball, "A Hypothesis Relating to the Nature of the Ether and Gravity," Messenger of Mathematics, 1891, 21: 20-24.

129. Rouse Ball, "A Hypothesis," p.22; Also in Mathematical Recreations and Problems of the Past and Present Times, (London: Macmillan, 1892; Third edition published in 1896; Sixth edition published in 1914).

130. C.H. Hinton, "The Recognition of the Fourth Dimension," read before the Society on 9 November 1901, Bulletin of the Philosophical Society of Washington, 1901, 14: 181-203; Reprinted in Speculations: 142-162.

131. Hinton, "Recognition," pp.201-203; The essay was reprinted in The Fourth Dimension, (London: Swann & Sonnenschein, 1904); Excerpts were reprinted in Speculations: 120-141.

132. C.H. Hinton, "The Geometrical Meaning of Cayley's Formulae of Orthogonal Transformations," read 29 November 1902, Proceedings of the Royal Irish Academy, 1902, 24: 59-65.

133. Karl Pearson, "Modern Physical Ideas," Chapter X in Grammar of Science, Part I, Physical, third edition, (London: Adam & Charles Black, 1911): 355-387.

134. R.S. Heath, "On the Dynamics of a Rigid Body," Philosophical Transactions of the Royal Society of London, 1884, 175: 281-324.

135. Homersham Cox, "Homogeneous Coordinates in Imaginary Geometry and their Applications to Systems of Forces," Quarterly Journal of Mathematics, 1881, 18: 178-215; "On the Application of Grassmann's Ausdehnungslehre to different kinds of Uniform Space," read 20 February 1882, Transactions of the Cambridge Philosophical Society, 1882, 13, Part II: 69-143.

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136. Cayley,"Presidential Address," pp.435-436.

137. Samuel Roberts, "Remarks on Mathematical Terminology, and the Philosophic Bearing of Recent Mathematical Speculations concerning the Realities of Space," Proceedings of the London Mathematical Society, 9 November 1882, 14: 5-15, on p.9.

138. Johann Bernhard Stallo, The Concepts and Theories of Modern Physics, edited by Percy W. Bridgman, (Cambridge, Massachusetts: The Belknap Press of Harvard University, 1960; Reprint of the second edition of 1884; First edition published in 1881), on pp.222-279, especially pp.225-228, 244 and 251.

139. Klein did not tackle the problem of Clifford's model of a non-Euclidean space until 1890. It is strange that Klein was supposed to have returned to a more physical conception of mathematics in the 1890's with his work on the theory of the top, at least according to the standard historical view. However, given Clifford's physical emphasis on geometry, it would be reasonable to conclude that Klein's rehabilitation of this geometrical model which Clifford developed in 1873 might have initiated Klein's new found interest in the physical aspects of geometry before his development of the theory of the top. Klein presented his theory of the top before an American audience in 1896. The Mathematical Theory of the Top, lectures delivered on the occasion of the Sesquicentennial Celebration of Princeton University, 12-15 October 1896, (New York: Scribners' Sons, 1897).

140. Poincaré, "Non-Euclidean," p.406.

141. Edward Kasner, "The Present Problems of Geometry," in Harry J. Rogers, editor, Congress of Arts and Sciences, Universal Exposition, St. Louis, 1904, Volume I, Philosophy and Mathematics, (Boston: Houghton & Mifflin, 1905): 559-586, on p.559; Reprinted in Bulletin of the American Mathematical Society, 1905, 11: 283-314, on p.559.

142. Kasner, "Present," p.562.

143. Corrado Segre, "On Some Tendencies in Geometric Investigations," Bulletin of the American Mathematical Society, 1904, 11: 442-468, on pp.446-447, or 462.

411

144. Enriques, Problems, p.232.

145. Enriques, Problems, p.194.

146. G.W. de Tunzelmann, A Treatise on Electrical Theory and the Problem of the Universe, (London: Charles Griffin, 1910), on pp.x-xiii.

147. De Tunzelmann, Treatise, pp.77-78.

148. De Tunzelmann, Treatise, p.78.

149. G.B. Halsted in Lobachewski, "Geometrical Researches on the Theory of Parallels," translated by Halsted, reprinted in Roberto Bonola, Non-Euclidean Geometry, translated by H.S. Carslaw, (New York: Dover, 1955; Reprint of La Salle: Open Court, 1912; From the original Italian of 1892), on pp.49-50. The pagination follows Halsted's original, not that of Bonola's book.

150. Vladimir Varicak, "Uber die nichteuklidische Interpretation der Relativtheorie," Jahrber. D. Math. Ver., 1914, 21: 103-127. This article is a late summary of Varicak's mathematical concepts.

151. Henry P. Manning, Geometry of Four Dimensions, (New York: Macmillan, 1914), on pp.11-12.

152. Henry P. Manning, editor, The Fourth Dimension Simply Explained, (London: Methuen, 1922: Originally published New York: Munn, 1910).

153. Edwin Bidwell Wilson and Gilbert N. Lewis, "The Space-Time Manifold of Relativity: The Non-Euclidean Geometry of Mechanics and Electrodynamics," Proceedings of the American Academy of Arts and Sciences, (Boston), November 1912, 43: 389-507, on p.389.

154. Edwin Bidwell Wilson, "The So-Called Foundations of Geometry," Grunert's Archiv der Mathematik und Physik, 1904, 3 Reihe VI: 104-122, on p.121-122.

155.Wilson,"So-Called,"p.122.

412

156. Harry Bateman, "The Conformal Transformations of a Space of Four Dimensions and their Applications to Geometrical Optics," Proceedings of the London Mathematical Society, 1909, 7: 70-89. This paper was submitted to the London Mathematical Society on 9 October 1908, only a few weeks after Minkowski presented his lecture on space-time, so it would seem that Bateman could have had no knowledge of Minkowski's work before he completed his own work.

157. Harry Bateman, "The Transformation of the Electrodynamical Equations," Proceedings of the London Mathematical Society, 1910, 8: 223-264.

158. Harry Bateman, "The Physical Aspects of Time," Memoirs and Proceedings of the Manchester Literary and Philosophical Society, 1910, 54: 1-13, on p.4-5, 10-11.

159. William Kingdon Clifford, Der Sinn der Exakten Wissenschaft, translated by Hans Kleinpeter, (Leipzig: Verlag von Johann Ambrosius Barth, 1913), on pp.233-234.

160. Ludwik Silberstein, "General Relativity without the Equivalence Hypothesis," written 25 March 1918, Philosophical Magazine, 1918, sixth series, 36: 94-128, on p.100.

161. Henry L. Brose in Erwin Freundlich, The Foundations of Einstein's Theory of Gravitation, translated by Henry L. Brose, with an introduction by H.H. Turner and a preface by A. Einstein, (Cambridge: at the University Press, 1920), on p.vii.

162. Oliver Lodge, "The New Theory of Gravity," The Nineteenth Century, December 1919, 86: 1189-1201, on p.1199.

163. G.W. de Tunzelmann, "Physical Relativity Hypotheses Old and New," Science Progress, 1918-1919, 13: 474-482; Continued as "The General Theory of Relativity and Einstein's Theory of Gravitation," Science Progress, 1918-1919, 13: 652-657, on pp.656-657.

164. Harry Bateman, "On General Relativity," letter to the editor dated 10 August 1918, Philosophical Magazine, 1919, sixth series, 37: 219-223, on p.219.

413

165. Willem de Sitter, "Space, Time, and Gravitation," The Observatory, October 1916, 39: 412-419, on p.412.

166. Herbert Dingle, The Sources of Eddington's Philosophy, the eighth Arthur Stanley Eddington Memorial Lecture, 2 November 1954, (Cambridge: Cambridge University Press, 1954), on p.5. Eddington makes roughly the same statement in The Mathematical Theory of Relativity, (Cambridge: at the University Press, 1963; Reprint of 1923 original), p.vi.

167. Arthur Eddington, Report on the Relativity of Gravitation, (London: Fleetway Press, 1918), on pp.83-84; "Proceedings of the Meeting of the Royal Astronomical Society", The Observatory, May 1917, 40: 180-187, on p.185.

168. Arthur S. Eddington, "Einstein's Theory of Space and Time," The Contemporary Review, July-December 1919, 116: 639-643, on p.640.

169. Eddington, Space, p.75. The quote from Clifford can be found in Common Sense, on pp.302-204, as the concluding remarks to Clifford's chapter on "Position."

170. Eddington, Space, p.152. The quote from Clifford can be found in "Postulates," in Lectures and Essays, Volume 1, on p.300.

171. Eddington, Space, p.153. The quote originally comes from Clifford, "Unseen Universe," p.788.

172. Edmund T. Whittaker, From Euclid to Eddington: A Study of Conceptions of the External World, (New York: Dover, 1958: Unaltered reprint of Cambridge: at the University Press, 1947), on p.204.

173. Howard K. Smokler, "W.K. Clifford," p.125. The book to which Smokler referred was Arthur S. Eddington, The Nature of the Physical World, (Cambridge: at the University Press, 1928).

174. Sir Arthur S. Eddington, Fundamental Theory, edited by E.T. Whittaker, (Cambridge: at the University Press, 1946), pp.269-270.

175. Eddington, Fundamental Theory, p.269.

414

176. Edmund T. Whittaker, History of the Theories of Aether and Electricity, Volume II: The Modern Theories, 1900-1926, (New York: Harper Torchbooks, 1953; Reprint of London: Nelson & Sons, 1951): on p.156.

177. Thomas Greenwood, "Geometry and Reality," delivered 12 June 1922 at the meeting of the Aristotelian Society, Proceedings of the Aristotelian Society, 1922, new series, 16: 189-204, on p.195-196.

178. Frank Kassel, Relativity and the Critical Philosophy, (Philadelphia: 1926), p.36.

415

416

On 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]

Abstract

The main macroscopic phenomena predicted by general relativity (the motion of Mercury’s perihelion, the bending of light in the vicinity of the sun, and the gravitational red shift of spectral lines) are studied in the framework of the sub microscopic concept that has recently been developed by the author. The concept is based on the dynamic inerton field that is induced by an object in the surrounding space considered as a tessellation lattice of primary balls (superparticles) of Nature. Submicroscopic mechanics says that the gravitational interaction between objects must consist of two terms: (i) the radial inerton interaction between two masses M and m, which results in classical Newton’s gravitational law rMmGU /−= , and (ii) the tangential inerton interaction between the masses, which is caused by the tangential component of the motion of the test mass m and which is characterized by the correction 222 /)()/( crrmMG ϕ&− . It is shown it is precisely this correction that is responsible for the three aforementioned macroscopic phenomena and the derived equations exactly coincide with those derived in the framework of the formalism of general relativity, which means that the latter must be reinterpreted as follows: the gravitational field of the resting central mass is flat, but the emergence of a test mass disturbs the field and its distribution exactly looks like the Schwarzschild metric prescribes.

417

1. Introduction

The general theory of relativity formally predicted such phenomena as the motion of Mercury’s perihelion, the bending of light by the gravitational field of the sun and the gravitational red shift of spectral lines (see, e.g. Refs. [1-3]). The predictions were verified experimentally and since then general relativity was widely recognized as the fundamental physical concept of the 20th century.

Since general relativity has all attributes of an action-at-a-distance theory, some researchers try to understand its deeper sense coming back to the old idea of retarded potentials, or velocity-depended potentials, which would account for a nature of the motion of the front of the gravitational potential.

Soares [4] considering light as classical massive corpuscles calculated the deflection of a light beam under the Sun’s gravitational force, which is described by the central force hyperbolic orbit; in the first approximation he obtained the so-called Newtonian deflection )/(2 Sun

2SunN RcGM=δ , though Einsteinian’s is still

NGR 2δδ = where SunM and SunR are the Sun’s mass and radius. Giné [5,6] reviewed tens of works dedicated to the study of the modified Newton’s

potential, among which there were such potentials as Weber’s, Gerber’s and others. Gené argues that Weber’s potential, which is a velocity dependent potential

rcrV /1)2/1( 22 ⋅−= & , allows one to introduce an additional force component. Such a component is the tangential component of the speed of a test particle in the gravitational field of a central mass M, which significantly influences the eccentricity of the hyperbolic orbit of the particle. Thus taking into account the finite propagation speed – the velocity of light c – he [5] concludes that the anomalous precession of the Mercury’s perihelion is associated with a second order delay of the retarded potential

( )

ctrtr

mV )( ττ −⋅−−⋅

−= .

As Giné [6] shows, at some fixed parameters the deflection of a light beam would reach that of derived by Einstein in 1916, i.e. )/(4 p

2GR rcGM=δ where pr is the

closest approach, i.e. perihelion of the beam. So far the mentioned phenomena have not been described on the basis of a

microscopic approach. Nevertheless, before applying such an approach to the study of the problem, one has to become familiar with major statements of the concept. However, let us initially consider general discrepancies between phenomenological and microscopic standpoints. General relativity, as a typical phenomenological theory, considers matter and space-time as two independent entities, which, however, can influence each other [7]: a matter curves space-time that is treated as a geometric

418

entity resting on the statement of constancy of the speed of light c; photons are massless, they form the world line of light ray. Thus with such an approach the microscopic peculiarities of the real space remain beyond the study of the problem.

Indeed, since photons transfer momentum, they physically have mass. But what is mass? At a scale comparative with the de Broglie wavelength λ of the quantum system in question, a phenomenological description has to make way for a quantum mechanical one. However, conventional quantum mechanics is constructed in an abstract phase space and hence it cannot be used to investigate the behaviour of matter at a sub microscopic size: in line with the theory the less scale, the more indeterminism… Therefore, to account for the behaviour of matter at extremely small scales we have to rely on a theory developed in the real physical space, which is able to operate at any microscopic scale.

For the first time Bounias and the author [8-12] proposed a detailed theory of the constitution of the real physical space. In line with those researches, which are based on topology, set theory and fractal geometry, the real space emerges as a tessellation lattice of primary topological balls (or primary entities, or superparticles of Nature) whose size can be estimated as the Planck’s one, 10-35 m. It has been shown how mathematical characteristics, such as length, surface, volume and fractal geometry generate in this tessel-lattice the basic physical notions, such as mass, particle, electric charge, the particle’s de Broglie wavelength, etc. and the corresponding fundamental laws. In particular, mass emerges from space as its local deformation, i.e. when a volumetric fractal deformation is created in the appropriate cell of the tessel-lattice. Hence matter is no longer separated from space, as it occurs in general relativity, but can reasonably appear at special conditions.

In the present paper we show in what way submicroscopic mechanics [13-19] developed in the real physical space [8-12] is capable of coping with the mentioned challenge, i.e. the (sub) microscopic description of three gravitational phenomena: the anomalous precession of Mercury’s perihelion, the bending of light and the red shift of spectral lines. We will see below how this difficult problem becomes really trivial in the framework of the sub-microscopic consideration based on the constitution of real space. Namely, we will see this is the motion of matter, which generates deformations of space around the matter: one component of such motion is responsible for the Newton gravitational term, the other component introduces a correction to Newton’s law, which we currently know as a curvature of space-time in general relativity.

2. Correction to Newton’s gravitational law

Submicroscopic mechanics [13-19] studies the motion of a particle in the densely packed tessel-lattice, which means the induction of the interaction between a moving particle and the tessel-lattice. As a result, a cloud of deformations of the space tessel-lattice is accompanying the particle. These elementary excitations that migrate from cell to cell of the tessel-lattice represent a resistance of space, i.e. inertia, and, because

419

of that, they have been called inertons. Thus, collision-like phenomena are produced: deformations of space (inertons) go from the particle to the surrounding space and then due to elastic properties of the tessel-lattice some come back to the particle. The Euler-Lagrange equations show the periodicity in the behaviour of the particle. Namely, the particle’s velocity oscillates between the initial value υ and zero along each section λ of the particle path and this section emerges as the de Broglie wavelength of the particle [13,14]. The amplitude of the particle’s cloud of inertons

υλ /c=Λ uncovers the physical meaning of the ψ -function: the latter, although determined in an abstract physical space, describes peculiarities of the range of space around the particle perturbed by the particle’s inertons.

The next stage is that inertons transfer not only inertial, or quantum mechanical properties of particles, but also gravitational properties, because they transfer fragments of the deformation of space (i.e. mass) induced by the particle. The corresponding study [18,19] shows that inertons move like a typical standing spherical wave that is specified by the dependency 1/r; it is this behaviour that allows the derivation of Newton’s static gravitational law, 1/r .

Thus inertons are carriers of both the inertial interaction (or, in other words, quantum mechanical’s including the so-called Casimir forces) and the gravitational interaction. Experimental evidence of the existence of inertons was carried out in Refs. [20-25]. The experiments described there were performed in micro and mesoscopic ranges. The inerton radiation, i.e. a flow of free inertons, carriers of mass, can be measured by a device designed by Didkovsky and the author [26] and, moreover, the inerton field allows a number of practical applications: for instance medical applications (so-called Teslar watch, see Refs. [23,24]), the manufacture of biodiesel [27], etc.

Thus, having such conclusive results, we can now try to apply the description of the macroscopic phenomena starting from the same submicroscopic standpoint.

Inertons moving by the hopping mechanism pass a local deformation, i.e. a fragment of mass, from cell to cell of the tessel-lattice. These quasi-particles can be either bound with an object or free (if they are emitted from the object’s inerton cloud). Any object, from a canonical particle to a star, is surrounded by its own inerton cloud. The inerton cloud oscillates in the vicinity of an object as a standing spherical wave and brings a tension to the surrounding space [17,18]; inerton waves of such central object are practically instant: they reach a test body with a speed no less than the velocity of light and, hence, these spherical waves are perceived by an outstanding observer as the static (Newtonian) gravitational potential:

V = −G M /r . (1)

In the case of a classical motionless object, its massive particles (atoms, etc.) oscillate at their equilibrium positions and the particles’ clouds of inertons overlap. If the object has a form close to spherical, the motion of the object’s inertons will happen only along radial lines and the velocity of the inertons will be characterized by the

420

radial component that is equal to the speed of light c (the tangential component of inerton motion averaged by all the particles and directions is reduced to zero).

When a test body falls within the inerton field of the central object, one can distinguish two components of the body’s inerton cloud. The components are: radial

radr& , which is parallel/antiparallel to the radial ray issued from the central object to the

test body; and tangential tanr& , which is transferral to the radial ray. It is interesting to refer to Poincaré [28]: What exactly did he indicate as the main

reasons for gravity a hundred of years ago? By Poincaré, the expression for the attraction should include two components: one is parallel to the vector that joins positions of both interacting objects and the second one is parallel to the velocity of the attracted object. Thus the velocity of an object must influence the value of its gravitational potential. Grand Poincaré was at the origin of topology, he understood how the generalized theory of space was important for physics. Now his ideas indeed have received further development in the studies of Bounias and the author [9-19].

Equating the radial component to the velocity of light c, i.e. cr =rad& [13-15], we obtain that the total velocity of the test body’s inertons ˆ c in the frame of reference associated with the central object is defined from the geometric relationship (compare with Ref. [18])

2tan

22ˆ rcc &+= (2)

Hence around the test body in the region Λ<r ( Λ is the amplitude of the body’s inerton cloud, which is huge for a macroscopic system [18]) inertons of the test body move with the velocity cc >ˆ .

Besides, relationship (2) shows that a test body does not fall exactly to the centre of mass of the central object, as expression (1) prescribes, but to a point distant from the centre of mass at a section calculated on the basis of expression (2). In other words, this means that the true gravitational attraction between a central heavy motionless object (mass M) and a test moving body (mass m) should be different from the Newton’s expression

rmMGU −= . (3)

Namely, the correct expression for the potential energy of gravitational attraction of the mass m to the central mass M should have the following form

⎟⎟⎠

⎞⎜⎜⎝

⎛+⋅−=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⋅−= 2

2tan

2rad

2tan 11

cr

rMmG

rr

rMmGU

&

&

& (4)

where tanr& is the tangential velocity of the body with the mass m, i.e. the body’s orbital velocity (because the projection of the velocity of body’s inertons on the

421

body’s path has the value of the velocity of the body, though in perpendicular directions the velocity of inertons can be compared with the speed of light c – in these directions the spatial tessel-lattice itself is guiding inertons [14-18]). We can see that the correction in the parentheses is very close to Weber’s for a velocity dependent potential (see Introduction) and such a correction indeed takes into account inner peculiarities of the system studied, which Weber and then Giné associated with the necessity to consider a short range action between interacting physical systems. In our case these are inertons that establish the direct interaction between distant masses M and m.

Corrected Newton’s gravitational law (4) can be applied now to study the anomalous precession of the Mercury’s perihelion, the bending of light and the red shift of spectral lines.

3. Motion of Mercury’s perihelion

Classical mechanics yields the following equations describing the motion of a body with a mass m in the gravitational field induced by a large central mass M (see, e.g. Refs. 1-3)

ϕ&2rmI = ; (5)

rmMGrmrmE −+= 22

212

21

cl. ϕ&& . (6)

Eqs. (5) and (6) are the classical integrals of the movement of momentum and the energy, respectively. However, as follows from the consideration above, in Eq. (6) we have to change the potential gravitation energy (3) to the corrected expression (4). Then the energy conservation law (6) is corrected, such that two equations (5) and (6) are transformed to

ϕ&2rmI = ; (7)

⎟⎟⎠

⎞⎜⎜⎝

⎛+⋅−+= 2

2222

212

21 1

cr

rmMGrmrmE ϕϕ

&&& . (8)

Note that here the dot over r and ϕ means the differentiation by the proper time t of the body, i.e. t is the natural parameter that is proportional to the body path [14-17]. The system of equations (7) and (8) are identical to the equations of motion of a body in the Schwarzschild field obtained in the framework of the general theory of relativity (see, e.g. Refs. [1-3]). The solution to Eqs. (7) and (8) are available in literature (see, e.g. Refs. [1-3]) and it shows that it is the last term in Eq. (8), which displaces the perihelion of the planetary orbit by amount

422

26cL

GMπϕ =Δ (9)

where L is the focal parameter.

4. Bending of a light ray

The energy E of a photon in the gravitational field induced by a large mass M can easily be written by recognizing that the photon is characterized by mass m [29,10]. However, the photon is not a canonical particle, but a quasi-particle, a local excitation of the tessel-lattice, which migrates in space by hopping from cell to cell. This means the photon does not possess its inerton cloud at all; it is itself similar to an inerton (also an elementary excitation of the tessel-lattice), though in addition to the inerton it has an electrically polarized surface [30].

Therefore, since a photon does not disturb the ambient space with a cloud of inertons, it cannot experience the radial component of the gravitational field of a heavy object (no overlapping with the inerton cloud of the heavy object). Hence, the radial component rGMm /− is absent in the interaction between the heavy object and the photon (recall that this Newton’s component emerges owing to the overlapping of inerton clouds of two interacting objects, the central object and the test body). Nevertheless, the tangential component 22 / crmMG ϕ&− associated with the true motion of the photon must still be preserved. That is why the behaviour of the photon in the gravitational field of mass M has to be defined by the following pair of equations

ϕ&2rmI = ; (10)

2

222

212

21

crmMGrmrmE ϕϕ&

&& −+= (11)

where the time t is treated as the natural parameter proportional to the photon path, which is very important for the invariance of the theory [14].

Again, Eqs. (10) and (11) are exactly the same input equations for the study of the bending of a light ray in the Schwarzschild field, which are obtained in the framework of the formalism of general relativity. As is well known (see, e.g. Ref. [1-3]) the solution to Eqs. (10) and (11) yields the following angle deviation of the ray from the direct line

Δϕ = 4 GMc2r

. (12)

423

5. Red-shift of spectral lines

Let us consider a simple task. Let l and m be, respectively, length and mass of a mathematical pendulum and let ϕ be the angle of the deviation of the pendulum from the equilibrium. The pendulum is found on the surface of a planet with the radius r. In this case the kinetic energy of the massive point is

2221 ϕ&lmK = (13)

and the potential energy is

⎟⎟⎠

⎞⎜⎜⎝

⎛+⋅

−⋅+−= 2

221

)cos1( cl

lrMmGU ϕ

ϕ&

(14)

(to write the expression, we have used corrected Newton’s law (4)). Because of the small variable ϕ one can write the energy E = K +U of the massive point as follows

⎟⎟⎠

⎞⎜⎜⎝

⎛+−⋅−−≅ 2

22222

21

2 cl

rl

rMmG

rMmGlmE ϕϕϕ

&& . (15)

In the case of the potential depending on the velocity the equation of motion is determined by the Euler-Lagrange equation [31]

0=+−−qU

qK

qU

tdd

qK

tdd

∂∂

∂∂

∂∂

∂∂

&&

where in our case q ≡ ϕ and t is the proper time of the oscillating massive point. In the explicit form it yields

02 2

22 =+⎟⎟

⎠

⎞⎜⎜⎝

⎛+ ϕϕ l

rMG

cl

rMGl && . (16)

If we designate (2πν0 )2 = 2GM /(rl), we can write instead of Eq. (16)

0)/(21

)2(2

20 =−

++ ϕνπϕ

rcGM&& . (17)

In Eq. (17) assuming the inequality r0 = 2GM /c2 << r , we acquire the renormalized frequency of the pendulum

424

ν ≈ 1−GMc2 r

⎛

⎝ ⎜

⎞

⎠ ⎟ ν0 . (18)

The scheme described above may easily be applied to vibrating atoms (ions) located on the surface of a star. This means that expression (18) determines the so-called gravitational red shift of spectral lines

δν ≅ −GMc2 r

ν0 . (19)

The result (19) is in complete agreement with that derived in the framework of general relativity (see, e.g. Refs. 1 and 2).

6. Discussion

To derive the equations of motion of the perihelion, Eqs. (7) and (8), the motion of light ray, Eqs. (10) and (11), and the shift of spectral lines, Eq. (17), we have started from very transparent ideas of classical physics and the sub-microscopic deterministic physical concept developed in works [8-27,29,30]. General relativity derives the same equations of motion, Eqs. (7), (8), (10) and (11), starting from the equations of motion in the form of a geodesic line (written in polar coordinates )(4 iξ )

02

42=Γ+

tdd

tdd

dtd σρ

μσρ

ξξξ (20)

for the investigation of the motion of the perihelion of a planet, and in addition takes into account the geodesic line for a light ray

0=sd

dsd

dgσρ

ρσξξ

. (21)

Here, the components of the metric tensor have the form

)/(21

1211 rcGM

g−

−= ; (22)

222 rg −= ; (23)

ϑ2233 cosrg −= ; (24)

)/(21 244 rcGMg −= . (25)

425

General relativity achieves the result (18), (19) from the relationship connecting the

coordinate frequency ν of oscillating atoms and their proper frequency ν0 ,

044νν g= (26)

where the time component of the metric tensor 44g is determined in expression (25). It is believed that the Schwarzschild metric (22)-(25) describes the space-time

around a spherically symmetric object, such as a point mass, a planet, a star (and a “black hole”).

In contrast, the submicroscopic concept deriving Newton’s gravitational law (3) [18,19] does not reveal the reasons for the emergence of the term )/(2 2rcGM in the metric of real space around a resting spherical object with mass M. From the sub microscopic viewpoint the metric of a resting mass object must be linear

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

−−

−

=

1000010000100001

ρσg (27)

The sub microscopic theory argues that an additional gravitational term appears in the equations of motion of a test body, Eqs. (8) and (11), owing to its interaction with the Newtonian gravitational field of the central mass M. In other words, it is the test body that perturbs the flat-space metric (27) of the resting object M in the place of the body’s motion. The perturbation introduces a correction to the Newtonian gravitation (see expression (4)), such that through the tangential velocity tanr& of the test body, the

additional term 2/2 cGM is added to the Newtonian one. Thus, if the submicroscopic approach is correct, a lack of correspondence should be

available in the interpretation of the Schwarzschild’s solution. Let us recall how the result (22)-(25) is obtained in general relativity (see, e.g. Ref. 1, sect. 58 and Ref. 2, chap. 13). The coordinate system is treated as undetermined identically. At the transformation that contains an arbitrary function f (r) (for instance, the turn of spatial coordinates ξ i round the axis that goes through the origin)

ii

rrf ξξ )(

=′ (28)

where

426

)()()()(,)()()( 222221232221 rfrr =′+′+′=′++= ξξξξξξ , (29)

the square of linear element

pq

pqpq

pq

qq

ddrDddrC

ddrBdrAsd

ξξχχξξδ

ξξχξ

)()(

)(2)( 4242

+−

+= (30)

has to preserve its form. A suitable transformation of the coordinates, a kind of normalization, allows one to reduce the number of unknown functions A, B, C and D, such that the problem still remains spherical and static. Coordinates change as follows

iirf ξξξξ =′+=′ ),(44 . (31)

It was convenient to consider the metric in the form

g44 = A, g4 q = 0, gqp = −Cδqp + Dχqχ p . (32)

The metric tensor components g4 q are transformed in line with equations

,4

4

444 qqq ggg +′

=′ξ∂ξ∂

kip

k

q

i

pq ggξ∂

∂ξξ∂

∂ξ′′

=′ . (33)

The further transformations reduced the number of unknown functions to two, A and D. The choice (32) and the rules of transformations (33) generate a special form of Christoffel symbols i

pqΓ in which a term proportional to 1/r appears. The time

component of metric tensor becomes rg /144 α−= , which after comparison with

Newton’s law allows one to write )/(21 244 rcGMg −= .

It is generally recognized that the transformations (22)-(25) and (28)-(33) are completely correct, because they are performed in line with the similar transformations conventional in the special theory of relativity in which the interval

222222 zyxtcs +++= is treated as invariant with respect to the Lorentz transformations. However, Lorentz’s transformations are associated with the introduction of a (relative) velocity υ to the system studied, which reduces the system

parameters in accordance with the Lorentz factor 22 /1 cυ− . Note the velocity υ is a foreign parameter for the system, which is imposed on the system from outside.

427

That is why if one wishes to search for invariance of the interval 2ds (30), the one constructs the element 2sd ′ introducing some foreign parameters in it looking for the conditions when the equality 22 sdds ′= is held. Such foreign parameters are available on the right hand side of expression (30) somewhere among functions A, B, C and D and also among coefficients iχ . Moreover, owing to the structure of these

coefficients, rii /ξχ = , i.e. their inverse dependency on distance r , we can

recognize them as possible sources of the outside gravitational field. Carrying out transformations (31)-(33) and so on until we reach the metric (22)-(25) (see, e.g. Refs. 1 and 2), we gradually add a perturbation to Newton’s gravitational potential of the central mass M on the side of a test mass. That is the crucial point! Therefore, a point mass at rest possesses the conventional Minkowski flat-space metric (27), but this metric disturbed by a smaller mass changes to the metric (22)-(25) in the place of the smaller mass location.

7. Conclusion

In the present work we have shown how the sub microscopic views allow us to solve the problems of the motion of Mercury’s perihelion, the bending of a light ray by the sun and the gravitational red shift of spectral lines. The solutions are exactly as those derived from the formalism of general relativity. This means that the Schwarzschild metric (22)-(25) is correct, however, the interpretation of the final result is different; namely, the Schwarzschild metric does not represent properties of the geometry of space-time of a point mass M at rest, but the geometry of space-time around this mass disturbed by a test smaller mass m.

The misunderstanding could not be resolved so far, because a sub microscopic theory of the real space was absent. The availability of such theory [8-27,29,30] has allowed us to look now at many problems of gravitational physics from a very new point of view. In particular, it is finally clear now that the idea of black holes is fiction, as the parameter 2

0 /2 cGMr = does not have the meaning of a critical radius at all (that was already accurately demonstrated by many researchers by means of using general relativity, especially see remarkable works by Angelo Loinger [32,33]). There are not also gravitational waves, because on the microscopic scale such role is played by inerton waves [21,18,19] (see also Loinger [32,33]). The presence of inertons allows us to talk about such discipline as inerton astronomy [26]. However, all this is only a first step of the sub microscopic deterministic concept of physics. The other steps promise to be even more exciting.

428

Acknowledgment

Thanks and appreciation to Dr. Michael C. Duffy, the founder and the organizer of the conferences “Physical Interpretation of Relativity Theory” at Imperial College London.

References

[1] W. Pauli, The theory of relativity (Nauka, Moscow, 1983), p. 228 (Russian translation).

[2] P. G. Bergmann, Introduction to the theory of relativity (Gosudarstvennoe izdatelstvo inostrannoy literatury, Moscow, 1947), p. 281 (Russian translation).

[3] S. Weinberg, Gravitation and cosmology: principles and application of general theory of relativity (Mir, Moscow, 1975), 191 (Russian translation).

[4] D. S. L. Soares, Newtonian gravitational deflection of light revisited, arXiv.org: physics/0508030.

[5] J. Giné, On the origin of the anomalous precession of Mercury’s perihelion, arXiv.org: physics/0510086.

[6] J. Giné, On the origin of the deflection of light, arXiv.org: physics/0512121. [7] C. W. Misner, K. S. Thorn and J. A. Wheeler, Gravitation (W. H. Freeman, San

Francisco, 1973). [8] M. Bounias and V. Krasnoholovets, Scanning the structure of ill-known spaces:

Part 1. Founding principles about mathematical constitution of space, Kybernetes: The Int. J. Systems and Cybernetics 32, no. 7/8, 945-975 (2003). arXiv.org: physics/0301049.

[9] M. Bounias and V. Krasnoholovets, Scanning the structure of ill-known spaces: Part 2. Principles of construction of physical space, ibid. 32, no. 7/8, 976-1004 (2003). arXiv.org: physics/0212004.

[10] M. Bounias and V. Krasnoholovets, Scanning the structure of ill-known spaces: Part 3. Distribution of topological structures at elementary and cosmic scales, ibid. 32, no. 7/8, 1005-1020 (2003). arXiv.org: physics/0301049.

[11] M. Bounias and V. Krasnoholovets, The universe from nothing: A mathematical lattice of empty sets, Int. J. Anticipatory Computing Systems 16, 3-24 (2004); Ed. D. Dubois, Publ. by CHAOS. arXiv.org: physics/0309102.

[12] V. Krasnoholovets, The tessel-lattice of mother-space as a source and generator of matter and physical laws, in: Einstein and Poincaré: The Physical Vacuum, Ed.: V. V. Dvoeglazov (Apeiron, Montreal, 2006), pp. 143-153.

[13]V. Krasnoholovets, Motion of a particle and the vacum, Phys. Essays 6, no. 4, 54-563 (1993). arXiv.org: quant-ph/9910023.

[14] V. Krasnoholovets, Motion of a relativistic particle and the vacuum, Phys. Essays 10, no. 3, 407-416 (1997). arXiv.org: quant-ph/9903077.

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[15] V. Krasnoholovets, On the nature of spin, inertia and gravity of a moving canonical particle, Ind. J. Theor. Phys. 48, no. 2, 97-132 (2000). arXiv.org: quant-ph/0103110.

[16] V. Krasnoholovets, Space structure and quantum mechanics, Spacetime & Substance 1, no. 4, 172-175 (2000). arXiv.org: quant-ph/0106106.

[17] V. Krasnoholovets, Submicroscopic deterministic quantum mechanics, Int. J. Computing Anticip. Systems 11, 164-179 (2002); Ed. D. Dubois, Publ. by CHAOS. arXiv.org: quant-ph/0109012.

[18] V. Krasnoholovets, Gravitation as deduced from submicroscopic deterministic quantum mechanics, arXiv.org: hep-th/ 0205196.

[19] V. Krasnoholovets, Reasons for the gravitational mass and the problem of quantum gravity, in Ether, Spacetime and Cosmology, Vol. 1, Eds.: M. Duffy, J. Levy and V. Krasnoholovets (PD Publications, Liverpool, 2008), pp. 419-450.

[20] V. Krasnoholovets, On the theory of the anomalous photoelectric effect stemming from a substructure of matter waves, Ind. J. Theor. Phys. 49, no. 1, 1-32 (2001). arXiv: quant-ph/9906091.

[21] V. Krasnoholovets and V. Byckov, Real inertons against hypothetical gravitons. Experimental proof of the existence of inertons, Ind. J. Theor. Phys. 48, no. 1, 1-23 (2000). arXiv: quant-ph/0007027.

[22] V. Krasnoholovets, Collective dynamics of hydrogen atoms in the KIO3·HIO3 crystal dictated by a substructure of the hydrogen atoms’ matter waves, arXiv: cond-mat/0108417.

[23] V. Krasnoholovets, O. Strokach and S. Skliarenko, On the behavior of physical parameters of aqueous solutions affected by the inerton field of Teslar technology, Int. J. Mod. Phys. B 20, no. 1, 1-14 (2006); The study of the influence of a scalar physical field on aqueous solutions in a critical range, J. Mol. Liquids 127, 50-52 (2006).

[24] E. Andreev, G. Dovbeshko and V. Krasnoholovets, The study of influence of the Teslar technology on aqueous solution of some biomolecules, Research Lett. Phys. Chem., vol. 2007, Article ID 94286, 5 pages (2007); http://www.hindawi.com/getarticle.aspx?doi=10.1155/2007/94286 .

[25] V. Krasnoholovets, N. Kukhtarev and T. Kukhtareva, Heavy electrons: Electron droplets generated by photogalvanic and pyroelectric effects, Int. J. Mod. Phys. B

20, no. 16, 2323-2337 (2006). [26] V. Didkovsky and V. Krasnoholovets, A first step of inerton astronomy,

submitted. [27] V. Dekhtiaruk, V. Krasnoholovets and J. Heighway, Biodiesel manufacture,

International patent No. PCT/GB2007/001957 (Intern. filing date 25 May 2007; London, UK).

[28] H. Poincaré, Sur la dynamique de l’électron, Rendiconti del Circolo matematico di Palermo 21, 129-176 (1906); also: Oeuvres, t. IX, pp. 494-550 (also in Russian translation: H. Poincaré, Selected Transactions, ed. N. N. Bogolubov (Nauka, Moscow, 1974), vol. 3, pp. 429-486).

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[29] V. Krasnoholovets, On the notion of the photon, Ann. Fond. L. de Broglie 27, no. 1, 93-100 (2002). arXiv.org: quant-ph/0202170.

[30] V. Krasnoholovets, On the nature of the electric charge, Hadronic J. Suppl. 18, no. 4, 425-456 (2003). arXiv.org: physics/0501132.

[31] D. ter Haar, Elements of Hamiltonian mechanics (Nauka, Moscow, 1974), p. 60 (Russian translation).

[32] A. Loinger, On black holes and graviational waves (La Goliardica Pavese, 2002). [33] A. Loinger, More on BH’s and GW’s. III (La Goliardica Pavese, 2007).

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432

CAN PHYSICS LAWS BE DERIVED FROM MONOGENIC

FUNCTIONS

José B. AlmeidaUniversidade do Minho, Physics Department

Braga, Portugal, e-mail: [email protected]

AbstractThis is a paper about geometry and how one can derive several fundamen-tal laws of physics from a simple postulate of geometrical nature. Themethod uses monogenic functions analysed in the algebra of 5-dimensionalspacetime, exploring the 4-dimensional waves that they generate. With thismethod one is able to arrive at equations of relativistic dynamics, quantummechanics and electromagnetism. Fields as disparate as cosmology and par-ticle physics will be influenced by this approach in a way that the paper onlysuggests. The paper provides an introduction to a formalism which showsprospects of one day leading to a theory of everything and suggests severalareas of future development.

1. Introduction

The editor’s invitation to write a chapter for this book about ether and the Universe ledme to think how my recent work had anything to do with ether, because the word wasnever used previously in my writings. It will become clear in the following sections thatthe concept of a privileged frame or absolute motion underlies all the argument. Whenone accepts the existence of a preferred frame, the question of attaching that frame tosome observable feature of the Universe is immediate. This question is addressed in Sec.8 but we can anticipate that galaxy clusters are fixed and can be seen as the anchors forthe preferred frame. This statement seems inconsistent with the observation that clustersof galaxies move relative to each other but it is resolved invoking an hypersphericalsymmetry in the Universe that is revealed by the choice of appropriate coordinates.

The relationship between geometry and physics is probably stronger in the GeneralTheory of Relativity (GTR) than in any other physics field. It is the author’s belief that

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mailto:[email protected]

a perfect theory will eventually be formulated, where geometry and physics becomeindistinguishable, so that the complete understanding of space properties, together withproper assignments between geometric and physical entities, will provide all necessarypredictions, not only in relativistic dynamics but in physics as a whole.

We don’t have such perfect theory yet, however the author intends to show that GTRand Quantum Mechanics (QM) can be seen as originating from monogenic functions inthe algebra of the 5-dimensional spacetime G4,1. These functions can generate a nulldisplacement condition, thus reducing the dimensionality by one to the number of di-mensions we are all used to. Besides generating GTR and QM, the same space generatesalso 4-dimensional Euclidean space where dynamics can be formulated and is quite of-ten equivalent to the relativistic counterpart; Euclidean relativistic dynamics resemblesFermat’s principle extended to 4 dimensions and is thus designated as 4-DimensionalOptics (4DO).

Our goal is to show how the important equations of physics, such as relativity equa-tions and equations of quantum mechanics, can be put under the umbrella of a commonmathematical approach[1, 2]. We use geometric algebra as the framework but introducemonogenic functions with their null derivatives in order to advance the concept. Fur-thermore, we clarify some previous work in this direction and identify the steps to takein order to complete this ambitious project.

Since A. Einstein formulated dynamics in 4-dimensional spacetime, this space is rec-ognized by the vast majority of physicists as being the best for formulating the laws ofphysics. However, mathematical considerations lead to several alternative 4D spaces.For example, the Euclidean 4-dimensional space of 4DO is equivalent to the 4D space-time of GTR when the metric is static, and therefore the geodesics of one space can bemapped one-to-one with those of the other. Then one can choose to work in the spacethat is more suitable. We build upon previous work by ourselves and by other authorsabout null geodesics, regarding the condition that all material particles must follow nullgeodesics of 5D space:

The implication of this for particles is clear: they should travel on null5D geodesics. This idea has recently been taken up in the literature, and hasa considerable future. It means that what we perceive as massive particlesin 4D are akin to photons in 5D.[3]

Accordingly, particles moving on null paths in 5D (dS2 = 0) will appearas massive particles moving on timelike paths in 4D (ds2 > 0) . . . [4]

We actually improve on these null displacement ideas by introducing the more fun-damental monogenic condition, deriving the former from the latter and establishing acommon first principle.

The only postulates in this paper are of a geometrical nature and can be summarizedin the definition of the space we are going to work with; this is the 4-dimensional null

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subspace of the 5-dimensional space with signature (−+ + + +). The choice of thisgeometric space does not imply any assumption for physical space up to the point wheregeometric entities like coordinates and geodesics start being assigned to physical quan-tities like distances and trajectories. Some of those assignments will be made very soonin the exposition and will be kept consistently until the end in order to allow the readersome assessment of the proposed geometric model as a tool for the prediction of phys-ical phenomena. Mapping between geometry and physics is facilitated if one choosesto work always with non-dimensional quantities; this is done with a suitable choice forstandards of the fundamental units. From this point onwards all problems of dimensionalhomogeneity are avoided through the use of normalizing factors listed below for all units,defined with recourse to the fundamental constants: h→ Planck constant divided by 2π ,G→ gravitational constant, c→ speed of light and e→ proton charge.

Length Time Mass Charge√Ghc3

√Ghc5

√hcG

e

This normalization defines a system of non-dimensional units (Planck units) with im-portant consequences, namely: 1) All the fundamental constants, h, G, c, e, becomeunity; 2) a particle’s Compton frequency, defined by ν = mc2/h, becomes equal to theparticle’s mass; 3) the frequent term GM/(c2r) is simplified to M/r.

5-dimensional space can have amazing structure, providing countless parallels to thephysical world; this paper is just a limited introductory look at such structure and par-allels. The exposition makes full use of an extraordinary and little known mathematicaltool called geometric algebra (GA), a.k.a. Clifford algebra, which received an importantthrust with the works of David Hestenes [5]. A good introduction to GA can be found inGull et al. [6] and the following paragraphs use basically the notation and conventionstherein. A complete course on physical applications of GA can be downloaded from theinternet [7]; the same authors published a more comprehensive version in book form [8].An accessible presentation of mechanics in GA formalism is provided by Hestenes [9].This is the subject of first section, where some essential GA concepts and notation areintroduced.

Section two deals with monogenic function in flat 5D spacetime, deriving special rela-tivity and the free particle Dirac equation from this simple concept. 4DO appears here asa perfect equivalent to special relativity, where trajectories can be understood as normalsto 4-dimensional plane-like waves. The following section improves on this by allowingfor curved space, introducing the notion of refractive index tensor. Section five examinesthe variational principle applied in both 4DO and GTR spaces to justify the equivalenceof geodesics between the two spaces for static metrics. Refractive index is then relatedto its sources and the sources tensor is defined. The case of a central mass is examined

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and the links to Schwarzschild’s metric are thoroughly discussed. Electromagnetism andelectrodynamics are formulated as particular cases of refractive index in section sevenand the sources tensor is here related to a current vector. The next section introduces thehypothesis of an hyperspherical symmetry in the Universe, which would call for the useof hyperspherical coordinates; the consequences for cosmology would include a com-plete dismissal of dark matter for a flat rate Hubble expansion. Before the conclusion,section nine shows how the monogenic condition is effective in generating an SU(4)symmetry group and makes some advances towards a relation with the standard modelof particle physics.

2. Introduction to geometric algebra

Geometric algebra is not usually taught in university courses and its presence in theliterature is scarce; good reference works are [5, 7, 8]. We will concentrate on thealgebra of 5-dimensional spacetime because this will be our main working space; thisalgebra incorporates as subalgebras those of the usual 3-dimensional Euclidean space,Euclidean 4-space and Minkowski spacetime. We begin with the simpler 5D flat spaceand progress to a 5D spacetime of general curvature.

The geometric algebra G4,1 of the hyperbolic 5-dimensional space we consider isgenerated by the coordinate frame of orthonormal basis vectors σα such that

(σ0)2 =−1,

(σi)2 = 1, (2.1)

σα ·σβ = 0, α 6= β .

Note that the English characters i, j, k range from 1 to 4 while the Greek charactersα,β ,γ range from 0 to 4. See the Appendix A for the complete notation conventionused.

Any two basis vectors can be multiplied, producing the new entity called a bivector.This bivector is the geometric product or, quite simply, the product; this product is dis-tributive. Similarly to the product of two basis vectors, the product of three differentbasis vectors produces a trivector and so forth up to the fivevector, because five is thedimension of space.

We will simplify the notation for basis vector products using multiple indices, i.e.σασβ ≡ σαβ . The algebra is 32-dimensional and is spanned by the basis

• 1 scalar, 1,

• 5 vectors, σα ,

• 10 bivectors (area), σαβ ,

436

• 10 trivectors (volume), σαβγ ,

• 5 tetravectors (4-volume), iσα ,

• 1 pseudoscalar (5-volume), i≡ σ01234.

Several elements of this basis square to unity:

(σi)2 = (σ0i)2 = (σ0i j)2 = (iσ0)2 = 1. (2.2)

It is easy to verify the equations above; suppose we want to check that (σ0i j)2 = 1. Startby expanding the square and remove the compact notation (σ0i j)2 = σ0σiσ jσ0σiσ j, thenswap the last σ j twice to bring it next to its homonymous; each swap changes the sign,so an even number of swaps preserves the sign: (σ0i j)2 = σ0σi(σ j)2σ0σi. From thethird equation (2.1) we know that the squared vector is unity and we get successively(σ0i j)2 = σ0σiσ0σi =−(σ0)2(σi)2 =−(σ0)2; using the first equation (2.1) we get finally(σ0i j)2 = 1 as desired.

The remaining basis elements square to −1 as can be verified in a similar manner:

(σ0)2 = (σi j)2 = (σi jk)2 = (iσi)2 = i2 =−1. (2.3)

Note that the pseudoscalar i commutes with all the other basis elements while being asquare root of −1; this makes it a very special element which can play the role of thescalar imaginary in complex algebra.

We can now address the geometric product of any two vectors a = aασα and b = bβ σβ

making use of the distributive property

ab =

(−a0b0 +∑

iaibi

)+ ∑

α 6=β

aαbβσαβ ; (2.4)

and we notice it can be decomposed into a symmetric part, a scalar called the inneror interior product, and an anti-symmetric part, a bivector called the outer or exteriorproduct.

ab = a ·b+a∧b, ba = a ·b−a∧b. (2.5)

Reversing the definition one can write inner and outer products as

a ·b =12

(ab+ba), a∧b =12

(ab−ba). (2.6)

The inner product is the same as the usual ”dot product,” the only difference being in thenegative sign of the a0b0 term; this is to be expected and is similar to what one finds inspecial relativity. The outer product represents an oriented area; in Euclidean 3-space itcan be linked to the "cross product" by the relation cross(a,b) = −σ123a∧b; here we

437

introduced bold characters for 3-dimensional vectors and avoided defining a symbol forthe cross product because we will not use it again. We also used the convention thatinterior and exterior products take precedence over geometric product in an expression.

When a vector is operated with a multivector the inner product reduces the gradeof each element by one unit and the outer product increases the grade by one. Wewill generalize the definition of inner and outer products below; under this generalizeddefinition the inner product between a vector and a scalar produces a vector. Given amultivector a we refer to its grade-r part by writing <a>r; the scalar or grade zero partis simply designated as <a>. By operating a vector with itself we obtain a scalar equalto the square of the vector’s length

a2 = aa = a ·a+a∧a = a ·a. (2.7)

The definitions of inner and outer products can be extended to general multivectors

a ·b = ∑α,β

⟨<a>α <b>β

⟩|α−β | , (2.8)

a∧b = ∑α,β

⟨<a>α <b>β

⟩α+β

. (2.9)

Two other useful products are the scalar product, denoted as < ab > and commutatorproduct, defined by

a×b = (ab−ba)/2. (2.10)

In mixed product expressions we will always use the convention that inner and outerproducts take precedence over geometric products, thus reducing the number of paren-thesis.

We will encounter exponentials with multivector exponents; two particular cases ofexponentiation are specially important. If u is such that u2 =−1 and θ is a scalar

euθ = 1+uθ − θ 2

2!−u

θ 3

3!+

θ 4

4!+ . . .

= 1− θ 2

2!+

θ 4

4!− . . .= cosθ+

+uθ −uθ 3

3!+ . . .= usinθ (2.11)

= cosθ +usinθ .

438

Conversely if h is such that h2 = 1

ehθ = 1+hθ +θ 2

2!+h

θ 3

3!+

θ 4

4!+ . . .

= 1+θ 2

2!+

θ 4

4!+ . . .= coshθ+

+hθ +hθ 3

3!+ . . .= hsinhθ (2.12)

= coshθ +hsinhθ .

The exponential of bivectors is useful for defining rotations; a rotation of vector a byangle θ on the σ12 plane is performed by

a′ = eσ21θ/2aeσ12θ/2 = RaR; (2.13)

the tilde denotes reversion and reverses the order of all products. As a check we makea = σ1

e−σ12θ/2σ1eσ12θ/2 =

(cos

θ

2−σ12 sin

θ

2

)σ1 ∗

∗(

cosθ

2+σ12 sin

θ

2

)(2.14)

= cosθσ1 + sinθσ2.

Similarly, if we had made a = σ2, the result would have been −sinθσ1 + cosθσ2.If we use B to represent a bivector whose plane is normal to σ0 and define its norm by|B|= (BB)1/2, a general rotation in 4-space is represented by the rotor

R≡ e−B/2 = cos(|B|2

)− B|B|

sin(|B|2

). (2.15)

The rotation angle is |B| and the rotation plane is defined by B. A rotor is defined as aunitary even multivector (a multivector with even grade components only) which squaresto unity; we are particularly interested in rotors with bivector components. It is moregeneral to define a rotation by a plane (bivector) then by an axis (vector) because thelatter only works in 3D while the former is applicable in any dimension. When the planeof bivector B contains σ0, a similar operation does not produce a simple rotation butproduces a boost, eventually combined with a rotation. Take for instance B = σ01θ/2and define the transformation operator T = exp(B); a transformation of the basis vector

439

σ0 produces

a′ = T σ0T = e−σ01θ/2σ0eσ01θ/2

=(

coshθ

2−σ01 sinh

θ

2

)σ0 ∗

∗(

coshθ

2+σ01 sinh

θ

2

)(2.16)

= coshθσ0 + sinhθσ1.

In 5-dimensional spacetime of general curvature, we introduce 5 coordinate framevectors gα , the indices follow the conventions set forth in Appendix A. We will alsoassume this spacetime to be a metric space whose metric tensor is given by

gαβ = gα ·gβ ; (2.17)

the double index is used with g to denote the inner product of frame vectors and not theirgeometric product. The space signature is (−++++), which amounts to saying thatg00 < 0 and gii > 0. A reciprocal frame is defined by the condition

gα ·gβ = δα

β . (2.18)

Defining gαβ as the inverse of gαβ , the matrix product of the two must be the identitymatrix; using Einstein’s summation convention this is

gαγgγβ = δα

β . (2.19)

Using the definition (2.17) we have(gαγgγ

)·gβ = δ

αβ ; (2.20)

comparing with Eq. (2.18) we determine gαwith

gα = gαγgγ . (2.21)

If the coordinate frame vectors can be expressed as a linear combination of the or-thonormed ones, we have

gα = nβασβ , (2.22)

where nβα is called the refractive index tensor or simply the refractive index; its 25

elements can vary from point to point as a function of the coordinates.[2, 10] Whenthe refractive index is the identity, we have gα = σα for the main or direct frame andg0 =−σ0, gi = σi for the reciprocal frame, so that Eq. (2.18) is verified. In this work wewill not consider spaces of general curvature but only those satisfying condition (2.22).

440

The first use we will make of the reciprocal frame is for the definition of two derivativeoperators. In flat space we define the vector derivative

∇ = σα

∂α . (2.23)

It will be convenient, sometimes, to use vector derivatives in subspaces of 5D space;these will be denoted by an upper index before the ∇ and the particular index useddetermines the subspace to which the derivative applies; For instance m∇ = σm∂m =σ1∂1 +σ2∂2 +σ3∂3. In 5-dimensional space it will be useful to split the vector derivativeinto its time and 4-dimensional parts

∇ =−σ0∂t +σi∂i =−σ0∂t + i

∇. (2.24)

The second derivative operator is the covariant derivative, sometimes called the Diracoperator, and it is defined in the reciprocal frame gα

D = gα∂α . (2.25)

Taking into account the definition of the reciprocal frame (2.18), we see that the covariantderivative is also a vector. In cases such as those we consider in this work, where thereis a refractive index, it will be possible to define both derivatives in the same space.

We define also second order differential operators, designated Laplacian and covari-ant Laplacian respectively, resulting from the inner product of one derivative operatorby itself. The square of a vector is always a scalar and the vector derivative is no excep-tion, so the Laplacian is a scalar operator, which consequently acts separately in eachcomponent of a multivector. For 4+1 flat space it is

∇2 =− ∂ 2

∂ t2 + i∇

2. (2.26)

One sees immediately that a 4-dimensional wave equation is obtained by zeroing theLaplacian of some function

∇2ψ =

(− ∂ 2

∂ t2 + i∇

2)

ψ = 0. (2.27)

This procedure will be used in the next section for the derivation of special relativity andwill be extended later to general curved spaces.

3. Monogenic functions and waves in flat space

It turns out that there is a class of functions of great importance, called monogenicfunctions,[8] characterized by having null vector derivative; a function ψ is monogenicif and only if

∇ψ = 0. (3.1)

441

A monogenic function has by necessity null Laplacian, as can be seen by dotting Eq.(3.1) with ∇ on the left. We are then allowed to write

∑i

∂iiψ = ∂00ψ. (3.2)

This can be recognized as a wave equation in the 4-dimensional space spanned by σi

which will accept plane wave type solutions of the general form

ψ = ψ0ei(pα xα+δ ), (3.3)

where ψ0 is an amplitude whose characteristics we shall not discuss for now, δ is a phaseangle and pα are constants such that

∑i(pi)2− (p0)2 = 0. (3.4)

By setting the argument of ψ constant in Eq. (3.3) and differentiating we can get thedifferential equation

pαdxα = 0. (3.5)

The first member can equivalently be written as the inner product of the two vectorsp · dx = 0, where p = σα pα . In 5D hyperbolic space the inner product of two vectorscan be null when the vectors are perpendicular but also when the two vectors are null;since we have established that p is a null vector, Eq. (3.5) can be satisfied either by dxnormal to p or by (dx)2 = 0. In the former case the condition describes a 3-volume calledwavefront and in the latter case it describes the wave motion. Notice that the wavefrontsare not surfaces but volumes, because we are working with 4-dimensional waves.

The condition describing wave motion can be expanded as

−(dx0)2 +∑(dxi)2 = 0. (3.6)

This is a purely scalar equation and can be manipulated as such, which means we areallowed to rewrite it with any chosen terms in the second member; some of those ma-nipulations are particularly significant. Suppose we decide to isolate (dx4)2 in the firstmember: (dx4)2 = (dx0)2−∑(dxm)2. We can then rename coordinate x4 as τ , to get theinterval squared of special relativity for space-like displacements

dτ2 = (dx0)2−∑(dxm)2. (3.7)

We have thus derived the space-like part of special relativity as a consequence of mono-geneity in 5D hyperbolic space and simultaneously justified the physical interpretationfor coordinates x0 and x4 as time and proper time, respectively.

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A different manipulation of Eq. (3.6) has great significance because it leads to the4DO concept.[11, 12] If we isolate (dx0)2 and replace x0 by the letter t, we see that timebecomes the interval in Euclidean 4D space

dt2 = ∑(dxi)2. (3.8)

From this we conclude that the monogenic condition produces plane waves whose wave-fronts are 3D volumes but can be represented by wavefront normals, just as it happensin standard optics with electromagnetic waves.

Several readers may be worried with the fact that proper time is a line integral and nota coordinate in special relativity and so dτ should not be allowed to appear on the rhs ofthe equation. To this we will argue that the manipulations we have done, collapsing 5Dspacetime into 4 dimensions through a null displacement condition and then promotingone of the coordinates into interval, is exactly equivalent to the process of defining alight cone in Minkowski spacetime and then applying Fermat’s principle to define anEuclidean 3D metric on the light cone; we have just upgraded the procedure by includingone extra dimension.

The Dirac equation can also be derived from the monogenic condition but since itappears formulated in terms of matrices in all textbooks we will have to rewrite Eq. (3.1)also in terms of matrices, so that our GA manipulations can also be understood as matrixoperations. This is easily achieved if we assign our frame vectors to Dirac matricesthat square to the the identity matrix or minus the identity matrix as appropriate; thefollowing list of assignments can be used but others would be equally effective1

σ0 ≡

i 0 0 00 −i 0 00 0 i 00 0 0 −i

, σ1 ≡

0 0 0 10 0 −1 00 −1 0 01 0 0 0

,

σ2 ≡

0 i 0 0−i 0 0 00 0 0 −i0 0 i 0

, σ3 ≡

0 1 0 01 0 0 00 0 0 10 0 1 0

,

σ4 ≡

0 0 0 −i0 0 i 00 −i 0 0i 0 0 0

.

(3.9)

There is no need to adopt different notations to refer to the frame vectors or to theirmatrix counterparts because the context will usually be sufficient to determine what ismeant.

1There are 16 possible 4∗4 Dirac matrices,[13] of which we must choose 5 such that (σ0)2 =−I, (σi)2 = Iand σα σβ = −σβ σα , for α 6= β ; the present choice will simplify our symmetry discussions furtheralong.

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We can check that matrices σα form an orthonormal basis of 5D space by definingthe inner product of square matrices as

A ·B =AB+BA

2. (3.10)

It will then be possible to verify that the inner product of any two different σ -matricesis null, (σ0)2 = −I and (σ i)2 = I; these are the conditions defining an orthonormalbasis expressed in matrix form. A more formal approach to this subject would lead usto invoke the isomorphism between the complex algebra of 4 ∗ 4 matrices and Cliffordalgebra G4,1, the geometric algebra of 5D spacetime.[14].

It will now be convenient to expand the monogenic condition (3.1) as (σ µ∂µ +σ4∂4)ψ =0. If this is applied to the solution (3.3) and the derivative with respect to x4 is evaluatedwe get

(σ µ∂µ +σ

4ip4)ψ = 0. (3.11)

Let us now multiply both sides of the equation on the left by σ4 and note that matrixσ4σ0 squares to the identity while the 3 matrices σ4σm square to minus identity; werename these products as γ-matrices in the form γµ = σ4σ µ . Rewriting the equation inthis form we get

(γµ∂µ + ip4)ψ = 0. (3.12)

The only thing this equation needs to be recognized as Dirac’s is the replacement of p4 bythe particle’s mass m; simultaneously we assign the energy E to p0 and 3D momentump to σm pm.

We turn now our attention to the amplitude ψ0 in Eq. (3.3) because we know thatthe Dirac equation accepts solutions which are spinors and we want to find out theirequivalents in our formulation. Applying the monogenic condition to Eq. (3.3) we seethat the following equation must be verified

ψ0(σα pα) = 0. (3.13)

If the σs are interpreted as matrices, remembering that p is null, the only way the equa-tion can be verified is by ψ0 being some constant multiplied by the matrix in parenthesis,which is a matrix representation of p. We can set the multiplying constant to unity andψ0 becomes equal to p; the wavefunction ψ can then be interpreted as a Dirac spinor.The wave function in Eq. (3.3) can now be given a different form, taking in considerationthe previous assignments

ψ = A(σ4m+p∓σ0E)eu(±Et+p·x+mτ+δ ); (3.14)

where A is the amplitude and x = σmxm is the 3-dimensional position.In order to separate left and right spinor components we use a technique adapted from

Ref. [8]. We choose an arbitrary 4×4 matrix which squares to identity, for instance σ4,

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with which we form the two idempotent matrices (I +σ4)/2 and (I−σ4)/2.2 These ma-trices are called idempotents because they reproduce themselves when squared. Theseidempotents absorb any σ4 factor; as can be easily checked (I + σ4)σ4 = (I + σ4) and(I−σ4)σ4 =−(I−σ4).

Obviously we can decompose the wavefunction ψ as

ψ = ψI +σ4

2+ψ

I−σ4

2= ψ+ +ψ−. (3.15)

This apparently trivial decomposition produces some surprising results due to the fol-lowing relations

eiθ (I +σ4) = (cosθ + i sinθ)(I +σ4)= (I cosθ + iσ4 sinθ)(I +σ4) (3.16)

= eiσ4θ (I +σ4).

and similarlyeiθ (I−σ4) = e−iσ4θ (I−σ4). (3.17)

If we had chosen a different idempotent the result would have been similar; we will seehow the various idempotents are arranged in a symmetry group and it has been arguedthat they may be related to elementary particles.[15]

4. Relativistic dynamics

When working in curved spaces the monogenic condition is naturally modified, replac-ing the vector derivative ∇ with the covariant derivative D. A generalized monogenicfunction is then a function that verifies the equation

Dψ = 0. (4.1)

Similarly to what happens in flat space, the covariant Laplacian is a scalar and a mono-genic function must verify the second order differential equation

D2ψ = 0. (4.2)

It is possible to write a general expression for the covariant Laplacian in terms of themetric tensor components (see [16, Section 2.11]) but we will consider only situationswhere that complete general expression is not needed.

When Eq. (4.1) is multiplied on the left by D, we are applying second derivatives tothe function, but we are simultaneously applying first order derivatives to the reciprocalframe vectors present in the definition of D itself. We can simplify the calculations if the

2Matrix σ4 is the same as matrix γ5 = iγ0γ1γ2γ3.

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variations of the frame vectors are taken to be much slower than those of function ψ sothat frame vector derivatives can be neglected. With this approximation, the covariantLaplacian becomes D2 = gαβ ∂αβ and Eq. (4.2) can be written

gαβ∂αβ ψ = 0. (4.3)

This equation can have a solution of the type given by Eq. (3.3) if again the derivativesof pα are neglected. This approximation is usually of the same order as the former oneand should not be seen as a second restriction. Inserting Eq. (3.3) one sees that it is asolution if

gαβ pα pβ = 0. (4.4)

This equation is the curved space equivalent to Eq. (3.4) and it means that the square ofvector p = gα pα is zero, that is, p is a vector of zero length; for this reason it is calleda null vector or nilpotent. Vector p is the momentum vector and should not be confusedwith 4-dimensional conjugate momentum vectors defined below.

We arrive again at Eq. (3.5) and the condition describing 4D wave motion can beexpanded as

gαβ dxαdxβ = 0. (4.5)

This condition effectively reduces the spatial dimension to four but the resulting spaceis non-metric because all displacements have zero length. We will remove this difficultyby considering two special cases. First let us assume that vector g0 is normal to the otherframe vectors so that all g0i factors are zeroed; condition (4.5) becomes

g00(dx0)2 +gi jdxidx j = 0. (4.6)

All the terms in this equation are scalars and we are allowed to rewrite it with (dx0)2 inthe lhs

(dx0)2 =−gi j

g00dxidx j. (4.7)

We could have arrived at the same result by defining a 4-dimensional displacement vector

dx0v =−1√

g00gidxi; (4.8)

and then squaring it to evaluate its length; v is a unit vector called velocity because itsdefinition is similar to the usual definition of 3-dimensional velocity; its components are

vi =dxi

dx0 . (4.9)

Being unitary, the velocity can be obtained by a rotation of the σ4 frame vector

v = Rσ4R. (4.10)

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The rotation angle is a measure of the 3-dimensional velocity component. A null anglecorresponds to v directed along σ4 and null 3D component, while a π/2 angle corre-sponds to the maximum possible 3D component. The idea that physical velocity can beseen as the 3D component of a unitary 4D vector has been explored in several papers butsee [17].

Equation (4.8) projects the original 5-dimensional space into an Euclidean signature4 dimensional space, where an elementary displacement is given by the variation ofcoordinate x0. In the particular case where g0 = σ0 the displacement vector simplifies todx0v = gidxi and we can see clearly that the signature is Euclidean because the four gi

have positive norm. Although it has not been mentioned, we have assumed that none ofthe frame vectors is a function of coordinate x0.

Returning to Eq. (4.6) we can now impose the condition that g4 is normal to the otherframe vectors in order to isolate (dx4)2 instead of (dx0)2, as we did before;

(dx4)2 =−gµν

g44dxµdxν . (4.11)

We have now projected onto 4-dimensional space with signature (+−−−), known asMinkowski signature. In order to check this consider again the special case with g0 = σ0and the equation becomes

(dx4)2 =1

g44(dx0)2− gmn

g44dxmdxn; (4.12)

the diagonal elements gii are necessarily positive, which allows a verification of Minkowskisignature. Contrary to what happened in the previous case, we cannot now obtain (dx4)2

by squaring a vector but we can do it by consideration of the bivector

dx4ν =

1√g44g44

gµg4dxµ . (4.13)

All the products gµg4 are bivectors because we imposed g4 to be normal to the otherframe vectors. When (dx4)2 is evaluated by an inner product we notice that g0g4 haspositive square while the three gmg4 have negative square, ensuring that a Minkowskisignature is obtained. Naturally we have to impose the condition that none of the framevectors depends on x4. Bivector ν is such that ν2 = νν = 1 and it can be obtained by aLorentz transformation of bivector σ04.

ν = T σ04T, (4.14)

where T is of the form T = exp(B) and B is a bivector whose plane is normal to σ4. Notethat T is a pure rotation when the bivector plane is normal to both σ0 and σ4.

In special relativity it is usual to work in a space spanned by an orthonormed frameof vectors γµ such that (γ0)2 = 1 and (γm)2 =−1, producing the desired Minkowski sig-nature [8]. The geometric algebra of this space is isomorphic to the even sub-algebra of

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G4,1 and so the area element dx4ν (4.13) can be reformulated as a vector called relativis-tic 4-velocity. The four γ bivectors are defined in a similar way to the γ matrices used inEq. (3.12), which is to be expected from the isomorphism between geometric and matrixalgebras already mentioned.

Equations (4.7) and (4.11) define two alternative 4-dimensional spaces, those of 4-dimensional optics (4DO), with metric tensor −gi j/g00 and general theory of relativity(GTR) with metric tensor−gµν/g44, respectively; in the former x0 is an affine parameterwhile in the latter it is x4 that takes such role. In fact Eq. (4.11) only covers the space-like part of GTR space, because (dx4)2 is necessarily non-negative. Naturally there isthe limitation that the frame vectors are independent of both x0 and x4, equivalent toimposing a static metric, and also that g0i = gµ4 = 0. Provided the metric is static, thegeodesics of 4DO can be mapped one-to-one with spacelike geodesics of GTR and wecan choose to work on the space that best suits us for free fall dynamics. For a phys-ical interpretation of geometric relations it will frequently be convenient to assign newdesignations to the 5D coordinates that acquire the role of affine parameter in the nullsubspace. We recall the assignments x0 ≡ t and x4 ≡ τ; total derivatives with respect tothese coordinates will receive a special notation: d f /dt = f and d f /dτ = f .

Unless otherwise specified, we will assume that the frame vector associated with co-ordinate x0 is unitary and normal to all the others, that is g0 = σ0 and g0i = 0. Recallingfrom Eq. (4.7), these conditions allow the definition of 4DO space with metric tensorgi j. Although we could try a more general approach, we would loose the possibility ofinterpreting time as a line element and this, as we shall see, provides very interestingand novel interpretations of physics’ equations. In many cases it is also true that g4 isnormal to the other frame vectors and we have seen that in those cases we can makemetric conversions between GTR and 4DO; as we shall see, electromagnetism requiresa non-normal g4 and so we leave this possibility open.

For the moment we will concentrate on isotropic space, characterized by orthogonalrefractive index vectors gi whose norm can change with coordinates but is the same forall vectors. Normally we relax this condition by accepting that the three gm must haveequal norm but g4 can be different. The reason for this relaxed isotropy is found inthe parallel we make with physics by assigning dimensions 1 to 3 to physical space.Isotropy in a physical sense need only be concerned with these dimensions and ignoreswhat happens with dimension 4. We will therefore characterize an isotropic space bythe refractive index frame g0 = σ0, gm = nrσm, g4 = n4σ4. Indeed we could also accepta non-orthogonal g4 within the relaxed isotropy concept but we will not do so for themoment.

Equation (4.7) can now be written in terms of the isotropic refractive indices as

dt2 = (nr)2∑m

(dxm)2 +(n4dτ)2. (4.15)

Spherically symmetric static metrics play a special role; this means that the refractive

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index can be expressed as functions of r if we adopt spherical coordinates. The previousequation then becomes

dt2 = (nr)2 [dr2 + r2(dθ2 + sin2

θdϕ2)]+(n4dτ)2. (4.16)

Since we have g4 normal to the other vectors we can apply metric conversion and writethe equivalent quadratic form for GTR

dτ2 =

(dtn4

)2

−(

nr

n4

)2 [dr2 + r2(dθ

2 + sin2θdϕ

2)]. (4.17)

In the case of a central mass, we can examine how the Schwarzschild metric in GTRcan be transposed to 4DO. The usual form of the metric is

dτ2 =

(1− 2M

χ

)dt2−

(1− 2M

χ

)−1

dχ2−

−χ2 (dθ

2 + sin2θdϕ

2) ; (4.18)

where M is the spherical mass and χ is the radial coordinate, not the distance to thecentre of the mass. This form is non-isotropic but a change of coordinates can be madethat returns the expression to isotropic form (see D’Inverno [18], section 14.7):

r =(

χ−M +√

χ2−2Mχ

)/2; (4.19)

and the new form of the metric is

dτ2 =

1− M2r

1+M2r

2

dt2−(

1+M2r

)4 [dr2− r2 (dθ

2 + sin2θdϕ

2)] . (4.20)

From this equation we immediately define two coefficients, which are called refractiveindex coefficients,

n4 =1+

M2r

1− M2r

, nr =

(1+

M2r

)3

1− M2r

. (4.21)

These refractive indices provide a 4DO Euclidean space equivalent to Schwarzschildmetric, allowing 4DO to be used as an alternative to GTR. Recalling that we derivedtrajectories from solutions (3.3) of a 4-dimensional wave equation (4.3), it becomes clearthat orbits can also be seen as 4-dimensional guided waves by what could be describedas a 4-dimensional optical fibre. Modes are to be expected in these waveguides and weshall say something about them later on.

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5. Fermat’s principle in 4 dimensions

Fermat’s principle applies to optics and states that the path followed by a light ray is theone that makes the travel time an extremum; usually it is the path that minimizes thetime but in some cases a ray can follow a path of maximum or stationary time. Thesesolutions are usually unstable, so one takes the view that light must follow the quick-est path. In Eq. (4.7) we have defined a time interval associated with a 4-dimensionalelementary displacement, which allows us to determine, by integration, a travel time as-sociated with displacements of any size along a given 4-dimensional path. We can thenextend Fermat’s principle to 4D and impose an extremum requirement in order to selecta privileged path between any two 4D points. Taking the square root to Eq. (4.7)

dt =√−

gi j

g00dxidx j. (5.1)

Integrating between two points P1 and P2

t =∫ P2

P1

√−

gi j

g00dxidx j =

∫ P2

P1

√−

gi j

g00xix j dt. (5.2)

In order to evaluate the previous integral one must know the particular path linking thepoints by defining functions xi(t), allowing the replacement dxi = xidt. At this stage it isuseful to define a Lagrangian

L =−gi j

2g00xix j. (5.3)

The time integral can then be written

t =∫ P2

P1

√2Ldt. (5.4)

Time has to remain stationary against any small change of path; therefore we envisagea slightly distorted path defined by functions xi(t)+ εχ i(t), where ε is arbitrarily smalland χ i(t) are functions that specify distortion. Since the distortion must not affect theend points, the distortion functions must vanish at those points. The time integral willnow be a function of ε and we require that

dt(ε)dε

∣∣∣∣ε=0

= 0. (5.5)

Now, the Lagrangian (5.3) is a function of xi, through gαβ and also an explicit functionof xi. Allowing for a path change, through ε makes t in Eq. (5.4) a function of ε

t(ε) =∫ P2

P1

√2L(xi + εχ i + xi + ε χ i)dt. (5.6)

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This can now be derived with respect to ε

dt(ε)dε

∣∣∣∣ε=0

=[∫ P2

P1

1√2L

(∂L∂ xi χ

i +∂L∂xi χ

i)

dt]

ε=0. (5.7)

Note that the first term on the rhs can be written∫ P2

P1

1√2L

∂L∂ xi χ

idt =∫ P2

P1

∂ (√

2L)∂ xi

χidt. (5.8)

This can be integrated by parts

∫ P2

P1

∂ (√

2L)∂ xi

χidt =

[∂ (√

2L)∂ xi

χi

]P2

P1

−∫ P2

P1

ddt

(∂ (√

2L)∂ xi

)χ

idt. (5.9)

The first term on the second member is zero because χ i vanishes for the end points;replacing in Eq. (5.7)

dt(ε)dε

∣∣∣∣ε=0

=1√2

∫ P2

P1

[ddt

(− 1√

L∂L∂ xi

)+

1√L

∂L∂xi

]χ

idt. (5.10)

The rhs must be zero for arbitrary distortion functions χ i, so we conclude that the fol-lowing set of four simultaneous equations must be verified

ddt

(1√L

∂L∂ xi

)=

1√L

∂L∂xi ; (5.11)

these are called the Euler-Lagrange equations.Consideration of Eqs. (4.8) and (4.11) allows us to conclude that the Lagrangian de-

fined by (5.3) can also be written as L = v2/2 and must always equal 1/2. From theLagrangian one defines immediately the conjugate momenta

vi =∂L∂ xi =

−gi j

g00x j. (5.12)

Notice the use of the lower index (vi) to represent momenta while velocity componentshave an upper index (vi). The conjugate momenta are the components of the conjugatemomentum vector

v =givi√−g00

(5.13)

and from Eq. (2.18) √−g00v = givi = gigi jx j = g jx j. (5.14)

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The conjugate momentum and velocity are the same but their components are referredto the reciprocal and refractive index frames, respectively.3 Notice also that by virtue ofEq. (3.4) it is also

vi =pi

p0. (5.15)

The Euler-Lagrange equations (5.11) can now be given a simpler form

vi = ∂iL. (5.16)

This set of four equations defines trajectories of minimum time in 4DO space as long asthe frame vectors gα are known everywhere, independently of the fact that they may ormay not be referred to the orthonormed frame via a refractive index. By definition thesetrajectories are the geodesics of 4DO space, spanned by frame vectors gi/

√−g00, with

metric tensor −gi j/g00.Following an exactly similar procedure we can find trajectories which extremize proper

time, defined by taking the positive square root of Eq. (4.11). The Lagrangian is nowdefined by

L =−12

gµν

g44xµ xν . (5.17)

Consequently the conjugate momenta are

νµ =∂L

∂ xµ=−gµν

g44xν . (5.18)

From Eq. (3.4) we have νµ = pµ/p4; the associated Euler-Lagrange equations are

νµ = ∂µL . (5.19)

"These are, by definition, spacelike geodesics of GTR with metric tensor −gµν/g44and we have thus defined a method for one-to-one geodesic mapping between 4DOand spacelike GTR. Recalling the conditions for this mapping to be valid, all the framevectors must be independent of both t and τ and g0 and g4 must be normal to the other3 frame vectors. In tensor terms, all the gαβ must be independent from t and τ andg0i = gµ4 = 0."

6. The sources of refractive index

The set of 4 equations (5.16) defines the geodesics of 4DO space; particularly in caseswhere there is a refractive index, it defines trajectories of minimum time but does not tellus anything about what produces the refractive index in the first place. Similarly the set

3In most cases g00 = −1, the velocity can be conveniently written v = gixi and conjugate momenta vi =gi j x j.

452

of equations (5.19) defines the geodesics of GTR space without telling us what shapesspace. In order to analyse this question we must return to the general case of a refractiveframe gα without other impositions besides the existence of a refractive index.

Considering the momentum vector

p = pαgα = pαnβα

σβ , (6.1)

with nαγnβ

γ = δβ

α , we will now take its time derivative. Using Eq. (B.4)

p = x · (Dp) = x ·G. (6.2)

By a suitable choice of coordinates we can always have g0 = σ0. We can then invokethe fact that for an elementary particle in flat space the momentum vector componentscan be associated with the concepts of energy, 3D momentum and rest mass as p =Eσ0 +p+mσ4 (see Sec. 3.) If this consequence is extended to curved space and to massdistributions, we write p = Eσ0 +p+mg4, where now E is energy density, p = pmgm is3D momentum density and m is mass density. The previous equation then becomes

Eσ0 + p+mg4 = x ·G. (6.3)

When the Laplacian is applied to the momentum vector the result is still necessarily avector

D2 p = S. (6.4)

Vector S is called the sources vector and can be expanded into 25 terms as

S = (D2nβα)σβ pα = Sβ

ασβ pα ; (6.5)

where pα = gαβ pβ . Tensor Sαβ contains the coefficients of the sources vector and we

call it the sources tensor. The sources tensor influences the shape of geodesics as weshall see in one particularly important situation. One important consequence that wedon’t pursue here is that by zeroing the sources vector one obtains the wave equationD2 p = 0, which accepts gravitational wave solutions.

If σ0 is normal to the other frame vectors we can write p = E(σ0 +v) in the reciprocalframe, with v a unit vector or p = E(−σ0 + v) in the direct frame. Equation (6.2) canthen be given the form

E(σ0 + v)+Ev = σ0 + v ·G. (6.6)

Since G can have scalar and bivector components, the scalar part must be responsiblefor the energy change, while the bivector part rotates the velocity v. The bivector part ofG is generated by D∧ p, which allows a simplification of the previous equation to

v = v · (D∧ v), (6.7)

453

if the frame vectors are independent of t. This equation is exactly equivalent to the setof Euler-Lagrange equations (5.16) but it was derived in a way which tells us when toexpect geodesic movement or free fall.

We will now investigate spherically symmetric solutions in isotropic conditions de-fined by Eq. (4.16); this means that the refractive index can be expressed as functions ofr. The vector derivative in spherical coordinates is of course

D =1nr

(σr∂r +

1r

σθ ∂θ +1

r sinθσϕ∂ϕ

)−σt∂t +

1n4

στ∂τ . (6.8)

The Laplacian is the inner product of D with itself but the frame vectors’ derivatives mustbe considered; all the derivatives with respect to t, r and τ are zero and the non-zero onesare

∂θ σr = σθ , ∂ϕσr = sinθσϕ ,∂θ σθ =−σr, ∂ϕσθ = cosθσϕ ,∂θ σϕ = 0, ∂ϕσϕ =−sinθ σr− cosθ σθ .

(6.9)

After evaluation the curved Laplacian becomes

D2 =1

(nr)2

(∂rr +

2r

∂r−n′rnr

∂r +1r2 ∂θθ +

+cotθ

r2 ∂θ +csc2 θ

r2 ∂ϕϕ

)−∂tt +

1(n4)2 ∂ττ . (6.10)

The search for solutions of Eq. (6.4) must necessarily start with vanishing secondmember, a zero sources situation, which one would implicitly assign to vacuum; thisis a wrong assumption as we will show. Zeroing the second member implies that theLaplacian of both nr and n4 must be zero; considering that they are functions of r we getthe following equation for nr

n′′r +

2n′rr− (n′r)

2

nr= 0, (6.11)

with general solution nr = bexp(a/r). It is legitimate to make b = 1 because the re-fractive index must be unity at infinity. Using this solution in Eq. (6.10) the Laplacianbecomes

D2 = e−a/r(

∂rr +2r

∂r +ar2 ∂r +

1r2 ∂θθ +

+cotθ

r2 ∂θ +csc2 θ

r2 ∂ϕϕ

)−∂tt +

1(n4)2 ∂ττ ; (6.12)

which produces the solution n4 = nr. So space must be truly isotropic and not relaxedisotropic as we had allowed. The solution we have found for the refractive index com-ponents in isotropic space can correctly model Newton dynamics, which led the author

454

to adhere to it for some time [17]. However if inserted into Eq. (4.11) this solution pro-duces a GTR metric which is verifiably in disagreement with observations; consequentlyit has purely geometric significance.

The inadequacy of the isotropic solution found above for relativistic predictions de-serves some thought, so that we can search for solutions guided by the results that areexpected to have physical significance. In the physical world we are never in a situationof zero sources because the shape of space or the existence of a refractive index mustalways be tested with a test particle. A test particle is an abstraction corresponding to apoint mass considered so small as to have no influence on the shape of space; in realitya point particle is a black hole in GTR, although this fact is always overlooked; onewonders how a black hole is postulated not to influence space geometry. A test particlemust be seen as source of refractive index itself and its influence on the shape of spaceshould not be neglected in any circumstances. If this is the case the solutions for van-ishing sources vector may have only geometric meaning, with no connection to physicalreality.

The question is then what should go into the second member of Eq. (6.4) in order tofind physically meaningful solutions. If we are testing gravity we must assume somemass density to suffer gravitational influence; this is what is usually designated as non-interacting dust, meaning that some continuous distribution of non-interacting particlesfollows the geodesics of space. Mass density is expected to be associated with S4

4; onthe other hand we are assuming that this mass density is very small and so we use flatspace Laplacian to evaluate it. We consequently make an ad hoc proposal for the sourcesvector in the second member of Eq. (6.4)

S =−∇2n4σ4. (6.13)

Equation (6.4) becomesD2x =−∇

2n4σ4; (6.14)

as a result the equation for nr remains unchanged but the equation for n4 becomes

n′′4 +

2n′4r− n′rn

′4

nr=−n

′′4 +

2n′4r

. (6.15)

When nr is given the exponential form found above, the solution is n4 =√

nr. Thiscan now be entered into Eq. (4.11) and the coefficients can be expanded in series andcompared to Schwarzschild’s for the determination of parameter a. The final solution,for a stationary mass M is

nr = e2M/r, n4 = eM/r. (6.16)

The equivalent GTR space is characterized by the quadratic form

dτ2 = e−2M/rdt2− e2M/r

∑m

(dxm)2. (6.17)

455

Expanding in series of M/r the coefficients of this metric one would find that the lowerorder terms are exactly the same as for Schwarzschild’s and so the predictions of themetrics are indistinguishable for small values of the expansion variable. Montanus [19]arrives at the same solutions with a different reasoning; Yilmaz was probably the firstauthor to propose this metric [20, 21, 22].

Equation (6.14) can be interpreted in physical terms as containing the essence of gravi-tation. When solved for spherically symmetric solutions, as we have done, the first mem-ber provides the definition of a stationary gravitational mass as the factor M appearing inthe exponent and the second member defines inertial mass as ∇2n4. Gravitational massis defined with recourse to some particle which undergoes gravitational influence and isanimated with velocity v and inertial mass cannot be defined without some field n4 act-ing upon it. Complete investigation of the sources tensor elements and their relation tophysical quantities is not yet done; it is believed that 16 terms of this tensor have stronglinks with homologous elements of stress tensor in GTR, while the others are related toelectromagnetic field.

7. Electromagnetism in 5D spacetime

Maxwell’s equations can easily be written in the form of Eq. (6.4) if we don’t imposethe condition that g4 should remain normal the other frame vectors; as we have seen insection 3 this has the consequence that there will be no GTR equivalent to the equationsformulated in 4DO.

We will consider the non-orthonormed reciprocal frame defined by

gµ = σµ , g4 =

qm

Aµσµ +σ

4; (7.1)

where q and m are charge and mass densities, respectively, and A = Aµσ µ is the electro-magnetic vector potential, assumed to be a function of coordinates t and xm but indepen-dent of τ . The associated direct frame has vectors

gµ = σµ −qm

Aµσ4, g4 = σ4; (7.2)

and one can easily verify that Eq. (2.18) is obeyed. The momentum vector in the recipro-cal frame is p = Eσ0 + pmσm +qAµσ µ +mσ4 and G in the second member of Eq. (6.2)is G = qDA. We will assume D ·A to be zero, as one usually does in electromagnetism;also D can be replaced by µ∇ because the vector potential does not depend on τ . It isconvenient to define the Faraday bivector F = µ∇A, similarly to what is done in Ref. [8];the dynamics equation then becomes

p+qA = qx ·F ; (7.3)

and rearrangingp = qx ·F−qA. (7.4)

456

The first term in the second member is the Lorentz force and the second term is due tothe radiation of an accelerated charge.

Recalling the wave displacement vector Eq. (B.1) we have now

dx = σαdxα − qm

Aµσ4dxµ . (7.5)

This corresponds to a refractive index tensor whose non-zero terms are

nαα = 1, n4

µ =− qm

Aµ . (7.6)

According to Eq. (6.5) the sources tensor has all terms null except for the following

S4µ =− q

mD2Aµ ; (7.7)

where D is the covariant derivative given by

D = gα∂α = σ

µ∂µ +(σ4 +

qm

Aµσµ)∂4. (7.8)

We can then define the current vector J verifying

µ∇

2A = µ∇F = J, (7.9)

whereJ =−m

qS4

µσµ . (7.10)

Please refer to [8, Chap. 7] or to [7, Part 2] to see how these equations generate classicalelectromagnetism.

In free space we make J = 0 and Eq. (7.9) accepts plane wave solutions for F whichare of course electromagnetic waves. Notice that these solutions propagate in direc-tions normal to proper time, which is perfectly consistent with the classical relativisticformulation.

The Dirac equation for a free particle has been derived from the 5-dimensional mono-genic condition in Sec. 3 but we are now in position to include the effects of an EM field.Because we are working in geometric algebra, our quantum mechanics equations will in-herit that character but the isomorphism between the geometric algebra of 5D spacetime,G4,1, and complex algebra of 4∗4 matrices, M(4,C), ensures that they can be translatedinto the more usual Dirac matrix formalism. Electrodynamics can now be implementedin the the same way used in Sec. 7 to implement classical electromagnetism. The mono-genic condition must now be established with the covariant derivative given by Eq. (7.8)

σµ

∂µψ +(

σ4 +

qm

Aµσµ

)∂4ψ = 0. (7.11)

457

Multiplying on the left by σ4 and taking ∂4ψ = imψ[γ

µ(∂µ + iqAµ)+ im]

ψ = 0. (7.12)

This equation can be compared to what is found in any quantum mechanics textbook..It is now adequate to say a few words about quantization, which is inherent to 5D

monogenic functions. We have already seen that these functions are 4-dimensionalwaves, that is, they have 3-dimensional wavefronts normal to the direction of propaga-tion. Whenever the refractive index distribution traps one of these waves a 4-dimensionalwaveguide is produced, which has its own allowed propagating modes. In the particularcase of a central potential, be it an atom’s or a galaxy’s nucleus, we expect sphericalharmonic modes, which produce the well known electron orbitals in the atom and haveunknown manifestations in a galaxy.

8. Hyperspherical coordinates

Deriving physical equations and predictions from purely geometrical equations is anexercise whose success depends on the correct assignment of coordinates to physicalentities; the same space will produce different predictions if different options are takenfor coordinate assignment. In the previous sections we assumed that empty space couldbe modelled by an assignment of time, three spatial directions and proper time to fiveorthogonal directions in 5D spacetime. We are now going to experiment with a differentassignment of flat space coordinates, which will explore the possibility that physics andthe Universe have an inbuilt hyperspherical symmetry. The exercise consists on assign-ing coordinate x4 = τ to the radius of an hypersphere and the three xm coordinates todistances measured on the hypersphere surface; time, x0, will still be measured alonga direction normal to all others. If the hypersphere radius is very large we will not beable to notice the curvature on everyday phenomena, in the same way as everyday dis-placements on Earth don’t seem curved to us. The Universe as a whole will manifestthe consequences of its hyperspherical symmetry; using the Earth as a 3-dimensionalanalogue of an hyperspherical Universe, although our everyday life is greatly unaffectedby Earth’s curvature the atmosphere senses this curvature and shows manifestations ofit in winds and climate. What we propose here is an exercise consisting of an arbitraryassignment between coordinates and physical entities; the validity of such exercise canonly be judged by the predictions it allows and how well they conform with observations.

Hyperspherical coordinates are characterized by one distance coordinate, τ and threeangles ρ,θ ,ϕ; following the usual procedure we will associate with these coordinatesthe frame vectors στ ,σρ ,σθ ,σϕ. The position vector for one point in 5D space is quitesimply

x = tσt + τστ . (8.1)

458

In order to write an elementary displacement dx we must consider the rotation of framevectors, but we don’t need to think hard about it because we can extend what is knownfrom ordinary spherical coordinates.

dx = σ0dt +σ4dτ + τσρdρ + τ sinρσθ dθ + τ sinρ sinθσϕdϕ. (8.2)

Just as before, we consider only null displacements to obtain time intervals;

dt2 = dτ2 + τ

2 [dρ2 + sin2

ρ(dθ

2 + sin2θdϕ

2)] . (8.3)

The velocity vector, v = x− σ0, can be immediately obtained from the displacementvector dividing by dt

v = σ0τ + τσρ ρ + τ sinρσθ θ + τ sinρ sinθσϕ ϕ. (8.4)

Geodesics of flat space are naturally straight lines, no matter which coordinate systemwe use, however it is useful to derive geodesic equations from a Lagrangian of the form(5.3); in hyperspherical coordinates the Lagrangian becomes

2L = v2 = τ2 + τ

2 [ρ

2 + sin2ρ(θ

2 + sin2θϕ

2)] . (8.5)

Because de Lagrangian is independent of ϕ we can establish a conserved quantity

Jϕ = τ2 sin2

ρ sin2θϕ. (8.6)

It may seem strange that any physically meaningful relation can be derived from thesimple coordinate assignment that we have made, that is, proper time is associated withhypersphere radius and the three usual space coordinates are assigned to distances onthe hypersphere radius. This unexpected fact results from the possibility offered byhyperspherical coordinates to explore a symmetry in the Universe that becomes hiddenwhen we use Cartesian coordinates. In the real world we measure distances betweenobjects, namely cosmological objects, rather than angles; we have therefore to define adistance coordinate, which is obviously r = τρ . It does not matter where in the Universewe place the origin for r and we find it convenient to place ourselves on the origin.

Radial velocities r measure movement in a radial direction from our observation point;we are particularly interested in this type of movement in order to find a link to theHubble relation. Applying the chain rule and then replacing ρ

r = ρτ + ρτ =τ

τr + ρτ. (8.7)

We expect objects that have not suffered any interaction to move along στ ; from (8.4) wesee that this implies ρ = θ = ϕ = 0 and then τ becomes unity. Replacing in the equationabove and rearranging

rr

=1τ. (8.8)

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What this equation tells us is exactly what is expressed by the Hubble relation. The valueof τ can be taken as constant for any given observation because the distance informationis carried by photons and these preserve proper time, as we have seen in our discussionabout electromagnetic waves.4 The first member of the equation is the definition of theHubble parameter and we can then write H = 1/τ . In this way we find the physicalmeaning of coordinate τ as being the Universe’s age.

Underlying the present discussion there is an assumption a preferred frame wherestillness means moving along στ ; there is no question of equivalent inertial frames here.This preferred frame is obviously attached to the observable still objects in the Universewhich are galaxy clusters, as much as we can tell. This is far from the orthodox pointof view, because galaxy clusters are seen as moving relative to each other and so can-not possible define a fixed frame. But in our formulation still objects move in straightlines along the proper time direction and keep their angular separations constant; thisis naturally perceived as increasing mutual distances. If there is any relation betweenour formulation and an ether it must be found in the fact that movement has an absolutemeaning, so it is defined relative to something that is fixed; we call the fixed reference apreferred frame while other authors call it ether.

How does the use of hyperspherical coordinates affect dynamics in our laboratoryexperiments? We would like to know if these coordinates need only be considered inproblems of cosmological scale or, on the contrary, there are implications for everydayexperiments. The answer implies rewriting (8.2) with distance rather than angle coordi-nates; replacing ρ ,

dx = σ0dt +(

σ4−rτ

σρ

)dτ +σρdr + r(σθ dθ + sinθσϕdϕ). (8.9)

Evaluating time intervals from the null displacement condition, as before

dt2 =[

1+( r

τ

)2]

dτ2−2

rτ

dτdr +dr2 + r2(dθ2 + sin2

θdϕ2). (8.10)

This would be a version of (3.8) in spherical coordinates, were it not for the extra termswith powers of r/τ in the second member. The coefficient r/τ implies a comparisonbetween the distance from the object to the observer and the size of the Universe; re-member that τ is both time and distance in non-dimensional units. We can say thatordinary special relativity will apply for objects which are near us, but distant objectswill show in their movement an effect of the Universe’s hyperspherical nature.

With Eqs. (6.16) we have established the refractive indices nr and n4 to account forthe dynamics near a massive sphere using Cartesian coordinates; since this is frequentlyapplied on a cosmological scale, we must find out how the dynamics is modified by

4In order to preserve proper time photons must travel on the hypersphere surface and thus don’t followgeodesics.

460

the use of hyperspherical coordinates. Using the refractive indices and hypersphericalcoordinates, noting that nr = n2

4, Eq. (4.7) becomes

dt2 = n24dτ

2 +n44τ

2dρ2. (8.11)

Dividing both members by dt2 and reversing the equation

n24τ

2 +n44τ

2ρ

2 = 1; (8.12)

and replacing τρ by r− rτ/τ

n24τ

2 +n44

[r2 +

(τ

τ

)2

r2−2τ rrτ

]= 1. (8.13)

Dividing both members by n44r2 and rearranging results in the equation(

rr

)2

=(

1n4

4− τ2

n24

)1r2 −

(τ

τ

)2

+2τ rτr

. (8.14)

As a further step we take the refractive index coefficients from Schwarzschild’s metric(4.21) or those of from the exponential metric (6.16) and expand the second member inseries of M/r taking only the two first terms.(

rr

)2

≈ 1− τ2

r2 +(2τ2−4)M

r3 −(

τ

τ

)2

+2τ rτr

. (8.15)

The previous equation applies to bodies moving radially under the influence of mass Mlocated at the origin which is, remember, the observer’s position. For comparison wederive the corresponding equation in Cartesian coordinates; starting with (8.12) it is now

n24τ

2 +n44r2 = 1; (8.16)

dividing by n44r2 and rearranging(

rr

)2

=(

1n4

4− τ2

n24

)1r2 ≈

1− τ2

r2 +(2τ2−4)M

r3 . (8.17)

If we want to apply these equations to cosmology it is easiest to follow the approachof Newtonian cosmology, which produces basically the same results as the relativisticapproach but presumes that the observer is at the centre of the Universe [18, 23]. In orderto adopt a relativistic approach we need equations that replace Einstein’s in 4DO. A setof such was proposed above Eq. (6.4) but their application in cosmology has not yet beentested, so we will have to defer this more correct approach to future work. The strategywe will follow here is to consider a general object at distance r from the observer, moving

461

away from the latter under the gravitational influence of the mass included in a sphereof radius r. If we designate by µ the average mass density in the Universe, then mass Min (8.15) is 4πµr3/3; this will have to be considered further down.

Friedman equation governs standard cosmology and can be derived both from Newto-nian and relativistic dynamics, with different consequences in terms of the overall size ofthe Universe and the observer’s privileged position. From the cited references we writeFriedman equation as (

rr

)2

=8π

3µ +

Λ

3− k

r2 ; (8.18)

with Λ a cosmological constant and k the curvature constant; the gravitational constantwas not included because it is unity in non-dimensional units and the equation is writtenin real, not comoving, coordinates. In order to compare (8.15) with Friedman equationthere is a problem with the last term because the Hubble parameter r/r does not appearisolated in the first member; we will find a way to circumvent the problem later on butfirst let us look at what (8.15) tells us when the mass density is zeroed. In this case n4 = 1and we find from (8.12) that τ is unity, unless ρ is non-zero, for which we can find noreasonable explanation. Replacing n4 and τ with unity in (8.15) we find that r/r = 1/τ ,confirming what had already been found in (8.8). Comparing with Friedman equation,this corresponds to a flat Universe with a critical mass density µ = µc; it is immediatelyobvious that µc = 3/(8πτ2). Let us not overlook the importance of this conclusionbecause it completely removes the need for a critical density if the Universe is flat;remember this is one of the main reasons to invoke dark matter in standard cosmology.Notice also that this conclusion does not depend on a privileged observer, because it isjust a consequence of space symmetry and not of dynamics.

Let us now see what happens when we consider a small mass density; here we aretalking about matter that is observed or measured in some way but not postulated matter.The matter density that we will consider is of the order of 1% of the presently acceptedvalue. It is therefore just a perturbation of the flat solution that we described above andthe fact that we are presuming a privileged observer has to be taken just for this pertur-bation. The first thing we note when we consider matter density is that τ < 1, becausethere is now a component of the velocity vector along σρ . Ideally we should solve theEuler-Lagrange equations resulting from (8.12) in order to find τ and ρ but this is a diffi-cult process and we shall carry on with just a qualitative discussion. Considering that weare discussing a perturbation it is legitimate to make r/r ≈ τ/τ and the two last terms inthe second member of (8.15) can be combined into one single term (τ/τ)2, the same aswe encountered for the flat solution, albeit with a numerator slightly smaller than unity.The first term has now become slightly positive and we can see from Friedman equationthat this corresponds to a negative curvature constant, k, and to an open Universe. Lastlythe second term includes the mass M of a sphere with radius r and can be simplified to8πµ(τ2−2)/3; this has the effect of a negative cosmological constant; the combined ef-

462

fect of the two terms is expected to close the Universe [23, 24]. The previous discussionwas done in qualitative terms, making use of several approximations, for which reasonwe must question some of the findings and expect that after more detailed examinationthey may not be quite as anticipated; in particular there is concern about the refractiveindices used, which were derived in Cartesian coordinates both by the author and thosethat preceded him in using an exponential metric; it may happen that the transposition tohyperspherical coordinates has not been properly made, with consequences in the pertur-bative analysis that was superimposed on the flat solution. The latter, however, is totallyindependent of such concerns and allows us to state that the assumption of hyperspheri-cal symmetry for the Universe dispenses with dark matter in accounting for the gross ofobserved expansion.

Dark matter is also called in cosmology to account for the extremely high rotationvelocities found in spiral galaxies [25, 26] and we will now take a brief look at howhyperspherical symmetry can help explain this phenomenon. Galaxy dynamics is anextremely complex subject, which we do not intend to explore here due to lack of spacebut most of all due to lack of author’s competence to approach it with any rigour; wewill just have a very brief outlook at the equation for flat orbits, to notice that an effectsimilar to the familiar Coriollis effect on Earth can arise in an expanding hypersphericalUniverse and this could explain most of the observed velocities on the periphery ofgalaxies. Let us recall (8.9), divide by dt and invoke null displacement to obtain thevelocity

v =(

σ4−rτ

σρ

)τ +σρ r + r(σθ θ + sinθσϕ ϕ). (8.19)

If orbits are flat we can make θ = π/2 and the equation simplifies to

v = τσ4 +(

r− rτ

τ

)σρ + rϕσϕ . (8.20)

Suppose now that something in the galaxy is pushing outwards slightly, so that the paren-thesis is zero; this happens if r/r = τ/τ and can be caused by a pressure gradient, forinstance. The result is that (8.20) now accepts solutions with constant rϕ , which is ex-actly what is observed in many cases; swirls will be maintained by a radial expansion ratewhich exactly matches the quotient τ/τ . In any practical situation τ will be very nearunity and the quotient will be virtually equal to the Hubble parameter; thus the expansionrate for sustained rotation is r/r ≈ H. If applied to our neighbour galaxy Andromeda,with a radial extent of 30 kpc, using the Hubble parameter value of 81 km s−1/Mpc, theexpansion velocity is about 2.43 km s−1; this is to be compared with the orbital velocityof near 300 kms−1 and probably within the error margins. An expansion of this sortcould be present in many galaxies and go undetected because it needs only be of theorder of 1% the orbital velocity.

463

9. Symmetries of G4,1 algebra

In this algebra it is possible to find a maximum of four mutually annihilating idem-potents, which generate with 0 an additive group of order 16; for a demonstration seeLounesto [14], section 17.5. Those idempotents can be generated by a choice of twocommuting basis elements which square to unity; for the moment we will use σ023 andσ014. The set of 4 idempotents is then given by

f1 =(1+σ023)(1+σ014)

4, f2 =

(1+σ023)(1−σ014)4

,

f3 =(1−σ023)(1−σ014)

4, f4 =

(1−σ023)(1+σ014)4

.

(9.1)

Using the matrices of Sec. 3 to make matrix replacements of σ023 and σ014 one canfind matrix equivalents to these idempotents; those are matrices which have only onenon-zero element, located on the diagonal and with unit value.

SU(3) symmetry can now be demonstrated by construction of the 8 generators

λ1 = σ02( f1 + f2) =σ3 +σ02

2,

λ2 = σ03( f1 + f2) =−σ2 +σ03

2,

λ3 = f1− f2 =σ014−σ1234

2,

λ4 =−σ1( f2 + f3) =−σ1−σ04

2,

λ5 =−σ4( f2 + f3) =−σ4 +σ01

2,

λ6 = σ012( f1 + f3) =σ012 +σ034

2,

λ7 =−σ024( f1 + f3) =σ013−σ024

2,

λ8 =f1 + f2−2 f3√

3=

2σ023 +σ014 +σ1234

2√

3.

(9.2)

464

These have the following matrix equivalents

λ1 ≡

0 1 0 01 0 0 00 0 0 00 0 0 0

, λ2 ≡

0 −j 0 0j 0 0 00 0 0 00 0 0 0

, λ3 ≡

1 0 0 00 −1 0 00 0 0 00 0 0 0

,

λ4 ≡

0 0 0 00 0 1 00 1 0 00 0 0 0

, λ5 ≡

0 0 0 00 0 −j 00 j 0 00 0 0 0

, λ6 ≡

0 0 1 00 0 0 01 0 0 00 0 0 0

λ7 ≡

0 0 −j 00 0 0 00 j 0 00 0 0 0

, λ8 ≡(

1/√

3)

1 0 0 00 1 0 00 0 −2 00 0 0 0

,

(9.3)

which reproduce Gell-Mann matrices in the upper-left 3∗3 corner [15, 27, 28]. Since thealgebra is isomorphic to complex 4 ∗ 4 matrix algebra, one expects to find higher ordersymmetries; Greiner and Müller [27] show how one can add 7 additional generatorsto those of SU(3) in order to obtain SU(4) and the same procedure can be adopted ingeometric algebra. We then define the following additional SU(4) generators

λ9 = σ1( f1 + f4) =σ1−σ04

2,

λ10 = σ4( f1 + f4) =σ4 +σ01

2,

λ11 =−σ012( f2 + f4) =−σ012−σ034

2,

λ12 = σ024( f2 + f4) =σ013 +σ024

2,

λ13 = σ3( f3 + f4) =σ3−σ02

2,

λ14 = σ2( f3 + f4) =σ2 +σ03

2,

λ15 =f1 + f2 + f3−3 f4√

6=

σ023−σ014−σ1234√6

.

(9.4)

Once again, making the replacements with Eq. (3.9) produces the matrix equivalent gen-

465

erators

λ9 ≡

0 0 0 10 0 0 00 0 0 01 0 0 0

, λ10 ≡

0 0 0 −j0 0 0 00 0 0 0j 0 0 0

, λ11 ≡

0 0 0 00 0 0 10 0 0 00 1 0 0

,

λ12 ≡

0 0 0 00 0 0 −j0 0 0 00 j 0 0

, λ13 ≡

0 0 0 00 0 0 00 0 0 10 0 1 0

, λ14 ≡

0 0 0 00 0 0 00 0 0 −j0 0 j 0

,

λ15 ≡(

1/√

6)

1 0 0 00 1 0 00 0 1 00 0 0 −3

.

(9.5)

The standard model involves the consideration of two independent SU(3) groups, onefor colour and the other one for isospin and strangeness; if generators λ1 to λ8 apply toone of the SU(3) groups we can produce the generators of the second group by resortingto the basis elements σ3 and σ04. The new set of 4 idempotents is then given by

f1 =(1+σ3)(1+σ04)

4, f2 =

(1+σ3)(1−σ04)4

,

f3 =(1−σ3)(1−σ04)

4, f4 =

(1−σ3)(1+σ04)4

.

(9.6)

Again a set of SU(3) generators can be constructed following a procedure similar to theprevious one

α1 = σ02( f1 + f2) =σ02 +σ023

2,

α2 = σ01( f1 + f2) =σ01 +σ013

2,

α3 = f1− f2 =σ04−σ034

2,

α4 = σ2( f2 + f3) =σ2 +σ024

2,

α5 =−σ1( f2 + f3) =−σ1−σ014

2,

α6 = σ4( f1 + f3) =σ4−σ03

2,

α7 = σ012( f1 + f3) =σ012 +σ1234

2,

α8 =f1 + f2−2 f3√

3=

2σ3 +σ04 +σ034

2√

3.

(9.7)

466

This new SU(3) group is necessarily independent from the first one because its matrixrepresentation involves matrices with all non-zero rows/columns, while the group gen-erated by λ1 to λ8 uses matrices with zero fourth row/column. In the following sectionwe will discuss which of the two groups should be associated with colour.

At the end of Sec. 3 we used one particular idempotent to split the wavefunction intoleft and right spinors and here we discuss how the different idempotents are related tothe symmetries discussed above, suggesting a relation between idempotents and the dif-ferent elementary particles. We have already established that each set of 4 idempotentsis generated by a pair of commuting unitary basis elements. Let any two such basis el-ements be denoted as h1 and h2; then the product h3 = h1h2 is itself a third commutingbasis element. For consistence we choose, as before,

h1 ≡ σ023, h2 ≡ σ014; (9.8)

to geth3 ≡ σ1234, (9.9)

which commutes with the other two as can be easily verified. The result of this exerciseis the existence of triads of commuting unitary basis elements but no tetrads of suchelements. We are led to state that a general unitary element is a linear combination ofunity and the three elements of one triad

h = a0 +a1h1 +a2h2 +a3h3. (9.10)

Since h is unitary and the three hm commute we can write

h2 =[(a0)2 +(a1)2 +(a2)2 +(a3)2]+2(a0a1−a2a3)h1+

+2(a0a2−a1a3)h2 +2(a0a3−a1a2)h3 = 1(9.11)

The only form this equation can be verified is if the term in square brackets is unitywhile all the others are zero. We then get a set of four simultaneous equations with atotal of sixteen solutions, as follows: 8 solutions with one of the aµ equal to ±1 andall the others zero, 6 solutions with two of the aµ equal to −1/2 and the other twoequal to 1/2 and 2 solutions with all the aµ simultaneously ±1/2. The aµ coefficientsplay the role of quantum numbers which determine the particular idempotent that goesinto Eq. (3.17); these unusual quantum numbers are expressed in terms of the SU(4)generators λ3, λ8 and λ15 in Table 1 in order to highlight the symmetries. We don’tpropose here any direct relationship between the various idempotents and the knownelementary particles, although the fact that the standard model gauge symmetry group isfound as direct consequence of the monogenic condition which itself generates the Diracequation is rather intriguing.

467

Table 1: Coefficients for the various unitary elements.

1 σ023 σ014 σ1234 λ3 λ8 λ15(a0) (a1) (a2) (a3)

1 0 0 0 0 0 00 1 0 0 0 2/

√3

√2/3

0 0 1 0 1 1/√

3 −√

2/30 0 0 1 −1 1/

√3 −

√2/3

−1 0 0 0 0 0 00 −1 0 0 0 −2/

√3 −

√2/3

0 0 −1 0 −1 −1/√

3√

2/30 0 0 −1 1 −1/

√3

√2/3

−1/2 −1/2 1/2 1/2 0 0 −√

3/2−1/2 1/2 −1/2 1/2 −1 1/

√3 1/

√6

−1/2 1/2 1/2 −1/2 1 1/√

3 1/√

61/2 −1/2 −1/2 1/2 −1 −1/

√3 −1/

√6

1/2 −1/2 1/2 −1/2 1 −1/√

3 −1/√

61/2 1/2 −1/2 −1/2 0 0

√3/2

1/2 1/2 1/2 1/2 0 2/√

3 −1/√

6−1/2 −1/2 −1/2 −1/2 0 −2/

√3 1/

√6

10. Conclusion and future work

Monogenic functions applied in the algebra of 5-dimensional spacetime have been shownto originate laws of fundamental physics in such diverse areas as relativistic dynamics,quantum mechanics and electromagnetism, with possible, still unclear, consequences forcosmology and particle physics. To say that those functions provide us with a theory ofeverything is certainly unwarranted at this stage but it is clear that there is a case formuch greater effort being invested in their study.

There are unanswered questions in the present work. For instance, how can we avoidan ad hoc definition of inertial mass or what is the true relation between the symme-tries generated by monogenic functions and elementary particles? In spite of its variousloose ends, the formalism is perfectly capable of unifying relativistic dynamics, quan-tum mechanics and electromagnetism, which in itself is no small achievement. Certaindevelopments seem relatively straightforward but they must be made, even if no newpredictions are expected. Applying monogenic functions to the Hydrogen atom shouldnot be difficult because the form of the Dirac equation we arrived at is perfectly equiva-lent to the standard one; one should then find the same solutions but in a GA formalism.

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In the same line one could try to solve the equation for a central gravitational potential,being certain to find quantum states. It is not clear how important these could be inplanetary mechanics or galaxy dynamics.

Gravitational waves are predicted by the monogenic function formalism as we pointedout but did not investigate. How important are they and what chance is there of thembeing detected by experiment? We don’t know the answer and we don’t know what dif-ficulties lie on the path of those who try to solve the equations; this is an open area. Thesources’ tensor must be clearly understood and directly related to geometry; at the mo-ment all densities, mass, electromagnetic energy, etc. must be inserted in the equationsbut one would expect that a perfect theory would produce such densities out of nothing.In previous papers we suggested that a recursive, non-linear, equation could be the an-swer to the problem but the concept has not yet been formalized and there are no clearideas for achieving such goal.

In conclusion, the present work opens the gate of a path that will possibly lead to anentirely new formulation and understanding of physics but this path is very likely to havemany hurdles to jump and several dead ends to avoid.

A. Indexing conventions

In this section we establish the indexing conventions used in the paper. We deal with 5-dimensional space but we are also interested in two of its 4-dimensional subspaces andone 3-dimensional subspace; ideally our choice of indices should clearly identify theirranges in order to avoid the need to specify the latter in every equation. The diagramin Fig. 1 shows the index naming convention used in this paper; Einstein’s summation

Figure 1: Indices in the range 0,4 will be denoted with Greek letters α,β ,γ. Indicesin the range 0,3 will also receive Greek letters but chosen from µ,ν ,ξ . Forindices in the range 1,4 we will use Latin letters i, j,k and finally for indicesin the range 1,3 we will use also Latin letters chosen from m,n,o.

convention will be adopted as well as the compact notation for partial derivatives ∂α =∂/∂xα .

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B. Time derivative of a 4-dimensional vector

If there is a refractive index the wave displacement vector can be written as

dx = gαdxα = nβασβ dxα . (B.1)

Because this vector is nilpotent, by virtue of Eq. (4.6), the five coordinates are not in-dependent and we can divide both members by dx0 = dt defining the nilpotent vector

x = g0 +gixi = nα0σα +nβ

iσβ xi. (B.2)

Suppose we have a 5D vector a = σαaα and we want to find its time derivative alonga path parameterized by t, that is all the xi are functions of t. We can write

a = ∂β aα xβσα ; (B.3)

where naturally x0 = 1. Remembering the definition of covariant derivative (2.25) andEq. (B.2) we can modify this equation to

a = xβ gβ ·gβ∂β aα

σα = x · (Da). (B.4)

We have expressed vector a in terms of the orthonormed frame in order to avoid vectorderivatives but the result must be independent of the chosen frame.

This procedure has an obvious dual, which we arrive at by defining

x = gµ xµ +g4. (B.5)

The proper time derivative of vector a is then

a = x · (Da). (B.6)

References

[1] J. B. Almeida, Choice of the best geometry to explain physics, 2005, arXiv:physics/0510179.

[2] J. B. Almeida, Monogenic functions in 5-dimensional spacetime used as first prin-ciple: Gravitational dynamics, electromagnetism and quantum mechanics, 2006,arXiv: physics/0601078.

[3] P. S. Wesson, In defense of Campbell’s theorem as a frame for new physics, 2005,arXiv: gr-qc/0507107.

[4] T. Liko, J. M. Overduin, and P. S. Wesson, Astrophysical implications of higher-dimensional gravity, Space Sci. Rev. 110, 337, 2003, arXiv: gr-qc/0311054.

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[5] D. Hestenes and G. Sobczyk, Clifford Algebras to Geometric Calculus. A UnifiedLanguage for Mathematics and Physics, Fundamental Theories of Physics (Reidel,Dordrecht, 1989).

[6] S. Gull, A. Lasenby, and C. Doran, Imaginary numbers are not real. —The geometric algebra of spacetime, Found. Phys. 23, 1175, 1993, URLhttp://www.mrao.cam.ac.uk/~clifford/publications/abstracts/imag_numbs.html.

[7] A. Lasenby and C. Doran, Physical applications of geometric algebra, hand-out collection from a Cambridge University lecture course, 2001, URL http://www.mrao.cam.ac.uk/~clifford/ptIIIcourse/index.html.

[8] C. Doran and A. Lasenby, Geometric Algebra for Physicists (Cambridge UniversityPress, Cambridge, U.K., 2003).

[9] D. Hestenes, New Foundations for Classical Mechanics (Kluwer Academic Pub-lishers, Dordrecht, The Netherlands, 2003), 2nd ed.

[10] J. B. Almeida, The null subspace of G(4,1) as source of the main physical theo-ries, in Physical Interpretations of Relativity Theory – IX (London, 2004), arXiv:physics/0410035.

[11] J. B. Almeida, K-calculus in 4-dimensional optics, 2002, arXiv:physics/0201002.

[12] J. B. Almeida, An alternative to Minkowski space-time, in GR 16 (Durban, SouthAfrica, 2001), arXiv: gr-qc/0104029.

[13] E. W. Weisstein, Dirac matrices, in Math World – A Wolfram Web Resource (1999),URL http://mathworld.wolfram.com/DiracMatrices.html.

[14] P. Lounesto, Clifford Algebras and Spinors, vol. 286 of London Mathematical So-ciety Lecture Note Series (Cambridge University Press, Cambridge, U.K., 2001),2nd ed.

[15] J. B. Almeida, Geometric algebra and particle dynamics, in 7th InternationalConference on Clifford Algebras, ICCA7, edited by P. Anglès (To be published,Toulouse, France, 2005), arXiv: math.GM/0504025.

[16] G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists (AcademicPress, N. Y., 1995), 4th ed.

[17] J. B. Almeida, 4-dimensional optics, an alternative to relativity, 2001, arXiv:gr-qc/0107083.

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[19] J. M. C. Montanus, Proper-time formulation of relativistic dynamics, Found. Phys.31, 1357, 2001.

[20] H. Yilmaz, New approach to general relativity, Phys. Rev. 111, 1417, 1958.

[21] H. Yilmaz, New theory of gravitation, Phys. Rev. Lett. 27, 1399+, 1971.

[22] M. Ibison, The Yilmaz cosmology, in 1st Crisis in Cosmology Conference, CCC–I,edited by E. Lerner and J. B. Almeida (American Institute of Physics, Monção,Portugal, 2005), to be published.

[23] J. V. Narlikar, Introduction to Cosmology (Cambridge University Press, Cam-bridge, U. K., 2002), 3rd ed.

[24] J. L. Martin, General Relativity: A Guide to its Consequences for Gravity andCosmology (Ellis Horwood Ltd., U. K., 1988).

[25] J. Silk, A Short History of the Universe (Scientific American Library, N. York,1997).

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[27] W. Greiner and B. Müller, Quantum Mechanics: Symmetries (Springer, Berlin,2001), 2nd ed.

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Volume 1 has already been published under the name “Modern ether concepts, relativity and geometry”. Abstracts of the papers published in volume 1 are given below. ABSTRACTS ETHER AS A DISCLOSING MODEL. M. C. Duffy, School of Computing & Technology, University of Sunderland, Sunderland, Great Britain, SR1 3SD, & PO Box 342, Burnley, Lancashire, GB, BB10 1XL. [email protected] ABSTRACT The modern ether concept is compatible with relativity, quantum mechanics, and non-classical geometrization. Misuse of the term "ether" in anti-Relativity polemics in former times causes many physicists to avoid the word and equivalent terms are used instead. The modern concept results from three development programmes. First, there was the evolution of Relativity, Relativistic Cosmology and Geometrodynamics which discarded the early 20th C passive, rigid, ether in favour of geometrized space-time. A non-classical ether, defined as field or space-time, was accepted by Einstein in his later years. This had two main aspects: static (or geometric) and dynamic (or frame-space perspective). Second, there was a Lorentzian programme, which provided a quasi-classical exposition of Relativity in terms of moving rod and clock readings. The Einstein-Minkowski and the Lorentzian programmes can be reconciled. The third development programme is associated with Quantum Mechanics and studies of the physical vacuum. A group of analogues based on the vortex sponge promises to unify these programmes of interpretation. The modern ether, from the smallest scale point of view, resembles a "sea of information", which demands new techniques for interpreting it, drawn from information science, computer science, and communications theory. Key Words: Analogues; Ether; Relativity; Space-Time Geometry; Physical Vacuum. EINSTEIN'S NEW ETHER 1916 - 1955 Ludwik Kostro Department for Logic, Methodology and Philosophy of Science, University of Gdańsk, ul. Bielańska 5, 80-851 Gdańsk, Poland E-mail: [email protected] Abstract In 1905 A. Einstein banished the ether from physics in connection with the formulation of his Special Relativity Theory. This is very well known but less known is the fact that in 1916 he reintroduced the ether in connection with his General Relativity. He denominated it “new ether” because, in opposition to the old one, the new one did not

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violate his Special and General Principle of Relativity. It didn’t violate it because the new ether is not conceived as a privileged reference frame but it is considered as an ultra-referential primordial material reality which is not composed of points (or particles) and not divisible in parts and to which therefore the notions of motion and rest are not applicable. The purpose of this paper is to present a short outline of the history of Einstein’s concepts on ether and to show which elements of the mathematical formalism of General Relativity were considered by Einstein as mathematical tools describing the relativistic ether, i.e. the ultra-referential space-time characterized with a certain kind of energy density. It will be indicated also that Einstein’s intuitions and ideas concerning the ultra-referential space-time have to be investigated in the framework of Connes’ non-commutative geometry, as the commutative geometries are not sufficient to do it. In Poland Michal Heller and his colleagues are trying to create an unification of General Relativity and Quantum Mechanics with the help of Connes’ non-commutative geometry. BASIC CONCEPTS FOR A FUNDAMENTAL AETHER THEORY Joseph Levy 4 square Anatole France, 91250 St Germain-lès-Corbeil, France E. mail: [email protected] ABSTRACT In the light of recent experimental and theoretical data, we go back to the studies tackled in previous publications [1] and develop some of their consequences. Some of their main aspects will be studied in further detail. Yet this text remains self- sufficient. The questions asked following these studies will be answered. The consistency of these developments in addition to the experimental results, enable to strongly support the existence of a preferred aether frame and of the anisotropy of the one-way speed of light in the Earth frame. The theory demonstrates that the apparent invariance of the speed of light results from the systematic measurement distortions entailed by length contraction, clock retardation and the synchronization procedures with light signals or by slow clock transport. Contrary to what is often believed, these two methods have been demonstrated to be equivalent by several authors [1]. The compatibility of the relativity principle with the existence of a preferred aether frame and with mass-energy conservation is discussed and the relation existing between the aether and inertial mass is investigated. The experimental space-time transformations connect co-ordinates altered by the systematic measurement distortions. Once these distortions are corrected, the hidden variables they conceal are disclosed. The theory sheds light on several points of physics which had not found a satisfactory explanation before. (Further important comments will be made in ref [1d]).

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AETHER THEORY AND THE PRINCIPLE OF RELATIVITY Joseph Levy 4 Square Anatole France, 91250 St Germain-lès-Corbeil, France E. Mail: [email protected] ABSTRACT This paper completes and comments on some aspects of our previous publications. In ref [1], we have derived a set of space-time transformations referred to as the extended space-time transformations. These transformations, which assume the existence of a preferred aether frame and the variability of the one-way speed of light in the other frames, are compared to the Lorentz-Poincaré transformations. We demonstrate that the extended transformations can be converted into a set of equations that have a similar mathematical form to the Lorentz-Poincaré transformations, but which differ from them in that they connect reference frames whose co-ordinates are altered by the systematic unavoidable measurement distortions due to length contraction and clock retardation and by the usual synchronization procedures, a fact that the conventional approaches of relativity do not show. As a result, we confirm that the relativity principle is not a fundamental principle of physics [i.e, it does not rigorously apply in the physical world when the true co-ordinates are used]. It is contingent but seems to apply provided that the distorted coordinates are used. The apparent invariance of the speed of light also results from the measurement distortions. The space-time transformations relating experimental data, therefore, conceal hidden variables which deserved to be disclosed for a deeper understanding of physics.

Ether theory of gravitation: why and how? Mayeul Arminjon Laboratoire “Sols, Solides, Structures, Risques” (CNRS & Universites de Grenoble), BP 53, F-38041 Grenoble cedex 9, France. Abstract Gravitation might make a preferred frame appear, and with it a clear space/time separation—the latter being, a priori, needed by quantum mechanics (QM) in curved space-time. Several models of gravitation with an ether are discussed: they assume metrical effects in an heterogeneous ether and/or a Lorentz-symmetry breaking. One scalar model, starting from a semi-heuristic view of gravity as a pressure force, is detailed. It has been developed to a complete theory including continuum dynamics, cosmology, and links with electromagnetism and QM. To test the theory, an asymptotic scheme of post-Newtonian approximation has been built. That version of the theory which is discussed here predicts an internal-structure effect, even at the point-particle limit. The same might happen also in general relativity (GR) in some gauges, if one would use a similar scheme. Adjusting the equations of planetary motion on an ephemeris leaves a residual difference with it; one should adjust the

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equations using primary observations. The same effects on light rays are predicted as with GR, and a similar energy loss applies to binary pulsars. A Dust Universe Solution to the Dark Energy Problem James G. Gilson [email protected] School of Mathematical Sciences Queen Mary University of London Mile End Road London E14NS December 19th 2005 Abstract Astronomical measurements of the Omegas for mass density, cosmological constant lambda and curvature k are shown to be sufficient to produce a unique and detailed cosmological model describing dark energy influences based on the Friedman equations. The equation of state Pressure turns out to be identically zero at all epochs as a result of the theory. The partial omega, for dark energy, has the exact value, minus unity, as a result of the theory and is in exact agreement with the astronomer’s measured value. Thus this measurement is redundant as it does not contribute to the construction of the theory for this model. Rather, the value of omega is predicted from the theory. The model has the characteristic of changing from deceleration to acceleration at exactly half the epoch time at which the input measurements are taken. This is a mysterious feature of the model for which no explanation has so far been found. An attractive feature of the model is that the acceleration change time occurs at a red shift of approximately 0.8 as predicted by the dark energy workers. Using a new definition of dark energy density it is shown that the contribution of this density to the acceleration process is via a negative value for the gravitational constant, -G, exactly on a par with gravitational mass which occurs via the usual positive value for G. EDDINGTON, ETHER AND NUMBER Raúl A. Simón LAMB, Santiago, CHILE Abstract For Eddington, the word “ether” was synonymous with de Sitter space- time,and as such it plays only an episodic role in his later work. Nevertheless, it is good to find out why he held such an opinion, for this leads us into most interesting physical – and not only historical – considerations. For this reason, in the present paper we have included the mathematical background necessary to make Eddington’s physics clearer. We have also included some of Eddington’s epistemological derivations of the “number of particles in the universe”, not only as a curiosity, but also as a means of understanding the general character of his later work.

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THE DYNAMICAL SPACE-TIME AS A FIELD CONFIGURATION IN A BACKGROUND SPACE-TIME A.N.Petrov Department of Physics and Astronomy, University of Missouri-Columbia, Columbia, MO 65211, USA; Sternberg Astron. Inst.,Universitetskii pr., 13 Moscow, 119992, RUSSIA; e-mail: [email protected] Abstract In this review paper, general relativity (GR) is presented in the field theoretical form, where gravitational field (metric perturbations) together with other physical fields are propagated in an auxiliary either curved, or flat background space time. Such a reformulation of GR is exact and equivalent to GR in the standard geometrical description. It is actively used for study of theoretical problems and in applications. Conserved currents are constructed on the basis of a symmetrical (with respect to a background metric) total energy-momentum tensor and are expressed through divergences of anti-symmetrical tensor densities (super- potentials). This form connects local properties of perturbations with the academic imagination on the quasi-local nature of the conserved quantities in GR. The gauge invariance is studied, its properties allow to consider the problem of non-localization of energy in GR in exact mathematical expressions. The Friedmann solution for a closed world and the Schwarzschild solution are presented as field configurations in Minkowski space, properties of which are analyzed. An original modification of the field formulation of GR is given by Babak and Grishchuk. Basing on this they have modified GR itself. The resulting theory includes massive terms" describing spin-2 and spin-0 gravitons with non-zero masses. We present and discuss their results. It is shown that all the local weak-field predictions of the massive theory are in agreement with experimental data. Otherwise, the exact non-linear equations of the new theory eliminate the black hole event horizons and replace a permanent power-law expansion of the homogeneous isotropic universe with an oscillator behavior. Locality and Electromagnetic Momentum in Critical Tests of Special Relativity Gianfranco Spavieri, Jesús Quintero, Arturo Sanchez, José Ayazo Centro de Física Fundamental, Universidad de Los Andes,Mérida, 5101-Venezuela ([email protected]). George T. Gillies Department of Mechanical and Aerospace Engineering, University of Virginia, P.O. Box 400746, Charlottesville, Virginia 22904, USA ([email protected]). ABSTRACT In this review of recent tests of special relativity it is shown that the elec-tromagnetic momentum plays a relevant role in various areas of classical and quantum physics. Crucial tests on the locality of Faraday’s law for “open” currents, on a modifed Trouton-Noble experiment, on non conservation of mechanical angular momentum, on the force on the magnetic dipole, and on a reciprocal Rowland.s experiment are outlined.

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Electromagnetic momentum provides a link also between quantum non local effects and light propagation in moving media. Since light waves in moving media behave as matter waves in nonlocal quantum effects, the flow of the medium does affect the phase velocity of light, but not necessarily the momentum of photons. Thus, Fizeau’s experiment is not suitable for testing the addition of velocities of special relativity. A crucial, non-interferometric experiment for the speed of photons in moving media, is described. PACS: 03.30.+p, 03.65.Ta, 42.15.-i KEYWORDS: electromagnetic momentum, Faraday’s law, nonlocality, light in moving media. CORRELATION LEADING TO SPACE-TIME STRUCTURE IN AN ETHER

J. E. Carroll Engineering Department, University of Cambridge, CB2 1PZ E-mail: [email protected] Abstract It is proposed that the ether behaves like a coordinate invariant system. By using the general theory of signals in systems, the paper describes a formalism similar to quantum theory, provides a rationale for Lagrangian methods and also discovers how geometric structures naturally form. From the concepts of convolution and correlation used in linear systems it is shown that the multi-vectors of the ‘Hestenes’ geometric algebra correspond with generalised correlation matrices that link an observer’s view of even and odd properties of incoming signals in the ether system. The analysis shows why three spatial dimensions is the lowest dimensionality to give a homogeneous space. Any fourth dimension, even if it were not time, has to behave differently from the other three spatial dimensions and cannot create a homogeneous space. A more speculative approach suggests that 3+1 space-time is embedded in a 3+3 space-time ether. Elsewhere it has been shown that Maxwell’s equations could be construed as a necessary consequence of this embedding process, while here a Dirac equation with vector potentials emerges from similar assumptions. Mass is created by correlations in a temporal plane that is transverse to the temporal axis. Future prospects for this generalised theory are discussed.

REASONS FOR GRAVITATIONAL MASS AND THE PROBLEM OF QUANTUM GRAVITY Volodymyr Krasnoholovets Institute for Basic Research, 90 East Winds Court, Palm Harbor, FL 34683, USA Abstract The problem of quantum gravity is treated from a radically new viewpoint based on a detailed mathematical analysis of what the constitution of physical space is, which has

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been curried out by Michel Bounias and the author. The approach allows the introduction of the notion of mass as a local deformation of space regarded as a tessellation lattice of founding elements, topological balls, whose size is estimated as the Planck one. The interaction of a moving particle-like deformation with the surrounding lattice of space involves a fractal decomposition process that supports the existence and properties of previously postulated inerton clouds as associated to particles. The cloud of inertons surrounding the particle spreads out to a range υλ/c=Λ from the particle where υ and c are velocities of the particle and light, respectively, and λ is the de Broglie wavelength of the particle. Thus the particle’s inertons return the real sense to the wave ψ-function as the field of inertia of the moving particle. Since inertons transfer fragments of the particle mass, they play also the role of carriers of gravitational properties of the particle. The submicroscopic concept has been verified experimentally, though so far in microscopic and intermediate ranges.

The web site of the program is given in the link below


All information relative to the program is given in the site, including data on the order of books.

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INSTRUCTIONS TO CONTRIBUTORS TO FUTURE VOLUMES Authors who are familiar with the modern ether concepts and who have founded ideas about the nature and the properties of the ether should follow the instructions given below: Review papers which give an overview of the development of the ether concept through time, or present the ideas of a physicist who has significantly contributed to the ether theory, can also be submitted. Papers can be submitted by E-mail.* Papers submitted for publication in the forthcoming volumes of “Ether space-time and cosmology” should obey the following technical instructions: The papers should be preferably submitted in Word format. Papers submitted directly in Tex Latex or PDF can be accepted provided that they strictly obey the following rules and that they are not numbered. The number of pages should not exceed 60 or 65 pages The size of the texts (written part) excluding page numbers should be about 185X135 cm. It is important to respect this format. The fonts should be Times new roman of size 11 point for the main text. Authors are requested not to begin their text with a table of contents. The figures can be placed anywhere, they should be neat with letters easily legible when printed. (A resolution of 300 ppi minimum is generally needed to obtain neat figures and is recommended if possible). Only black and white colours are admitted. The figures should be numbered and briefly explained by a caption. The space above the title should be about 6.5 cm when printed in a format A4 page. The title of the papers should be in Times new roman in boldface and size 14 font. The references should be put at the end of the papers. Papers accepted for publication will be numbered by the editors. Papers formatted in PDF must be embedded by the authors. Papers formatted in Word will be converted in PDF and embedded by the editors. Failure to comply with these instructions in an article may lead to a postponement of its publication. Additional information, if necessary, is given in the web site of the PIRT Meeting.


* Contributions to future volumes should be sent to: Dr J Levy [email protected] Dr M C Duffy [email protected]

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Documents

[Michael C. Duffy, Joseph Levy] Ether Space-time a(BookZZ.org)