“Horsehead nebula”
Apeiron
ETHER SPACE-TIME & COSMOLOGY Volume 2 NEW INSIGHTS INTO A KEY PHYSICAL MEDIUM
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Michael C. Duffy and Joseph Levy Editors
A book dealing with experimental and theoretical studies devoted to the exploration of the modern ether concept, evidence of its reality and implications for modern physics.
Apeiron Montreal
Published by C. Roy Keys Inc. 4405, rue St-Dominique Montreal, Quebec H2W 2B2 Canada http://redshift.vif.com Copyright © 2009 by C. Roy Keys Inc. All rights reserved No parts of this book may be reproduced stored in a retrieval system and transmitted in any form or by any means without the written permission of the copyright owner.
First Published 2009 Library and Archives Canada Cataloguing in Publication Ether space-time and cosmology : new insights into a key physical medium / Michael C. Duffy and Joseph Lévy, editors. Includes bibliographical references. ISBN 978-0-9732911-8-6 1. Ether (Space). 2. Cosmology. 3. Relativity (Physics). 4. Space and time. I. Lévy, Joseph, 1936- II. Duffy, Michael Ciaran QC177.E84 2009
530.11
C2009-900610-3
Illustration, front: Horsehead Nebula by Robert Gendler
http://www.robgendlerastropics.com
ETHER SPACE-TIME & COSMOLOGY Volume 2 New insights into a key physical medium EDITORS M. C. Duffy & J. Levy EDITORIAL BOARD M. Arminjon, Laboratoire “Sols, Solides, Structures, Risques” CNRS & Universites de Grenoble France. J. Carroll, Engineering Department, University of Cambridge, Cambridge, United Kingdom G.T. Gillies, Department of aerospace and mechanical engineering, University of Virginia, USA J. G. Gilson, School of Mathematical Sciences, Queen Mary University of London, Mile End Road, London E14NS V. O. Gladyshev, Moscow Bauman State Technical University, Moscow, Russia. A. L. Kholmetskii, Department of Physics, Belarusian State University, Minsk Belarus A. N. Petrov Department of Physics and Astronomy, University of MissouriColumbia, Columbia, MO 65211, USA; Sternberg Astron. Inst.,Universitetskii pr.,13 Moscow, 119992, RUSSIA; P. Rowlands, University of Liverpool, Liverpool, United Kingdom. F. Selleri, Dipartimento di Fisica, Università di Bari, INFN, Sezione di Bari, Bari, Italy. G. Spavieri, Centro de fisica fundamental, University de los Andes, Mérida, 5101, Venezuela T. Suntola, www.sci.fi/~suntola,Finland L. Székely, Institut for philosophical research of the Hungarian Academy of sciences. Web Site of the Program
http://www.physicsfoundations.org/Ether_spacetime/book.htm
Contents of Volume 2 -1-
Introduction Michael C. Duffy & Joseph Levy -3Relativity in Terms of Measurement and Ether Lajos Jánosssy’s Ether-Based Reformulation of Relativity Theory László Székely Institute for Philosophical Research of the Hungarian Academy of Sciences Postal address: HU-1398 Budapest, Post Box: 594
[email protected] -37AETHER THEORY CLOCK RETARDATION vs. SPECIAL RELATIVITY TIME DILATION
Joseph Levy 4 Square Anatole France, 91250, St Germain lès Corbeil, France E-mail:
[email protected] -53RELATIVITY AND AETHER THEORY A CRUCIAL DISTINCTION
Joseph Levy 4 Square Anatole France, 91250 St Germain lès Corbeil, France E-mail: :
[email protected] -67The Dynamic Universe Zero-energy balance restores absolute time and space Tuomo Suntola Vasamatie 25, 02630 Espoo, Finland E-mail:
[email protected] -135DYNAMICAL 3-SPACE: A REVIEW
Reginald T. Cahill School of Chemistry, Physics and Earth Sciences, Flinders University, Adelaide 5001, Australia
-201Relativistic physics from paradoxes to good sense - 1 F. Selleri Dipartimento di Fisica, Università di Bari INFN, Sezione di Bari
-267Photon-like solutions of Maxwell’s equations John Carroll, Joseph Beals IV, Ruth Thompson Centre for Advanced Photonics and Electronics, Engineering Department, University of Cambridge, 9 JJ Thomson Avenue, Cambridge, CB3 0FA, UK
-323Cosmological Coincidence and Dark Mass Problems in Einstein Universe and Friedman Dust Universe with Einstein’s Lambda James G. Gilson
[email protected] School of Mathematical Sciences Queen Mary University of London Mile End Road London E14NS
-355THE SPACE-CURVATURE THEORY OF MATTER AND ETHER 1870-1920
James E. Beichler P.O. Box 624, Belpre, Ohio, 45714, USA
[email protected] -417On microscopic interpretation of the phenomena predicted by the formalism of general relativity Volodymyr Krasnoholovets Indra Scientific bvba, Square du Solbosch 26 B-1050, Brussels, Belgium* E-mail address:
[email protected] -433CAN PHYSICS LAWS BE DERIVED FROM MONOGENIC FUNCTIONS
José B. Almeida Universidade do Minho, Physics Department Braga, Portugal, e-mail:
[email protected] -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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Relativity in Terms of Measurement and Ether Lajos Jánosssy’s Ether-Based Reformulation of Relativity Theory László Székely Institute for Philosophical Research of the Hungarian Academy of Sciences Postal address: HU-1398 Budapest, Post Box: 594
[email protected] Abstract In his monograph Theory of Relativity Based on Physical Reality, Hungarian physicist Lajos Jánossy develops the complete Einsteinian formalism of relativity theory by analysing the process of measurement, the systems of measures created in this process and experimental data expressed in terms of measures. He demonstrates that based on a simple principle (which he calls the Lorentz principle) and its generalization the whole formalism of the original theory may be developed in conformity with the notions of common sense without mathematizing physical reality, so that the new way of development is of the same heuristic power as the original one. His analysis makes it clear that the allegedly revolutionary new notions of space and time follow not from physical experiences but from Einstein’s positivist philosophical commitments. Having established the place and role of a privileged (but not absolute) reference system, at the second level of his theory Jánossy connects this system to the carrier of electromagnetic phenomena which he also assumes to be the carrier of the gravitational and other physical fields. Although he uses the term ‘ether’, he explicitly rejects the old theories of this entity and attributes to it dynamic properties. In the last section of the paper Einstein’s and Jánossy’s ether concepts are compared and it is argued that despite the parallelism between the two concepts, from Jánossy’s point of view Einstein’s ether is too mathematical to cure the inverted relation between mathematics and physics characteristic for Einstein’s relativity. Key words: relativity, ether, propagation of light, privileged reference system, space-time, measurement, ideal solid rod, ideal clock, common sense in physics, mathematics in physics, physical reality, Einstein, Lorentz, Jánossy
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1. Introduction In Physical Relativity, a monograph published by Clarendon Press in 2005, Harvey Brown criticizes the received view of Einstein’s theory and argues for a physical interpretation of relativistic phenomena. [Brown 2005] Both Brown’s book and the regular conferences on the interpretations of relativity theory organized by Michel Duffy [Duffy 1988, 1990 ….2006] clearly indicate that the long tradition of considering the original, Einsteinian-Minkowskian notion of relativity theory too mathematical and claiming that it blurs (or even turns into its opposite) the epistemological relation between mathematics and physics is alive even today, more than 100 years after Einstein’s famous paper. In the introduction to his book Brown mentions the Hungarian physicist Lajos Jánossy as one of his forerunners inspiring his ideas. [Brown 2005, vii.] Jánossy was an important figure in the tradition of alternative interpretations of Einstein’s theory, who (following Lorentz’s ideas) elaborated a comprehensive alternative (“physical”) relativity. He, along with American Herbert Ives (who belonged to a former generation of physicists) and Prokhovnik (a contemporary of Jánossy) may be considered as one of the classics of the field. However, while on the basis of personal communications it seems that his work on relativity theory was well known by those who did research in the topic in the last decades, he (in contrast with Ives and Prokhovnik) is only rarely cited in the literature. (M. Duffy mentions Jánossy’s work in his recent paper [Duffy 2008] and Bell in his famous study How Teach Relativity? also expresses his appreciation for Jánossy’s contribution to the topic [Bell 1976].) The aim of this paper is to give a brief review of Jánossy’s reformulation of relativity theory, which deserves more recognition than it has received until now. 2. Lajos Jánossy’s career Lajos Jánossy was born in Mátyásföld (then a village near Budapest, now part of the Hungarian capital) in 1912. His father Imre Jánossy was an astronomer who died relatively young in 1920. After the death of her husband, his mother, Gertrud Borstrieber (a mathematician belonging to the first generation of Hungarian women with a university degree) married the Hungarian philosopher George Lukács, who was considered by the French philosopher Lucian Goldman the first representative of the existentialist philosophy, but who later gave up his youthful enthusiasm for Kierkegaard and became a famous and highly controversial Marxist philosopher of the 20th century, oscillating permanently between communist movement discipline and sovereign philosophical thought and causing many a disturbance for the party leadership. After the fall of the Hungarian Soviet Republic in 1919 Lajos Jánossy’s family moved to Austria and later to Berlin. Instead of following his stepfather in politics or philosophy, Jánossy became a physicist. He studies physics at the Humbold University in Berlin where he was a student of Edwin Schrödinger whose
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metatheoretical considerations on physics had a determinative influence on him. In the 1930s Jánossy became also a university professor and read physics (and especially relativity theory) at Manchester University (while his stepfather left Hitler’s Berlin for Stalin’s Moscow and lived there with his political and moral compromises). His main research field being cosmic radiations, he became an internationally respected scientist in the field, and his monograph on the topic belongs to the basic literature on the subject [Jánossy 1948, 1950]. After Word War Two George Lukács returned to Hungary and in 1950 Lajos Jánossy (then a professor at the Institute for Advanced Studies in Dublin and a colleague of his former professor, Schrödinger) followed him. While his stepfather was never a “pet” (or with the good German word a “Liebling”) of the communist party, party leaders needed his international respect, as well as Lajos Jánossy’s scientific knowledge. So the latter became head of the Central Institute for Physical Research, a grand new research institute established on a Soviet model. It is generally held that Einstein’s theory of relativity was deemed by the official Soviet ideologists as a bourgeois theory, so Jánossy’s criticism of the Einsteinian notion of relativity may appear in this context as a version of the Soviet criticism of the theory, but it is not the case. On the one hand, although several attempts were made in the Soviet Union to discredit relativity theory as a prototype of false, idealistic physics, and at the turn of the forties to the fifties of the last century a fierce campaign was waged against the theory, the attempts never resulted in its official denunciation. On the contrary, after the death of Stalin, Einstein‘s Soviet followers won the debate and Einstein’s theory came to be glorified as a true dialectical theory, which as such fully corresponds to Marxism-Leninism. [See e.g. Graham 1972, 111-138; Székely 1987] On the other hand, and quite importantly, Lajos Jánossy had never taken part in the antirelativistic campaign. The greater part of his critical considerations on relativity theory was published in a period when official Soviet ideology endorsed Einstein. Hence, beside the criticism his concept received from orthodox Einsteinian physicists, Jánossy’s notion of the relativity theory also became a target of official philosophers of the Soviet block. Although the Hungarian Academy Press undertook the publication of his comprehensive work „Theory of Relativity Based on Physical Reality” [Jánossy 1971], in his last years he was considered by the orthodox Einsteinian physicists who were then dominating the Hungarian physics scene as an anti-relativist dinosaur and (while formally preserving his university position) he was gradually displaced from Hungarian scientific life. He died in 1978. Whereas the ideological, political and sociological contexts of Lajos Jánossy’s scientific work would also offer interesting topics, this contribution will be restricted to reviewing his concept of the theory of relativity only from the point of view of physics and the philosophy of science.
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3. The metatheoretical foundation 3.1. The relation between mathematics and physics and the norm of common sense As indicated in the title of his monograph, Lajos Jánossy characterizes his notion of relativity theory as a theory based on physical reality. This title expresses both a critical and a confirmative aspect. On the one hand, Jánossy argues that the Einsteinian theory is not based on physical reality: while it is an effective mathematical tool for handling the results of measurements and for making predictions, it does not provide an appropriate theory of physical reality. On the other hand, he affirms that the mathematical formulas of Einstein’s theory are correct in the sense that they are in correspondence with observation and empirical data and are able to give correct predictions about the behaviour of physical reality. Of course, Jánossy sees clearly that what Einstein offers us is not only mere mathematics but a definite physical theory. He insists, however, that Einstein turns the relation between mathematics and physics into its opposite: in his view the German physicist projects mathematical formulae into the physical world and in this way constructs physical reality by hypostatisation of mathematical ideas. Consequently in the context of his criticism the so called “geometrization of physics” which is often praised as a great achievement of relativity theory appears as a result of hypostatisation and Jánossy focuses his criticism on this element of the theory: “The theory of relativity in its original formulation is certainly not a mere attempt to describe phenomena by suitable mathematical expressions – the theory is a far reaching attempt to give a theory of space and time. Our criticism of the theory is just connected with this latter feature. We think that the theory reflects correctly certain general physical laws, but these laws – in our opinion – have nothing to do with the “general structure of space and time”. Therefore our attempt is to give a physical interpretation of relativistic formulae, which is different from old one.” [Jánossy 1971, 13] But how do we know that the view Einstein offers of the physical world is inappropriate? An incorrect methodology does not necessarily imply the incorrectness of the theory. Does the theory have any independent, non-methodological features which might make it problematic? In answering this question, Lajos Jánossy represents a view which is typical in the criticism of Einstein’s relativity and which can be characterized as “common sense criticism”.
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“I got acquainted with the theory of relativity at a comparatively early age – I read the famous popular book written by Einstein. Reading the latter I had difficulties with some of Einstein’s concepts: however, having been young and enthusiastic, I convinced myself in the end that I could understand those concepts – to prove this I tried to explain the theory to everybody who was interested. In the course of such attempts I learned the ‘language of relativity’ and I gradually ‘got used’ to the theory. …. Many years later I read several years in succession a course of physics at the university of Manchester. My course contained also the special theory of relativity. As the years went on I developed a technique of presenting the subject so that in the end I could convince my students that they really understood the theory. However, as my technique presenting the theory improved, my own belief in the adequateness of the concepts vanished. In the end I became convinced that from the philosophical point of view the concepts had to be changed. Since about 1950 I have struggled with the problem of the reformulation of the theory and the results of my deliberations are found in this volume.” [Jánossy 1971, 14] As Descartes’s narrative about his schools and education in his Discours de la Methode (Discourse on the Method) expresses a radical criticism of the philosophical views of the epoch and his personal style functions as endorsement and authentication of the criticism, here, in Jánossy’s reminiscence we also encounter a radical philosophical criticism. Jánossy challenges the generally received view that relativity theory requires us to give up our common sense terms. We should not be mislead, he argues, but recognize that there is really something disturbing in Einstein’s theory and the correct attitude is not to suppress this disturbing factor by blaming our common sense for incapacity to grasp physical reality but to face and eliminate it by reformulating the theory. In other writings he is more sanguine and characterizes the received attitude of modern physics to common sense as a cult of irrationality, in the context of which contradiction with common sense becomes a virtue and the scientific character of a theoretical claim is measured by the extent of its absurdity. Rejecting this approach, he insists that “[a] scientific way of thinking cannot be but the refinement, deepening and further development of everyday thought” and that “the whole complex of the theory of relativity can be built up by means of natural methods in conformity with everyday thought”. [Jánossy and Elek 1963, 9, the original is in Hungarian] (Jánossy, influenced by the philosophy of his stepfather, prefers the term “everyday thought” to “common sense” but in his argument the former functionally corresponds to the latter.)
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To summarize, the metatheoretical foundation of the criticism and reformulation of relativity theory by Jánossy consists of two interlaced moments, namely, the priority of physics regarding the mathematical formalism and the conscious acceptance of the terms of common sense as a norm for theory construction. Whereas these moments are common to criticisms of Einstein’s theory, the metatheoretical foundation of the criticism is only seldom formulated so explicitly and definitely as in his case, and this is especially true regarding the role of common sense. The requirement of conformity with the basic notions of common sense as a norm for theory construction emphasized so resolutely by Jánossy may be regarded as Jánossy’s thesis and considered as one of the most important metatheoretical theses concerning modern physics. [Székely 1987; Székely 1988] 3.2. Measures, measurement and relativity theory Metatheoretical norms and principles, however excellent, cannot have any significance if one cannot find the way of their correct application in concrete theories. Jánossy’s main achievement regarding relativity theory is not simply the formulation of the metatheoretical foundation of the criticism but a complete and consistent reformulation of the theory in physical and mathematical terms. In the following parts of our paper Jánossy’s version of relativity theory will be often contrasted with the Einsteinian one. To avoid misunderstandings, it is important to emphasize that in doing so we will always use the terms „Einstein’s theory” or “Einsteinian relativity” in the sense of the version of the theory as it was presented in Einstein’s original (physical) papers and as it is generally taught at universities and presented in textbooks. That is, in our usage the term „Einstein’s theory” will not include any of the metatheoretical and physical reflections made by the German physicist after the publication of the theory. The relation of Jánossy’s notion of relativity theory to Einstein’s subsequent, out-of-theory reflections (which cast a new light on his original formulation of the theory and leave room for a reading which might suggest its reformulation in the direction represented by Jánossy) will be considered at the end of this paper. As a consequence of the heated ideological debates, late in his scientific career Jánossy abandoned philosophical categories regarding relativity theory. Thus in his comprehensive monograph “Theory of Relativity Based on Physical Relativity” published in 1971 we cannot find even such ideologically neutral categories as “common sense” or “everyday thought”. Instead of using philosophical categories he identifies the indicated disturbing aspect of Einstein’s theory in terms of measurement theory. According to him, “ [i]n our approach of physics in general and the theory of relativity in particular we think it very important always to remember that we are dealing with objective physical quantities
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and that we attempt to describe the latter in terms of measures.” [Jánossy 1971, 15] Furthermore, “ [a]n objective physical process develops according to its own laws and it can be described in arbitrary measures.” [Jánossy 1971, 14] Distinguishing measures from things measured, Jánossy definitely commits himself to the traditional concept of physical reality, according to which there exists something „out there” with its own laws and thus he rejects the positivist approach. But emphasizing the arbitrariness of the measures used by physics, he also opposes naive, metaphysical realism which maintains that the investigated objects and the theoretical entities directly correspond to each other (or – in a weaker version – considers the latter the approximations or conceptual pictures of the formers). In his concept physical quantities as characteristics of physical entities are outside of physical theories, while measures (and theoretical construction, so coordinate systems built up of these measures) are the representations of these quantities which physicists can chose arbitrarily. [Jánossy 1971, 72] Consequently, in Jánossy’s interpretation space and time coordinates, as well as their transformations lose the mystical character conferred them by relativity theory: “We may write x=r,t for a four-coordinate of an event. Changing from one system of reference to another we can introduce transformed coordinates x’=f(x) (1) where f(x) is some reversible four-function of its variable x. If the coordinates x are suitable to describe events, then the transformed coordinates are also suitable. Introducing particular measures x or x’ for events we give some kind of names to the events with the help of which we recognize them. … The fact that a transformation type (1) mixes the measures of time and space coordinates does not seem to be of particular importance and it does not imply any properties of space and time.” [Jánossy 1971, 14] This view of physical quantities and their measures is open to contention. However, it is based on acceptable and justified metatheoretical postulates well established in the history of physics which may serve as a foundation for physical theories. Furthermore, it is clear that these postulates contradict the Machianpositivist philosophical background of Einstein’s notion of relativity and thus their definite formulation by Jánossy makes it evident that that notion is not neutral from
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the point of view of physics: it does not follow from the nature of the physical world but rather is a consequence of Einstein’s metatheoretical commitments. But if measures are only names or signs arbitrarily chosen by physicists, how is it possible to know anything about physical reality that is supposed to exist outside physics, a system of human theories? Lajos Jánossy answers this problem by introducing the concept of distinguished measures. While a physical quantity can be described by an infinite number of systems of measures, the majority of the possible descriptions do not contain any information about the quantity in question. Distinguished measures are particular classes of measures which “reflect clearly certain properties of quantities” [Jánossy 1971, 72] Therefore, one of the most important tasks of theoretical research is to find distinguished measures for the quantities under scrutiny, that is, to attempt to find for the description of particular quantities numbers which reflect adequately certain physical properties [ibid.]. To elucidate the concept in more detail, in Chapter III of his monograph Jánossy analyses the measurement of electric charges and then (taking into account that relativity theory is strongly connected to the so called space and time coordinates) in Chapter IV he works out distinguished measures for space and time. According to his analysis distinguished measures are characterized by the fact that in general both their sum and product (or in certain special cases at least their sum) express significant physical quantities, that is, their sum and product also appear in our measurements and/or in the established physical laws. For example, the sum of the usual measures of two electric charges E1 and E2 (say measures e1 and e2) will be equal to the measure we receive measuring the joint charge, while the product of e1 and e2 appears in Columb’s Law. (In fact, Jánossy designates physical quantities with Gothic letters while their measures with Roman letters, so he designates a physical charge with a Gothic e while its measures with a Roman e. For technical reasons we do not follow his notation here.) A physicist used to the usual notation and language of physics may find this terminology rather curious, since physical texts do not usually distinguish the charge and its measure but designate both by the same symbol (say e). However, in the metatheoretical context established by Jánossy it is clear that the charges as objective physical entities do not determine directly the measures to be constructed in the process of measurement and hence it is not at all evident that the measure of joint charges should be the sum of the measures of the two original ones. In Jánossy’s words, “[i]n practice there seems to be no point in introducing nonadditive scales for quantities if there is a possibility of introducing also additive representations. It must be emphasized, however, that it is not trivial that for certain quantities additive measures can be introduced. Whether or not such measures can
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be introduced in a particular case is a question which can be decided experimentally….” [Jánossy 1971, 78]. Of course, the question of measurement is a very complex topic and in his monograph on relativity theory Jánossy could only briefly outline his respective ideas. A more detailed presentation can be found in his earlier monograph Theory and Practice of Evaluation of Measurements which contains a comprehensive presentation of his theory of measurement. That book should be consulted by those interested in this aspect of Jánossy’s theory. [Jánossy 1965] What follows is a brief sketch of Jánossy’s reformulation of relativity theory based on the metatheoretical commitments outlined above. We will attempt to reproduce the logic and the conceptual structure of his theory and will set aside the technical-mathematical details that are essentially the same as the well known textbook formulation of Maxwellian electrodynamics, the formulae of Lorentz transformation and the Einsteinian formalism of the special and general theory of relativity. 3.2. Measures and relativity 3.2.1. Measures of space and time based on rigid rods and physical laws. The definition of ideal clocks. While in his famous paper Einstein firstly introduces a scale of length with the help of rigid rods and then “defines time” (ie, in Jánossy’s terms, “introduces distinguished temporal measures“) with the help of clocks and light signals and so he establishes a “hybrid” scale of space and time, Jánossy separates the rigid rod method from the light signal method and introduces two independent systems of measures: one based on rigid rods, another on light signals. As we have seen, for Jánossy it is not at all trivial that additive length measures can be introduced. The use of additive length scales in everyday practice is based on the fact that with the help of rods considered in every day life as “solid” additive length measures can be obtained. According to Jánossy, science can introduce the term of ideal solid rods only because we are given this experience and he defines a rod to be an ideal solid rod if with its help an additive scale of length can be obtained. [Jánossy 1971, 79] On the other hand, Jánossy emphasizes that with the help of periodical processes (such as mechanical clocks, planetary motions etc) we can complete our system of length measures to set up a combined system of length and temporal measures in terms of which physical phenomena obey certain rules. As measures in general, temporal measures in particular can be obtained in several ways and there is no a priori guarantee that these ways will all result in the same measures (or that measures arrived at in different ways will coincide). However, considering that the
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aim of physics is to discover rules in the behaviour of the physical world and formulate them as physical laws, from the point of view of science it is rational to attempt to complete our length scale with a temporal scale in such a way that certain fundamental and in the practice well confirmed laws, for example, Newton’s first law be fulfilled. At first sight, perhaps, this approach may seem to be logically circular, since physical rules may appear only if we have already a joint scale of length and time, while Jánossy want to complete the length scale with a temporal scale with the help of already known laws. Is this not a vicious circle? Taking a closer look at the issue reveals that the approach is correct. In the history of physics we are given physical rules (for example, Newton’s first law) which seem to work if we use our everyday length and time measures or measures established in the history of physics. These rules appear in terms of measures, which are intuitive and without reflection (or are based on metaphysical commitments as for example in Newton’s case) and therefore it cannot be excluded that they are to a certain extent consequences of our choice of measures. To enlighten the nature of these rules we need an a priori analysis of the applied measures and in this analysis (while suspending the validity of the concerned rules regarding physical reality) we may introduce a hypothetical world in which these rules are assumed to be fulfilled, and Jánossy follows this methodology. Thus we may assume a region where Newton’s first law is valid in terms of a given (but yet unknown) system of measures. Provided that we already have a length scale, in such a region we no longer need Einstein’s radar method to synchronize clocks: it will suffice to observe the motion of free particles and to adjust the local measures of time showed on the local clocks in such a way that Newton’s first law be fulfilled. (To observe the path of a particle we need not use light signals: every observer can measure with the help of his own clock and make a note of the time when his own position is crossed by a moving particle and then the notes can be collected and analysed in order to synchronize the clocks.) Exploiting this a priori possibility, Jánossy introduces the term of „ideal clock”. According to his definition a clock is ideal when it gives immediately (without correction) the distinguished temporal measures based on Newton’s first law. [Jánossy 1971, 95-96] The rate of an ideal clock is by definition constant and our physical practice definitely shows that there are regions in the real world which allow us to introduce good approximations of a system of measures based on ideal solids and ideal clocks. (Otherwise Newton’s first law would not be applicable in practice.) Similarly, we may introduce temporal measures using planetary motions or the rotation of the Earth around its axis and assuming the validity of the law of gravitation and it is also possible to use atoms as clocks and taking into account the physical theories of atoms. Of course, it is not evident that all these scales will correspond to the first, mechanical or ‘ideal’ temporal scale; neither is it evident that the non-mechanical (planetary, sideric or atomic) scales will be adjustable to each
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other. In this respect Jánossy’s definition of ideal clocks is a metatheoretical norm requesting a physical explanation in any case when an applied time scale deviates from the ideal one. (Incidentally, since Newton’s first law is deducible from Leibniz’s principle of sufficient reason, Jánossy’s definition of ideal clocks may be deduced from this fundamental Leibnizian thesis. On the other hand, it can be also shown that the Einsteinian version of special relativity does not fulfil the Leibnizian principle. Thus Jánossy’s version of relativity theory - despite its empirical orientation - can be seen as a reformulation of the original Einsteinian theory, with the aim of satisfying Leibnitz’s principle. Furthermore, Jánossy’s method of definitions of ideal solid rods and ideal clocks, a beautiful example of the application of everyday experiences in physics, follows – unconsciously – the logic of the so called “hermeneutic circle” emphasized by Heideggers’ philosophy and indicates how promising a possible Heideggerian metatheory of physics may be.) 3.2.2. Measures by radar method without rods Jánossy also shows that it is possible to attempt to introduce length and time scales using only light signals, provided that we assume that light is propagated isotropically and with a constant velocity relative to a given reference system, say K. It is clear that similarly to the rigid rod scale, we do not have any a priori guarantee of success in this case either. It is a matter of practice whether a coherent system of space and time coordinates can be constructed in such a way and if we succeed and a system of coordinates introduced by this method passes the test of coherence, then this fact „can be taken to support the hypothesis about the mode of propagation of light in K”. [Jánossy 1971, 99] The introduction of such a scale follows the same logic as the rod scale without the radar method: first an ideal region is assumed where light is propagated isotropically and the measures are defined for this ideal region, then, as the second step, experience will show whether these measures can or cannot be applied in the real world. 4. Lorentz transformations and Jánossy’s theorem 4.1. Lorentz trasformations as transformations of measures Applying the conceptual basis introduced above, Jánossy demonstrates that: if there is a system of coherent measures M of length and time in terms of which light appears to be propagated isotropically and with the velocity c relative to a reference system S, then there exists a group of mathematical transformations of that system of measures with the following characteristic: - each members of the group transforms the system of measures M into another system of measures M’ in whose terms light appears to be propagated isotropically
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and with the velocity c relative to another reference system S’ which is in rectilinear and even motion relative to the original reference system S; - vice versa, for any reference system S’ in rectilinear and even motion relative to the original reference system S there exist a member of the group of transformation above, which transforms the system of measures M into a system of measures M’ so that in the reference system S’ light will appear to be propagated isotropically in terms of M’. [Jánossy 1971, 100-105] Anyone familiar with relativity theory will see that the group of transformations which Jánossy found is the well known group of the Lorentz transformations. That is, he did not discover transformation of a new kind but deduced the famous ones in a new way different from both the Einsteinian and the Lorentzian deductions. However, what is important for us is not simply the new deduction but the new meaning of the transformations. Whereas in Einstein Lorentz transformations are deduced as transformations which connects inertial reference systems so that Einstein’s two axioms be satisfied, in Jánossy they emerge in an investigation of the propagation of light in terms of various systems of measures without referring to the concept of inertia and their existence are stated in the form of an a piori, mathematical theorem. We will refer to this theorem as “Jánossy’s theorem” and (following his terminology) call the reference systems relative to which light appears to be propagated isotropically in terms of a particular system of measures “Lorentz systems”. Notice, that Jánossy’s theorem is not about inertial systems: it is valid independently of whether Lorentz systems are inertial or not. 4.2. The analysis of Jánossy’s theorem Jánossy’s theorem imposes two a priori constraints upon physical reality. A) On the one hand, if rods and clocks are never deformed when in motion with respect to any Lorentz system (that is they preserve their shape and pace), then i) (on simple geometrical grounds) there will be only one Lorentz system in which the system’s own Lorentz measures (that is, the measures in terms of which light appears to be propagated isotropically relative to the system) and measures based on rods and clocks without light signals will coincide; consequently ii) the relative velocity of any other Lorentz system with respect to this special system will be determinable with the help of rods and clocks and light signals, since in terms of measures established with the help of these rods and clocks light will not appear to be propagated isotropically relative to these systems. (This simply follows from the fact that Lorentz measures are connected with Lorentz transformations which change the measures of length and time, while the unchanged rods and clocks will establish the same system of measures independently of their motion relative to any Lorentz system.)
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B) On the other hand, if we observe that Lorentz measures and measures based on rods and clocks co-moving with the systems will always coincide, then this observation will indicate that i) there is a definite Lorentz system in physical reality which may be called as the basic system, and ii) rods and clocks moving relatively to this basic system suffer deformation according to the formulae of the Lorentz transformations. The observed relativistic effects (that is, the relativistic contraction of lengths and the slowing down of physical processes according to Lorentz’s formulae) show that in physical reality the second possibility is the case, thus on the basis of Jánossy’s theorem as an a priori theorem these effects necessarily imply the existence of a basic physical system in which rods and clocks at rest are not deformed, while in motion relative to this system they suffer deformation according to Lorentz’s formulae. 4.3. The hidden epistemological and logical background of Einstein’s special theory The a priori analysis of Jánossy’s theorem makes clear that Einstein’s special theory of relativity is based on two mathematical “boundary-conditions”. On the one hand, special relativity is only possible because Jánossy’s theorem is valid, that is, Lorentz transformations exist and they transform a system of measures in term of which light appears to be propagated isotropically into another system of measures with the same characteristic. On the other hand, the Einsteinian version of the theory, that is, the version in which – in contrast to the implication of Jánossy’s theorem –, the existence of any privileged systems is rejected, can only escape logical contradiction because Einstein implicitly rejects that the spatial relations of the physical entities of a given region form a definite, consistent spatial configuration. To enlighten the latter moment of Einstein’s theory, let us recall that the claim about the isotropic propagation of light in any inertial system is perhaps the most paradoxical ingredient of the Einsteinian theory of special relativity. Namely, if a physical effect is propagated in a given reference system isotropically, then it cannot (on geometrical grounds) be propagated in a similar way in other systems moving rectilinearly and evenly with respect to the former. How is it possible that Einstein succeeded in working out a consistent theory incorporating this geometrically impossible characteristic of the propagation of light? Jánossy’s theorem helps to explore the hidden conceptual background which makes possible for Einstein to avoid the contradiction. Namely, geometry excludes the simultaneous isotropic propagation of light relative two different (physical) reference systems in motion at a constant velocity relative to each other only if the following two premises are fulfilled: i) the space and time relations of the physical entities of the concerned region define a common, definite space in which the investigated systems move, and
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ii) length and time are measured in both systems with the same measures. Consequently, we can construct a consistent physical theory in which light appears to be propagated isotropocilly with respect to two different reference systems in motion relative to each other only if we reject at least one of these premises. Now, Jánossy explains relativistic phenomena with the help of the assumption that rods and clocks in motion relative to the basic system are deformed according to Lorentz’s formulae. Consequently, in the case of two Lorentz systems in motion at different rate relative to the basic system these measuring tools will suffer different deformations and so the systems of measures introduced with their help will be also different. Thus in Jánossy’s conceptual framework it is premise ii) that is not fulfilled, and the function of the assumed deformations of rods and clocks are exactly to give an explanation of the change of measures that takes place despite the use of the same rods and clocks when we change the systems. Of course this explanation – as any Lorentzian kind approach – breaks the ontological symmetry of the relativistic effects: in its context the contraction of rods observed from a system moving faster relative to the basic system than the observed rods is only an apparent phenomenon since the latter suffer smaller contraction than the measuring rod of the observer and hence in reality they are longer than the observer’s rod. Since Einstein’s theory excludes the existence of any basic system and assumes relativistic effects to be symmetric that does not allow to speak about real, physical deformations of measuring tools, his theory can be consistent only if the first premise is rejected. However, if we assume that physical entities are definite entities with definite spatial relations, then these relations will form a definite physical space in which these entities exist and move. So the rejection of premise i) amounts to rejecting that physical entities have definite spatial relations independently of the applied measures and in Einstein’s special relativity this really is the case. Due to Einstein’s neopositivist attitude, in his theory physical entities exist and move not in a common physical space but inside relative co-ordinate spaces, that is, (using Jánossy’s term) inside spaces of different systems of measures and it can’t be introduced any common system of spatial relations that could be independent of our measures. Put differently, Einstein’s axiom of special relativity by exclusion of the existence of any privileged reference system also excludes the possibility of any definite physical configuration formed by the spatial relations of the physical entities, and so, in the words of Hungarian philosopher Melchior Palágyi, it fragments physical reality into an infinite number of reference systems. [Palágyi 1914, 59-60; see also: Székely 1996]
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5. Ether and Lorentz principle 5.1. Ether and Lorentz deformations Jánossy calls the deformations of clocks and rods in motion relative to the basic system “Lorentz deformations”. Taking into account that these deformations emerge when rods and clocks are in motion with respect to the basic system, it is natural to assume that the basic system is connected to some physical entity (such as a background physical field) and the deformation is somehow caused by this entity. Furthermore, considering that the classical concepts of the ether have a function similar to that of this entity, the latter can be called “ether” without any commitment to the notions of the classical ether theories. However, it is not necessary to use this term. What is important is only that if one distinguishes measures as representations from the measured things as parts of physical reality, then the observed relativistic phenomena discussed in the special theory of relativity will imply the existence of such a background entity as well as the Lorentz deformations of clocks and rods in motion relative to it. Now Jánossy identifies this background entity with the electromagnetic ether which he introduces on common sense grounds. According to him “From Maxwell’s theory it follows that light in particular and all electromagnetic action in general is propagated with a velocity c= c’, where c’ is the critical velocity. …. The question cannot be avoided relative to what are electromagnetic waves propagated with velocity c?….A simple answer to this question could be obtained claiming that light is propagated with the velocity c relative to its source. The latter assumption contradicts, however, the well established theory of Maxwell and seems also to be contradicted directly by experiments…. Electromagnetic perturbation once it has left its source is propagated thus with a velocity c independently of how the perturbation comes about. The only reasonable interpretation of this is to assume that the perturbation moves with a velocity c relative to its carrier. The carries may be denoted using Maxwell’s terminology, ether. We shall in accord with the ideas of Maxwell also assume that light is propagated with a velocity c relative to the ether.” [Jánossy 1971, 48] That is, for him the existence of a basic system is granted in advance, independently of the Lorentz transformation and an analysis of relativistic phenomena, on the basis of Maxwell’s theory. So the logic of his presentation does not follow strictly our a priori analysis above. We have made a small change in the
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presentation of his ideas and deduced the existence of a basic system from the observed relativistic phenomena with the help of his theorem just to indicate the heuristic power of his approach. 5.2. The Lorentz principle Relying on the null results of the experiments aiming to determine the translation velocity of the Earth relative to the ether (such as the Michelson-Morley and the Kennedy-Thorndike experiments) and on the observation of the perpendicular Doppler effect, Jánossy finds it reasonable to introduce the following general principle which he calls “the Lorentz principle”: “The law of nature is such that provided S is a real physical system, then the Lorentz deformed systems S* are possible systems obeying the same laws as S.” [Jánossy 1971, 120] It is evident that this is a reformulation of Einstein’s principle of special relativity in physical terms, implying the same observational predictions and the same modifications of classical physics as Einstein’s principle does. It is a frequently repeated argument against Lorentzian-type interpretations that they are ad hoc in contrast to Einstein’s beautiful axiomatic theory. Now Jánossy has definitely showed that this is not the case. On the one hand, the Lorentz transformation can be deduced in a train of thought of simple considerations about measurements and measures. On the other hand, the Lorentz principle as a simple idea based on observational data completely substitutes Einstein’s axiom of the equivalence of inertial systems and predicts the relativistic phenomena in a similarly simple and coherent way as Einstein’s axiom does. Furthermore, if we want to compare the two approaches using the term “ad hoc”, we must conclude that it is Einstein’s theory and not Jánossy’s reformulation that is ad hoc in the particular sense that it states the equivalence of inertial systems as an unexplainable and non-deducible axiom, while Jánossy’s Lorentz principle and the concept of Lorentz deformations are based on an analysis of physical measurement and measures, a problem that Einstein’s positivist attitude prevents even to address. 6. Jánossy’s general relativity Jánossy does not stop at the reformulation of the special theory of relativity, but also reconsiders the general one. His notion of general relativity is based on two ideas: i) the concept of measures applied in the reformulation of the special theory and the term of rigid bodies defined with the help of these measures;
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ii) the generalization of the Lorentz principle originally introduced by him in the context of the special theory. 6.1. Ideal solid rods and the exclusion of the space-time metaphysics of general relativity As we have seen, Jánossy defines ideal solid rods as rods with the help of which a consistent additive length scale may be obtained. In a further step Jánossy introduces a system of space co-ordinate vectors (that is, the usual space coordinate system) with the help of measures determined by rigid measuring rods and defines the distance of two points in this co-ordinate space by the formula (Ri-Rk)G(Ri-Rk)=Rik2 (formula F) where Ri and Rk are the coordinate vectors of points Pi and Pk, G is a positive definite symmetric matrix, and Rik is the distance. It is clear that according to this definition for any N+1 points P0, P1 ….. P(N+1) we are given N(N+1)/2 equations for the 3xN components of the co-ordinate vectors, thus we will have an overdetermined system of equations which does not have necessary solutions. Jánossy applies this fact to an extended definition of ideal solid rods: if in a system of space co-ordinates which has been established by measuring rods the distance formula above will work for any number of points (that is, the system of equations defined by the formula F for any points P0, P1…Pk …..Pn will have solutions), then we may consider our rods to behave as ideal solid rods. [Jánossy 1971, 81] Consequently, if we observe that the system of equations according to the formula F does not have a solution at any set of points, then this fact will indicate that the rods we have used in the construction of our co-ordinate system have been deformed in the process of measurement (that is, they are not ideal solid rods). It is clear that this extended notion of ideal solid rods introduced by Jánossy aims to exclude any word usage about non-Euclidean physical spaces and is in full agreement with Poincare’s idea of the relation of physics to geometry. Our hypothesis is neither on geometry nor on physics in itself but on geometry and physics together, Poincaré emphasizes, and Jánossy commits himself to a connection of physics and geometry in which the structure of space (at least in the Einstenian sense) loses its meaning. If formula F does not work consistently (that is, our co-ordinate space is not Euclidean), that will only inform us about the behaviour of measuring rods but will have nothing to do with the “structure of physical space”: “The above statements can also be formulated in another way. If the measured distances rik between the points of a set can be expressed by a quadratic form (F), then one might conclude the space in which the points are situated is ‘Euclidean’. Or if no consistent co-ordinate measures can be obtained one might conclude that the space is ‘non- Euclidean‘.
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We do not think, however, that such a conclusion has any meaning. The fact that the overdetermined system (F) poses solutions Rk k=0,1, 2 …… n seems to us to reflect upon the method of measurement of the distances rik and in particular upon the measuring rods used. Roughly speaking one may conclude from the consistency of measures that the measuring rods made use of are behaving like rigid bodies, i.e. if the measuring rods are turned or shifted they do not change their length.” [Jánossy 1971, 86. Italics mine: Sz. L.] It is to be noted that this conceptual scheme (whereas it radically opposes Einstein’s view of the relation between geometry and experience presented in his paper of 1921 [Einstein, 1921]) is more than a clever trick to prevent any talk about non-Euclidean physical spaces. On the contrary, it is based on a correct epistemological presentation of the practice of physics and the relations among measures, the measured characteristics of physical entities and measuring tools. Jánossy’s concept of the ideal solid rod makes it once again clear that non-Euclidean spaces in Einstein’s theory are only implications of Einstein’s positivist philosophical commitment which neglects that measures and theoretical spaces built up of them are only human constructs which do not correspond directly to physical entities or their characteristics. On the other hand, this positivist washing away of the difference between physical reality and human representations may turn into its opposite and result in a metaphysics of space-time if (as is the general case in the university teaching of relativity theory) we assume that the metric of space-time appearing in the general theory is the cause of the gravitational phenomena. Namely, in this interpretation the structures and characteristics of co-ordinate spaces will appear as objective properties of the physical world and thus Einstein’s positivist starting point will results in a theory of “objective” curvature of space-time which determines the behaviour of physical phenomena. If we have a feeling that in Einstein the cart is put before the horses [see: Balashov and Jansen 2003, 340; Brown 2005, 133-134]), then Jánossy’s analysis will explain the reason for this feeling. Measures as human constructions are numbers, so co-ordinate systems, coordinate spaces etc. constructed with their help are necessarily of a mathematical-geometrical nature. Washing away the difference between these human constructions and physical reality necessarily transforms physical reality into mathematics. 6.2. The extension of the Lorentz principle from homogeneous regions to inhomogeneous ones On the face of it Jánossy’s notion of general relativity may seem disturbing. While Einstein introduces the principle of special relativity as the equivalence of
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inertial systems and arrives at the general theory by extending that principle to arbitrary systems, in Jánossy’s reformulation the special theory deals with the propagation of light and not with inertial systems. However, this apparent difference can be easily resolved, since (as Jánossy shows) the Lorentz principle implies that Lorentz systems are inertial systems and vice versa. This implication is eventually equivalent to the claim that the two independent systems of measures introduced by Jánossy (that is, the system of measures based on rods without light signals and that established with the help of light signals by means of the radar method) are equivalent. So Jánossy would also be able to introduce the general theory as the extension of the special theory from inertial to arbitrary systems. Nevertheless, he does not follow this path, but continues to investigate the problem in terms of measures and the propagation of light. While he demonstrates that Lorentz systems and inertial systems coincide and thus a Lorentz system can be identified with the help of inertial phenomena, in his approach inertia remains only a secondary characteristic of these systems. The primary characteristic of a Lorentz system is for him the existence of a special system of measures in whose terms light appears to be propagated isotropically and with constant velocity relative to the system itself. Since such systems can only exist in physical regions where light appears to be propagated homogeneously, Lorentz systems are connected to such regions and Jánossy formulates the problems of general relativity with the help of this fact: “In the special theory of relativity only such regions are considered in which light is propagated homogeneously. The laws governing the motion of physical systems inside such regions obey symmetries which can be expressed by the Lorentz principle. In reality light can nowhere be assumed to be propagated strictly homogeneously, as we have reason to believe that the propagation of light is affected by gravitation and regions entirely free of gravitation do not exist. The Lorentz principle can be therefore taken to be valid only to such an approximation as gravitational effects can be neglected. The question arises, how the Lorentz principle should be generalized so as to apply to regions containing not negligible gravitational fields.” [Jánossy 1971, 214] Although his terminology considerably differs from that of Einstein’s, Jánossy’s train of thought is mathematically parallel to the consideration of the German physicist. So he shows that the sufficient and necessary condition of the homogeneous propagation of light in a given physical region is the existence of a straight (that is Euclidean) representation of the region established with the help of light signals, a criterion which is mathematically equivalent to the criterion that the
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Riemann-Christoffel tensor formed of the propagation tensor of light expressed in any measures of coordinate is equal to zero. [Jánossy 1971, 218-220] “We see thus that using signals of light only we are in a position to examine whether or not light is propagated homogeneously in the region we are investigating, and if the propagation of light proves to be homogeneous, we are in a position to construct a straight system of reference with the help of the signals of light.” [Jánossy 1971, 222] The Lorentz principle implies that homogeneous regions obey the same physical laws even if a system is Lorentz deformed, so physical laws of homogeneous regions are Lorentz invariant. Jánossy generalizes this fact along the following train of thoughts: i) From a mathematical point of view the laws valid for homogeneous regions may have several generalizations for inhomogeneous regions even if a) we restrict the possibilities of generalizations by requiring that the Lorentz principle originally valid for homogenous regions should also be valid for sufficiently small inhomogeneous regions [Jánossy 1971, 230], and b) we prescribe that the laws of homogenous regions should be contained as limiting cases by the generalized laws. [Ibid 264] ii) Since i) allows an unlimited number of possibilities for generalization, we ought to seek further restrictions and it seems that the most rational and heuristically most fruitful restriction is to seek only generalizations that can be expressed in tensors and covariant operators. Jánossy introduces the latter requirement as the extended (that is, generalized) Lorentz principle. [Ibid.] Thus in contrast to Einstein’s general principle of relativity which forms a definite claim on the nature of physical reality, his general theory of relativity is based on a methodological principle involving only a vague ontological element: namely the conjecture, that physical reality is such that this principle can be successfully applied to it. There is no place and perhaps it is not even necessary to give a more detailed presentation of Jánossy’s development of the general theory, since it is easy to see that it mathematically corresponds to Einstein’s considerations. What is important for us is the physical meaning of his presentation which considerably differs from Einstein’s. a) Firstly, Jánossy’s version of general relativity primarily is about the propagation of light. As a consequence, in his presentation the metric tensor of the general theory primarily appears as the propagation tensor of light. It emerges only in a later phase of the development of the theory that this tensor coincides with the metric tensor of the gravitation field [Jánossy 1971, 242-256, 266] (a coincidence which requires an explanation since the extended Lorentz principle as a heuristic
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principle cannot explain anything). Similarly, the equivalence of the gravitational and inertial masses (on which Einstein’s theory is based) loses its fundamental role and appears only as a secondary implication of the theory [241, 263]. b) Secondly, since the generalized Lorentz principle is for Jánossy only a heuristic principle, it is not at all granted that laws constructed with its help really are natural laws: “Sometimes suggestions are made to the effect as if the generalizations of the laws of nature which lead to the forms of the laws in gravitational fields could be obtained in a priori considerations. According to this view the laws thus obtained are logically more or less the only possible ones …. Such considerations are at fault; we shall show in the following that relativistic laws are based on well-defined physical hypotheses concerning the structure of matter and gravitation. It is a question of fact as to what extent these hypotheses give a correct description of real nature. ” [Jánossy 1971, 213-214] “So as to find the form of various physical laws in inhomogeneous regions it is useful to see how the mathematical form of such laws, valid in homogeneous regions, can be generalized. It is a question of experiment to find out whether or not the generalizations which suggest themselves are in accord with experiment.” [Jánossy 1971, 235] In any particular case it remains thus to be decided by experiment which of the generalized form of the physical law describes correctly the observed phenomenon. However, we have to go further: it is also a question to be decided by experiment whether or not the law describing a particular phenomenon correctly is an invariant one?” [Jánossy 1971, 264] (Notice that in Jánossy’s terminology a law is characterized as ‘invariant’ if it can be expressed in terms of tensors and covariant operators.) c) Thirdly, while non-Euclidean spaces appear in both Einstein’s and Jánossy’s theory, Jánossy argues that they are only spaces of measures, that is, theoretical spaces constructed by human beings to represent physical reality. The primary physical terms for Jánossy are homogeneous and inhomogeneous regions of the propagation of light, of which the first can but the second cannot be represented with straight coordinates. As a consequence, with Jánossy straight coordinates always indicate homogeneous regions and so regarding such regions they should be
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considered as privileged representations which directly characterize the regions, while in Einstein there are no privileged representations. d) Lastly, in Jánossy the four dimensional space-time is only a construction built up of measures, that is of representations constructed by human beings. So the interpretation that real physical bodies move on their geodetic paths in the four dimensional space-time is meaningless. “… it seems to us that it is a play with words if we suppose the geodetic line to be a ‘straight line in four dimensions’. ….. the solutions [of Einstein’s field equations]…. include among others Kepler’s ellipses along which planets move. – If we call those orbits ‘straight‘ then we lose completely the meaning of what is usually called straight.” [Jánossy 1971, 241] 7. The nature of Jánossy’s ether. 7.1. The antenna problem. The erroneous claim on the simplicity of Einstein’s theory Jánossy’s reformulation of relativity theory in terms of measurement arrives at a mathematical formalism equivalent to that of Einstein’s theory. Jánossy might as well stop at this point since his version of the theory does everything that the original version does. However, since in his conceptual framework the term „structure of space-time” as an entity existing “out there” in the physical world is devoid of meaning, he is firmly opposed to using it to explain relativistic phenomena. In his interpretation it is not the metric of space-time that determines the behaviour of other physical entities but the latter imply the former: the relations and rules of the physical world are such as to permit theoretical representation with the help of this term. As a consequence, in Jánossy‘s conceptual framework the original version of relativity theory appears as a merely phenomenological theory, while the ether-based interpretation of its mathematical formalism serves as its completion with a second, explanatory level. And at this point we arrive at the heart of any ether-based issue: what is the nature of the ether and what is the mechanism by which it impacts physical phenomena? It is often argued in favour of Einstein that his theory needs no such mechanism and so it is incomparably simpler and more elegant as any ether-based approach. A common counterargument (present also in Jánossy) is that simplicity and elegance are no criteria of truth since nature does not have to respect these human qualifications. As a matter of fact, even this counterargument is unnecessary since Einstein does not give us any explanation of how the assumed mathematical properties of space-time can influence physical phenomena or, put differently, how it is possible
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that physical entities follow geodetic lines. Is there an influence, a constraint exercised by the space-time (or the ether) on physical entities determining their behaviour or do the latter have an innate inclination to follow geodesics? Here we face the so called “antenna problem” well known in the literature. [Nerlich 1976, 264; DiSalle 1994; Brown 2005, 24-25] The main point is not, however, the problem, but the fact that the classical formulation of Einstein’s theory does not even attempt to answer the problem. Now, a theory that ignores and fails to address a crucial point of its subject and is, in this respect, incomplete, is highly likely to be simpler than another theory, which not only deals with the issues addressed by the first theory but also confronts problems ignored by the other one. The claim that Einstein’s theory is simpler and more elegant than the ether-based approaches is mere tautology. (Put ironically, the null theory is the simplest theory as it sees no problem and thus only declares that there is nothing to be solved.) Consequently, the ether-based explanation of relativistic phenomena is not an unnecessary and clumsy alternative to the original, Einsteinian explanation but, on the contrary, is a completion of the latter proposing a physical-causal explanation of the phenomena described mathematically by the original one. 7.2. The nature of Jánossy’s ether and Jánossy’s hypothesis on the mechanism of the Lorentz deformation Turning to Jánossy’s views on the physical nature of the ether, it should be emphasized that the main objective of Jánossy’s monograph on relativity theory is to reformulate Einstein’s theory on correct epistemological and physical grounds and to elucidate the logical and physical place of a privileged physical system in the context of a theory of relativistic phenomena. As such, the work does not aim at a complete theory of the ether. It lays down only the basic principles and outlines a few provisional hypotheses in order to assist and orientate further work on the topic. So the Hungarian physicist emphasizes that using the term “ether” he does not want to commit himself to any traditional theory: “ [...] as to avoid misconceptions we wish to emphasize that we regard the ether merely as the carrier of electromagnetic waves and possibly the waves associated with other fields and of elementary particles.” [Jánossy 1971, 48] He also rejects the macroscopic-mechanical models of the ether and its notion as a reference frame at absolute rest: “Einstein’s polemic against the ether concerned mainly the assumption that the ether is at ‘absolute rest’. Thus Einstein
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denied the existence of a system K0 which is at ‘absolute rest’. ” [Jánossy 1971, 49] “We think that the assumption that electromagnetic waves possess a carrier has nothing to do with the question of absolute rest. The concept of ‘absolute rest’ is a metaphysical concept which must be rejected. However, the concept of the ether as the carrier of electromagnetic and other phenomena is quite a different one….. Whether or not the ether, i.e. the carrier of electromagnetic waves, is at rest or at ‘absolute rest’ is a question which does not arise here and certainly has no significance in relation to our problems…. For our consideration it is also immaterial whether or not various parts of the ether move relative to each other. It seems quite plausible that considered on a cosmic scale distant parts of the ether are streaming with various velocities ….” [Jánossy 1971, 49-50] On the other hand, as an affirmative feature of his concept, besides being primarily the carrier of electromagnetic interactions, the ether also appears as an entity causing the Lorentz deformations. Jánossy assumes that the deformation emerge when physical entities accelerate relative to the ether. If the acceleration is slow enough and proceeds step by step, then the accelerated physical system will have time after all consecutive phases to settle down into newer and newer configurations. However, if the acceleration is continuous, the system will lag behind the configuration corresponding to the achieved velocity, and it will settle down into the latter only after a certain small temporal interval following the acceleration. So in the latter case the process of the deformation is (at least theoretically) observable, since there is a minor temporal interval during which the deformation has not yet taken place and thus the states of measuring tools do not coincide with the states expected according to the Lorentz transformation. [Jánossy, 127-128] Jánossy illustrates this hypothesis with the help of a practically solid rod. A rod is a configuration of its atoms and these latter are in a state of dynamic equilibrium. The forces causing the acceleration disturb this state, but after the acceleration has ceased, the atoms - now moving relatively to the ether - will establish a new equilibrium [Ibid, 127]. (Of course, in the case of deceleration inverse processes occur.) These hypothetical processes also constitute a physical explanation for the Lorentz principle. Whereas the principle declares the form of possible physical systems, the mechanism of deformations caused by the acceleration relative to the ether explains how and when such systems come to exist. To express the connection between these processes and the Lorentz principle, Jánossy formulates a dynamic
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version of the principle, which he considers to be one “compatible with the originally formulated Lorentz principle and […] an addition to it” [Jánossy 1971, 126]: “If a connected physical system is carefully accelerated [with respect to the ether] then, as a result of the acceleration, it suffers a Lorentz deformation” [Ibid.]. In contrast with Jánossy, Harvey Brown opines that this principle is only a simple implication of the original formulation of the Lorentz principle [Brown 123124]. However, the original formulation leaves open the question about the concrete physical cause of the Lorentz deformations, since from an a priori point of view it is not necessary to connect these deformations to the acceleration. So one may assume that the deformations are caused by permanent pressure of the ether during the rectilinear and even motion. By connecting the Lorentz deformations to the process of acceleration the dynamic principle excludes the alternative explanations and hence it really contains an additional element with respect to the original formulation. In the context of the general theory Jánossy assigns other characteristics to the ether. So it may have different physical states, it contains inhomogeneous structures, strains, etc. and functions as the seat of different physical fields (such as the field of gravitation). While Einstein explains the phenomena of the general theory with the help of the metric of space-time, for Jánossy the clue is the state of the ether which is represented with the metric tensor: “G (the metric tensor) represents some physical field which appears when observing very different physical phenomena – the propagation of light is only one of many such phenomena. The usually accepted interpretation of G is that it represents the ‘metric of the space-time continuum’. We do not think the latter interpretation to be a fortunate one. We would rather suggest that G represents the state of the ether which is the carrier of all physical fields.” [Jánossy 1971, 266] Before closing this section, we would like to make two brief remarks. The first concerns the above citation which shows some ambiguity. Namely, Jánossy speaks here about the tensor G as a representation but uses a Gothic G rather than a Roman G to denote it, which seems to run contrary to the convention introduced earlier in his book, according to which representations are notated by Roman Gs. However, from the context it is clear, that the word “representation” here does not mean the representation in our theories but a characterization of the state of the ether by physical quantities as, for example, the quantities of volume, temperature, pressure etc. “represent” – that is characterize – the state of a gas cloud. Jánossy’s refers here with the term „metric tensor” to a complex of physical quantities
27
(or briefly to a “tensor quantity”) which is not a mathematical entity and, therefore, does not consist of numerical values or mathematical functions but may be considered as a “tensor” only in the sense that we need mathematical tensors to represent it. Consequently, its notation by a Gothic G is correct and the ambiguity of Jánossy’s text follows not from the notation but from the fact that he uses the word “represent” in two different senses. Our second remark is about a critical reflection by Harvey Brown on the relation between Jánossy’s Lorentz principle and the Lorentz covariance of the laws of physics. “[The] ambiguity in the formulation of the principle would be removed if Jánossy just equated it with the Lorentz covariance of the fundamental laws of physics, and it is hard to see why he didn’t. It is almost as if Jánossy intends the Lorentz principle to stand over Lorentz covariance. At the start of the mentioned discussion of Maxwell’s equations, he announces that ‘Physically new statements are obtained if we apply the Lorentz principle to Maxwell’s equations’. But of course what emerges in the discussion is simply the Lorentz covariance of these equations.” [Brown 2005, 123] Brown is formally right. Lorentz covariance really may substitute Jánossy’s Lorentz principle. However, the requirement of Lorentz covariance in itself is only a formal requirement which allows different physical interpretations. Whereas in the original Einsteinian theory Lorentz convariance relates to our representations depending of the chosen reference system and later is connected to the‘structure of space-time’, in Jánossy the term ‘structure of space-time’ denotes only an element of our theoretical representation and Lorentz covariance is rather connected to physical deformations emerging independent of our reference frames. The function of Jánossy’s Lorentz principle is to exclude the Einsteinian interpretation and to give a physical interpretation to the relativistic phenomena; a function that cannot be served by the formal requirement of the Lorentz covariance of physical laws. (May be I am wrong but it seems to me that Brown overlooks this important moment of Jánossy’s notion since he – despite his excellent analysis of the relation between the mathematical formalism and the physical meaning of relativity theory and his valuable reflections on Jánossy’s theory – ignores the measurement theory aspect of Jánossy’s investigations.) 7.3. The emergence of gravitation forces Jánossy closes his reformulation of general relativity with an interesting hypothesis on the emergence of gravitational forces. According to his hypothesis
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closed physical systems are kept together by internal forces which are propagated with the velocity of light in the ether. When a gravitation field is present, it disturbs the homogeneous propagation of these forces (as it also disturbs the propagation of light). Due to this disturbance a new force emerges which tends to accelerate the system just like the Newtonian gravitational force is expected to do. Consequently, “[t]he gravitational force observed phenomenologically is equal to the self force with which a closed system acts upon itself, if the propagation of the internal forces is made inhomogeneous by the gravitational field.” [Jánossy 1971, 263] In this context we also receive an physical explanation of the equivalence of inertial and free gravitational motion, which forms a basis pillar of Einstein’s general relativity: “in a free falling particle the propagation of inner forces is nearly homogeneous relative to the particle itself, therefore in the free fall no resultant self force is present” [Ibid.]. Jánossy illustrates the applicability of his hypothesis on the example of an electric charge, but he unfortunately does not proceed further in this direction. However, the author of the present paper think that his hypothesis is of heuristic value and worth for further consideration even in this preliminary form and even if one agrees with Brown that several aspects of Jánossy’s idea of the ether are too traditional. If one feels similar to Brown then one must be aware of that we face a problem here characteristic for any ether-based approach. Namely, it is not so easy to find how to satisfy all the requirements we expect from a modern theory of the ether without turning it into a ‘mathematical ghost’ as Walter Ritz had characterized yet not the Einsteinian but the Lorentzian ether 100 years ago [Ritz 1908], which, in turn, was considered by Einstein several years later as still too mechanical. 7.4. Einstein’s and Jánossy’s ether It is well known that after publishing his general theory of relativity, Einstein made several metatheoretical assertions which shed new light on the problem of relativity. So he reintroduced the concept of the ether in the interpretation of the metric field of the general theory, and (as a more far reaching change with respect to his early ideas) he also indicated that his special theory needed a completion concerning the dynamical mechanism of the deformation of rods and clocks. These assertions clearly indicated that after the publication of his theory Einstein had a feeling that it was not sufficiently complete but needed a second, physical level. That is, in contrast to many current representatives of Einstein’s theory at universities and research institutes (who are more Einsteinian in this respect than the German
29
physicists himself was) Einstein did not consider the published version of his theory to be necessarily final and did not exclude an ether-based interpretation of relativistic phenomena and a physical-dynamical theory of the deformation of rods and clocks. [See e.g. Einstein 1920; 1921, 127; 1924; 1949, 22-23; Kostro 2000; 2008; Brown 2005, 113-114] So it is not at all an unfounded reference to Einstein’s authority when at the beginning of Chapter II of his discussed book Jánossy cites an important fragment of Einstein on the ether and insists that his reformulation of the relativity theory is based on similar ideas as the ideas expressed there by the German physicist. The key assertion of the citation runs as follows: “Dass es in der allgemeinen Relativitaetstheorie keine bevorzugten, mit der Metrik eindeutig verknüpften raumzeitlichen Koordinaten gibt, ist mehr für die matematischen Form dieser Theorie als für ihren physikalishen Gehalt charakteristisch.”[Jánossy 1971, 49; the original: Einstein 1924, 90-91] In Jánossy’s translation: “The fact, that in the framework of the general theory of relativity, there are no distinguished space-time representations connected in an unambiguous manner with metric - is rather a characteristic of the mathematical methods of the theory than a characteristic of its physical contents.” [Ibid.] Although this study deals with Jánossy and not with Einstein, it seems necessary to emphasize that this is a very serious statement, which seems to withdraw the principle of general relativity, or more adequately, to degrade it to a mere consequence of the applied methodology. If taken seriously, the assertion will imply that there is a definite, privileged (bevorzugt) metric structure of physical reality (the structure carried by the ether) while the metric structures of a given system of “raumzeitlichen Koordinaten” (and so the “relativity” of the possible systems of coordinates) are only an implication of the “matematischen Form” of the theory. Now it is easy to see that Jánossy translates Einstein’s German terms into his own English terminology (so “bevorzugten … Koordinaten” into “distinguished representations”) but even in the absence of his tendentious translation the German original allows us to perceive a definite parallelism between Einstein’s view expressed here and Jánossy’s ether based notion of relativity. If both the original Einsteinian formulation and the received view of the relativity theory may be characterized by a washing away of the difference between the theoretical-mathematical representation as a human product and the represented physical reality, then here, in this discussion of the problem of the
30
ether Einstein recaptures this difference and represents a view similar to that of Jánossy. So despite the contrast between the original formulation of relativity theory and Jánossy’s reformulation, Einstein’s out-of-theory reflections seem to be near to Jánossy’s view. However, at a closer look it will be also clear that the parallelism between Einstein and Jánossy is limited. Firstly, whereas Jánossy’s ether is the carrier both of the electromagnetic waves and the gravitational field, for Einstein the ether is only the “gravitational ether”. [See for example: Kostro 2008. 52-53] Secondly, for Jánossy the united space-time or the space-time continuum is only a human construction, a human representation of physical reality. Consequently, he opposes the view according to which the difference between space and time is a mere appearance due to the shortcomings of our senses, as it is claimed in Minkowski’s paper introducing the concept of the four dimensional space-time [Minkowski 1909] and then many times endorsed by Einstein. Whereas Einstein claims that the separation of space and time is without ‘objective meaning’ [i.e. Einstein 1949, 22; 1949a, 99-100; Kostro 2008, 57] for Jánossy it is an evident and “objective” physical fact appearing in the radical difference between the measuring tools of length and time. As a consequence, Jánossy’s ether is definitely a three dimensional spatial entity, while in the case of Einstein it is hard to see how his ether could be imagined otherwise than a mystical four dimensional space-time continuum. It also can be easily seen that the problem of four dimensional space-time is closely connected to Jánossy’s thesis on the importance of common sense regarding physical theories. Taking seriously the ontological priority of the EinsteinianMinkowskian space-time, a temporal interval, for example, that between the birth and the death of a person will appear of the same nature as the spatial distance between, say, Budapest and London, and the difference between translational motion and aging will disappear. Considering these consequences we may see that Jánossy’s thesis on the role of common sense is considerably more than a naive insistence on our accustomed everyday habits and judgments; it concerns our most ultimate ontological experiences, such as our experience of life and death. But these consequences also show that Einstein’s claim on the ontological, “objective” priority of the four dimensional space-time has significant metaphysical implications and transforms Einstein’s positivist starting point into a metaphysics of a four dimensional space time. Lastly, whereas Einstein verbally acknowledges that the ether has physical properties and speaks only about its deprivation of “mechanical” characteristics, it is clear that with the term “mechanical” he refers to all traditional physical properties including pressure, strain, density etc. In contrast with Einstein, Jánossy characterizes the state of the ether with the help of these terms. Of course, in doing so he is using the latter only in a metaphorical sense and he does not mean to claim that the ether has exactly the same properties as macroscopic entities. However, the application of
31
these terms definitely indicates that in his view the ontological nature of the ether is basically similar to the macroscopic physical entities. The difference between Einstein’s and Jánossy’s notions of the ether cannot be reduced even if we assume that in the context of physics Einstein also uses the term “geometry of space-time” metaphorically. The metaphorical use does not change the fact that Jánossy’s terms come from physics and they attribute to the ether physical characteristics even if they are used metaphorically, while Einstein’s term is transferred into physics from mathematics and hence its application necessarily results in a mathematization of physical reality. Therefore the conversion of the German physicist to the concept of the ether does not cure the epistemologically inverted relation between mathematics and physics characterizing his theory. In this respect it is often argued that the Einsteinian turn of physics brought about not only a theory change but also transformed the conceptual framework of physics and, as part of this transformation, it gave a new meaning to the word “physical”. The properties of Einstein’s ether are not “physical” if we use the old meaning of the word but in the new conceptual framework they become definitely physical. However, this argument is invalid since the point is exactly whether one accepts or refuses the conceptual change. The new meaning of “physical” is a consequence of the mathematization of physics by the Einsteinian version of relativity theory, the main target of Jánossy’s reformulation of the theory. Dubbing mathematical terms and properties as physical will not change their real nature. On the contrary, physical reality should first be attributed a mathematical nature in order to characterize such properties as physical. And conversely, if we really think that the latter are truly “physical”, then this will amount to transforming the nature of physical reality from physical to mathematical. Or is it possible that the mathematization of physical reality, criticized so vehemently by Lajos Jánossy (and more recently by H. Brown) regarding the theory of relativity, but also present in quantum mechanics, is more than a pure consequence of a methodological mistake? Is it possible that in its ultimate ontology the world around us is not of a physical but a mathematical nature? Maybe the cart is put before the horses not only by the received interpretation of relativity theory but also in physical reality? These are far reaching metaphysical questions that surely do not belong to relativity theory, and especially not to the topic of the present review of Jánossy’s interpretation of relativity theory, but still concern so intensively the whole interpretational problem of the theory that they must be raised at the close of this paper. 8. Summary We have seen that Jánossy’s theory of relativity consists of two levels. At the first level he reformulates Einstein’s theory in terms of measurement, while at the second level he outlines an ether-based explanation of relativistic effects. His
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reformulation of the relativity theory not only elucidates the relation between the mathematical formalism of the theory and physical reality and establishes an etherbased interpretation of relativistic phenomena, but also gives a deep insight into the hidden conceptual background of the Einsteinian version of the theory. In our days when the relation between physics and mathematics in relativity theory has become a topical issue again, Jánossy’s analysis of the relativistic phenomena and his deduction of the formalism of the theory in terms of measurement are especially significant both from a physical and a philosophical point of view. We have seen furthermore that his consideration about the role of the ether in the explanation of relativistic phenomena as well as his hypotheses about the nature of this entity are of high heuristic value and may give significant stimulation for further research in the direction of a dynamical theory of the ether. Acknowledgement The author expresses his gratefulness to the Hungarian Scientific Research Fund (OTKA) for the support granted to his research (Project Number T 046261)
References Balashov, Y. and Janssen M. 2003, “Critical Notices: Presentism and Relativity”, British Journal for the Philosophy of Science, 54. 327-346. J. S. Bell 2001, The Foundation of Quantum Mechanics, M. Bell, K. Gottfried and M. Veltman (eds.) World Scientific. J. S. Bell 1987, Speakable and Unspeakable in Quantum Mechanics. (1. Edition) Cambridge University Press, Cambridge. J. S. Bell 1976, How to Teach Special Relativity, Progress in Scientific Culture, Vol. 1, No. 2, reprinted in [Bell 1987] and [Bell 2001] H. Brown 2005, Physical Relativity. Clarendon Press, Oxford. DiSalle, R. 1995, “On Dynamics, Indiscernability and Space-Time Ontology”, British Journal for the Philosophy of Science, 45, No 1 (Mart 1994) 265-287 Duffy, M.C. 2008, “Ether as a Disclosing Model.” In: Duffy M. C. and Levy J. 2008. Duffy M. C. and Levy J. (eds.) 2008, Ether, Space-Time and Cosmology. Volume 1. Liverpool. Duffy, M. C. (ed.) 1988-2006, Physical Interpretations of Relativity Theory IXIV. Proceedings of the Conferences 1988, 1990, 1992, 1994,1996, 1998, 2000, 2002, 2004, 2006. The British Society for the Philosophy of Science, London.
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Einstein, A. 1920, Äther und Relativitätstheorie. Verlag von J. Springer, Berlin. Einstein, A. 1921, „Geometrie und Erfahrung.” Sitzungsberichte der königlichen Preussische Akademie der Wissenschaften. 123-130. o. Einstein, A. 1924, „Über den Äther.” Verhandlungen der Schweizerischen Naturforschenden Gesellschaft. 105. Teil 2. 85-93. Einstein, A.1949, „Autobiographisches“. In: Schilpp 1949, 1-35. Einstein, A. – Infeld L. 1949a, Die Physik als Abenteur der Erkenntnis. Sijthofs Witgeversmaatschappij, Leiden. Graham, L. R. 1972, Science and Philosophy in the Soviet Union. Knopf, New York. Jánossy L. 1948, 1950, Cosmic Rays. Clarendon Press, Oxford. Jánossy L. 1963, „Foreword 1.” In: Jánossy, L. and Elek, T, A relativitáselmélet filozófiai problémái. (The Philosophical Problems of Relativity Theory.) Budapest: Akadémiai Kiadó. 9-11. (In Hungarian) Jánossy L. 1965, Theory and Practice of the Evaluation of Measurements. Clarendon Press, Oxford. Jánossy, L. 1971, Theory of Relativity Based on Physical Reality. Akadémiai Kiadó, Budapest. Kostro, L. 2000, Einstein and Ether. Aperion, Montreal. Kostro, L. 2008, „Einstein’s New Ether 1916 – 1955.” In: Duffy M. C. and Levy J. 2008. Minkowski, H. 1909, „Raum und Zeit“, Physicalishce Zeitschrift 10. 104-111. Nerlich G. 1976, Cambridge.
The
Shape of Space. Cambridge University Press,
Palágyi, M. 1914, Die Relativitaetstheorie in der modernen Physik. Berlin: 1914. (Also in Palágyi 1925. pp 34-83.) Palágyi, M. 1925, Ausgewaehlte Werke, Band III. Zur Weltmechanik (Beitraege zur Metaphysik der Physik). Leipzig: Johann Ambrosius Barth. Ritz, W. 1908, „Du rôle de l'éther en physique“ Scientia 1908, Vol 3. Nr. VI. 260-274. Republished in Karl Dürr’s German translation as „Über die Rolle des Aethers in Physik” in Ritz 1963. Ritz, W. Theorien Über Aether, Gravitation, Relativitaet und Elektrodynamik.” Bern und Badisch-Reihnfelden: Schritt Verlag 1963.
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Schilpp P. A. (ed.) 1949, Albert Einstein als Philosoph und Naturforscher. Kohlhammer Verlag, Stuggart. Székely, L. 1987, „Physical Theory and Philosophical Values.” Doxa 9. (Published by The Institute for Philosophy of the Hungarian Academy of Sciences.) 159-181. Székely, L. 1988, „A Hungarian Interpretation of Relativity Theory.” In: Duffy (ed.) 1988. Székely, L. 1996, "Melchior Palágyi's Space-Time” Ultimate Reality and Meaning. (Interdisciplinary Studies in the Philosophy of Understanding.) Vol. 19. No. 1. 3-15.
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AETHER THEORY CLOCK RETARDATION vs. SPECIAL RELATIVITY TIME DILATION Joseph Levy 4 Square Anatole France, 91250, St Germain-lès-Corbeil, France E-mail:
[email protected] ABSTRACT Assuming a model of aether non-entrained by the motion of celestial bodies, one can provide a rational explanation of the experimental processes affecting the measurement of time when clocks are in motion. Contrary to special relativity, aether theory does not assume that the time itself is affected by motion; the reading displayed by the moving clocks results from two facts: 1/ Due to their movement through the aether, they tick at a slower rate than in the aether frame. 2/ The usual synchronization procedures generate a synchronism discrepancy effect. These facts give rise to an alteration of the measurement of time which, as we shall show, exactly explains the experimental results. In particular, they enable to solve an apparent paradox that special relativity cannot explain (see chapter 4). When the measurement distortions are corrected, the time proves to be the same in all co-ordinate systems moving away from one another with rectilinear uniform motion. These considerations strongly support the existence of a privileged aether frame. The consequences concern special relativity (SR) as well as general relativity (GR) which is an extension of SR. We should note that Einstein himself became conscious of the necessity of the aether from 1916, in contrast with conventional relativity. Yet the model of aether presented here differs from Einsein’s in that it assumes the existence of an aether drift, in agreement with the discoveries of G.F. Smoot and his co-workers listed in Smoot’s Nobel Lecture, December 8th 2006. Although it makes reference to previous studies, this text remains self-sufficient.
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1. INTRODUCTION In the present text, the points of view of special relativity and aether theory regarding the measurement of time in moving co-ordinate systems are successively presented and compared. The measurement concerns the two way transit time of light along a rod perpendicular to the direction of motion. We show that the approach of aether theory we have developed in Ref [1-4], can give a rational explanation of the experimental processes affecting this measurement, but, contrary to special relativity, these processes do not result from time dilation, but rather from the slowing down of clocks moving through the aether and from the synchronism discrepancy effect caused by the standard synchronization procedures. After correction of these measurement distortions the true value of time in moving co-ordinate systems is rediscovered. This study gives an illustration, in a specific example, of the differences existing between special relativity and aether theory. (We should bear in mind for the reader not informed of our approach, that the concept of aether assumed in this text conforms to the Lorentz views: it is associated with a privileged aether frame and is not entrained by the motion of bodies. It is this approach that we shall refer to as “aether theory” all through the text). This study does not question the experimental results brought about by relativity theory since, as we shall see, at least in the cases studied here, it predicts the same clock readings as SR provided that we use the standard measurement procedures. It nevertheless gives another interpretation of the experimental data (demonstrating that the procedures used entail measurement distortions and that the results obtained conceal hidden variables). This different interpretation and the disclosing of hidden variables should have important consequences for the future development of physics insofar as it concerns not only SR, but also GR. An important argument supporting our approach is that it solves an apparent paradox related to reciprocity that SR cannot explain (see chapter 4). Let us bear in mind that, contrary to what is often believed, Einstein did not definitively reject the concept of aether. He assumed, no later than 1916, that the consistency of general relativity needed recognition of the aether, an opinion which he recorded in an address he delivered on May 5th 1920 in the University of Leyden [5]. But as Einstein declared at the end of this address, “the idea of motion may not be applied to this model of aether…”, and, therefore, it cannot explain the discoveries of G.F. Smoot who, in a report done at the university of California, declared: “The motion of the Earth with respect to the distant matter (“aether drift”) was measured, and the homogeneity and isotropy of the universe (“the cosmological principle”) was probed. This recognition of an aether drift was confirmed in his Nobel lecture, December 8th 2006 [6, 7]. On the contrary, our model assumes an aether drift in agreement with the experimental studies performed by Smoot, Gorenstein and their co-workers.
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2. TIME DILATION ACCORDING TO RELATIVITY THEORY Let us consider two inertial coordinates systems S0 (x,, y, z) and S1 (x’, y’ ,z’) receding from one another along the x-axis of the co-ordinate system S0 , and suppose that a light ray starts from a point M fixed to the coordinate system S1 , and travels along a rod L=MB, perpendicular to the x’-axis (see fig 1). After reflection in a mirror placed in B, the signal returns to point M. In the coordinate system S1 , the two-way transit time of light along the rod is 2 t1 = 2L/C. But, viewed from S0 , the light ray starts from a point A in this co-ordinate system, and after reflection in B returns to point A’. The total duration of the cycle in S0 will be labelled 2t0 .According to relativity, the speed of light is C in all inertial frames and in all directions of space. Let v01 refers to the real relative speed separating S0 and S1 (measured with non contracted standards). When the light ray has covered the distance AB, S1 has moved away from S0 a distance AM = v01t0
S0
O
B
S1
O’
A M
A’
x, x’
FIG 1. In the coordinate system S1 , the light ray travels from point M to the mirror B and, after reflection, returns to M. In S0 the signal starts from point A, and after reflection in B returns to A’. According to an observer attached to S0 the transit time of light t0 along AB is given 2 2 by: C 2t02 − v01 t0 = L2 , therefore:
t0 =
L 2 C 1 − v01 / C2
.
Replacing L/C by its value t1 this expression reduces to:
t0 =
t1 2 1 − v01 /C2
39
.
This classical formula is interpreted as time dilation by special relativity, (an expression obtained because the position of the clock in S1 remains fixed relative to this co-ordinate system) 3. CLOCK RETARDATION ACCORDING TO AETHER THEORY The theory on which this study is based, assumes the existence of a preferred frame in which the aether is at rest. The one-way speed of light is C in the aether frame, and different from C in all other co-ordinate systems moving with respect to the aether frame. Yet, as we saw in Ref [1-4] and [8], due to measurement distortions (that will be evoked in the text which follows), it appears to be of magnitude C in all ‘inertial’ frames and in all directions of space. Contrary to relativity theory, the motion of bodies does not affect the time, but the motion through the aether causes a slowing down of the moving clocks. The real two-way transit time of light, along a rod attached to a certain ‘inertial’ frame, is the same for the observers of all frames, but, due to clock retardation, the reading displayed by clocks moving relative to the rod will depend on their speed with respect to the rod [2, 8, 9]. Although the variables used in this study should be difficult to determine experimentally, our approach, as we shall see, allows an exact theoretical comparison of the concepts of time assumed by the two theories. In this section we shall study successively two different cases: in section 3.1. the clock reading in a moving ‘inertial’ co-ordinate system is compared to the time in the aether frame; this case introduces to the section 3.2. which puts forward exhaustively the differences between aether theory and relativity. The paradox inherent in conventional relativity when we assume a complete symmetry between frames will be examined in section 4. 3.1. Comparison of the clock readings displayed in frames S 0 and S1 In this section we shall compare the clock readings in two ‘inertial’ co-ordinate systems as we did in section 2, but from the point of view of aether theory. The only difference is that the co-ordinate system S 0 is assumed to be at rest in the aether frame where the clock reading is not altered by motion (and which can be regarded as the basic time or, by definition, the real time) (Fig 1). Since the line AB is the path of the light signal in the aether frame, the speed of light is C along this line. Referring to the transit time of light along AB, that would be displayed by a clock attached to frame S0 , as t0 , we have: 2 2 C 2t02 − v01 t0 = L2 ,
and therefore:
t0 =
L 2 C 1 − v01 / C2
40
.
According to the aether theory under consideration, clock retardation is defined with respect to the aether frame; the ratio between the time in the aether frame and the reading displayed by clocks moving at absolute speed v is assumed to be equal to (1 − v 2 / C 2 ) −1 / 2 . This assumption will be justified a posteriori; its experimental implications will be studied in the text that follows. Therefore, the clocks attached to the co-ordinate system S1 tick at a slower rate and display the reading t1app = L/C. Thus,
t0 =
t1app 2 1 − v01 / C2
,
(1)
where the suffix ‘app’ means apparent. This formula assumes the same mathematical form as the time dilation formula of special relativity; yet its meaning is quite different because, contrary to special relativity, t1app is not the true time in the co-ordinate system S1 , it is the clock reading displayed by clocks slowed down by motion. (We bear in mind that, if we assume the existence of a preferred aether frame, then, real frames attached to bodies, even if they are not submitted to physical influences other than the aether drift, are never perfectly inertial. The term ‘inertial’ is an approximation which must be limited to the cases where the absolute speed of the frames under consideration is low compared to the speed of light. See Ref [2]). 3.2. Case of two co-ordinate systems moving away from the aether frame We now propose to study a different case: we shall determine the clock retardation formula between two co-ordinate systems S1 and S 2 receding with rectilinear uniform motion with respect to the co-ordinate system S0 which is attached to the aether frame. The direction of motion is the x-axis (see fig 2). This case is that to which we usually deal with in practice. In relativity, there is no preferred frame, therefore the co-ordinate system S0 is inexistent and the time dilation formula between the systems S1 and S 2 takes the form:
T1 =
t2 1 − v122 / C 2
,
(2)
where v12 refers to the relative speed between the coordinate systems S1 and S 2 . In aether theory, things are very different. Let v01 , v02 and v12 refer to the real relative speeds between the three co-ordinate systems (obtained in the absence of measurement distortions). The rod MB perpendicular to the x”-axis is firmly fixed to the co-ordinate system S 2 . We propose to compare the apparent times (displayed by the clocks attached to S1 and S 2 ) which are needed by the light signal to achieve a
41
cycle (from M to B and to M again in the system S 2 and from A to B and to A’ in the system S1 ).
S0
O
S1
O’
B
S2
O’’
A
M
A’
x, x’, x’’
FIG 2. The co-ordinate systems S1 and S 2 recede from S0 along the common x-axis. The light ray travels along the rod MB which is at rest in S 2 (from M to B and to M again). With respect to S1 it starts from point A, is reflected in B and then returns to A’, (where A and A’ are two points at rest in the co-ordinate system S1 ). During a cycle of the signal, S 2 has moved with respect to S1 a distance AA’. The real value of the one-way speed of light along AB is not equal to C since the coordinate system S1 is not at rest with respect to the aether frame. 3.2.1. We shall first assume that the clocks placed at points A and A’ are exactly synchronized. Let us label as 2 t0 the two-way transit time of the light signal that would be displayed by clocks attached to the co-ordinate system S0 . Due to clock retardation the clock readings in S1 and S 2 are related to t0 as follows: 2 t1app = t0 1 − v01 /C2 ,
(3)
2 t2 app = t0 1 − v02 /C2 .
(4)
and (Note nevertheless that the true time, needed for half a cycle, measured with clocks not slowed down by motion, is t0 for all observers). From (3) and (4) we infer:
t1app = t2 app
2 1 − v01 / C2 2 1 − v02 / C2
.
Assuming that v02 / C te ) travelling through space, and then retrospectively assigns the time and distance of a distant event B according to (ignoring directional information for simplicity) 1 c TB = (tr + te ) , DB = (tr − te ) , (43) 2 2 where each observer is now using the same numerical value of c. The event B is then plotted as a point in an individual geometrical construct by each observer, known as a spacetime record, with coordinates (DB , TB ). This is the same as an historian recording events according to some agreed protocol. Unlike historians, who don’t confuse history books with reality, physicists do so. We now show that because of this protocol and the absolute motion dynamical effects, observers will discover on comparing their historical records of the same events that the expression 1 2 2 2 τAB = TAB − 2 DAB , (44) c is an invariant, where TAB = TA − TB and DAB = DA − DB are the differences in times and distances assigned to events A and B using the Einstein measurement protocol (43), so long as both are sufficiently small compared with the scale of inhomogeneities in the velocity field. To confirm the invariant nature of the construct in (44) one must pay careful attention to observational times as distinct from protocol times and distances, and this must be done separately for each observer. This can be tedious. We now demonstrate this for the situation illustrated in Fig. 3. 0 = v0 , By definition the speed of P 0 according to P is v00 = DB /TB and so vR 0 where TB and DB are the protocol time and distance for event B for observer P P )2 = T 2 − 1 D 2 since according to (43). Then using (44) P would find that (τAB B c2 B v 02
P )2 = (1 − R )T 2 = (t0 )2 where the both TA = 0 and DA =0, and whence (τAB B B c2 last equality follows from the time dilation effect on the P 0 clock, since t0B is the
150
P (v0 = 0)
Figure 3:
P 0 (v00 )
tr H Y HHγ T TB
HH
γ
H H 0 * B (tB )
te
A
DB
Here T − D is the spacetime construct (from the Einstein measurement protocol) of a special observer P at rest wrt space, so that v0 = 0. Observer P 0 is moving with speed v00 as determined by observer P , and therefore with speed 0 vR = v00 wrt space. Two light pulses are shown, each travelling at speed c wrt both P and space. Event A is when the observers pass, and is also used to define zero time for each for convenience.
D
time of event B according to that clock. Then TB is also the time that P 0 would 0 compute for event B when correcting for the time-dilation effect, as the speed vR 0 0 of P through the quantum foam is observable by P . Then TB is the ‘common time’ for event B assigned by both observers. For P 0 we obtain directly, also from P 0 )2 = (T 0 )2 − 1 (D 0 )2 = (t0 )2 , as D 0 = 0 and T 0 = t0 . (43) and (44), that (τAB B B B B B B c2 Whence for this situation P 2 P0 2 ) , (45) (τAB ) = (τAB and so the construction (44) is an invariant. While so far we have only established the invariance of the construct (44) when one of the observers is at rest in space, it follows that for two observers P 0 and P 00 both in absolute motion it follows that they also agree on the invariance of (44). This is easily seen by using the intermediate step of a stationary observer P: P0 2 P 2 P 00 2 (τAB ) = (τAB ) = (τAB ) . (46) Hence the protocol and Lorentzian absolute motion effects result in the construction in (44) being indeed an invariant in general. This is a remarkable and subtle result. For Einstein this invariance was a fundamental assumption, but here it is a derived result, but one which is nevertheless deeply misleading. Explicitly indicating small quantities by ∆ prefixes, and on comparing records retrospectively, an ensemble of nearby observers agree on the invariant ∆τ 2 = ∆T 2 −
1 ∆D2 , c2
(47)
for any two nearby events. This implies that their individual patches of spacetime records may be mapped one into the other merely by a change of coordinates, and that collectively the spacetime patches of all may be represented by one
151
pseudo-Riemannian manifold, where the choice of coordinates for this manifold is arbitrary, and we finally arrive at the invariant ∆τ 2 = gµν (x) ∆xµ ∆xν ,
(48)
with xµ = {D1 , D2 , D3 , T }. Eqn. (48) is invariant under the Lorentz transformations x0µ = Lµν xν , (49) where, for example for relative motion in the x direction, Lµν is specified by x − vt , 1 − v 2 /c2
x0 = p
y 0 = y,
z 0 = z,
t − vx/c2 t0 = p 1 − v 2 /c2
(50)
So absolute motion and special relativity effects, and even Lorentz symmetry, are all compatible: a possible preferred frame is hidden by the Einstein measurement protocol. The experimental question is then whether or not a supposed preferred frame actually exists or not — can it be detected experimentally? The answer is that there are now eight such consistent experiments. The notion that the special relativity formalism requires that the speed of light be isotropic, that it be c in all frames, has persisted for most of the last century. The actual situation is that it only requires that the round trip speed be invariant. This means that the famous Einstein light speed postulate is actually incorrect. This is discussed in [29, 30, 31, 32, 33].
8 Deriving the General Relativity Formalism As discussed above the generalised Dirac equation gives rise to a trajectory determined by (25), which may be obtained by extremising the time-dilated elapsed time (32). Z
τ [r0 ] =
v2 dt 1 − R c2
!1/2
(51)
This happens because of the Fermat least-time effect for quantum matter waves: only along the minimal time trajectory do the quantum waves remain in phase under small variations of the path. This again emphasises that gravity is a quantum effect. We now introduce a spacetime mathematical construct according to the metric ds2 = dt2 − (dr − v(r, t)dt)2 /c2 = gµν dxµ dxν (52)
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Then according to this metric the elapsed time in (51) is s
Z
τ=
dt gµν
dxµ dxν , dt dt
(53)
and the minimisation of (53) leads to the geodesics of the spacetime, which are thus equivalent to the trajectories from (51), namely (25). Hence by coupling the Dirac spinor dynamics to the 3-space dynamics we derive the geodesic formalism of General Relativity as a quantum effect, but without reference to the HilbertEinstein equations for the induced metric. Indeed in general the metric of this induced spacetime will not satisfy these equations as the dynamical space involves the α-dependent dynamics, and α is missing from GR. So why did GR appear to succeed in a number of key tests where the Schwarzschild metric was used? The answer is provided by identifying the induced spacetime metric corresponding to the in-flow in (3) outside of a spherical matter system, such as the earth. Then (52) becomes 1 ds2 = dt2 − 2 (dr + c
s
2GM (1 + r
α 2
+ ..)
dt)2 −
1 2 2 r (dθ + sin2 (θ)dφ2 ), c2
(54)
Making the change of variables t → t0 and r → r0 = r with 2 t0 = t − c
s
2GM (1+ α2 + . . .)r 4 GM (1+ α2 + . . .) + tanh−1 c2 c3
s
2GM (1+ α2 + . . .) c2 r (55)
this becomes (and now dropping the prime notation) 2GM (1 + α2 + ..) 1 1− dt2 − 2 r2 (dθ2 + sin2 (θ)dφ2 ) 2 c r c !
ds2 =
− c2
dr2 !. 2GM (1 + α2 + ..) 1− c2 r
(56)
which is one form of the the Schwarzschild metric but with the α-dynamics induced effective mass shift. Of course this is only valid outside of the spherical matter distribution, as that is the proviso also on (3). As well the above particular change of coordinates also introduces spurious singularities at the event horizon4 , 4
The event horizon of (56) is at a different radius from the actual event horizon of the black hole solutions that arise from (1).
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but other choices do not do this. Hence in the case of the Schwarzschild metric the dynamics missing from both the Newtonian theory of gravity and General Relativity is merely hidden in a mass redefinition, and so didn’t affect the various standard tests of GR, or even of Newtonian gravity. Note that as well we see that the Schwarzschild metric is none other than Newtonian gravity in disguise, except for the mass shift. While we have now explained why the GR formalism appeared to work, it is also clear that this formalism hides the manifest dynamics of the dynamical space, and which has also been directly detected in gas-mode interferometer and coaxial-cable experiments. Nevertheless we now show [1] that in the limit α → 0 the induced metric in (52), with v from (1), satisfies the Hilbert-Einstein equations so long as we use relativistic corrections for the matter density on the RHS of (1). This means that (1) is consistent with for example the binary pulsar data - the relativistic aspects being associated with the matter effects upon space and the relativistic effects of the matter in motion through the dynamical 3-space. The agreement of GR with the pulsar data is implying that the α-dependent effects are small in this case, unlike in black holes and spiral galaxies. The GR equations are 1 8πG Gµν ≡ Rµν − Rgµν = 2 Tµν , 2 c
(57)
where Gµν is the Einstein tensor, Tµν is the energy-momentum tensor, Rµν = α Rµαν and R = g µν Rµν and g µν is the matrix inverse of gµν . The curvature tensor is ρ Rµσν = Γρµν,σ − Γρµσ,ν + Γρασ Γαµν − Γραν Γαµσ , (58) where Γαµσ is the affine connection Γαµσ
1 = g αν 2
∂gνµ ∂gνσ ∂gµσ + − σ µ ∂x ∂x ∂xν
.
(59)
Let us substitute the metric in (52) into (57) using (58) and (59). The various components of the Einstein tensor are then found to be X
G00 =
vi Gij vj − c2
i,j=1,2,3
Gi0 = −
X
X
G0j vj − c2
j=1,2,3 2
Gij vj + c Gi0 ,
X
vi Gi0 + c2 G00 ,
i=1,2,3
Gij = Gij ,
i, j = 1, 2, 3.
j=1,2,3
where the Gµν are given by G00 =
1 1 ((trD)2 − tr(D2 )), Gi0 = G0i = − (∇ × (∇ × v))i , i = 1, 2, 3. 2 2
154
(60)
Gij
=
d 1 1 (Dij − δij trD) + (Dij − δij trD)trD − δij tr(D2 ) + (ΩD − DΩ)ij , dt 2 2 i, j = 1, 2, 3. (61)
In vacuum, with Tµν = 0, we find from (57) and (60) that Gµν = 0 implies that Gµν = 0. We see that the Hilbert-Einstein equations demand that (trD)2 − tr(D2 ) = 0
(62)
but it is these terms in (1) that explain the various gravitational anomalies. This simply corresponds to the fact that GR does not permit the ‘dark matter’ effect, and this happens because GR was forced to agree with Newtonian gravity, in the appropriate limits, and that theory also has no such effect. As well in GR the energy-momentum tensor Tµν is not permitted to make any reference to absolute linear motion of the matter; only the relative motion of matter or absolute rotational motion is permitted, contrary to the experiments. It is very significant to note that the above exposition of the GR formalism for the metric in (52) is exact. Then taking the trace of the Gij equation in (61) we obtain, also exactly, and in the case of zero vorticity, and outside of matter so that Tµν = 0, ∂ (∇.v) + ∇.((v.∇)v) = 0 (63) ∂t which is the Newtonian ‘velocity field’ formulation of Newtonian gravity outside of matter, as in (1) but with α = β = 0. So GR turns out to be Newtonian gravity in a grossly overstructured mathematical formalism.
9 Experimental and Observational Phenomena I We now briefly review the extensive range of light speed experiments that have detected that the speed of light is not isotropic - the speed is different in different directions when measured in a laboratory experiment on earth, as predicted by the generalised Maxwell equations, Sect.5. The most famous of these experiments was that of Michelson and Morley in 1887. Contrary to often repeated claims, this experiment decisively detected the anisotropy. The cause of the misunderstanding surrounding this experiment is that the Newtonian based theory Michelson used for the calibration of the experiment is simply wrong, and of course not unexpectedly. Clearly as the Michelson interferometer is a 2nd order v/c experiment its calibration requires a ‘relativistic’ analysis, in particular one must take account of arm contractions and also the Fresnel drag effect. The
155
Michelson-Morley fringe shift data then gives a speed in excess of 300km/s, as first discovered by Cahill and Kitto in 2002 [17].
9.1 Anisotropy of the Speed of Light That the speed of light in vacuum is the same in all directions, i.e. isotropic, for all observers has been taken as a critical assumption in the standard formulation of fundamental physics, and was introduced by Einstein in 1905 as one of his key postulates when formulating his interpretation of Special Relativity. The need to detect any anisotropy has challenged physicists from the 19th century to the present day, particularly following the Michelson-Morley experiment of 1887. The problem arose when Maxwell in 1861 successfully computed the speed of light c from his unified theory of electric and magnetic fields: but what was the speed c relative to? There have been many attempts to detect any supposed light-speed anisotropy and there have so far been 8 successful and consistent such experiments, and as well numerous unsuccessful experiments, i.e. experiments in which no anisotropy was observed. The reasons for these different outcomes is now understood: any light-speed anisotropy produces not only an expected ‘direct’ effect, being that which is expected to produce a ‘signal’, but also affects the very physical structure of the apparatus, and with this effect usually overlooked in the design of some detectors. In some designs these effects exactly cancel. As already stated there is overwhelming evidence from 8 experiments that the speed of light is anisotropic, and with a large anisotropy at the level of 1 part in 103 : so these experiments show that a dynamical 3-space exists, and that the spacetime concept was only a mathematical construct - it does not exist as an entity of reality, it has no ontological significance. These developments have lead to a new physics in which the dynamics of the 3-space have been formulated, together with the required generalisations of the Maxwell equations (as first suggested by Hertz in 1890 [14]), and of the Schr¨odinger and Dirac equations, which have lead to the new emergent theory and explanation of gravity, with numerous confirmations of that theory from the data from black hole systematics, light bending, spiral galaxy rotation anomalies, bore hole anomalies, etc. This data has revealed that the coupling constant for the self-interaction of the dynamical 3-space is none other than the fine structure constant ≈ 1/137 [39, 40, 41, 42], which suggests an emerging unified theory of quantum matter and a quantum foam description of the dynamical 3-space. The most influential of the early attempts to detect any anisotropy in the speed of light was the Michelson-Morley experiment of 1887, [2]. Despite that, and its influence on physics, its operation was only finally understood in 2002 [16,
156
17, 18]. The problem has been that the Michelson interferometer has a major flaw in its design, when used to detect any light-speed anisotropy effect5 . To see this requires use of Special Relativity effects. The Michelson interferometer compares the round-trip light travel time in two orthogonal arms, by means of interference fringe shifts measuring time differences, as the device is rotated. However if the device is operated in vacuum, any anticipated change in the total travel times caused by the light travelling at different speeds in the outward and inward directions is exactly cancelled by the Fitzgerald-Lorentz mirror-supporting-arm contraction effect - a real physical effect. Of course this is precisely how Fitzgerald and Lorentz independently arrived at the idea of the length contraction effect. In vacuum this means that the round-trip travel times in each arm do not change during rotation. This is the fatal design flaw that has confounded physics for over 100 years. However the cancellation of a supposed change in the round-trip travel times and the Lorentz contraction effect is merely an incidental flaw of the Michelson interferometer. The critical observation is that if we have a gas in the light path, the round-trip travel times are changed, but the Lorentz arm-length contraction effect is unchanged, and then these effects no longer exactly cancel. Not surprisingly the fringe shifts are now proportional to n − 1, where n is the refractive index of the gas. Of course with a gas present one must also take account of the Fresnel drag effect, because the gas itself is in absolute motion. This is an important effect, so large in fact that it reverses the sign of the time differences between the two arms, although in operation that is not a problem. As well, since for example for air n = 1.00029 at STP, the sensitivity of the interferometer is very low. Nevertheless the Michelson-Morley experiment as well as the Miller interferometer experiment of 1925/1926 [3] were done in air, which is why they indeed observed and reported fringe shifts. As well Illingworth [19] and Joos [20] used helium gas in the light paths in their Michelson interferometers; taking account of that brings their results into agreement with those of the air interferometer experiment, and so confirming the refractive index effect. Jaseja et al. [21] used a He-Ne gas mixture of unknown refractive index, but again detected fringe shifts on rotation. A re-analysis of the data from the above experiments, particularly from the enormous data set of Miller, has revealed that a large lightspeed anisotropy had been detected from the very beginning of such experiments, where the speed is some 430 ± 20km/s - this is in excess of 1 part in 103 , and the Right Ascension and Declination of the direction was determined by Miller [3] long ago. We also briefly review the RF coaxial cable speed experiments of 5
Which also severely diminishes its use in long-baseline terrestrial interferometers built to detect gravitational waves.
157
Torr and Kolen [22], DeWitte [23] and Cahill [24], which agree with the gas-mode Michelson interferometer experiments.
9.2 Michelson Gas-mode Interferometer Let us first consider the new understanding of how the Michelson interferometer works. This brilliant but very subtle device was conceived by Michelson as a means to detect the anisotropy of the speed of light, as was expected towards the end of the 19th century. Michelson used Newtonian physics to develop the theory and hence the calibration for his device. However we now understand that this device detects 2nd order effects in v/c to determine v, and so we must take account of relativistic effects. However the application and analysis of data from various Michelson interferometer experiments using a relativistic theory only occurred in 2002, some 97 years after the development of Special Relativity by Einstein, and some 115 years after the famous 1887 experiment. As a consequence of the necessity of using relativistic effects it was discovered in 2002 that the gas in the light paths plays a critical role, and that we finally understand how to calibrate the device, and we also discovered, some 76 years after the 1925/26 Miller experiment, what determines the calibration constant k that Miller had determined using the Earth’s rotation speed about the Sun to set the calibration. This, as we discuss later, has enabled us to now appreciate that gas-mode Michelson interferometer experiments have confirmed the reality of the Fitzgerald-Lorentz length contraction effect: in the usual interpretation of Special Relativity this effect, and others, is usually regarded as an observer dependent effect, an illusion induced by the spacetime. But the experiments are to the contrary showing that the length contraction effect is an actual observer-independent dynamical effect, as Fitzgerald and Lorentz had proposed. The Michelson interferometer compares the change in the difference between travel times, when the device is rotated, for two coherent beams of light that travel in orthogonal directions between mirrors; the changing time difference being indicated by the shift of the interference fringes during the rotation. This effect is caused by the absolute motion of the device through 3-space with speed v, and that the speed of light is relative to that 3-space, and not relative to the apparatus/observer. However to detect the speed of the apparatus through that 3-space gas must be present in the light paths for purely technical reasons. The post relativistic-effects theory for this device is remarkably simple. Consider here only the case where the arms are parallel/anti-parallel to the direction of absolute motion. The relativistic Fitzgerald-Lorentz contraction effect causes the arm AB
158
C ?
L (a)
(b) 6
-
-
A ? D
L
-
B
C C C C v C CW C α C C- B A1 A2C C D
Figure 4: Schematic diagrams of the Michelson Interferometer, with beamsplitter/mirror at A and mirrors at B and C on arms from A, with the arms of equal length L when at rest. D is a screen or detector. In (a) the interferometer is at rest in space. In (b) the interferometer is moving with speed v relative to space in the direction indicated. Interference fringes are observed at the detector D. If the interferometer is rotated in the plane through 90o , the roles of arms AC and AB are interchanged, and during the rotation shifts of the fringes are seen in the case of absolute motion, but only if the apparatus operates in a gas. By measuring fringe shifts the speed v may be determined. parallel to the absolute velocity to be physically contracted to length (see Fig.4) s
L|| = L
1−
v2 . c2
(64)
The time tAB to travel AB is set by V tAB = L|| + vtAB , while for BA by V tBA = L|| − vtBA , where V = c/n is the speed of light, with n the refractive index of the gas present. For simplicity we ignore here the Fresnel drag effect, an effect caused by the gas also being in absolute motion, see [1]. The Fresnel drag effect is actually large, and results in a change of sign in (67) and (68). For the total ABA travel time we then obtain tABA = tAB + tBA
2LV = 2 V − v2
s
1−
v2 . c2
(65)
For travel in the AC direction we have, from the Pythagoras theorem for the right-angled triangle in Fig.4 that (V tAC )2 = L2 + (vtAC )2 and that tCA = tAC . Then for the total ACA travel time 2L tACA = tAC + tCA = √ . (66) V 2 − v2 Then the difference in travel time is (n2 − 1) L v 2 v4 ∆t = + O c c2 c4
159
!
.
(67)
after expanding in powers of v/c. This clearly shows that the interferometer can only operate as a detector of absolute motion when not in vacuum (n = 1), namely when the light passes through a gas, as in the early experiments (in transparent solids a more complex phenomenon occurs). A more general analysis [1] with the arms at angle θ to v gives ∆t = k 2
LvP2 cos(2(θ − ψ)), c3
(68)
where ψ specifies the direction of v projected onto the plane of the interferometer relative to the local meridian, and where k 2 ≈ n(n2 − 1). Neglect of the relativistic Fitzgerald-Lorentz contraction effect gives k 2 ≈ n3 ≈ 1 for gases, which is essentially the Newtonian theory that Michelson used. However the above analysis does not correspond to how the interferometer is actually operated. That analysis does not actually predict fringe shifts for the field of view would be uniformly illuminated, and the observed effect would be a changing level of luminosity rather than fringe shifts. As Miller knew, the mirrors must be made slightly non-orthogonal with the degree of non-orthogonality determining how many fringe shifts were visible in the field of view. Miller experimented with this effect to determine a comfortable number of fringes: not too few and not too many. Hicks [27] developed a theory for this effect – however it is not necessary to be aware of the details of this analysis in using the interferometer: the non-orthogonality reduces the symmetry of the device, and instead of having period of 180◦ the symmetry now has a period of 360◦ , so that to (68) we must add the extra term a cos(θ − β) in ∆t = k 2
L(1 + eθ)vP2 cos(2(θ − ψ)) + a(1 + eθ) cos(θ − β) + f c3
(69)
The term 1 + eθ models the temperature effects, namely that as the arms are uniformly rotated, one rotation taking several minutes, there will be a temperature induced change in the length of the arms. If the temperature effects are linear in time, as they would be for short time intervals, then they are linear in θ. In the Hick’s term the parameter a is proportional to the length of the arms, and so also has the temperature factor. The term f simply models any offset effect. Michelson and Morley and Miller took these two effects into account when analysing his data. The Hick’s effect is particularly apparent in the Miller and Michelson-Morley data. The interferometers are operated with the arms horizontal. Then in (69) θ is the azimuth of one arm relative to the local meridian, while ψ is the azimuth of the absolute motion velocity projected onto the plane of the interferometer,
160
with projected component vP . Here the Fitzgerald-Lorentz contraction is a real dynamical effect of absolute motion, unlike the Einstein spacetime view that it is merely a spacetime perspective artifact, and whose magnitude depends on the choice of observer. The instrument is operated by rotating at a rate of one rotation over several minutes, and observing the shift in the fringe pattern through a telescope during the rotation. Then fringe shifts from six (Michelson and Morley) or twenty (Miller) successive rotations are averaged to improve the signal to noise ratio, and the average sidereal time noted.
9.3 Michelson-Morley Experiment 1887 Page 340 of the Michelson-Morley 1887 paper reporting the observed fringe shifts is reproduced in Fig.5. Each row of the table is the average from six successive rotations. In the graphs Michelson and Morley are noting that the fringe shifts are much smaller than expected. But they were using Newtonian physics to calibrate the device. We now know that the detector is nearly 2000 times less sensitive than given by that calibration, and that these fringe shifts correspond to a speed in excess of 300km/s. Michelson and Morley implicitly assumed the Newtonian value k=1, while Miller used an indirect method to estimate the value of k, as he understood that the Newtonian theory was invalid, but had no other theory for the interferometer. Of course the Einstein postulates, as distinct from Special Relativity, have that absolute motion has no meaning, and so effectively demands that k = 0. Using k = 1 gives only a nominal value for vP , being some 8–9 km/s for the Michelson and Morley experiment, and some 10 km/s from Miller; the difference arising from the different latitudes of Cleveland and Mt. Wilson, and from Michelson and Morley taking data at limited times. The results from fitting the form in (69) to the data is shown in Fig.6. Most significantly we see that the projected speed and direction vary considerably for the same times on successive days. This effect was seen in later experiments. These are the ‘gravitational waves’ of the induced metric in (52). So we now understand that Michelson and Morley in 1887 detected a dynamical 3-space, and one in which the 3-space velocity fluctuations, the ‘gravitational waves’, were indeed apparent.
9.4 Miller Experiment 1925/26 The Michelson and Morley air-mode interferometer fringe shift data was based upon a total of only 36 rotations in July 1887, revealing the nominal speed of some 8–9 km/s when analysed using the prevailing but incorrect Newtonian theory which has k = 1 in (69), and this value was known to Michelson and Morley.
161
Figure 5: Page 340 from the 1887 Michelson-Morley paper [2] showing the table of observed fringe shifts, measured here in divisions of the telescope screw thread, and which is analysed using (69) with the results shown in Fig.6.
162
Figure 6: Analysis of the Michelson-Morley fringe shift data from the table in Fig.5. The plots are for the sidereal times and days indicated, and each plot arises from averaging six successive rotations, i.e. only 36 rotations were performed in July 1887. The data was fitted with (69) by a 6 parameter least-squares-fit by varying vP , ψ, a, β, e and f . Only vP and ψ are of physical interest, and are shown in each plot. ψ is measured clockwise from North. After these parameters have been determined the Hicks and temperature terms were subtracted from the data, and plotted above together with the cos 2(θ − ψ) expression. This makes the fringe shifts more easily seen. We see that four of the plots show a good fit to the expected form, while the other two give a poor fit. We also see that at the same time on successive days the speed and direction are significantly different. These are ‘gravitational wave’ effects, and were seen in later experiments as well.
163
Figure 7: Typical Miller rotation-induced fringe shifts from average of 20 rotations, measured every 22.5◦ , in fractions of a wavelength ∆λ/λ, vs arm azimuth θ(deg), measured clockwise from North, from Cleveland Sept. 29, 1929 16:24 UT; 11:29 hrs average local sidereal time. The curve is the best fit using the form in (69), and then subtracting the Hick’s cos(θ − β) and temperature terms from the data. Best fit gives ψ = 158◦ , or 22◦ measured from South, and a projected speed of vP = 315 km/s. This plot shows the high quality of the Miller fringe shift observations. In the 1925/26 run of observations the rotations were repeated some 8,000 times.
Including the Fitzgerald-Lorentz dynamical contraction effect as well as the effect of the gas present as in (69) we find that nair = 1.00029 gives k 2 = 0.00058 for air, which explains why the observed fringe shifts were so small. They rejected their own data on the sole but spurious ground that the value of 8 km/s was smaller than the speed of the Earth about the Sun of 30km/s. What their result really showed was that (i) absolute motion had been detected because fringe shifts of the correct form, as in (69), had been detected, and (ii) that the theory giving k 2 = 1 was wrong, that Newtonian physics had failed. Michelson and Morley in 1887 should have announced that the speed of light did depend of the direction of travel, that the speed was relative to an actual physical 3-space. However contrary to their own data they concluded that absolute motion had not been detected. This has had enormous implications for fundamental theories of space and time over the last 100 years. It was Miller [3] who recognised that in the 1887 paper the theory for the Michelson interferometer must be wrong. To avoid using that theory Miller introduced the scaling factor k, even though he had no theory for its value. He then used the effect of the changing vector addition of the Earth’s orbital velocity and the absolute galactic velocity of the solar system to determine the numerical value of k, because the orbital motion modulated the data, as shown in Fig.8. By making some 8,000 rotations of the interferometer at Mt. Wilson in 1925/26
164
August
40
April
40 20 0 - 20 - 40
20 0 - 20 - 40 0
5
10
15
20
25
0
40 September
40
20
20
0
0
- 20
- 20
- 40
- 40 0
5
10
15
20
5
10
15
20
15
20
February
0
5
10
Figure 8: Miller azimuths ψ, measured from south and plotted against sidereal time in hrs, showing both data and best fit of theory giving v = 433 km/s in the direction (α = 5.2hr , δ = −670 ), using n = 1.000226 appropriate for the altitude of Mt. Wilson. The variation form month to month arises from the orbital motion of the earth about the sun: in different months the vector sum of the galactic velocity of the solar system with the orbital velocity and sun in-flow velocity is different. As shown in Fig.9 DeWitte using a completely different experiment detected the same direction and speed.
Miller determined the first estimate for k and for the absolute linear velocity of the solar system. Fig.7 shows typical data from averaging the fringe shifts from 20 rotations of the Miller interferometer, performed over a short period of time, and clearly shows the expected form in (69). In Fig.7 the fringe shifts during rotation are given as fractions of a wavelength, ∆λ/λ = ∆t/T , where ∆t is given by (69) and T is the period of the light. Such rotation-induced fringe shifts clearly show that the speed of light is different in different directions. The claim that Michelson interferometers, operating in gas-mode, do not produce fringe shifts under rotation is clearly incorrect. But it is that claim that lead to the continuing belief, within physics, that absolute motion had never been detected, and that the speed of light is invariant. The value of ψ from such rotations together lead to plots like those in Fig.8, which show ψ from the 1925/1926 Miller [3] interferometer data for four different months of the year, from which the RA = 5.2 hr is readily apparent. While the orbital motion of the Earth about the Sun slightly affects the RA in each month, and Miller used this effect to determine the value of
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k, the new theory of gravity required a reanalysis of the data , revealing that the solar system has a large observed galactic velocity of some 420±30 km/s in the direction (RA = 5.2 hr, Dec =−67◦ ). This is different from the speed of 369 km/s in the direction (RA = 11.20 hr, Dec =−7.22◦ ) extracted from the Cosmic Microwave Background (CMB) anisotropy, and which describes a motion relative to the distant universe, but not relative to the local 3-space. The Miller velocity is explained by galactic gravitational in-flows [1]. An important implication of the new understanding of the Michelson interferometer is that vacuum-mode resonant cavity experiments should give a null effect, as is the case [28].
9.5 Other Gas-mode Michelson Interferometer Experiments Two old interferometer experiments, by Illingworth [19] and Joos [20], used helium, enabling the refractive index effect to be recently confirmed, because for helium, with n = = 1.000036, we find that k 2 = 0.00007. Until the refractive index effect was taken into account the data from the helium-mode experiments appeared to be inconsistent with the data from the air-mode experiments; now they are seen to be consistent. Ironically helium was introduced in place of air to reduce any possible unwanted effects of a gas, but we now understand the essential role of the gas. The data from an interferometer experiment by Jaseja et al. [21], using two orthogonal masers with a He-Ne gas mixture, also indicates that they detected absolute motion, but were not aware of that as they used the incorrect Newtonian theory and so considered the fringe shifts to be too small to be real, reminiscent of the same mistake by Michelson and Morley. While the Michelson interferometer is a 2nd order device, as the effect of absolute motion is proportional to (v/c)2 , as in (69), but 1st order devices are also possible and the coaxial cable experiments described next are in this class.
9.6 Coaxial Cable Speed of EM Waves Anisotropy Experiments Rather than use light travel time experiments to demonstrate the anisotropy of the speed of light, another technique is to measure the one-way speed of radio waves through a coaxial electrical cable. While this not a direct ‘ideal’ technique, as then the complexity of the propagation physics comes into play, it provides not only an independent confirmation of the light anisotropy effect, but also one which takes advantage of modern electronic timing technology.
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20 15 10
ns
5 0 -5 - 10 - 15 0
10
20
30 40 Sidereal Time
50
60
70
Figure 9: (a) Variations in twice the one-way travel time, in ns, for an RF signal to travel 1.5 km through a coaxial cable between Rue du Marais and Rue de la Paille, Brussels. An offset has been used such that the average is zero. The cable has a North-South orientation, and the data is ± difference of the travel times for NS and SN propagation. The sidereal time for maximum effect of ∼ 5 hr and ∼ 17 hr (indicated by vertical lines) agrees with the direction found by Miller. Plot shows data over 3 sidereal days and is plotted against sidereal time. The fluctuations are evidence of turbulence of gravitational waves. (b) Shows the speed fluctuations, essentially ‘gravitational waves’ observed by De Witte in 1991 from the measurement of variations in the RF coaxial-cable travel times. This data is obtained from that in (a) after removal of the dominant effect caused by the rotation of the Earth. Ideally the velocity fluctuations are three-dimensional, but the De Witte experiment had only one arm. This plot is suggestive of a fractal structure to the velocity field. This is confirmed by the power law analysis in [11, 23].
9.7 Torr-Kolen Coaxial Cable Anisotropy Experiment The first one-way coaxial cable speed-of-propagation experiment was performed at the Utah University in 1981 by Torr and Kolen. This involved two rubidium clocks placed approximately 500 m apart with a 5 MHz radio frequency (RF) signal propagating between the clocks via a buried EW nitrogen-filled coaxial cable maintained at a constant pressure of 2 psi. Torr and Kolen found that, while the round-trip speed time remained constant within 0.0001% c, as expected from Sect.7, variations in the one-way travel time were observed. The maximum effect occurred, typically, at the times predicted using the Miller galactic velocity, although Torr and Kolen appear to have been unaware of the Miller experiment. As well Torr and Kolen reported fluctuations in both the magnitude, from 1–3 ns, and the time of maximum variations in travel time. These effects are interpreted as arising from the turbulence in the flow of space past the Earth.
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9.8 De Witte Coaxial Cable Anisotropy Experiment During 1991 Roland De Witte performed a most extensive RF coaxial cable travel-time anisotropy experiment, accumulating data over 178 days. His data is in complete agreement with the Michelson-Morley 1887 and Miller 1925/26 interferometer experiments. The Miller and De Witte experiments will eventually be recognised as two of the most significant experiments in physics, for independently and using different experimental techniques they detected essentially the same velocity of absolute motion. But also they detected turbulence in the flow of space past the Earth — none other than gravitational waves. The De Witte experiment was within Belgacom, the Belgium telecommunications company. This organisation had two sets of atomic clocks in two buildings in Brussels separated by 1.5 km and the research project was an investigation of the task of synchronising these two clusters of atomic clocks. To that end 5MHz RF signals were sent in both directions through two buried coaxial cables linking the two clusters. The atomic clocks were caesium beam atomic clocks, and there were three in each cluster: A1, A2 and A3 in one cluster, and B1, B2, and B3 at the other cluster. In that way the stability of the clocks could be established and monitored. One cluster was in a building on Rue du Marais and the second cluster was due south in a building on Rue de la Paille. Digital phase comparators were used to measure changes in times between clocks within the same cluster and also in the one-way propagation times of the RF signals. At both locations the comparison between local clocks, A1-A2 and A1-A3, and between B1-B2, B1-B3, yielded linear phase variations in agreement with the fact that the clocks have not exactly the same frequencies together with a short term and long term phase noise. But between distant clocks A1 toward B1 and B1 toward A1, in addition to the same linear phase variations, there is also an additional clear sinusoidal-like phase undulation with an approximate 24 hr period of the order of 28 ns peak to peak, as shown in Fig. 9. The experiment was performed over 178 days, making it possible to measure with an accuracy of 25 s the period of the phase signal to be the sidereal day (23 hr 56 min). Changes in propagation times were observed over 178 days from June 3 to November 27, 1991. A sample of the data, plotted against sidereal time for just three days, is shown in Fig.9. De Witte recognised that the data was evidence of absolute motion but he was unaware of the Miller experiment and did not realise that the Right Ascensions for minimum/maximum propagation time agreed almost exactly with that predicted using the Miller’s direction (RA = 5.2 hr, Dec =−67◦ ). In fact De Witte expected that the direction of absolute motion should have been in the CMB direction, but that would have given the data a to-
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tally different sidereal time signature, namely the times for maximum/minimum would have been shifted by 6 hrs. The declination of the velocity observed in this De Witte experiment cannot be determined from the data as only three days of data are available. The De Witte data is analysed in [24] and assuming a declination of 60◦ S a speed of 430 km/s is obtained, in good agreement with the Miller speed and Michelson-Morley speed. So a different and non-relativistic technique is confirming the results of these older experiments. This is dramatic. De Witte reported the sidereal time of the ‘zero’ cross-over time, that is in Fig.9 for all 178 days of data. That showed that the time variations are correlated with sidereal time and not local solar time. A least-squares best fit of a linear relation to that data gives that the cross-over time is retarded, on average, by 3.92 minutes per solar day. This is to be compared with the fact that a sidereal day is 3.93 minutes shorter than a solar day. So the effect is certainly galactic and not associated with any daily thermal effects, which in any case would be very small as the cable is buried. Miller had also compared his data against sidereal time and established the same property, namely that the diurnal effects actually tracked sidereal time and not solar time, and that orbital effects were also apparent, with both effects apparent in Fig.8. The dominant effect in Fig.9 is caused by the rotation of the Earth, namely that the orientation of the coaxial cable with respect to the average direction of the flow past the Earth changes as the Earth rotates. This effect may be approximately unfolded from the data leaving the gravitational waves shown in Fig.9, [11, 23]. This is the first evidence that the velocity field describing the flow of space has a complex structure, and is indeed fractal. The fractal structure, i. e. that there is an intrinsic lack of scale to these speed fluctuations, is demonstrated by binning the absolute speeds and counting the number of speeds within each bin, as discussed in [11, 23]. The Miller data also shows evidence of turbulence of the same magnitude. So far the data from four experiments, namely Miller, Torr and Kolen, De Witte and Cahill, show turbulence in the flow of space past the Earth. This is what can be called gravitational waves. This can be understood by noting that fluctuations in the velocity field induce ripples in the mathematical construct known as spacetime, as in (52). Such ripples in spacetime are known as gravitational waves.
9.9 Cahill Coaxial Cable Anisotropy Experiment During 2006 Cahill [24] performed another RF coaxial cable anisotropy experiment. This detector uses a novel timing scheme that overcomes the limitations associated with the two previous coaxial cable experiments. The intention in such
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Figure 10: Top: De Witte data, with sign reversed, from the first sidereal day in Fig.9. This data gives a speed of approximately 430km/s. The data appears to have been averaged over more than 1hr, but still shows wave effects. Middle: Absolute projected speeds vP in the Miller experiment plotted against sidereal time in hours for a composite day collected over a number of days in September 1925. Maximum projected speed is 417 km/s. The data shows considerable fluctuations. The dashed curve shows the non-fluctuating variation expected over one day as the Earth rotates, causing the projection onto the plane of the interferometer of the velocity of the average direction of the space flow to change. If the data was plotted against solar time the form is shifted by many hours. Note that the min/max occur at approximately 5 hrs and 17 hrs, as also seen by De Witte and in the Cahill experiment. Bottom: Data from the Cahill experiment [24] for one sidereal day on approximately August 23, 2006. We see similar variation with sidereal time, and also similar wave structure. This data has been averaged over a running 1hr time interval to more closely match the time resolution of the Miller experiment. These fluctuations are believed to be real wave phenomena of the 3-space. The new experiment gives a speed of 418 km/s. We see remarkable agreement between all three experiments.
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experiments is simply to measure the one-way travel time of RF waves propagating through the coaxial cable. To that end one would apparently require two very accurate clocks at each end, and associated RF generation and detection electronics. However the major limitation is that even the best atomic clocks are not sufficiently accurate over even a day to make such measurements to the required accuracy, unless the cables are of order of a kilometre or so in length, and then temperature control becomes a major problem. The issue is that the time variations are of the order of 25 ps per 10 meters of cable. To measure that requires time measurements accurate to, say, 1 ps. But atomic clocks have accuracies over one day of around 100 ps, implying that lengths of around 1 kilometre would be required, in order for the effect to well exceed timing errors. Even then the atomic clocks must be brought together every day to resynchronise them, or use De Witte’s method of multiple atomic clocks. The new experiment is based on the notion that optical fibers respond differently to coaxial cable with respect to the speed of propagation of EM radiation. Some results are shown in Fig.10 (bottom), and show the earth rotation and wave effects.
9.10 Cahill Optical Fiber Anisotropy Experiment To measure v(r, t) more easily and more accurately a new optical-fiber detector design has been developed by Cahill [25]. The device is very small, very cheap and easily assembled from readily available opto-electronic components. The schematic layout of the detector is given in Fig.11. The detector relies on the phenomenon where the 3-space velocity v(r, t) affects differently the light travel times in the optical fibers, depending on the projection of v(r, t) along the fiber directions. The differences in the light travel times are measured by means of the interference effects in the beam joiner. The difference in travel times is given by ∆t = k 2
LvP2 cos(2θ) c3
where
(70)
(n2 − 1)(2 − n2 ) n is the instrument calibration constant, obtained by taking account of the three key effects: (i) the different light path trajectories, (ii) Lorentz contraction of the fibers, an effect depending on the angle of the fibers to the flow velocity, and (iii) the refractive index effect, including the Fresnel drag effect. Only if n 6= 1 is there a net effect, otherwise when n = 1 the various effects actually cancel. So in this regard the Michelson interferometer has a serious design flaw. This problem has k2 =
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o
ARM 1 He-Ne laser
2x2 beamsplitter --
-
data photodiode 2x2 beamlogger detector joiner
ARM 2 o o
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100mm
Figure 11:
Schematic layout of the interferometric optical-fiber light-speed anisotropy/gravitational wave detector. This is essentially an optical-fiber version of the Michelson interferometer, see Fig.4. Coherent 633nm light from the a He-Ne Laser is split into two lengths of single-mode polarisation preserving fibers by the 2x2 beam splitter. The two fibers take different directions, ARM1 and ARM2, after which the light is recombined in the 2x2 beam joiner, in which the phase differences lead to interference effects that are indicated by the outgoing light intensity, which is measured in the photodiode detector/amplifier, and then recorded in the data logger. The length of one straight section is 100mm, which is the center to center spacing of the plastic turners. The relative travel times, and hence the output light intensity, are affected by the varying speed and direction of the flowing 3-space, by affecting differentially the speed of the light, and hence the net phase difference between the two arms.
been overcome by using optical fibers. Here n = 1.462 at 633nm is the effective refractive index of the single-mode optical fibers (Fibercore SM600, temperature coefficient 5 × 10−2 fs/mm/C). Here L ≈ 200mm is the average effective length of the two arms, and vP (r, t) is the projection of v(r, t) onto the plane of the detector, and the angle θ is that of the projected velocity onto the arm. The interferometer operates by detecting the travel time difference between the two arms as given by (70). The cycle-averaged light intensity emerging from the beam joiner is given by I(t) ∝
2
Re(E1 + E2 eiω(τ +∆t )
ω(τ + ∆t) = 2|E| cos 2 ≈ a + b∆t 2
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2
(71)
Figure 12: D1 photodiode output voltage data (mV), recorded every 5 secs, from 5 successive days, starting September 22, 2007, plotted against local Adelaide time (UT= local time + 9.5hrs). Day sequence may be determined by identifying identical values at 0 and 24hrs. Dominant minima and maxima is earth rotation effect. Fluctuations from day to day are evident as are fluctuations during each day - these are caused by wave effects in the flowing space. Changes in RA cause changes in timing of min/max, while changes in magnitude are caused by changes in declination and/or speed. Blurring effect is caused by laser noise. These day plots correspond to the plots in Fig.10, there plotted against local sidereal time, and also inverted.
Here Ei are the electric field amplitudes and have the same value as the fiber splitter/joiner are 50%-50% types, and having the same direction because polarisation preserving fibers are used, ω is the light angular frequency and τ is a travel time difference caused by the light travel times not being identical, even when ∆t = 0, mainly because the various splitter/joiner fibers will not be identical in length. The last expression follows because ∆t is small, and so the detector operates in a linear regime, in general, unless τ has a value equal to modulo(T ), where T is the light period. The main temperature effect in the detector, so long as a temperature uniformity is maintained, is that τ will be temperature dependent. The temperature coefficient for the optical fibers gives an effective fractional fringe shift error of 3 × 10−2 /mm/C, for each mm of length difference. The photodiode detector output voltage V (t) is proportional to I(t), and so finally linearly related to ∆t. The detector calibration constants a and b depend on k, τ and the laser intensity and are unknown at present.
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Figure 13:
Photodiode data (mV) on October 4, 2007, from detectors D1 and D2 operating simultaneously with D2 located 1.1km due north of D1. A low-pass FFT filter (f ≤ 0.25mHz, Log10 [f(mHz)] ≤ -0.6) was used to remove laser noise. D1 arm is aligned 50 anti-clockwise from local meridian, while D2 is aligned 110 anti-clockwise from local meridian. The alignment offset between D1 and D2 causes the dominant earth-rotation induced minima to occur at different times, with that of D2 at t = 7.6hrs delayed by 0.8hrs relative to D1 at t = 6.8hrs. This is a fundamental test of the detection theory and of the phenomena. As well the data shows a simultaneous sub-mHz gravitational wave correlation at t ≈ 8.8hrs and of duration ≈ 1hr. This is the first observed correlation for spatially separated gravitational wave detectors. Two other wave effects (at t ≈ 6.5hrs in D2 and t ≈ 7.3hrs in D1) seen in one detector are masked by the stronger earth-rotation induced minimum in the other detector.
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Two detectors were used with each detector located inside a sealed air-filled bucket located inside an insulated container containing some 90kg of water for temperature stabilisation. Using two detectors enabled the confirmation of expected phenomena, as a test of the detector theory, and also enabled the simultaneous observations of wave phenomena. Detector D1 was in the School of Chemistry, Physics and Earth Sciences, with an arm orientation of 50 anticlockwise to the local meridian. Detector D2 was located 1.1km North of D1 in the Australian Science and Mathematics School. This detector had an arm orientation of 110 anti-clockwise to the local meridian. Fig.12 shows data from D1 over 5 days. Fig.13 shows an effect caused by D1 and D2 having different arm orientations and, as well, a simultaneous sub-mHz gravitational wave correlation at t ≈ 8.8hrs and of duration ≈ 1hr. This is the first observed correlation for spatially separated gravitational wave detectors. There are now at least 11 detections of the velocity field v(r, t) and of these 6 have observed the 3-space wave/turbulence effect.
10 Experimental and Observational Phenomena II 10.1 Gravitational Phenomena We have shown above that the dynamics of 3-space involves two constants: G and α. When generalising the Schr¨odinger and Dirac equations to take account of this 3-space we discovered that we arrive at an explanation for the phenomenon of gravity including the equivalence principle, as well as an explanation for the spacetime formalism. Here we explore various consequences of this new explanation for gravity particularly those effects which reveal the effects of the α-dependent dynamics, in particular the bore hole anomaly which gives us the best estimate for the value of α from several bore hole experiments. The dynamical 3-space also gives a completely new account of black holes; an account completely different from the putative black holes of GR. In particular these new black holes generate an acceleration g that varies essentially as 1/r, rather than as 1/r2 as in Newtonian gravity (NG) and GR. This is a dramatic difference. It explains immediately the rotation of spiral galaxies, for which the rotation speed is essen√ tially constant at the outer limits, whereas NG and GR predict a 1/ r Keplerian form. It was this dramatic failure of NG and GR, and also in galactic clusters, that lead to the introduction of ‘dark matter’ - to generate a greater gravitational acceleration. The new theory of 3-space does not need this ‘dark matter’. The black hole phenomena is complex, with minimal black holes induced by matter, to primordial black holes that attract matter. In the former case, and where the
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matter, in the form of stars and so on, has an essentially spherically symmetric distribution, it is possible to compute the effective mass of the induced minimal black holes. Observational data from these systems confirms the prediction. Other effects discussed are the gyroscope precession effect caused by the vorticity of the flow of 3-space past the earth. Finally we also discuss the cosmological Hubble expansion that arises from the 3-space dynamics. This gives an excellent parameter-free account of the redshift data from supernovae and gamma-ray bursts. GR requires ‘dark energy’ to fit that data, so here we see that the new 3-space dynamics does away with the need for ‘dark energy’. Not discussed herein are anomalies in the Cavendish-like experiments to determine G [38], the gravitational lensing effects predicted by the generalised Maxwell equations, and also a re-analysis of the precession of elliptical orbits, particularly that of Mercury, and various other gravitational effects, see [1].
10.2 Bore Hole Anomaly and the Fine Structure Constant We now show that the Airy method [34] originally proposed for measuring G actually gives a technique for determining the value of α from earth based bore hole gravity measurements. For a time-independent velocity field (7) may be written in the integral form |v(r)|2 = 2G
Z
d3 r 0
ρ(r0 ) + ρDM (r0 ) . |r − r0 |
(72)
When the matter density of the earth is assumed to be spherically symmetric, and that the velocity field is now radial6 (72) becomes v(r)2 =
8πG r
Z
r
s2 [ρ(s) + ρDM (s)] ds + 8πG
Z
0
∞
s [ρ(s) + ρDM (s)] ds,
(73)
r
where, with v 0 = dv(r)/dr, v2 vv 0 + 2r2 r
α ρDM (r) = 8πG
!
.
(74)
Iterating (73) once we find to 1st order in α that α ρDM (r) = 2 2r 6
∞
Z
sρ(s)ds + O(α2 ),
r
This in-flow is additional to the observed velocity of the earth through 3-space.
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(75)
so that in spherical systems the ‘dark matter’ effect is concentrated near the centre, and we find that the total ‘dark matter’ is Z ∞ Z 4πα ∞ 2 α 2 r ρDM (r)dr = r ρ(r)dr + O(α2 ) = M + O(α2 ) (76) MDM ≡ 4π 2 2 0 0 where M is the total amount of (actual) matter. Hence to O(α) MDM /M = α/2 independently of the matter density profile. This turns out to be a very useful property as complete knowledge of the density profile is then not required in order to analyse observational data. As seen in Fig.14 the singular behaviour of both v and g means that there is a black hole7 singularity at r = 0. From (2), which is also the acceleration of matter [11], the gravity acceleration8 is found to be, to 1st order in α, and using that ρ(r) = 0 for r > R, where R is the radius of the earth, α (1 + )GM 2 , r > R, rZ2 ! g(r) = (77) Z R Z r 4πG 2παG r 0 2 0 0 s ρ(s )ds ds, r < R. s ρ(s)ds + r2 r2 s 0 0 This gives Newton’s ‘inverse square law’ for r > R, even when α 6= 0, which explains why the 3-space self-interaction dynamics did not overtly manifest in the analysis of planetary orbits by Kepler and then Newton. However inside the earth (77) shows that g(r) differs from the Newtonian theory, corresponding to α = 0, as in Fig.14, and it is this effect that allows the determination of the value of α from the Airy method. Expanding (77) in r about the surface, r = R, we obtain, to 1st order in α and for an arbitrary density profile, but not retaining any density gradients at the surface,
g(r) =
GN M 2GN M (r − R), 2 − R R3
r > R, (78)
GN M 2GN M α − 4π(1 − )GN ρ (r − R), 2 − 3
R
R
2
r R, 2 R R3 gN (r) = (80) GN M 2GN M − 4πGN ρ (r − R), r < R 2 − R R3 Assuming Newtonian gravity (80) then means that from the measurement of difference between the above-ground and below-ground gravity gradients, namely 4πGN ρ, and also measurement of the matter density, permit the determination of GN . This is the basis of the Airy method for determining GN [34]. When analysing the bore hole data it has been found [35, 36] that the observed difference of the gravity gradients was inconsistent with 4πGN ρ in (80), in that it was not given by the laboratory value of GN and the measured matter density. This is known as the bore hole g anomaly and which attracted much interest in the 1980’s. The bore hole data papers [35, 36] report the discrepancy, i.e. the anomaly or the gravity residual as it is called, between the Newtonian prediction and the measured below-earth gravity gradient. Taking the difference between (78) and (80), assuming the same unknown value of GN in both, we obtain an expression for the gravity residual (
∆g(r) ≡ gN (r) − g(r) =
0, r > R, 2παGN ρ(r − R), r < R.
(81)
When α 6= 0 we have a two-parameter theory of gravity, and from (78) we see that measurement of the difference between the above ground and below ground gravity gradients is 4π(1− α2 )GN ρ, and this is not sufficient to determine both GN and α, given ρ, and so the Airy method is now understood not to be a complete measurement by itself, i.e. we need to combine it with other measurements. If we now use laboratory Cavendish experiments to determine GN , then from the bore hole gravity residuals we can determine the value of α, as already indicated in [39, 40]. These Cavendish experiments can only determine GN up to corrections of order α/4, simply because the analysis of the data from these experiments assumed the validity of Newtonian gravity [1]. So the analysis of the bore hole residuals will give the value of α up to O(α2 ) corrections, which is consistent with the O(α) analysis reported above. Gravity residuals from a bore hole into the Greenland Ice Shelf were determined down to a depth of 1.5 km by Ander et al. [35] in 1989. The observations were made at the Dye 3 2033 m deep bore hole, which reached the basement
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Figure 15:
The data shows the gravity residuals for the Greenland Ice Shelf [35] Airy measurements of the g(r) profile, defined as ∆g(r) = gN ewton −gobserved , and measured in mGal (1mGal = 10−3 cm/s2 ) and plotted against depth in km. The bore hole effect is that Newtonian gravity and the new theory differ only beneath the surface, provided that the measured above surface gravity gradient is used in both theories. This then gives the horizontal line above the surface. Using (81) we obtain α−1 = 137.9 ± 5 from fitting the slope of the data, as shown. The non-linearity in the data arises from modelling corrections for the gravity effects of the irregular sub ice-shelf rock topography.
Figure 16: Gravity residuals from two of the Nevada bore hole experiments [36] that give a best fit of α−1 = 136.8 ± 3 on using (81). Some layering of the rock is evident.
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rock. This bore hole is 60 km south of the Arctic Circle and 125 km inland from the Greenland east coast at an elevation of 2530 m. It was believed that the ice provided an opportunity to use the Airy method to determine GN , but now it is understood that in fact the bore hole residuals permit the determination of α, given a laboratory value for GN . Various steps were taken to remove unwanted effects, such as imperfect knowledge of the ice density and, most dominantly, the terrain effects which arises from ignorance of the profile and density inhomogeneities of the underlying rock. The bore hole gravity meter was calibrated by comparison with an absolute gravity meter. The ice density depends on pressure, temperature and air content, with the density rising to its average value of ρ = 920 kg/m3 within some 200 m of the surface, due to compression of the trapped air bubbles. This surface gradient in the density has been modelled by the author, and is not large enough the affect the results. The leading source of uncertainty was from the gravitational effect of the bedrock topography, and this was corrected for using Newtonian gravity. The correction from this is actually the cause of the non-linearity of the data points in Fig.15. A complete analysis would require that the effect of this rock terrain be also computed using the new theory of gravity, but this was not done. Using GN = 6.6742 × 10−11 m3 s−2 kg−1 , which is the current CODATA value, we obtain from a least-squares fit of the linear term in (81) to the data points in Fig.15 that α−1 = 137.9±5, which equals the value of the fine structure constant α−1 = 137.036 to within the errors, and for this reason we identify the constant α in (81) as being the fine structure constant. The first analysis [39, 40] of the Greenland Ice Shelf data incorrectly assumed that the ice density was 930 kg/m3 which gave α−1 = 139 ± 5. However trapped air reduces the standard ice density to the ice shelf density of 920 kg/m3 , which brings the value of α immediately into better agreement with the value of α = e2 /¯hc known from quantum theory. Thomas and Vogel [36] performed another bore hole experiment at the Nevada Test Site in 1989 in which they measured the gravity gradient as a function of depth, the local average matter density, and the above ground gradient, also known as the free-air gradient. Their intention was to test the extracted Glocal and compare with other values of GN , but of course using the Newtonian theory. The Nevada bore holes, with typically 3 m diameter, were drilled as a part of the U.S. Government tests of its nuclear weapons. The density of the rock is measured with a γ − γ logging tool, which is essentially a γ-ray attenuation measurement, while in some holes the rock density was measured with a coreing tool. The rock density was found to be 2000 kg/m3 , and is dry. This is the density used in the analysis herein. The topography for 1 to 2 km beneath the surface is dominated by
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Figure 17: The data shows Log10 [MBH ] for the black hole masses MBH for a variety of spherical matter systems with masses M , plotted against Log10 [M ], in solar masses M0 . The straight line is the prediction from (83) with α = 1/137. See [42] for references to the data.
a series of overlapping horizontal lava flows and alluvial layers. Gravity residuals from two of the bore holes are shown in Figs.16. All gravity measurements were corrected for the earth’s tide, the terrain on the surface out to 168 km distance, and the evacuation of the holes. The gravity residuals arise after allowing for, using Newtonian theory, the local lateral mass anomalies but assumed that the matter beneath the holes occurs in homogeneous ellipsoidal layers. We see in Fig.16 that the gravity residuals are linear with depth, where the density is the average value of 2000 kg/m3 , but interspersed by layers where the residuals show non-linear changes with depth. It is assumed here that these non-linear regions are caused by variable density layers. So in analysing this data we have only used the linear regions, and a simultaneous least-squares fit of the slope of (81) to the slopes of these four linear regions gives α−1 = 136.8 ± 3, which again is in extraordinary agreement with the value of 137.04 from quantum theory. Here we again used GN = 6.6742 × 10−11 m3 s−2 kg−1 , as for the Greenland data analysis. Zumberge et al. [37] performed an extensive underwater Airy experiment, but failed to measure the above water g, so their results cannot be analysed in the above manner.
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10.3 Black Hole Masses and the Fine Structure Constant Equation (1) (with β = −α) has ‘black hole’ solutions. The generic term ‘black hole’ is used because they have a compact closed event horizon where the inflow speed relative to the horizon equals the speed of light, but in other respects they differ from the putative black holes of General Relativity - in particular their gravitational acceleration is not inverse square law. The evidence is that it is these new ‘black holes’ from (1) that have been detected. There are two categories: (i) an in-flow singularity induced by the flow into a matter system, such as, herein, a spherical galaxy or globular cluster. These black holes are termed minimal black holes, as their effective mass is minimal, (ii) primordial naked black holes which then attract matter. These result in spiral galaxies, and the effective mass of the black hole is larger than required merely by the matter induced in-flow. These are therefore termed non-minimal black holes. These explain the rapid formation of structure in the early universe, as the gravitational acceleration is approximately 1/r rather than 1/r2 . This is the feature that also explains the so-called ‘dark matter’ effect in spiral galaxies. We consider now the minimal black holes. Equation (1) has exact analytic ‘black hole’ solutions where ρ = 0 (actually a one-parameter family - but we write in this form for comparison with the next section) 1/2 α 1 Rs 2 1 v(r) = K + (82) r Rs r where the 1/r term can only arise if matter is present, and the 2nd term is the ‘black hole’ effect. The consequent ‘black hole’ contribution to the total acceleration can be attributed to an effective mass MDM , which we now also call MBH . To O(α) this effective mass is independent of the matter density profile, and is given by (76), Z
MBH = MDM = 4π
∞
r2 ρDM (r)dr =
0
α M + O(α2 ) 2
(83)
This solution is applicable to the black holes at the centre of spherical star systems, where we identify MDM as MBH . So far black holes in 19 spherical star systems have been detected and together their masses are plotted in Fig.17 and compared with (83) [41, 42]. This result applies to any spherically symmetric matter distribution. This means that the bore hole anomaly is indicative of an in-flow singularity at the centre of the earth. This contributes some 0.4% of the effective mass of the earth,
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200
Figure 18:
175 150
V
125 100 75 50 25
5
10
15 r
20
25
Data shows the non-Keplerian rotation-speed curve vO for the spiral galaxy NGC 3198 in km/s plotted against radius in kpc/h. Lower curve is the rotation curve from the Newtonian theory for an exponential disk, √ which decreases asymptotically like 1/ r. The upper curve shows the asymptotic form from (85), with the decrease determined by the small value of α. This asymptotic form is caused by the primordial black holes at the centres of spiral galaxies, and which play a critical role in their formation. The spiral structure is caused by the rapid in-fall towards these primordial black holes.
as defined by Newtonian gravity. However in star systems this minimal black hole effect is more apparent, and we label MDM as MBH . Essentially even in the nonrelativistic regime the Newtonian theory of gravity, with its ‘universal’ Inverse Square Law, is deeply flawed.
10.4 Spiral Galaxies and the Rotation Anomaly Equation (82) gives also a direct explanation for the spiral galaxy rotation anomaly. For a non-spherical system numerical solutions of (1) are required, but sufficiently far from the centre we find an exact non-perturbative two-parameter class of analytic solutions as in (82). There K and Rs are arbitrary constants in the ρ = 0 region, but whose values are determined by matching to the solution in the matter region. Here Rs characterises the length scale of the non-perturbative part of this expression, and K depends on α, G and details of the matter distribution. From (4) and (82) we obtain a replacement for the Newtonian ‘inverse square law’, α 2 K 1 α Rs 2 g(r) = (84) 2 + , 2 r 2rRs r
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in the asymptotic limit. The centripetal acceleration relation for circular orbits p vO (r) = rg(r) gives a ‘universal rotation-speed curve’
vO (r) =
K 1 α + 2 r 2Rs
Rs r
α
1/2
2
(85)
Because of the α dependent part this rotation-velocity curve falls off extremely slowly with r, as is indeed observed for spiral galaxies. An example is shown in Fig.18. It was the inability of the Newtonian gravity and GR to explain these observations that led to the notion of ‘dark matter’. So ‘dark matter’ is not a part of reality. For the spatial flow in (82) we may compute the effective ‘dark matter’ density from (74) (1 − α)α K 2 Rs 2+α/2 ρDM (r) = (86) 16πG Rs3 r We see the standard 1/r2 behaviour usually attributed to ‘dark matter’ in spiral galaxies. It should be noted that the Newtonian component of (82) does not contribute, and that ρDM (r) is exactly zero in the limit α → 0. So supermassive black holes and the spiral galaxy rotation anomaly are all α-dynamics phenomena.
10.5 Lense-Thirring Effect and the GPB Gyroscope Experiment The Gravity Probe B (GP-B) satellite experiment was launched in April 2004. It has the capacity to measure the precession of four on-board gyroscopes to unprecedented accuracy [44, 45, 46, 47]. Such a precession is predicted by GR, with two components (i) a geodetic precession, and (ii) a ‘frame-dragging’ precession known as the Lense-Thirring effect. The latter is particularly interesting effect induced by the rotation of the earth, and described in GR in terms of a ‘gravitomagnetic’ field. According to GR this smaller effect will give a precession of 0.042 arcsec per year for the GP-B gyroscopes. Here we show that GR and the new theory make very different predictions for the ‘frame-dragging’ effect, and so the GP-B experiment will be able to decisively test both theories. While predicting the same earth-rotation induced precession, the new theory has an additional much larger ‘frame-dragging’ effect caused by the observed translational motion of the earth. As well the new theory explains the ‘frame-dragging’ effect in terms of vorticity in a ‘substratum flow’. Herein the magnitude and signature of this
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Figure 19: Shows the earth (N is up) and vorticity vector field component ω ~ induced by the rotation of the earth, as in (87). The polar orbit of the GP-B satellite is shown, S is the gyroscope starting spin orientation, directed towards the guide star IM Pegasi, RA = 22h 530 2.2600 , Dec = 160 500 28.200 , and VE is the vernal equinox.
S VE
new component of the gyroscope precession is predicted for comparison with data from GP-B when it becomes available. Here we consider one difference between the two theories, namely that associated with the vorticity part of (12), leading to the ‘frame-dragging’ or LenseThirring effect. In GR the vorticity field is known as the ‘gravitomagnetic’ field B = −c ω ~ . In both GR and the new theory the vorticity is given by (10) but with a key difference: in GR vR is only the rotational velocity of the matter in the earth, whereas in the 3-space dynamics vR is the vector sum of the rotational velocity and the translational velocity of the earth through the substratum. First consider the common but much smaller rotation induced ‘frame-dragging’ or vorticity effect. Then vR (r) = w × r in (12), where w is the angular velocity of the earth, giving G 3(r.L)r − r2 L ω ~ (r) = 4 2 , (87) c 2r5 where L is the angular momentum of the earth, and r is the distance from the centre. This component of the vorticity field is shown in Fig.19. Vorticity may be detected by observing the precession of the GP-B gyroscopes. The vorticity term in (87) leads to a torque on the angular momentum S of the gyroscope, Z
~τ =
d3 rρ(r) r × (~ ω (r) × vR (r)),
(88)
where ρ is its density, and where vR is used here to describe the rotation of the gyroscope. Then dS = ~τ dt is the change in S over the time interval dt. In the
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S VE
v
Figure 20: Shows the earth (N is up) and the much larger vorticity vector field component ω ~ induced by the translation of the earth, as in (90). The polar orbit of the GP-B satellite is shown, and S is the gyroscope starting spin orientation, directed towards the guide star IM Pegasi, RA = 22h 530 2.2600 , Dec = 160 500 28.200 , VE is the vernal equinox, and V is the direction RA = 5.2h , Dec = −670 of the translational velocity vc .
above case vR (r) = s × r, where s is the angular velocity of the gyroscope. This gives 1 ~ ×S (89) ~τ = ω 2 ~ /2 is the instantaneous angular velocity of precession of the gyroscope. and so ω This corresponds to the well known fluid result that the vorticity vector is twice the angular velocity vector. For GP-B the direction of S has been chosen so that this precession is cumulative and, on averaging over an orbit, corresponds to some 7.7 × 10−6 arcsec per orbit, or 0.042 arcsec per year. GP-B has been superbly engineered so that measurements to a precision of 0.0005 arcsec are possible. However for the unique translation-induced precession if we use vR ≈ vC = 430 km/s in the direction RA = 5.2hr , Dec = −670 , namely ignoring the effects of the orbital motion of the earth, the observed flow past the earth towards the sun, and the flow into the earth, and effects of the gravitational waves, then (12) gives 2GM vC × r ω ~ (r) = . (90) c2 r3 This much larger component of the vorticity field is shown in Fig.20. The maximum magnitude of the speed of this precession component is ω/2 = gvC /c2 = 8 × 10−6 arcsec/s, where here g is the gravitational acceleration at the altitude of the satellite. This precession has a different signature: it is not cumulative, and is detectable by its variation over each single orbit, as its orbital average is zero, to first approximation. Fig.21 shows ∆Θ = |∆S(t)|/|S(0)| over one orbit, where,
187
0.016 0.014
arcsec
0.012 0.01 0.008 0.006 0.004 0.002 0
20
40 60 orbit - minutes
80
Figure 21: Predicted variation of the precession angle ∆Θ = |∆S(t)|/|S(0)|, in arcsec, over one 97 minute GP-B orbit, from the vorticity induced by the translation of the earth, as given by (91). The orbit time begins at location S. Predictions are for the months of April, August, September and February, labeled by increasing dash length. The ‘glitches’ near 80 minutes are caused by the angle effects in (91). These changes arise from the effects of the changing orbital velocity of the earth about the sun. The GP-B expected angle measurement accuracy is 0.0005 arcsec.
as in general, Z
∆S(t) = 0
t
1 dt0 ω ~ (r(t0 )) × S(t0 ) ≈ 2
Z 0
t
1 dt0 ω ~ (r(t0 )) × S(0). 2
(91)
Here ∆S(t) is the integrated change in spin, and where the approximation arises because the change in S(t0 ) on the RHS of (91) is negligible. The plot in Fig.21 shows this effect to be some 30× larger than the expected GP-B errors, and so easily detectable, if it exists as predicted herein. This precession is about the instantaneous direction of the vorticity ω ~ (r((t)) at the location of the satellite, and so is neither in the plane, as for the geodetic precession, nor perpendicular to the plane of the orbit, as for the earth-rotation induced vorticity effect. Because the yearly orbital rotation of the earth about the sun slightly effects vC [16], predictions for four months throughout the year are shown in Fig.21. Such yearly effects were first seen in the Miller [3] experiment, see Fig.8.
10.6 Cosmology: Expanding 3-Space and the Hubble Effect We now examine the predictions for the global expansion of the 3-space that follows from (1) (with β = −α). We shall see that the solution gives an excellent
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parameter-free fit to the supernovae and gamma-ray bursts magnitude - redshift data [48]. This implies that there is no need to have a cosmological constant or ‘dark energy’, which are required by GR in order to fit this data. These also lead to the prediction that the universe expansion will accelerate in the future. This effect is also not required by the new 3-space dynamics. So, like ‘dark matter’, ‘dark energy’ is an unnecessary and spurious notion. Let us now explore the expanding 3-space from (1). Critically, and unlike the GR-FLRW model, the 3-space expands even when the energy density is zero. Suppose that we have a radially symmetric effective density ρ(r, t), modelling EM radiation, matter, cosmological constant etc, and that we look for a radially symmetric time-dependent flow v(r, t) = v(r, t)ˆr from (1) (with β = −α). Then ∂v(r, t) v(r, t) satisfies the equation, with v 0 = , ∂r ∂ ∂t
2v vv 0 α + v 0 + vv 00 + 2 + (v 0 )2 + r r 4
v 2 2vv 0 + r2 r
!
= −4πGρ(r, t)
(92)
Consider first the zero energy case ρ = 0. Then we have a Hubble solution v(r, t) = H(t)r, a centreless flow, determined by α H˙ + 1 + H2 = 0 4
(93)
dH with H˙ = . We also introduce in the usual manner the scale factor a(t) dt 1 da according to H(t) = . We then obtain the solution a dt 1 t0 t = H0 ; a(t) = a0 α (1 + 4 )t t t0
H(t) =
4/(4+α)
(94)
where H0 = H(t0 ) and a0 = a(t0 ). Note that we obtain an expanding 3-space even where the energy density is zero - this is in sharp contrast to the GR-FLRW model for the expanding universe, as shown below. We can write the Hubble function H(t) in terms of a(t) via the inverse function t(a), i.e. H(t(a)) and finally as H(z), where the redshift observed now, t0 , relative to the wavelengths at time t, is z = a0 /a − 1. Then we obtain H(z) = H0 (1 + z)1+α/4
(95)
To test this expansion we need to predict the relationship between the cosmological observables, namely the relationship between the apparent energy-flux magnitudes and redshifts. This involves taking account of the reduction in photon
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count caused by the expanding 3-space, as well as the accompanying reduction in photon energy. To that end we first determine the distance travelled by the light from a supernova or GRB event before detection. Using a choice of embeddingspace coordinate system with r = 0 at the location of a supernova/GRB event the speed of light relative to this embedding space frame is c + v(r(t), t), i.e. c wrt the space itself, as noted above, where r(t) is the embedding-space distance from the source. Then the distance travelled by the light at time t after emission at time t1 is determined implicitly by Z
t
r(t) =
dt0 (c + v(r(t0 ), t0 ),
(96)
t1
which has the solution on using v(r, t) = H(t)r Z
t
r(t) = ca(t) t1
dt0 . a(t0 )
(97)
This distance gives directly the surface area 4πr(t)2 of the expanding sphere and so the decreasing photon count per unit of that surface area. However also because of the expansion the flux of photons is reduced by the factor 1/(1 + z), simply because they are spaced further apart by the expansion. The photon flux is then given by LP FP = (98) 4πr(t)2 (1 + z) where LP is the source photon-number luminosity. However usually the energy flux is measured, and the energy of each photon is reduced by the factor 1/(1 + z) because of the redshift. Then the energy flux is, in terms of the source energy luminosity LE , LE LE FE = ≡ (99) 4πr(t)2 (1 + z)2 4πrL (t)2 which defines the effective energy-flux luminosity distance rL (t). Expressed in terms of the observable redshift z this gives an energy-flux luminosity effective distance Z z dz 0 rL (z) = (1 + z)r(z) = c(1 + z) (100) 0 0 H(z ) The dimensionless ‘energy-flux’ luminosity effective distance is then given by Z
dL (z) = (1 + z) 0
190
z
H0 dz 0 H(z 0 )
(101)
Figure 22: Plot of the scale factor R(t) vs t, with t = 0 being ‘now’ with R(0) = 1, for the four cases discussed in the text, and corresponding to the plots in Figs.23 and 24: (i) the upper curve is the ‘dark energy’ only case, resulting in an exponential acceleration at all times, (ii) the bottom curve is the matter only prediction, (iii) the 2nd highest curve (to the right of t = 0) is the best-fit ‘dark energy’ plus matter case showing a past deceleration and future exponential acceleration effect. The straight line plot is the dynamical 3-space prediction showing a slightly older universe compared to case (iii). We see that the best-fit ‘dark energy’ - matter curve essentially converges on the dynamical 3-space result. All plots have the same slope at t = 0, i.e. the same value of H0 . If the age of the universe is inferred to be some 14Gyrs for case (iii) then the age of the universe is also some 14Gyr for case (iv).
and the theory distance modulus is defined by µ(z) = 5 log10 (dL (z)) + m.
(102)
Using the Hubble expansion (95) in (101) and (102) we obtain the curve shown in Figs.23 and 24, yielding an excellent agreement with the supernovae and GRB data. Note that because α/4 is so small it actually has negligible effect on these plots. Hence the dynamical 3-space gives an immediate account of the universe expansion data, and does not require the introduction of a cosmological constant or ‘dark energy’, but which will be nevertheless discussed next. When the energy density is not zero we need to take account of the dependence of ρ(r, t) on the scale factor of the universe. In the usual manner we thus write ρ(r, t) =
ρm ρr + +Λ 3 R(t) R(t)4
(103)
for matter, EM radiation and the cosmological constant or ‘dark energy’ Λ, respectively, where the matter and radiation is approximated by a spatially uniform
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] (dots without error bars for clarity - see Fig.24 for error bars) and the Gamma-Ray Bursts data (with error bars) from Schaefer [52]. Upper curve is ‘dark energy’ only ΩΛ = 1, lowest curve is matter only Ωm = 1. Two middle curves show best fit of ‘dark energy’-matter and dynamical 3space prediction, and are essentially indistinguishable. However the theories make very different predictions for the future and for the age of the universe. We see that the best-fit ‘dark energy’ - matter curve essentially converges on the dynamical 3-space prediction.
192
Figure 24:
Hubble diagram as in Fig.23 but plotted logarithmically to reveal details for z < 2, and without GRB data. Upper curve is ‘dark energy’ only ΩΛ = 1. Next curve is best fit of ‘dark energy’-matter. Lowest curve is matter only Ωm = 1. 2nd lowest curve is dynamical 3-space prediction.
193
(i.e independent of r) equivalent matter density. We argue here that Λ - the dark energy density, like dark matter, is an unnecessary concept. Then (92) becomes for R(t) ¨ α R˙ 2 R 4πG ρm ρr =− + 4 +Λ + R 4 R2 3 R3 R
(104)
giving 8πG R˙ 2 = 3
ρm α ρr + 2 + ΛR2 − R R 2
Z
R˙ 2 dR R
(105)
In terms of R˙ 2 this has the solution !
8πG ΛR2 ρm ρr R = + + +bR−α/2 α α 3 (1 − 2 )R (1 − 4 )R2 (1 + α4 ) ˙2
(106)
which is easily checked by substitution into (105), and where b is an arbitrary integration constant. Finally we obtain from (106) Z
R
t(R) = R0
dR s
8πG 3
ρm ρr + 2 + ΛR2 + bR−α/2 R R
(107)
where now we have re-scaled parameters ρm → ρm /(1 − α2 ), ρr → ρr /(1 − α4 ) and Λ → Λ/(1 + α4 ). When ρm = ρr = Λ = 0, (107) reproduces the expansion in (94), and so the density terms in (107) give the modifications to the dominant purely spatial expansion dynamics, which we have noted above already gives an excellent account of the data. From (107) we then obtain H(z)2 = H0 2 (Ωm (1 + z)3 + Ωr (1 + z)4 + ΩΛ + Ωs (1 + z)2+α/2 )
(108)
with Ωm + Ωr + ΩΛ + Ωs = 1.
(109)
Using the Hubble function (108) in (101) and (102) we obtain the plots in Figs.23 and 24 for four cases: (i) Ωm = 0, Ωr = 0, ΩΛ = 1, Ωs = 0, i.e a pure ‘dark energy’ driven expansion, (ii) Ωm = 1, Ωr = 0, ΩΛ = 0, Ωs = 0 showing that a matter only expansion is not a good account of the data, (iii) from a least squares fit with Ωs = 0 we find Ωm = 0.28, Ωr = 0, ΩΛ = 0.68 which led to the suggestion that ‘dark energy’ effect was needed to fix the poor fit from (ii), and finally (iv) Ωm = 0, Ωr = 0, ΩΛ = 0, Ωs = 1, as noted above, that the spatial expansion dynamics alone gives a good account of the data. Of course the EM
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radiation term Ωr is non-zero but small and determines the expansion during the baryogenesis initial phase, as does the spatial dynamics expansion term because of the α dependence. If the age of the universe is inferred to be some 14Gyrs for case (iii) then, as seen in Fig.22, the age of the universe is also some 14Gyr for case (iv). We see that the best-fit ‘dark energy’ - matter curve essentially converges on the dynamical 3-space result. The induced effective spacetime metric in (52) is for the Hubble expansion ds2 = gµν dxµ dxν = dt2 − (dr − H(t)r)dt)2 /c2
(110)
The occurrence of c has nothing to do with the dynamics of the 3-space - it is related to the geodesics of relativistic quantum matter, as shown in Sect.8. Changing variables r → R(t)r we obtain ds2 = gµν dxµ dxν = dt2 − R(t)2 dr2 /c2
(111)
which is the usual Friedman-Robertson-Walker (FRW) metric in the case of a flat spatial section. However when solving for R(t) using the Hilbert-Einstein GR equations the Ωs term (with α → 0) is usually only present when the spatial curvature is non-zero. So some problem appears to be present in the usual GR analysis of the FRW metric. However above we see that that term arises in fact even when the embedding space is flat.
11 Conclusions We have briefly reviewed the extensive evidence for a dynamical 3-space, with the minimal dynamical equation now known and confirmed by numerous experimental and observational data. This 3-space has been repeatedly detected since the Michelson-Morley experiment of 1887, and they also detected ‘gravitational waves’, which are just 3-space velocity fluctuations. As well the dynamical 3-space has been indirectly detected by means of the dynamical equation explaining diverse phenomena. We have shown that this equation has a Hubble expanding 3-space solution that in a parameter-free manner manifestly fits the recent supernovae and gamma-ray bursts redshift data. All of these successes imply that ‘dark energy’ and ‘dark matter’ are unnecessary notions. This Hubble solution leads to a uniformly expanding universe, and so without acceleration: the claimed acceleration is merely a spurious artifact related to the unnecessary ‘dark energy’ notion. This result gives an age for the universe of some 14Gyr, and resolves as well various problems such as the fine turning problem, the horizon problem and
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other difficulties in the current modelling of the universe. We have also shown why the spacetime formalism appeared to be so successful, despite having no ontological status. One key discovery has been that Newton’s theory of gravity is flawed, except in the very special case of planets in orbit about a sun, which is of course the restricted manifestation of gravity that was available to Newton. At a deeper level the occurrence of α in (1) suggests that 3-space is actually a quantum system, and that (1) is merely a phenomenological description of that at the ‘classical’ level. In which case the α-dependent dynamics amounts to the detection of quantum space and quantum gravity effects, although clearly not of the form suggested by the quantisation of General Relativity. At a deeper level the information-theoretic Process Physics has given insights into the possible nature of reality as a limited self-referential system, in which quantum space and quantum matter are emergent phenomena, with both exhibiting non-local effects. In particular it implies that we have a ‘universal’ process time, as distinct from the current prevailing geometrical modelling of time. These results all suggest that a radically different paradigm for reality is emerging, and in which we see a unification of quantum space and quantum matter, and with gravity an emergent phenomenon. Thanks to Tim Eastman, Erich Weigold, Igor Bray and Lance McCarthy for ongoing support.
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[21] Jaseja T.S. et al. Phys. Rev. A 133, 1221, 1964. [22] Torr D.G. and Kolen P. in Precision Measurements and Fundamental Constants, Taylor, B.N. and Phillips, W.D. eds. Natl. Bur. Stand. (U.S.), Spec. Pub., 617, 675, 1984. [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 Motion and Gravitational Waves Detected, Progress in Physics, 4, 73-92, 2006. [25] Cahill R.T. Optical-Fiber Gravitational Wave Detector: Dynamical 3-Space Turbulence Detected, Progress in Physics, 4, 63-68, 2007. [26] Cahill R.T. and Stokes F. Correlated Detection of sub-mHz Gravitational Waves by Two Optical-Fiber Interferometers, Progress in Physics, 2, 103110, 2008. [27] Hicks W. M.On the Michelson-Morley Experiment Relating to the Drift of the Ether. Phil. Mag., v. 3, 9–42, 1902. [28] M¨ uller, H. et al. Modern Michelson-Morley Experiment using Cryogenic Optical Resonators. Phys. Rev. Lett. 91(2), 020401-1, 2003. [29] Cahill R.T. The Michelson and Morley 1887 Experiment and the Discovery of 3-Space and Absolute Motion, Australian Physics, 46, 196-202, Jan/Feb 2006. [30] Cahill R.T. The Speed of Light and the Einstein Legacy: 1905-2005, Infinite Energy, 10(60), 28-27, 2005. [31] Cahill R.T. The Einstein Postulates 1905-2005: A Critical Review of the Evidence, in Einstein and Poincar´e: The Physical Vacuum, 129-141, ed V. Dvoeglazov, Apeiron, Montreal 2006. [32] Levy J. From Galileo to Lorentz...and Beyond, Apeiron, Montreal, 2003. [33] Guerra V. and de Abreu R. Relativity Einstein’s Lost Frame, Extra]muros, 2005. [34] Airy G.B. Philos. Trans. R. Soc. London, 146, 297; v.146, 343, 1856.
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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, O 2 , ... O n 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− v 2 /c 2 = 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 + π0 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 / c 2
(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 energy plus rest mass energy) of a particle having rest mass total energy E (kinetic r m and velocity u relative to a frame of reference S is
E =
(
r r m c 2 1− u ⋅ v /c 2
(
r r 1− u ⋅ v /c 2
r
) −u 2
)
2
/c
(3.3) 2
if 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 valuer being the one calculated in the privileged isotropic inertial system. Notice that if 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 spatiotemporal 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 = t B , 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 S 0 (x 0 ,y 0 ,z 0 ,t 0 ) and S(x,y,z, t) , and the “delay“ t − t 0 (positive, null or negative) of S over S 0 could depend not only on the time t 0 , but
216
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 t 1 , is reflected backwards by a mirror placed at point B at time t 2 and finally returns at A at time t 3 . Of course t 1 and t 3 are marked by a clock near A , while t 2 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
t 2 − t1 =
1 (t 3 − t 1 ) 2
(5.1)
In this way one defines the time t 2 of the B clock in terms of the times t 1 and t 3 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
t 2 − t1 = ε (t 3 − 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 cWS , c ST .
W
We now can easily show that the mystery of the “Sagnac correction” of Earth physics disappears completely if one adopts the inertial transformations.
θ O
250
r
T
Using the definition β = ω r / c we can write ΔtT =
ω r (LWS cos α WS + LST cos α ST ) /c 2
but
r LWS sinθW + r LST sinθT = 2 A E where A E is the area of the quadrangle OWSTO of Fig. 10. We have thus provided 2 a full physical justification of the Sagnac correction 2A E ω / c . 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 S 0 and assume it is isotropic so that the one-way velocity of light relative to S 0 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 S 0 ) which rotates uniformly around its axis with angular velocity ω and peripheral velocity v = ω r . On its rim, consider a single clock C Σ (marking the time t ) and assume it to be set as follows: When a clock of the laboratory momentarily very near C Σ shows time t 0 = 0 then also C Σ is set at time t = 0 .
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When the platform is not rotating, C Σ constantly shows the same time as the laboratory clocks. When it rotates, however, motion modifies the pace of C Σ and the relationship between the times t and t 0 is taken to have the general form
t 0 = t F (v , ...)
(10.1) 2
where F is a function of velocity v and eventually acceleration a = v r and higher derivatives of position (not shown). Eq. (10.1) is a consequence of the isotropy of S 0 . 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 t 1 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 C Σ , are related to the corresponding platform times via (10.1): t0 i = ti F (v, ...) (i = 1,2,3) . The circumference length is assumed to be L 0 and L , measured in the laboratory S 0 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 )
=
t 2 − t1 L0 /L = F c (1 + β ) L
From (10.2) one has
;
1 t −t L0 /L = 3 1 = (10.2) c˜ (0) F c (1 − β ) L
1 + β c˜ (π ) = ˜c(0) 1 − 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 ω1, ω 2 , ... ω i, ... such that with angular velocities ω1 r1 = ω 2 r2 = ... = ω i ri = ... = 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
B r O
A
φ
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
1 + β c1( π ) = c1(0) 1 − 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 :
⎡β ⎤ 1 1 = + ⎢ + e1 R ⎥ cos θ ⎣c ⎦ c c1 (θ )
(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
1 + β + c e1R c1 ( π ) = c1 (0) 1 − β − c e1R
254
(10.6)
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.
ρ (c+v)/(c−v ) 1
acceleration 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)!
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: e 1 = 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 d 0 from one another. Their clocks are synchronous with those of S0 . At time t 0 = 0 they start accelerating in the + x 0 direction, and they do so in the same identical way, in such a way as to have the same velocity v (t 0 ) 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 + x 0 ; 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: t0
x 0 A (t 0 ) = x 00 A + ∫ dt 0′ v (t 0′ ) + (t 0 − t 0 )v 0
t0
x 0 B (t 0 ) = x 00 B + ∫ dt 0′ v (t 0′ ) + (t 0 − t 0 )v 0
(11.1) so that
x 0 B (t 0 ) − x 0 A (t 0 ) = x 0 B (0) − x 0 A (0 ) = d 0
256
(11.2)
Eq. (11.2) implies that the motion of A and B does not modify the distance d 0 between the spaceships as seen from S 0 . 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 S 0 will check: (i) That the distance d 0 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 S 0 ; (iii) That the observer in S using his shortened rod to measure the A - B distance finds a value 2
2
larger by a factor 1 / 1 − v / c than found before departure when the spaceships were at rest in S 0 . 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 S 0 sees him finding, namely a distance between A and B larger by a factor
1 / 1− v 2 / c 2 , as given by (11.3). It will next be shown that the transformation relating S 0 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 S 0 . 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 S 0 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 S 0 ) 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 tB − t A
257
(11.4)
where d is given by (11.3). Now define:
t 0 A : time of emission of light signal from A as seen in S 0 ; t 0 B : time of its arrival at B as seen in S 0 ( t 0 B > t 0 A );
τ:
delay generated by velocity up to time t 0 at which acceleration stops ( t 0 B > t 0 A > t 0 ).
Since for all t 0 ≥ t 0 time dilation in S is due to the constant velocity v, one has:
t A = t 0 − τ + (t 0A − t 0 ) 1− v 2 /c 2 2
t B = t 0 − τ + (t 0B − t 0 ) 1− v /c
2
(11.5)
By subtracting the first equation from the second, one gets
t B − t A = (t 0 B − t 0 A ) 1 − v 2 / c 2
(11.6)
Eq. (11.6) is the clock retardation formula for the travel time of the light pulse. The point x˜ 0 B of S 0 where the light is absorbed by B must satisfy:
x˜ 0B = x 0A + c(t 0B − t 0A ) x˜ 0B = x 0B + v(t 0B − t 0A )
(11.7)
if x 0 A and x 0 B are the positions of the spaceships A and B respectively at time t 0 A of emission of the light signal. In fact, while the light signal goes from x 0 A to x˜ 0 B with velocity c, spaceship B moves from x 0 B to x˜ 0 B with velocity v. From (11.7) it follows: (11.8) x 0B − x 0A = c (1− v c )( t 0B − t 0A ) Remembering that (11.2) holds for all times t 0 and using (11.3) and (11.6), Eq. (11.8) gives
tB − tA 1+ v c = d c
258
(11.9)
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:
A
B
After:
A
B
Figure 13. Two identical spaceships A and B are initially ar rest on the x 0 axis of the inertial system S 0 . 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 S 0 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 S 0 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 S 0 . 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
Tˆ A marking Einstein' s time
tˆA
PB has a first clock
TB marking the natural time
tB
PB has a second clock
Tˆ B marking Einstein' s time
tˆB
After a certain initial time interval during which t A = ˆt A = t B = ˆt B only Tˆ 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 Tˆ 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 S 0 one will have
t B = t A , but ˆt B ≠ ˆt A . The simultaneity of Tˆ A and Tˆ 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 Tˆ are shown, they discover having had the same age at the same time t, but different ages at the same time ˆt . 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 S 0 to S for describing the time of inertial systems concretely produced.
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A. Einstein, ÄTHER UND RELATIVITÄTSTHEORIE: Rede gehalten am 5. May 1920 an der Reichs-Universität zu Leiden, Springer, Berlin (1920). W.L. Craig and Q. Smith, eds., EINSTEIN, RELATIVITY ABSOLUTE SIMULTANEITY, Routledge, London (2008). R.A. Bertlmann and A. Zeilinger, QUANTUM [UN]SPEAKABLES,
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Springer, Berlin (2002). C. Møller, THE THEORY OF RELATIVITY, Clarendon Press, Oxford (1957). [DB] A. George (a cura di), LOUIS DE BROGLIE PHYSICIEN ET PENSEUR, “Les Savants et le monde”, A. Michel, Paris (1953). [DG] R. de Abreu and V. Guerra, RELATIVITY EINSTEIN'S LOST FRAME, Extra]muros[, Lisboa (2005). [DS] Henri Bergson, DURATION AND SIMULTANEITY, The Library of Liberal Arts, Indianapolis (1965). [EB] THE BORN-EINSTEIN LETTERS. Correspondence between Albert Einstein and Max and Hedwig Born: 1916-1955, Walker & Co., New York (1971). [EF] A. Einstein, Prefazione, in: CINQUANT’ANNI DI RELATIVITA’ 1905-1955, Editrice universitaria, Firenze (1955). [EI] A. Einstein, L. Infeld, THE EVOLUTION OF PHYSICS, Schuster, New York (1961) [EL] E. Lerner, THE BIG BANG NEVER HAPPENED, First Vintage Book Ed. (1992). [EP] V. Dvoeglazov, ed., EINSTEIN AND POINCARE': THE PHYSICAL VACUUM, Apeiron, Montreal (2006). [EU] S. Prokhovnik, LIGHT IN EINSTEIN'S UNIVERSE, Kluwer, Dordrecht (1985). [EW] E. T. Whittaker, A HISTORY OF THE THEORIES OF AETHER AND ELECTRICITY, Hodges, Figgis, & Co., Ltd., Dublin (1910). [FP] A. De Angelis, F. Honsell and B.G. Sidharth, eds., FRONTIERS OF FUNDAMENTAL PHYSICS, Springer, Berlin (2005). [FS] F. Selleri, LEZIONI DI RELATIVITA’ DA EINSTEIN ALL’ETERE DI LORENTZ, Progedit, Bari (2003). English translation in preparation. [GR] A. Held, ed., GENERAL RELATIVITY AND GRAVITATION, Plenum New York (1980). [HA] H. Arp, QUASARS, REDSHIFTS AND CONTROVERSIES, Interstellar Media, Berkeley (1987). [HL] H.A. Lorentz, THE THEORY OF ELECTRONS AND ITS APPLICATIONS TO THE PHENOMENA OF LIGHT AND RADIANT HEAT, Dover, New York (1952). [HR] H. R. Reichenbach, THE PHILOSOPHY OF SPACE AND TIME, Dover, New York (1958). [HV] T.E. Phipps, Jr, HERETICAL VERITIES: MATHEMATICAL THEMES IN PHYSICAL DESCRIPTION, Classic non-fiction library, Urbana (1986). [KR] K. Rudnicki, ed., GRAVITATION, ELECTROMAGNETISM AND [CM]
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COSMOLOGY, Apeiron, Montreal (2001). J. Jeans, PHYSICS AND PHILOSOPHY, Cambridge Univ. Press (1943). L. Kostro, EINSTEIN AND THE ETHER, Apeiron, Montreal (2000). L.D. Landau and E.M. Lifshitz, THE CLASSICAL THEORY OF FIELDS, Butterworth-Heinemann, Oxford (1996). [LS] A. Einstein, LETTRES A MAURICE SOLOVINE (1906-1955), Gauthiers-Villars, Paris (1956). [MB] M. Born, PHYSICS IN MY GENERATION, Springer, New York (1969). [MD] M. C. Duffy, ed., PHYSICAL INTERPRETATIONS OF RELATIVITY THEORY, IX, British Soc. Philosophy of Science, University of Sunderland (2004). [MJ] M. Jammer, CONCEPTS OF SIMULTANEITY, The Johns Hopkins University Press, Baltimore (2006). [MM] A. Einstein, Grundgedanken und Methoden der Relativitätstheorie in ihrer Entwicklung dargestellt (Morgan Manuscript), EA 2070. [MR] A. Einstein, THE MEANING OF RELATIVITY, Princeton University Press (1950). [OI] A. Ghosh, ORIGIN OF INERTIA, Apeiron, Montreal (2000). [OQ] F. Selleri, ed., OPEN QUESTIONS IN RELATIVISTIC PHYSICS, Apeiron, Montreal (1998). [PE] M. Klein, PAUL EHRENFEST, North-Holland, Amsterdam (1970). [PF] P. Frank, PHILOSOPHY OF SCIENCE, Prentice-Hall, Englewood Cl. (1957). [PR] A. Einstein, H.A.Lorentz ..., THE PRINCIPLE OF RELATIVITY, Dover, NY (1952). [PS] Paul A. Schilpp, ed., ALBERT EINSTEIN: PHILOSOPHER-SCIENTIST, Open Court (1949). [PV] S. Sauders and H.R.Brown, eds, THE PHILOSOPHY OF VACUUM, Clarendon Press, Oxford (1991). [RF] G. Rizzi and M.L. Ruggero, eds., RELATIVITY IN ROTATING FRAMES, Kluwer, Dordrecht (2004). [RS] Karl R. Popper, REALISM AND THE AIM OF SCIENCE, Vol. I of THE POSTSCRIPT, Rowman and Littlefield (1983). [SC] H. Dingle, SCIENCE AT THE CROSSROADS, Western Printing Services, Bristol (1972). [SD] Karl R. Popper, THE LOGIC OF SCIENTIFIC DISCOVERY, Hutchinson, London (1980). [SG] A. Einstein, RELATIVITY, THE SPECIAL, THE GENERAL THEORY, Chicago (1951). [SL] S.C. Tiwari, SUPERLUMINAL PHENOMENA IN MODERN PERSPECTIVE, Rinton Press, Princeton (2003). [SS] R. Sexl and H.K. Schmidt, RAUM-ZEIT-RELATIVITÄT, Rowohlt [JJ] [LK] [LL]
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Taschenbuch Verlag, Hamburg (1978). J.S. Bell, SPEAKABLE AND UNSPEAKABLE IN QUANTUM MECHANICS, Cambridge Univ. Press (1987). G. Toraldo di Francia, ed., PROBLEMS IN THE FOUNDATIONS OF PHYSICS, Società Italiana di Fisica, Bologna, and North Holland, Amsterdam (1979). Karl Popper, UNENDED QUEST AN INTELLECTUAL 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 crosssection. 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.
267
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
268
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 ET O
cBz / Ez
n
z
In general waves are mixtures of TE (Ez = 0) and/or TM (Bz = 0) waves.
cBT
(vgroup < c)
Diverging Electromagnetic TE / TM Wave
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.
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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 RiemannSilberstein 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
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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:
271
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: ⎡ Fx ⎤ ⎡ E x + icBx ⎤ FT = ⎢ ⎥ = ⎢ ⎥ ⎣ F y ⎦ ⎣ Fy + icB y ⎦
2.2
The matrix transverse gradient vector is defined from: ⎡∂ x ⎤ ∇T = ⎢ ⎥ ⎣∂ y ⎦
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: ⎡0 − 1⎤ ϕ =⎢ ⎥ ⎣1 0 ⎦
;
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 =F T / 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 = (∂x2 +∂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 − ∂z2 ]Fz
2.10
A typical solution used here with frequency ω and wave-vector k is: Fz = Fzo exp[i(kz − ωt)] 2
2
2.11
2
(ω/c) − k = κ
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 |ETm|/|c BTm| = |∇T∂zEz| / |ϕ ∇T∂ct Ez |
= kc/ω < 1
2.16 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.
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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: r (1/√2)(∂z −∂ct) = ∂τf ; (1/√2)(∂z +∂ct) = ∂τ 3.1 3.2 (1/√2)(z − ct) = τ f ; (1/√2)(z + ct) = τr For variations as in equation 2.11, define light-cone propagation constants from:
kf = (1/√2)[k + (ω/c)] ; k r = (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 + k r τr )]
3.4
Envelopes moving with the group velocity vg = c2 k/ω vary as: exp[i(ω/c)(z −vg t)] = exp{i[(ω/c) z − k ct)]} = exp[i(kf τ f − k r τr )] 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
k r ' = k r exp(α) ; k f '= k f exp(−α)
3.6b
Any term with a relevant superscript is said to lie on the forward branch of the lightcone 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 superand sub-scripts are then automatically Lorentz frame invariant e.g. ∂τ τr = 1 = ∂τf τ f ; (k f τ f + kr τr ) = (kz − ωt) ; r
(k f 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
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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) = − ∂τ [(1/√2) Fz]
3.11
r ∇T(1/√2)Fz = ∂τ (r FT) + ∂τf (f FT)
3.12
r
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 = ∂τ ∇T / (r FT) + ∂τf ∇T / (f FT) r
3.13
Eliminate ∇T / (r FT) and ∇T / (f FT) using equations 3.10-11: κ2 Fz = 2∂τ ∂τf Fz r
3.14
For variations as in equation 3.4 one requires: κ2 = − 2 k f k r = −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 r do not change with axial Lorentz transformations. The differentials ∂τf and ∂τ f respectively transform as τr or τ 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 lightcone 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 ∂τ ) (1/√2)Fz /(κ2) r
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: ⎡ Fx − iFy ⎤ r FT = ⎢ ⎥ ; f FT = ⎣ Fy + iFx ⎦
(rFT)x = −i(rFT)y
⎡ Fx + iFy ⎤ ⎢ F − iF ⎥ x⎦ ⎣ y
3.17
(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 ∂τ Fzp + r ∇T∂τf Fzq) + (r ∇T∂τf Fzp + f ∇T ∂τ Fzq)} /(√2κ2) r
r
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) ; k f '= kf exp(−α)
3.20
k f ' = (1/√2)(ω/c)[tanh(α) + 1]exp(−α) = (√2)(ω/c)/cosh(α)
3.21
k r ' =(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: r (Θ∇T) / ( f ΘFT) = − ∂τ [(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 ∂τ (r FT) + Θ ∂τf (f FT) r
= ∂τ (r ΘFT) + ∂τf (f ΘFT) − R r
4.4 4.5
where R = (∂τ Θ) (r FT ) + (∂τf Θ) (f FT ) r
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∂τ (r FT) + Θ ∂τf (f FT)] = ∂τ ∇T / (r FT) + ∂τf ∇T / (f FT) r
r
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) / [∂τ (rΘ FT) + ∂τf (f ΘFT)] = ∂τ ∇T / (r FT) + ∂τf ∇T / (f FT) r
r
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 τr
Provided also that Θ (∂τf Θ) and Θ (∂ Θ) are scalar numbers then, taking R from equation 4.6, one can re-arrange equation 4.9 to give: /
/
Θ / (∂τ Θ) ∇T / (r FT ) + Θ / (∂τf Θ) ∇T / (f FT ) r r = − Θ / (∂τ Θ) ∂τf [(1/√2) Fz] − Θ / (∂τf Θ) ∂τ [(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: r
Θ / (∂τ Θ) kf + Θ / (∂τf Θ) k r = 0 r
r
Θ = exp[W ϕ(k f τ − k τr)] = exp[ϕ W (ω/c)(z −vg t)] f
4.11 4.12
Notice how, in the light-cone formulation, P(k f τ f + k r τ r ) represents a function P moving at the phase velocity of the wave while Q(k f τ f − k r τ r ) represents a function Q moving at the group velocity: both (k f τ f +/− k r τ 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.
1
Standard rotations
2
3
Distributed spin rotations
1
2
4
Ox
3
4
E
Oy
cB
Ox
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 exp{i[k f exp(δ) τ f + k r exp(−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| 0 term that he introduced to prevent his theoretical universe from collapsing under the gravitational pull of its material contents, 8πGρr2 /3 = r˙ 2 + (k − Λr2 /3)c2 −8πGP r/c2 = 2¨ r + r˙ 2 /r + (k/r − Λr)c2 .
(1.1) (1.2)
Einstein’s preferential universe was of the closed variety which involves the curvature parameter being unity, k = 1 and with a positively valued Λ, we have, 8πGρr2 /3 = r˙ 2 + (1 − Λr2 /3)c2 −8πGP r/c2 = 2¨ r + r˙ 2 /r + (1/r − Λr)c2 .
(1.3) (1.4)
To get a static universe from these equations that holds for some finite time interval we have to impose the non expansion condition, v = r˙ = 0, together with the none acceleration condition a = v˙ = r¨ = 0 and if, additionally, we
324
choose the dust universe condition, P = 0, we get (1.5) and (1.6). 8πGρr2 /3 = (1 − Λr2 /3)c2 0 = (1/r − Λr)c2 .
(1.5) (1.6)
Einstein identified his cosmological constant Λ as arising from a density of dark 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 the Einstein universe given by (1.9), 8πG(ρ + ρΛ ) = 3c2 /rE2 8πG(ρΛ ) = c2 /rE2 rE = Λ−1/2 .
(1.7) (1.8) (1.9)
From (1.7) and (1.8) it follows that 8πGρ = 2c2 /rE2 ρΛ = ρ/2 ρ = 2ρΛ = ρ†Λ .
(1.10) (1.11) (1.12)
Equation (1.12) is the generic version of the so called cosmological coincidence problem. I think that Einstein would not have recognised the relationship between ρ and ρΛ at (1.12) as a problem in the early years after discovering it. He probably thought that the 2 factor was interesting and needed explaining but did not see it as a problem. In those early years he was convinced the universe was a time static entity and had no vision of the possibility that the relation might have a different coefficient from the integer 2 which could come about by the now recognised and accepted expansion process. Only after expansion was accepted does the question following arise. If at time now equation (1.12) holds in an expanding universe 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 must be 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 of the universe. Einstein’s generic cosmological coincidence problem is completely resolved by the cosmological model introduced in references A [23], B,[24] and C [32] as I shall next explain. However, there is one important reservation about this claim that will be discussed in the next section. I
325
call this first cosmological coincidence critical because it involves the integer point value number, 2, which would have zero probability of occurring in any finite time ranged variable quantity. Such coincidences need to be explained in any structure. The model introduced in those papers reveals the true nature of dark energy material and that is the clue to resolving the generic coincidence problem. One conclusion from those papers was that the dark energy density, contrary to Einstein’s identification, should be theoretically and physically measured as ρ†Λ (1.12) rather than as ρΛ . The second conclusion from those papers was that dark energy has positive mass density but is characterised by carrying a negative gravitational value of the gravitational constant, −|G|. Thus equation (1.12) achieves Einstein’s purpose of stopping the gravitation collapse of the universe by choosing conditions such that the positive mass material, ρ + ρ†Λ , within the universe is gravitationally neutral, Gρ + (−G)ρ†Λ = 0. Thus although that could have happened some time or other it would not necessarily hold for ever as in a constant universe or indeed occur at the time now. The model I am suggesting is a flat universe with, k = 0, and the actual time when such conditions apply is denoted by tc and can be calculated. At that time v(tc ) 6= 0 contrary to the what is implied in the Einstein universe where v = 0 given above. The time tc is the important time greatly in the past and recognised recently by astronomers when the acceleration of the universe changes through zero from negative to positive or when dark energy takes over from normal mass energy. The critical coincidence in the generic Einstein universe is completely resolved by the conceptual aspects of the Friedman dust universe that I have been proposing. This reinterpreted old and modified model which is closely related to an early Lemaˆıtre model has a structure that has identified the cause of the Einstein critical coincidence. The nature of this coincidence can be described as, mistaking the Einstein radius for a possible constant present time radius. This mistake is completely excusable on the grounds that Einstein did not recognise that the universe radius was in truth a variable with time quantity and he was completely unaware that at some time in the past the dark energy density as he defined it was exactly half the normal mass density. The explanation of the root cause of the critical Einstein coincidence can be used to identified the cause of another critical time coincidence between the present time t† and time tc , t† = 2tc , in the Friedman dust universe. This will be explained in the next section.
326
2
Coincidence in Friedman Dust Universe
The coincidence in the Friedman dust universe model involves, t† , the time now and, tc , the time when the universe changed from deceleration to acceleration. t† = 2tc .
(2.1)
This equation involves again the exact numerical integer value, 2. This is clearly critical because if two events over time are so related, then there must be some physical explanation because the probability of two such time-point events on any finite time line range is zero. The generic Einstein coincidence was critical in the same sense. This coincidence seems obviously related to the generic Einstein coincidence which suggests it is also totally explainable. The reservation I mentioned earlier is that you might see it as ironic that a model with a coincidence can completely solve the coincidence in an earlier model. This can be explained by the fact that theoretical structures involve patterns of abstract symbols as one aspect and numerical constants as another aspect when they are applied to physical situations. The new model is correct in the first aspect but in the second aspect, the numerical values have not all been associated with the measurement time, t† , but rather some with a conceptual time, t0 , the time that would be associated with the centre of the values given by the astronomical measurements. There is some subtlety in this situation because in this model, it seemed that t† should be equal to t0 . However, this equality created the degeneracy that led to the coincidence. It can all be resolved by using the formula for Hubble’s constant, the formula for the radius and the formula for the constant C, H(t) = (c/RΛ ) coth(3ct/(2RΛ ))
(2.2)
2/3
(2.3) (2.4)
r(t) = b sinh (3ct/(2RΛ )) C = ΩM,0 H 2 (t0 )r3 (t0 ).
These expressions involve the numerical parameter, RΛ . It is necessary to find the correct value for this parameter that is to be associated with these formulae. To make this step, we need the astronomical measurements of the Ωs. The accelerating universe astronomical observational workers [1] give
327
measured values of the three Ωs, and wΛ to be ΩM,0 ΩΛ,0 Ωk,0 ωΛ
= = = =
8πGρ0 /(3H02 ) = 0.25+0.07 −0.06 Λc2 /(3H02 ) = 0.75+0.06 −0.07 −kc2 /(r02 H02 ) = 0, ⇒ k = 0 PΛ /(c2 ρΛ ) = −1± ≈ 0.3.
(2.5) (2.6) (2.7) (2.8)
From these equations assumed to hold at a conceptual time, t0 , when the universe passes through the centre value of the measurement ranges, we get the formulae, t0 RΛ tc t0 /tc
= = = =
(2RΛ /(3c)) cosh−1 (2) 3ct0 /(2 cosh−1 (2)) (2RΛ /(3c) coth−1 (31/2 ) cosh−1 (2)/ coth−1 (31/2 ) = 2.
(2.9) (2.10) (2.11) (2.12)
Having found RΛ in terms of t0 this value of RΛ can be substituted into the formula for Hubble’s constant, (2.2), to find the value of the time now , t† . H(t† ) = (c/RΛ ) coth(3ct† /(2RΛ )) t† = (2RΛ /(3c)) coth−1 (RΛ H † /c) t0 3t0 H † −1 = coth cosh−1 (2) 2 cosh−1 (2) 6tc H † 2tc −1 coth , = cosh−1 (2) 2 cosh−1 (2)
(2.13) (2.14) (2.15) (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 problem. If we write (2.16) in the form 2 6tc H † −1 † t /tc = coth (2.17) cosh−1 (2) 2 cosh−1 (2) t† /tc = 2f (2tc ), (2.18) where f (2tc ) gives the deviation of the ratio t† /t0 from the value unity and removes the degeneracy. Expressed in another way it is the multiplicative function that breaks the coincidence at (2.12) and converts the integer 2 to a much less notable non integral value. However, we can give the formulae
328
(2.17) and (2.18) together an interpretation in terms of the uncertainties of the measurement process. This is achieved by defining the measurement deviation function dmeas (t0 ) as follows, dmeas (t0 ) = t† /t0 − f (t0 ) 1 3t0 H † −1 f (t0 ) = . coth cosh−1 (2) 2 cosh−1 (2)
(2.19) (2.20)
The function (2.19) is a dimensionless measure of how much the central Ω values from astronomy assumed to have occurred at t0 differ from the time now measurement from the Hubble variable quantity H(t† ) taken at time now, t† . It is sufficient to assume that the event at t0 is still yet to occur, t0 > t† , then we see that the function dmeas passes through zero when the full degeneracy holds at t0 = t† and it has a maximum at t0 ≈ 0.643 × 1018 s when t† and t0 assume the approximate maximum deviation, 0.17. When t0 = 0.643×1018 , t† can be assumed constant at the coincidence value 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 of t† ≈ 0.6757t0 = 1.3514tc . Thus the coincidence is decisively removed with t† 6= t0 = 2tc .
3
Conclusions
It has been shown that the generic Einstein coincidence problem can be resolved in terms of a correction in the value of the density he associated with his cosmological constant Λ and a rethink about the significance of the radius of his model. This solution then points clearly to resolution of the coincidence in the recent dust universe model as essentially the same concepts are involved. The conceptual centre Ω value measurements from the astronomers can not necessarily be assumed to occur at exactly the same epoch time t0 as the measurement of the value of the Hubble constant at epoch time now, t† . The usually assumed degeneracy t0 = t† can be removed to find the true range of values within which t† has to reside so that the integer 2 aspect of the same degeneracy t† = 2tc sees the 2 replaced with a
329
less mysterious non integer. The time tc is when the expansion acceleration changes from negative to positive. Acknowledgements I am greatly indebted to Professors Clive Kilmister and Wolfgang Rindler for help, encouragement and inspiration over many years.
4
Appendix 1 Fundamental Dark Mass, Dark Energy Time Relation in a Friedman Dust Universe and in a Newtonian Universe with Einstein’s Lambda Abstract
In this appendix, it is shown that the cosmological model that was introduced in a sequence of three earlier papers under the title A Dust Universe Solution to the Dark Energy Problem can be used to recognise a fundamental time dependent relational process between dark energy and dark mass. It is shown that the formalism for this process can also be obtained from Newtonian gravitational theory with only the additional assumption that Newtonian space contains a constant universal dark energy density distribution dependant on Einstein’s Lambda, Λ. It thus seems that the process is independent of general relativity and applies in more contexts than just the expansion of the entire universe. It is suggested that the process can be thought of as a local space and time packaging for dark mass going through part transmutations into locally condensed visible material. The process involves a contracting and then expanding sphere of conserved dark matter. At two stages in the process at special times before and after a singularity at time zero, the spherical package goes through a condition of gravitational neutrality of very low mass density which could be identified as cosmological voids. 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 cosmological model introduced in the papers A Dust Universe Solution to the Dark Energy Problem [23], Existence of Negative Gravity Material. Identification of Dark Energy [24] and Thermodynamics of a Dust Universe [32]. Further references will be mentioned as necessary. Application of the cosmological model introduced in the papers A [23], B,[24] and C, [32], is to be found in the paper D, ([34]), to the extensively discussed and analysed Cosmological Constant Problem. In the next section a relation between Dark Mass and Dark 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) 0 = G− ΓZ (t) + G+ ∆Z (t),
(6.1) (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 ∆ and Γ substances which together are assumed to form all the time conserved material of the universe, P∆B /c2 = ρ∆B,νc (t)ω∆ (t) PΓB /c2 = ρΓB,νc (t)ωΓ (t) P∆Z /c2 = ρ∆Z,νc (t)ω∆ (t) PΓZ /c2 = ρΓZ,νc (t)ωΓ (t)
= ∆B (t) = ΓB (t) = ∆Z (t) = ΓZ (t).
(6.3) (6.4) (6.5) (6.6)
The Z subscript above denotes zero-point values. Let us now consider the Einstein cosmological constant, Λ, in relation to the Friedman equations, 8πGρr2 /3 = r˙ 2 + (k − Λr2 /3)c2 −8πGP r/c2 = 2¨ r + r˙ 2 /r + (k/r − Λr)c2 .
(6.7) (6.8)
Einstein introduced a physical explanation for his Λ term by associating it 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 density the Friedman equations can be written with the Hubble function of epoch time H(t) as, 8πGρr2 /3 = r˙ 2 + (k − 8πGρΛ r2 /3)c2 −8πGP r/c2 = 2¨ r + r˙ 2 /r + (kc2 /r − 8πGρΛ r) H(t) = r(t)/r(t) ˙ = (c/(RΛ )) coth(3ct/(2Rλ )).
(6.9) (6.10) (6.11)
Thus the first friedman equation can be expressed as 8πG(ρ + ρΛ )/3 = H 2 (t) + (kc2 /r2 ) 8πGρTE = 3(H 2 (t) + kc2 /r2 ) ρTE = ρ + ρΛ ,
(6.12) (6.13) (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 ) = 3H 2 (t) 8πGρΛ kc2 8πGρ + − = 1. 3H 2 (t) 3H 2 (t) r2 H 2 (t)
(6.15) (6.16)
The three Omegas which the astronomers use to display their measurements are defined using the three terms on the left hand side of (6.16) according to which they have to add up to unity, ΩM (t) ΩΛ (t) Ωk (t) ΩM (t) + ΩΛ (t) + Ωk (t)
= = = =
8πGρ/(3H 2 (t)) 8πGρΛ /(3H 2 (t)) −kc2 /(r2 H 2 (t)) 1.
(6.17) (6.18) (6.19) (6.20)
There is a very strong case (A,B,C,D,E) for identifying the dark energy mass density that should account for Einstein’s constant Λ term as given by twice the density introduced by Einstein, ρ†Λ = 2ρΛ
(6.21)
ρ†Λ
(6.22)
ρT † = ρ +
and this implies the formula (6.22) for the total amount of physical mass density 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 ) = 3H 2 (t) + 8πGρΛ 8πGρ + + c2 Λ
3H 2 (t)
8πGρ†Λ 3H 2 (t) + c2 Λ
(6.23)
2
−
3kc r2 (3H 2 (t)
+ c2 Λ)
= 1.
(6.24)
Thus we now have three new Omegas Ω†M (t) = 8πGρ/(3H 2 (t) + c2 Λ)
Ω†M (t) + Ω†Λ (t)
Ω†Λ (t) Ω†k (t) + Ω†k (t)
=
8πGρ†Λ /(3H 2 (t) 2 2 2
(6.25)
2
+ c Λ)
(6.26)
2
= −k3c /(r (3H (t) + c Λ))
(6.27)
= 1.
(6.28)
Here I shall be mostly concerned with the flat space case k = 0 so that the two possible and equivalent sets of Omegas satisfy the relations ΩM (t) + ΩΛ (t) = 1
(6.29)
Ω†M (t)
(6.30)
+
Ω†Λ (t)
= 1.
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 time when ΩM (t0 ) = 1/4 ΩΛ (t0 ) = 3/4
(6.31) (6.32)
and this event will happen regardless of any measurements. I have assumed that the epoch time of this event in the history of the universe is given by t0 . Thus the usual use of the subscript 0 to denote time now has been abandoned and time now will in future be denoted by t† . The corresponding and more realistic time t0 relation between non-dark energy materials and dark energy will with a simple calculation be represented in terms of the dagger Omegas by Ω†M (t0 ) = 1/7
(6.33)
Ω†Λ (t0 )
(6.34)
= 6/7.
This implies that about 85.7% of the universe mass is dark energy rather than the usually assumed 75%, a substantially changed assessment. If this
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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 to be present within the total mass of the universe. The ratio dark mass to visible mass is often taken to be 4 to 1. Thus the percentage of dark mass in the universe according to (6.33) and (6.34) would become reduced to 20 × (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 the total mass. If it is taken that we know nothing about the dark elements, as is often suggested, then our actual knowledge of the universe is mass wise abysmal. However, fortunately it is not true that we have no knowledge of the dark elements. We do have indirect knowledge of these aspects. The theory associated with this model give a definite relation between dark energy and dark mass this relation can be read off from the gravitation polarisation equations (6.1, 6.2) repeated next GρΛ = G− ΓB (t) + G+ ∆B (t) 0 = G− ΓZ (t) + G+ ∆Z (t) ρ(t) = ρ∆,νc + ρΓ,νc .
(6.35) (6.36) (6.37)
The third equation above expresses the total time conserved density ρ(t) in terms of the CM B mass density, ρΓ,νc , and the rest of the universe mass density ρ∆,νc . The νc subscript indicates that zero point energies are included in these terms. The second equation above defines the zero-point energy of the dark energy as being zero, effectively defining energy zero for this cosmology theory. The total energy density for this model equation (6.22) can thus be written as (6.41) ρ†Λ ρT † (t) ρT † (t) ρT † (t) ρT † (t) ρ˜∆,νc ρ˜Γ,νc
= = = = = = =
2ρΛ ρ(t) + 2ρΛ ρ∆,νc + ρΓ,νc + 2(∆B (t) − ΓB (t)) ρ∆,νc + 2∆B (t) + ρΓ,νc − 2ΓB (t) ρ˜∆,νc + ρ˜Γ,νc ρ∆,νc + 2∆B (t) ρΓ,νc − 2ΓB (t).
(6.38) (6.39) (6.40) (6.41) (6.42) (6.43) (6.44)
The tilde versions of the basic two densities are the resultants of a gravitational vacuum polarisation process in which the basic Γ and ∆ densities
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induce, via their pressures and coexistence, the two polarisation densities 2ΓB (t) and 2∆B (t) which together represent the dark energy density ρΛ , equation (6.35). This process takes place through the equations of motion of the two components. Thus from this point of view dark energy within the universe boundary is a vacuum polarisation consequence of the of the existence of the basic Γ and ∆ fields in interaction under general relativity. The dark energy density also exists outside the universe boundary but in an un-polarised condition. Thus the polarisation within the universe is constrained by the constant value that exists everywhere. To examine the weight of this gravitational vacuum polarisation on the none polarised fields separately at time t† using the numerical results from (A,B,C) 2ω∆ (t† ) ≈ 6 2ωΓ (t† ) = 2/3
(6.45) (6.46)
they must be expressed in terms off the none polarised fields as in (6.47) and (6.48) 2∆B (t† ) 2ΓB (t† ) ρ∆B,νc (t† ) ρΓB,νc (t† ) ρΛ ρΓB,νc (t† )
≈ = ≈ ≈ = =
6ρ∆B,νc (t† ) (2/3)ρΓB,νc (t† ) (104 /1.9)ρΓB,νc (t† ) 1.9 × 10−4 ρ∆B,νc (t† ) Λc2 /(8πG) ≈ 7.3 × 10−27 aT 4 (t† ) ≈ 4.66 × 10−31 .
(6.47) (6.48) (6.49) (6.50) (6.51) (6.52)
The relation (6.49) also comes from (A,B,C). Thus we can express the positively weighted ∆B and negative weighted ΓB vacuum polarisation density poles as 2∆B (t† ) 2ΓB (t† ) 2∆B (t† ) 2ΓB (t† )
≈ ≈ ≈ ≈
(6 × 104 /1.9)ρΓB,νc (t† ) (2/3)1.9 × 10−4 ρ∆B,νc (t† ) 3 × 104 ρΓB,νc (t† ) 1.26 × 10−4 ρ∆B,νc (t† ).
(6.53) (6.54) (6.55) (6.56)
Returning to the gravitational vacuum polarisation equation (6.1) repeated here for convenience, GρΛ = G− ΓB (t) + G+ ∆B (t) 0 = G− ΓZ (t) + G+ ∆Z (t),
335
(6.57) (6.58)
we can do a spot numerical check using the values above and without the G factor as follows 7.3 × 10−27 ≈ = ≈ ≈ ≈ ≈ ≈
ρΛ = ∆B (t† ) − ΓB (t† ) ρ∆ ω∆ − ρΓ ωΓ (3 × 104 − (1/3))ρΓ (3 × 104 )ρΓ (3 × 104 ) × 4.66 × 10−31 (13.98/1, 9) × 10−27 7.3 × 10−27 .
(6.59) (6.60) (6.61) (6.62) (6.63) (6.64) (6.65)
This is just a rough check that does give a good though approximate result while showing that the induced ∆ and induced Γ fields in the form of a difference are the source of the dark energy density within the universes boundaries. At step (6.61), the −1/3 term from the Γ field is abandoned because 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 contributor because it occurs as multiplicatively weighted by the ∆ factor, 104 . As the ∆ field is all the conserved universe field density less the CM B the induced delta field ∆ is all the induced conserved density universe field less the induced CM B field. The ∆ field includes the so called dark matter as its major contributor of about 80% with normal visible mass making a smaller percentage of about a 20% contribution. Thus the important conclusion is that dark energy value within the universe is a direct consequence of the induced mass from the ∆ field which itself is largely dark mass. Briefly, dark energy within the universe is numerically very close in value to the vacuum polarised dark mass and if the Γ field is also classified as dark the closeness becomes coincidence. From the preceding discussion and equation (6.57) it should not be inferred that dark mass is a primary source of dark energy. I think the reverse is nearer to the truth and equation (6.57) is the direct result of a mechanical equilibrium between pressure equivalent induced density from the CM B and the sum of the pressure induced densities from the ∆ and Λ field at the boundary and within the universe. Thus this mechanical equilibrium effectively transfers the dark energy pressure from outside the universe to its boundary and hence by homogeneity to inside the universe. The P EID concept will be explained in the next section on 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 Equivalent Induced Density, PEID, in relations to the equations of state associated with specific subsystems of the total system. For example, suppose one subsystem is called the ∆ system with the equation of state, P∆ (t) = c2 ρ∆ ω∆ (t) ∆(t) = ρ∆ (t)ω∆ (t) = P∆ (t)/c2 ,
(7.1) (7.2) (7.3)
then I take the definition for the P EID, ∆(t), to be given by equation (7.2). Thus ∆(t) has the same dimensions as density because in common with all the omegas, ω∆ (t), is dimensionless and it is derived from ρ∆ (t) through the multiplicative action of the inducing function, ω∆ (t). From (7.3) it is clearly essentially a pressure with the dimensions of density. It represents this pressure in the form of the mass density, ∆(t). I am not aware that the P EID slant on equations of state has any important part elsewhere in physics but it seems that it does play an essential role in cosmology in relation to the understanding of dark energy and its connection to other key densities. This is clear from inspection of equation (6.1) again with and without the G weightings, GρΛ = G− ΓB (t) + G+ ∆B (t) ρΛ = ∆B (t) − ΓB (t).
(7.4) (7.5)
Thus from equation (7.5) the source of dark energy density within the universe is just the difference of the P EIDs for the ∆ and Γ fields which together constitute all the conserved mass of the universe. Thus the mystery of 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 resolved by 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 status of the dark energy density is its dominant characteristic. However, it seems to me that dark energy with approximately an equivalent density of 5 hydrogen atoms per cubic meter would not be visible anyway. The formula (7.5) can also be used to show a simple relation between dark mass and dark energy but before discussing that aspect it is useful to consider in the next paragraph the way this theory structure has developed and can
337
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 Friedman equations. The result subsequently turned out to be a reincarnation of the first model introduced by Lemaˆıtre [25] but with substantially different interpretations and additional details. The version of the model in A and B , like most cosmological models, involved the assumption that the mass density of the universe only depended on time and so was space-wise homogeneous. However, the structure unearthed in that version of the model was completely adequate to describe cosmological expansion and its change from deceleration to acceleration at some time tc in the past and various other new understandings of the cosmological process, all in complete agreement with up to date measurement. Thus this basic structure did not depend on differentiating the mass density into separate components to represent various contributory fields such as the electromagnetic or heavy particle contributions. The dark energy contribution was involved in that version of the 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 hyperspace into which the universe expands. I shall here denote that model by U0 = UΛ (DM ), meaning that it can be assumed to only contain an energy conserved 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 enveloping 3D-hyperspace. The conserved mass density, ρ(t) ∼ ΩM (t), in this model must represent all the dark mass, if we assume that none of this dark mass has 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 model UΛ (DM ), can be conceived as not containing any visible hadronic matter, which as we know can only be present in a very small proportion anyway and it would also likely be none uniformly distributed. It follows that the model U0 = UΛ (DM ) can be regarded as a very bland, over all time, approximation to the actual universe and which can be built up in stages to represent the universe with increasing accuracy. I emphasise the usual cosmological basic assumption that the model’s density function is space-wise homogeneous means that if the model contains any dark mass within its boundaries then it contains only uniform dark mass and together with the uniformly distributed dark energy background. The next stage in the build up process in which the cosmic microwave back ground was added was published in C and will be denoted by U1 = UΛ (DM = ∆(t) ∪ Γ(t)). This
338
means that the fixed amount of dark mass in the first version is now able to transform into time dependent components ∆(t) for one part and Γ(t) for the complementary part, the CM B, with the same total mass quantity as the original dark mass. The next stage of complexity is the introduction of the possibility that part of the ∆ mass, MU can transform into visible mass, often called hadronic mass. This universe can be represented by U2 = 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 by the D and V subscripts. Clearly the increasing complexity procedure can continue to produce universes with lower homogeneity described by U3 and so on. Let us now return to discussing the relation between dark mass and dark energy.
8
Dark Mass, Dark Energy Ratio
Consider firstly the basic universe type Friedman dust universe, U0 . The model in this basic case is an excellent representation of the modern astronomical measurements. However the basic density function is assumed to be rigorously homogeneous and contains only conserved with time dark mass and the hyperspace permeating dark energy. The density functions for the dark mass, dark energy and the ratio, rΛ,DM (t), of dark energy to dark mass as functions of time are respectively represented by ρ(t) = (3/(8πG))(c/RΛ )2 sinh−2 (3ct/(2RΛ )) ρ†Λ
2
= (3/(4πG))(c/RΛ ) ρ†Λ /ρ(t) 2
2
rΛ,DM (t) = = 2 sinh (3ct/(2RΛ )) rΛ,DM (±tc ) = 2 sinh (±3ctc /(2RΛ )) = 1.
(8.1) (8.2) (8.3) (8.4)
Equation (8.3) is a general result but in the case of a U0 universe it can be expressed differently by using equation (7.5) with the Γ term taken zero as ρΛ = ∆B,0 (t) = ρ(t)ω∆,0 (t),
(8.5) (8.6)
the zero subscripts having been added to differentiate the functions concerned from those in the U1 version. From paper C, we know that 3(c/RΛ )2 ρ−1 (t) MΓ ω∆ (t) = + /(1 − MΓ /MU ). (8.7) 3MU 8πG
339
Thus the zero Γ version for U0 is given by 3(c/RΛ )2 ρ−1 (t) . ω∆,0 (t) = 8πG
(8.8)
Substituting this into equation (8.6) confirms the validity of (8.6). Thus the rather trivial equation (8.6) gives the all time dependent relation between dark energy and dark mass for the nontrivial model U0 . However, trivial or not, the dark energy and dark mass densities are strongly numerically related 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 dark mass in the case of a universe in which the homogeneity has been broken by the addition of the cosmic microwave background, replacing some by the CM B. 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 ρ†Λ rΛ,DM,1 (t) = = 2 sinh2 (3ct/(2RΛ )). ρ∆ (t) + ρΓ (t)
(8.11)
The denominator of the ratio remains unchanged as also does the second equality because the numerical values are unchanged. It might be thought that the left and right sides of the first equality do not now agree because only the ∆ part contains dark mass, that which is left from the U0 universe case after some has converted to CM B. Numerically there is no problem as the quantity of dark mass is presumably shared between the ∆ and Γ fields. However, the terminology might be questioned. Arguably, the CM B is composed of photons which are not visible and therefore the CM B can be classified as dark mass equivalent material. Of course photons convey information about other visible materials to the eye but photons themselves are not seen in the usual meaning of the word. I have added the extra subscript 1 in the U1 ratio so that no confusion can arise if the case I have just made is not accepted. The dark energy dark mass ratio in either form above represents a fundamental time conditioned relation between dark mass and
340
dark energy. This result and the formula (7.5) both of which hold inside and on the boundary of the universe show how totally interdependent are the two dark facets. The ratio rΛ,DM (t) is of great generality and could play an important part in helping to understand cosmological voids, a recent astronomical discovery. This ratio has come out of general relativity but it can be shown that it is independent of general relativity and its existence only depends on some simple assumptions added to Newtonian gravitational theory. The very basic and major significance of this ratio will be discussed and demonstrated in the next section by showing that it is directly derivable from Newtonian gravitational theory. It will be indicated how this implies a context for its significance within smaller regions of space within the universe’s boundary.
9
Newtonian Dark Mass and Dark Energy
Consider an infinitely extended 3-dimensional Euclidean space such as that in which Newtonian gravity is usually considered to act between objects having the physical characteristic called mass. I shall make the usual assumption that Newtonian gravity acts between enclosed regions of space of spherical shape that enclose a uniform density distribution of mass that can change with time but retaining an overall fixed quantity with respect to time of the usual positive gravitational mass within it boundary, an amount M , say. Usually there will be some moving gravitational centroid at which the gravitation force between objects will be thought to be acting. I also only use configurations in which this centroid is the centre of a sphere. The difference from Newtonian theory that I am about to introduce is the assumption that this Euclidean space is filled uniformly throughout all its extent by a positively mass density field of negatively characterised gravitational 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 = c2 Λ/(4πG) just as in my double version of the Einstein theory quantity, ρΛ = c2 Λ/(8πG). Consider now a spherical region of this space of radius r about the origin of this space as centre. Suppose this sphere contains a total amount of dark mass, M , with its positive gravitation characteristic, G. The sphere will also contain an amount of negative gravity, −G, dark
341
energy given by MΛ = ρ†Λ V (t) V (t) = 4πr3 (t)/3.
(9.1) (9.2)
Thus the total gravitational acceleration caused by the sphere’s contents at its surface will be given by the Newtonian gravitational formula, r¨(t) = MΛ† G/r2 (t) − M G/r2 (t) =
4πr3 ρ†Λ G/(3r2 ) − C/(2r2 ) 4πrρ†Λ G/3 − C/(2r2 ) 2 2
= = rc Λ/3 − C/(2r ).
(9.3) (9.4) (9.5) (9.6)
If we multiply equation (9.5) through by r, ˙ we obtain 2 r¨r˙ = 4πrrρ ˙ †Λ G/3 − C r/(2r ˙ ) d 2 2 d d 2 r˙ /2 = r Λc /6 + C r−1 /2 dt dt dt 2 2 −1 r˙ = (rc) Λ/3 + Cr C = 2M G.
(9.7) (9.8) (9.9) (9.10)
The constant of integration that could occur in integrating (9.8) can be taken to be zero under the conditions that r(t) ˙ is taken to be infinite with r(t) = 0 at t = 0. Thus the spherical region expands with high speed from the origin, r = 0 at time t = 0. The solution to equation (9.9) was obtained in paper A in the form r(t) RΛ b C
= = = =
b sinh2/3 (3ct/(2RΛ )) (3/Λ)1/2 (RΛ /c)2/3 C 1/3 2M G
(9.11) (9.12) (9.13) (9.14)
where M here is any dark mass value. It follows that the dark mass density of the spherical region containing total dark mass, M , is as in (8.1) given by ρ(t) = M/(4πr3 (t)/3) = M sinh−2 (3ct/(2RΛ ))/b3 = (3/(8πG))(c/RΛ )2 sinh−2 (3ct/(2RΛ )).
342
(9.15) (9.16)
Thus the ratio of dark energy mass density to dark mass density within this region 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 energy to dark mass, rΛ,DM (t), depends only on the dark energy mass density through t and RΛ . The time variable origin t = 0 depends only on where the sphere expansion is assumed to have started from with radius zero, an arbitrarily chosen space origin r(t) = 0 at time t = 0, in Euclidean three space. Thus it seems that this is a fundamental formula governing a time evolutionary process relating dark energy and dark mass. The consequence of this situation is that we can visualise, quite independently of relativity, such mixed mass region expansions. They can take place over time from anywhere in astro-space and apparently originate from a point quantity of dark mass, M , with infinite density. Further, the formula is time reversible so that it suggests that spherical contractions of spherical dark mass regions can also be visualised as a possible cosmological sequence of events resulting in the appearance of a point dark mass, M, with infinite density locally. As such an expansion proceeds the spherical region picks up dark energy mass from the enveloping Newtonian space, the expansion continuing with the expanding region having then a mixture of the two gravitational types of mass, ±G. An important event in the history of such an expansion is when there are equal quantities of the two mass types within the sphere. At this event occurring, the sphere will be gravitationally neutral. The sphere will at that time exert no gravitational force on material outside its boundary, it will be gravitationally isolated from any material exterior to itself. If we denote the time when the sphere is so isolated by tc this time can be found from 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 ρ†Λ = ρ(tc ) sinh2 (3ctc /(2RΛ )) = 1/2 ⇒ tc = ±(2RΛ /(3c)) sinh−1 (1/21/2 )
(9.18) (9.19) (9.20) (9.21)
and, curiously, the times ±tc do not depend on the amount of dark mass within the expanding sphere but only depends on the value of the cosmological constant, Λ. It follows that the time tc has exactly the same value
343
as the relativistic epoch time when the universe changes from deceleration to acceleration. The time tc is a fundamental universal time interval in the cosmological context. It is important to note that, as the process is time reversal invariant, the contraction sequence, in negative time, with mass M can be immediately followed by an expansion sequence with the same mass M , in positive time, so that conservation of mass is assured and mass is neither created from nothing nor is it destroyed at the singular event when t = 0. The non dependence of the process on the amount of dark mass within the boundary of the contracting or expanding sphere of dark mass has a surprising explanation. The process conforms exactly to the principle of equivalence. Just as the acceleration of a falling mass in a gravitational field does not depend on the value of the falling mass so the acceleration r¨Λ,DM (t) of the collapsing sphere process does not depend on its mass. The collapsing sphere in its own gravitational field conforms exactly too and is a manifestation of the principle of equivalence. It can occur locally and is a basic part of the description of the whole universe motion with epoch time. Recognition of this fundamental process in relation to other physical processes in cosmology will be discussed in the final section.
10
Appendix 1 Conclusions
The cosmological model introduced in references A, B, C and applied to the finding of solutions to the cosmological constant problem in D has here been applied to unravelling the dark mass problem. Here it has been shown that a fundamental time moving relation holds between dark energy and dark mass. This relation was first shown to hold at the scale of the whole universe by using the Friedman equations from Einstein’s general relativity and involving his positively valued cosmological constant Λ. Here it has been shown that the same relation can be derived from Newtonian gravitation theory with only the addition of a constant and universally distributed density of dark energy, ρ†Λ = 2ρΛ , twice the Einstein value ρΛ , in Newtonian space and only subject to Newtonian gravity theory. This result implies that the formula relating dark mass and dark energy is independent of general relativity and the way it is derived also show that it can have applications at a much smaller scale than that of the entire universe. It can describe local space and time small scale movements of dark mass in relation to dark 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 just empty voids [35] in the cosmological fabric. As we have seen, there are five main events in the time sequences evolution of this dark energy dark mass process, E0 , E±1 , E±∞ , say. They involve E0 when some definite random quantity of dark mass M is located at some definite point in three space at some definite time labelled as t = 0 for the process. At that time the dark mass is by itself because a point cannot contain any of the uniform and finite constant density of dark energy mass. Thus in space around the point mass it will own a Newtonian gravitational potential field −M G/r. At both the events E±1 at times ±tc because of the time reversal invariance the contracting or expanding sphere will contain equal quantities of the dark mass and dark energy so that the sphere will be gravitationally neutral. It will thus be isolated gravitationally and so not own any gravitational potential. However the total mass density within the spheres boundaries will be ρ(tc ) + ρ†Λ , a numerically very small value ≈ 9 proton masses per cubic meter. I think that such a sphere being gravitationally isolated and of such low density could qualify for the title cosmological void . At the events E±∞ , the sphere will own a gravitation potential at points within its surface involving both the dark energy and dark mass within concentric spheres of radius r < ∞ but dominated by the repulsive dark mass for relatively large values of r. The contraction phase between E−∞ and E0 might represent a moving platform for an original dark mass concentration to convert from pure dark mass to becoming dark mass contaminated with visible mass while its volume descends to occupying some relatively small region containing a group of visible galaxies or, a single galaxy or even a single particle. In other words, the descending spherical volume could represent a time dependant packaging process for cosmological clumping. A final remark about the relation of this theory structure to aether theory is appropriate. It is dark energy rather than dark mass that seems to play a role much like the all pervading aether which has been used to give a physical explanation for electromagnetic wave motion in so called empty space. The dark energy density is certainly an all-pervading effect in this cosmological theory as has been shown in this article and as it is also perceived in the present day arena of astronomical observations. It seems to be an everywhere present background reference level against which many astrophysical and quantum problems 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 effect either universally or locally of a boundary motion at an interface between dark mass and dark energy described by the inverse of the ratio, rΛ,DM (t).
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Appendix 2 Expanding Boundary Pressure Process All Pervading Dark Energy Aether in a Friedman Dust Universe with Einstein’s Lambda Abstract
In this appendix, the cosmological model that was introduced in a sequence of three earlier papers under the title A Dust Universe Solution to the Dark Energy Problem is used make a more detailed study of the role of dark energy mass as a conserved with time substance that permeates the expanding universe. It shown that if dark energy is to be conserved over all time it has to satisfy the cosmological vacuum polarisation equation over the presingularity range of contraction and the post-singularity range of expansion in order for it to remain in a self mechanical equilibrium inside and outside the boundary of the expanding universe and so be able to be always and everywhere permeable to the positive gravity dark mass and visible mass material within the universe.
12
Effect of Boundary Pressures
In the paper D, [34], it was shown that the quantum vacuum polarisation idea can be seen to play a central role in the Friedman dust universe model introduced by the author. An essential part of that role involves the relations between three pressures at the boundary of the expanding universe. In particular of fundamental importance is a relation between pressure from the CM B, PΓ , pressure from all the rest of the universe which is not CM B and not dark energy, P∆ , and pressure from dark energy itself, PΛ . This
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relation takes the form PΓ = P∆ + PΛ PΛ = c2 ρΛ ωΛ = −c2 ρΛ ωΛ = −1.
(12.1) (12.2) (12.3)
Equation (12.2) together with equation (12.3) is the well known specification implying negative pressure, PΛ , from the dark energy density in the Einstein form ρΛ = Λc2 /(8πG). In earlier work, I have referred to the equation (12.1) as representing a mechanical equilibrium between the Γ field and the ∆ and Λ fields combined. I now think that designation while remaining formally correct should be presented with a changed interpretation because of the negative pressure associated with the dark energy field, Λ. In the usual specification of a mechanical equilibrium two pressures P1 = P2 are said to be equal where there is no complication of possible negative parts for either of them. Pressures on either side of a boundary between non-miscible liquids for example are said to be in mechanical equilibrium if the boundary is not accelerating. In such a case, although the pressures act at the boundary in opposite directions they are both taken as positive. Mechanical equilibrium in thermodynamics is a very contentious area of research so that my explanation in this context is very minimal. As a result of these complications it is desirable to express equation (12.1) in the alternative form using a modulus sign, | |. The term boundary of the universe at time t refers to a conceptual sphere of radius given by the function r(t) defined earlier (A). PΓ = P∆ − |PΛ | P∆ = PΓ + |PΛ |.
(12.4) (12.5)
In this form, all the pressures are expressed as positive quantities and the mechanical equilibrium between these three field can now be more safely reinterpreted as a mechanical equilibrium between the ∆ field and the Λ and Γ fields combined. This version of the equilibrium condition at the boundary of the expanding universe makes good sense physically for at least two reasons. The first reason is that dark energy material exists on both sides of the expanding boundary of the universe so the ±|PΛ | versions refer to the pressure direction on the boundary from dark energy material on one side or the other, whilst the gravitational pressure, P∆ , is directed towards 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 gravitational theory. Taking the total pressure, P , in the Friedman equations as positive is a rather anomalous convention which has caused much confusion which I attempt here to unravel. The second reason is that the form (12.4) rearranged as in (12.6) PΛ = −|PΛ | = PΓ − P∆ P (t) = +PG + PΛ = P∆ (t) − PΓ (t) + PΛ ≡ 0 ∀ t.
(12.6) (12.7)
clearly expresses the physics of the equilibrium condition. It is that the negative outward pressure PΛ that would be exerted on the boundary from the dark energy inside the universe is equal to the difference of the outward CM B pressure, PΓ , less the inward pressure P∆ exerted on the universe boundary from within by the positive G or normal gravitating material within the universe. The equations (12.6) and (12.2) firmly identify both the pressure PΛ and the mass density ρΛ of the dark energy as coming from the quantities PΓ and P∆ both defined with meanings within the universe. Elsewhere, I have expressed the equation (12.6) using the P EID form which explains it in terms of the mass densities, Γ(t), ∆(t) rather than the equivalent pressures, ρΛ GρΛ G+ G−
= = = =
∆(t) − Γ(t). G+ ∆(t) + G− Γ(t) +G −G.
(12.8) (12.9) (12.10) (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 whole history of the universe in this model is that of a dust universe. Clearly then the total pressure P (t) cannot be responsible for the acceleration. This conclusion agrees with what was noted earlier that the acceleration is accurately determined by a generalisation of the Newtonian gravitation theory only involving adding to the inverses square law a linear law term involving Einstein’s cosmological constant Λ. The realisation that the various pressures that we have been discussing earlier do not determine the dynamical behaviour of the system generates the question, what is this complicated pressure structure all about?
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The immediate answer to the question at the end of the last paragraph is that the pressure structure that derives from the equilibrium relation between the three pressures, PΛ , P∆ and PΓ generates the important relation, (12.8) or (12.9). This relation shows that within the spherical volume of the universe the constant valued dark energy density, ρΛ , is determined by the dark mass dominated quantity pair ∆(t) and Γ(t). I emphasise within because both these quantities are part of the constant space and time conserved energy of the universe, MU . However, the space-time constant dark energy mass density ρΛ , by initial assumption, exists everywhere in the universe’s enveloping space with the same definite numerical value outside as inside. Thus the following subsidiary question presents itself: If the dark energy density inside the universe is given by (12.8) in terms of the internal constituents, ∆ and Γ, how is it that outside the universe involving regions which will not have been reached by the internal constituents of the expanding universe, the dark energy density ρΛ exist in its own right by assumption, with the same constant value as inside and apparently not generated by any internal influence? The unique character of this model does allow a satisfactory answer to this question which depends on the model’s strict conformance to the principle of conservation of energy in contrast with the standard big bang model . This model involves two basic positive types of mass defined by their gravitational character, which is determined by whether the mass appears in the theory multiplied with G+ = +G or G− = −G, where the gravitational constant G is always constant, G > 0. Dark mass and normal mass belongs to the G+ category and dark energy belongs to the G− category. An important way in which this model differs from 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 of time occurs at t = +∞. This can be interpreted as this universe lasts for ever . The reader may prefer to regard this infinite time scale as just one out of a possible infinite number of infinite contiguous periodic time scales and so reinforcing the lasting for ever concept. This last extension can be usefully incorporated in the theory, see paper (C). Let us now consider the situation at and after the start of time taken as t = −∞ + t , t ≈ +|0| at this stage the radius of the universe is infinitely large, r(−∞ + t ), and will decrease with advancing time. In other words, near the beginning of time the universe is a sphere occupying almost the whole of hyperspace and so 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 dark energy density ρΛ , though this density is itself small it adds up in total over the whole universe volume to a very large amount of dark energy mass. The universe also contains a much smaller density value, ρ(−∞+t ), of conserved positive gravitational mass, MU , so that ρ(−∞ + t )VU (−∞ + t ) = MU . This conserved mass contains the mass of the universe that we see. The basic assumption in this model is that dark energy density, ρΛ (t), exists everywhere 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 outside and takes in all the hyper-universe so that all the dark energy is enclosed. After the small time elapse, t , the universe will have acquired a small outside volume and a slightly smaller inside volume than it had initially. It follows that in principle there are two types of simple likely possibilities. Firstly, the contracting universe leaves no dark energy density outside as it evolves in time and keeps the original value inside at the same value as given by formula (12.8). Secondly, as it evolves in time it leaves outside sufficient dark energy density to keep to the uniform constant density condition everywhere and so keeping the dark energy density inside at the same value given by formula (12.8) as outside. The first option means that the dark energy within the universe would decrease with decreasing volume consequently losing density to no recognisable sink and so implying dark energy is not conserved. This would also violate the assumption that dark energy density is constant everywhere and at all time. Thus we are left with only the second possibility and consequently the actual scenario has to be that as the universe contracts the internal pressure process described by the internal ∆ and Γ fields precipitates the right amount of dark energy material outside in the space produce by the contracting universe. This process will continue for all time and, in particular, past the singularity at t = 0 when the volume is zero. Thus after the singularity, when the universe is in an expanding mode, it will encounter the pre-singularity dark energy density outside its boundaries precipitated in its contracting mode. Thus the main role of formula (12.9) is to keep the conserved and bounded dark mass within the universe freely permeable to or non interacting with the dark energy in which it swims by maintaining the self mechanical equilibrium of the dark energy mass density in the form, PΛ,in = PΛ,out ρΛ,in = ρΛ,out ,
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(12.12) (12.13)
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. Equation (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 that the pre-singularity negative time phase is a necessary adjunct to making sense of this theory. The conclusion associated with this section is that the formula (12.9) together with the full time history of this model assures that dark energy and dark mass are both conserved over all time. The above argument is not meant to be mathematically rigorous but rather a plausibility construction. No doubt the reader can think of various improvements.
References [1] R. A. Knop et al. arxiv.org/abs/astro-ph/0309368 New Constraints on ΩM , ΩΛ and ω from an independent Set (Hubble) of Eleven High-Redshift Supernovae, Observed with HST [2] Adam G. Riess et al xxx.lanl.gov/abs/astro-ph/0402512 Type 1a Supernovae Discoveries at z > 1 From The Hubble Space Telescope: Evidence for Past Deceleration 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 and The Fine Structure Constant, Speculations in Science and 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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[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 Physics World Scientific Ltd. [10] Kilmister, C. W. 1994 , Eddington’s search for a Fundamental Theory, 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 Packing and 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.org The fine structure constant [15] McPherson R., Stoney Scale and Large Number Co-incidences, to be published in Apeiron, 2007 [16] Rindler, W. 2006, Relativity: Special, General and 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.pdf A Sketch for a Quantum Theory of Gravity: [21] J. G. Gilson, arxiv.org/PS cache/physics/pdf/0504/0504106v1.pdf Dirac’s Large Number Hypothesis and Quantized Friedman Cosmologies
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[22] Narlikar, J. V., 1993, Introduction to Cosmology, CUP [23] Gilson, J.G. 2005, A Dust Universe Solution to the Dark Energy Problem, 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 Gravity Material, Identification of Dark Energy, arxiv.org/abs/physics/0603226 [25] G. Lemaˆıtre, 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, 2006 http://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.pdf Presentation 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 in a 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 Einstein Universe and in a Friedman Dust Universe with Einstein’s Lambda
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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 nonEuclidean 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 nonEuclidean.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 nonEuclidean 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 nonEuclidean 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 "SpaceTheory" 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 "SpaceTheory" 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 nonEuclidean 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 spacecrumplers 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 "spacecrumplers" 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 spacetheory 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 threedimensional 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 threedimensional space was probably considered a boundary between two fourdimensional 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 locusin-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 nonEuclidean 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 nonEuclidean 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 nonEuclidean 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 ethersquirts, 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 nonEuclidean 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 nonEuclidean 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 nonEuclidean 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 nonEuclidean 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 nonEuclidean 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 fourdimensional 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 ethersquirts 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 nonEuclidean 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 nonEuclidean 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 nonEuclidean 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 nonEuclidean 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 nonEuclidean 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 "SpaceTheory 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 nonEuclidean 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 nonEuclidean 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 nonEuclidean 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 spacetime. 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 fourdimensional 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 nonEuclidean, 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 nonEuclidean 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 nonEuclidean geometries. These were no mean accomplishments and Greenwood did not need to mention Clifford's name within the context of the prerelativistic 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 nonEuclidean. 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 2026, 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.
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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: 381395; 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, JanuaryJune 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).
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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: 111, 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.
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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: 667687, 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
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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.
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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: 127132, 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, 18841886), 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.
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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.446447, or 462.
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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.
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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, 19181919, 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.
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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.
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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.
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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 U = −G Mm / r , 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 − G ( M m / r )(r 2ϕ& 2 ) / c 2 . 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.
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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 δ N = 2GM Sun /(c 2 R Sun ) , though Einsteinian’s is still
δ GR = 2δ N where M Sun and R Sun 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
V = (1 − r& 2 / 2c 2 ) ⋅1 / r , 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
V =−
m
r ⋅ (t − τ ) r ⋅ (t − τ ) − c
.
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. δ GR = 4GM /(c 2 rp ) where rp 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
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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 tessellattice 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
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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 Λ = λ c / υ 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
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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 r&rad , which is parallel/antiparallel to the radial ray issued from the central object to the test body; and tangential r& tan , 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. r& rad = c [13-15], we obtain that the total velocity of the test body’s inertons cˆ in the frame of reference associated with the central object is defined from the geometric relationship (compare with Ref. [18]) 2 cˆ 2 = c 2 + r& tan
(2)
Hence around the test body in the region r < Λ ( 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 cˆ > c . 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
U = −G
Mm . r
(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
U = −G
2 2 Mm ⎛⎜ r& tan ⎞⎟ Mm ⎛⎜ r& tan ⋅ 1 + 2 = −G ⋅ 1+ 2 r ⎜⎝ r&rad ⎟⎠ r ⎜⎝ c
⎞ ⎟ ⎟ ⎠
(4)
where r& tan 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
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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)
I = m r 2ϕ& ;
(5)
Ecl. = 12 mr& 2 + 12 m r 2ϕ& 2 − G
Mm . r
(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
I = m r 2ϕ& ;
(7)
E = 12 m r& 2 + 12 m r 2ϕ& 2 − G
M m ⎛ r 2ϕ& 2 ⎞ ⋅ ⎜1 + 2 ⎟⎟ . r ⎜⎝ c ⎠
(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
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Δϕ = 6π
GM Lc 2
(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 − GMm / r 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 − G M m r ϕ& 2 / c 2 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
I = m r 2ϕ& ;
(10)
E = 12 m r& 2 + 12 m r 2ϕ& 2 − G
M m r ϕ& 2 c2
(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
GM . c2r
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(12)
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
K = 12 m l 2ϕ& 2
(13)
and the potential energy is
U = −G
⎛ l 2ϕ& 2 ⎞ Mm ⋅ ⎜⎜1 + 2 ⎟⎟ r + l ⋅ (1 − cos ϕ ) ⎝ c ⎠
(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
E ≅ m l ϕ& 2 − G 1 2
2
Mm Mm ⎛ lϕ 2 l 2ϕ& 2 ⎞ −G ⋅⎜− + 2 ⎟⎟ . r r ⎜⎝ 2 r c ⎠
(15)
In the case of the potential depending on the velocity the equation of motion is determined by the Euler-Lagrange equation [31]
d ∂ K d ∂U ∂ K ∂U − − + =0 d t ∂ q& d t ∂ q& ∂ q ∂ q where in our case q ≡ ϕ and t is the proper time of the oscillating massive point. In the explicit form it yields
⎛ 2 M l2 ⎞ M ⎜ l + 2G ⎟ ϕ&& + G lϕ = 0 . 2 ⎟ ⎜ r c ⎠ r ⎝
(16)
If we designate (2 π ν 0 )2 = 2GM /(rl) , we can write instead of Eq. (16)
ϕ&& +
(2πν 0 ) 2 − ϕ = 0. 1 + 2GM /(c 2 r )
(17)
In Eq. (17) assuming the inequality r0 = 2GM / c2 0) . . . [4] We actually improve on these null displacement ideas by introducing the more fundamental monogenic condition, deriving the former from the latter and establishing a common first principle. The only postulates in this paper are of a geometrical nature and can be summarized in 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 this geometric space does not imply any assumption for physical space up to the point where geometric entities like coordinates and geodesics start being assigned to physical quantities like distances and trajectories. Some of those assignments will be made very soon in the exposition and will be kept consistently until the end in order to allow the reader some assessment of the proposed geometric model as a tool for the prediction of physical phenomena. Mapping between geometry and physics is facilitated if one chooses to work always with non-dimensional quantities; this is done with a suitable choice for standards of the fundamental units. From this point onwards all problems of dimensional homogeneity 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
r
r
r
G¯h c3
G¯h c5
h¯ c G
Charge e
This normalization defines a system of non-dimensional units (Planck units) with important consequences, namely: 1) All the fundamental constants, h¯ , G, c, e, become unity; 2) a particle’s Compton frequency, defined by ν = mc2 /¯h, becomes equal to the particle’s mass; 3) the frequent term GM/(c2 r) is simplified to M/r. 5-dimensional space can have amazing structure, providing countless parallels to the physical world; this paper is just a limited introductory look at such structure and parallels. The exposition makes full use of an extraordinary and little known mathematical tool called geometric algebra (GA), a.k.a. Clifford algebra, which received an important thrust with the works of David Hestenes [5]. A good introduction to GA can be found in Gull et al. [6] and the following paragraphs use basically the notation and conventions therein. A complete course on physical applications of GA can be downloaded from the internet [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 are introduced. Section two deals with monogenic function in flat 5D spacetime, deriving special relativity and the free particle Dirac equation from this simple concept. 4DO appears here as a perfect equivalent to special relativity, where trajectories can be understood as normals to 4-dimensional plane-like waves. The following section improves on this by allowing for curved space, introducing the notion of refractive index tensor. Section five examines the variational principle applied in both 4DO and GTR spaces to justify the equivalence of geodesics between the two spaces for static metrics. Refractive index is then related to 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 and electrodynamics are formulated as particular cases of refractive index in section seven and the sources tensor is here related to a current vector. The next section introduces the hypothesis of an hyperspherical symmetry in the Universe, which would call for the use of hyperspherical coordinates; the consequences for cosmology would include a complete 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 model of particle physics. 2. Introduction to geometric algebra Geometric algebra is not usually taught in university courses and its presence in the literature is scarce; good reference works are [5, 7, 8]. We will concentrate on the algebra of 5-dimensional spacetime because this will be our main working space; this algebra incorporates as subalgebras those of the usual 3-dimensional Euclidean space, Euclidean 4-space and Minkowski spacetime. We begin with the simpler 5D flat space and progress to a 5D spacetime of general curvature. The geometric algebra G4,1 of the hyperbolic 5-dimensional space we consider is generated by the coordinate frame of orthonormal basis vectors σα such that (σ0 )2 = −1, (σi )2 = 1, σα · σβ = 0,
(2.1) α 6= β .
Note that the English characters i, j, k range from 1 to 4 while the Greek characters α, β , γ range from 0 to 4. See the Appendix A for the complete notation convention used. 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 distributive. Similarly to the product of two basis vectors, the product of three different basis vectors produces a trivector and so forth up to the fivevector, because five is the dimension of space. We will simplify the notation for basis vector products using multiple indices, i.e. σα σβ ≡ σαβ . The algebra is 32-dimensional and is spanned by the basis • 1 scalar, 1, • 5 vectors, σα , • 10 bivectors (area), σαβ ,
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• 10 trivectors (volume), σαβ γ , • 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. Start by expanding the square and remove the compact notation (σ0i j )2 = σ0 σi σ j σ0 σi σ j , then swap 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 = σ0 σi (σ j )2 σ0 σi . From the third equation (2.1) we know that the squared vector is unity and we get successively (σ0i j )2 = σ0 σi σ0 σi = −(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 a square root of −1; this makes it a very special element which can play the role of the scalar imaginary in complex algebra. We can now address the geometric product of any two vectors a = aα σα and b = bβ σβ making use of the distributive property ! ab =
−a0 b0 + ∑ ai bi + i
∑ aα bβ σαβ ;
(2.4)
α6=β
and we notice it can be decomposed into a symmetric part, a scalar called the inner or interior product, and an anti-symmetric part, a bivector called the outer or exterior product. 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 =
1 (ab + ba), 2
a∧b =
1 (ab − ba). 2
(2.6)
The inner product is the same as the usual ”dot product,” the only difference being in the negative sign of the a0 b0 term; this is to be expected and is similar to what one finds in special relativity. The outer product represents an oriented area; in Euclidean 3-space it can be linked to the "cross product" by the relation cross(a, b) = −σ123 a ∧ b; here we
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introduced bold characters for 3-dimensional vectors and avoided defining a symbol for the cross product because we will not use it again. We also used the convention that interior and exterior products take precedence over geometric product in an expression. When a vector is operated with a multivector the inner product reduces the grade of each element by one unit and the outer product increases the grade by one. We will generalize the definition of inner and outer products below; under this generalized definition the inner product between a vector and a scalar produces a vector. Given a multivector a we refer to its grade-r part by writing < a >r ; the scalar or grade zero part is simply designated as < a >. By operating a vector with itself we obtain a scalar equal to 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 = ∑ < a >α < b >β |α−β | , (2.8) α,β
a∧b =
∑
< a >α < b >β α+β .
(2.9)
α,β
Two other useful products are the scalar product, denoted as < ab > and commutator product, defined by a × b = (ab − ba)/2. (2.10) In mixed product expressions we will always use the convention that inner and outer products take precedence over geometric products, thus reducing the number of parenthesis. We will encounter exponentials with multivector exponents; two particular cases of exponentiation are specially important. If u is such that u2 = −1 and θ is a scalar euθ
θ2 θ3 θ4 −u + +... 2! 3! 4! θ2 θ4 = 1− + − . . . {= cos θ } + 2! 4! θ3 +uθ − u + . . . {= u sin θ } 3! = cos θ + u sin θ .
= 1 + uθ −
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(2.11)
Conversely if h is such that h2 = 1 ehθ
θ2 θ3 θ4 +h + +... 2! 3! 4! θ2 θ4 = 1+ + + . . . {= cosh θ } + 2! 4! θ3 +hθ + h + . . . {= h sinh θ } 3! = cosh θ + h sinh θ . = 1 + hθ +
(2.12)
The exponential of bivectors is useful for defining rotations; a rotation of vector a by angle θ on the σ12 plane is performed by ˜ a0 = eσ21 θ /2 aeσ12 θ /2 = RaR;
(2.13)
the tilde denotes reversion and reverses the order of all products. As a check we make a = σ1 θ θ e−σ12 θ /2 σ1 eσ12 θ /2 = cos − σ12 sin σ1 ∗ 2 2 θ θ (2.14) ∗ cos + σ12 sin 2 2 = cos θ σ1 + sin θ σ2 . Similarly, if we had made a = σ2 , the result would have been − sin θ σ1 + cos θ σ2 . If we use B to represent a bivector whose plane is normal to σ0 and define its norm by ˜ 1/2 , a general rotation in 4-space is represented by the rotor |B| = (BB) |B| B |B| −B/2 R≡e = cos − sin . (2.15) 2 |B| 2 The rotation angle is |B| and the rotation plane is defined by B. A rotor is defined as a unitary even multivector (a multivector with even grade components only) which squares to unity; we are particularly interested in rotors with bivector components. It is more general to define a rotation by a plane (bivector) then by an axis (vector) because the latter only works in 3D while the former is applicable in any dimension. When the plane of bivector B contains σ0 , a similar operation does not produce a simple rotation but produces a boost, eventually combined with a rotation. Take for instance B = σ01 θ /2 and define the transformation operator T = exp(B); a transformation of the basis vector
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σ0 produces a0 = T˜ σ0 T = e−σ01 θ /2 σ0 eσ01 θ /2 θ θ cosh − σ01 sinh σ0 ∗ = 2 2 θ θ ∗ cosh + σ01 sinh 2 2 = cosh θ σ0 + sinh θ σ1 .
(2.16)
In 5-dimensional spacetime of general curvature, we introduce 5 coordinate frame vectors gα , the indices follow the conventions set forth in Appendix A. We will also assume 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 their geometric product. The space signature is (− + + + +), which amounts to saying that g00 < 0 and gii > 0. A reciprocal frame is defined by the condition gα · gβ = δ α β .
(2.18)
Defining gαβ as the inverse of gαβ , the matrix product of the two must be the identity matrix; using Einstein’s summation convention this is gαγ gγβ = δ α β .
(2.19)
gαγ gγ · gβ = δ α β ;
(2.20)
Using the definition (2.17) we have
comparing with Eq. (2.18) we determine gα with gα = gαγ gγ .
(2.21)
If the coordinate frame vectors can be expressed as a linear combination of the orthonormed ones, we have gα = nβ α σβ , (2.22) where nβ α 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] When the refractive index is the identity, we have gα = σα for the main or direct frame and g0 = −σ0 , gi = σi for the reciprocal frame, so that Eq. (2.18) is verified. In this work we will not consider spaces of general curvature but only those satisfying condition (2.22).
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The first use we will make of the reciprocal frame is for the definition of two derivative operators. In flat space we define the vector derivative ∇ = σ α ∂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 used determines the subspace to which the derivative applies; For instance m∇ = σ m ∂m = σ 1 ∂1 +σ 2 ∂2 +σ 3 ∂3 . In 5-dimensional space it will be useful to split the vector derivative into its time and 4-dimensional parts ∇ = −σ0 ∂t + σ i ∂i = −σ0 ∂t + i∇.
(2.24)
The second derivative operator is the covariant derivative, sometimes called the Dirac operator, 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 covariant derivative is also a vector. In cases such as those we consider in this work, where there is 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 covariant Laplacian respectively, resulting from the inner product of one derivative operator by itself. The square of a vector is always a scalar and the vector derivative is no exception, so the Laplacian is a scalar operator, which consequently acts separately in each component of a multivector. For 4 + 1 flat space it is ∇2 = −
∂2 i 2 +∇ . ∂t 2
(2.26)
One sees immediately that a 4-dimensional wave equation is obtained by zeroing the Laplacian of some function ∂2 (2.27) ∇2 ψ = − 2 + i∇2 ψ = 0. ∂t This procedure will be used in the next section for the derivation of special relativity and will 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 monogenic functions,[8] characterized by having null vector derivative; a function ψ is monogenic if and only if ∇ψ = 0. (3.1)
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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
∑ ∂ii ψ = ∂00 ψ.
(3.2)
i
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 = ψ0 ei(pα x
α +δ )
,
(3.3)
where ψ0 is an amplitude whose characteristics we shall not discuss for now, δ is a phase angle and pα are constants such that
∑(pi )2 − (p0 )2 = 0.
(3.4)
i
By setting the argument of ψ constant in Eq. (3.3) and differentiating we can get the differential equation pα dxα = 0. (3.5) The first member can equivalently be written as the inner product of the two vectors p · dx = 0, where p = σ α pα . In 5D hyperbolic space the inner product of two vectors can 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 dx normal to p or by (dx)2 = 0. In the former case the condition describes a 3-volume called wavefront and in the latter case it describes the wave motion. Notice that the wavefronts are 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 are allowed to rewrite it with any chosen terms in the second member; some of those manipulations are particularly significant. Suppose we decide to isolate (dx4 )2 in the first member: (dx4 )2 = (dx0 )2 − ∑(dxm )2 . We can then rename coordinate x4 as τ, to get the interval squared of special relativity for space-like displacements dτ 2 = (dx0 )2 − ∑(dxm )2 .
(3.7)
We have thus derived the space-like part of special relativity as a consequence of monogeneity in 5D hyperbolic space and simultaneously justified the physical interpretation for 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 the 4DO concept.[11, 12] If we isolate (dx0 )2 and replace x0 by the letter t, we see that time becomes the interval in Euclidean 4D space dt 2 = ∑(dxi )2 .
(3.8)
From this we conclude that the monogenic condition produces plane waves whose wavefronts are 3D volumes but can be represented by wavefront normals, just as it happens in standard optics with electromagnetic waves. Several readers may be worried with the fact that proper time is a line integral and not a coordinate in special relativity and so dτ should not be allowed to appear on the rhs of the equation. To this we will argue that the manipulations we have done, collapsing 5D spacetime into 4 dimensions through a null displacement condition and then promoting one of the coordinates into interval, is exactly equivalent to the process of defining a light cone in Minkowski spacetime and then applying Fermat’s principle to define an Euclidean 3D metric on the light cone; we have just upgraded the procedure by including one extra dimension. The Dirac equation can also be derived from the monogenic condition but since it appears 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 matrix operations. This is easily achieved if we assign our frame vectors to Dirac matrices that square to the the identity matrix or minus the identity matrix as appropriate; the following list of assignments can be used but others would be equally effective1 i 0 0 0 0 0 0 1 0 −i 0 0 , σ 1 ≡ 0 0 −1 0 , σ0 ≡ 0 0 i 0 0 −1 0 0 0 0 0 −i 1 0 0 0 0 1 0 0 0 i 0 0 1 0 0 0 −i 0 0 0 3 (3.9) σ2 ≡ 0 0 0 −i , σ ≡ 0 0 0 1 , 0 0 1 0 0 0 i 0 0 0 0 −i 0 0 i 0 σ4 ≡ 0 −i 0 0 . i 0 0 0 There is no need to adopt different notations to refer to the frame vectors or to their matrix counterparts because the context will usually be sufficient to determine what is meant. 1 There are 16 possible 4∗4 Dirac matrices,[13] of which we must choose 5 such that (σ )2 = −I, (σ )2 = I i 0
and σα σβ = −σβ σα , for α 6= β ; the present choice will simplify our symmetry discussions further along.
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We can check that matrices σ α form an orthonormal basis of 5D space by defining the 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 -matrices is null, (σ 0 )2 = −I and (σ i )2 = I; these are the conditions defining an orthonormal basis expressed in matrix form. A more formal approach to this subject would lead us to invoke the isomorphism between the complex algebra of 4 ∗ 4 matrices and Clifford algebra 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 +σ 4 ∂4 )ψ = 0. If this is applied to the solution (3.3) and the derivative with respect to x4 is evaluated we get (σ µ ∂µ + σ 4 ip4 )ψ = 0. (3.11) Let us now multiply both sides of the equation on the left by σ 4 and note that matrix σ 4 σ 0 squares to the identity while the 3 matrices σ 4 σ m square to minus identity; we rename these products as γ-matrices in the form γ µ = σ 4 σ µ . Rewriting the equation in this form we get (γ µ ∂µ + ip4 )ψ = 0. (3.12) The only thing this equation needs to be recognized as Dirac’s is the replacement of p4 by the particle’s mass m; simultaneously we assign the energy E to p0 and 3D momentum p to σ m pm . We turn now our attention to the amplitude ψ0 in Eq. (3.3) because we know that the Dirac equation accepts solutions which are spinors and we want to find out their equivalents in our formulation. Applying the monogenic condition to Eq. (3.3) we see that the following equation must be verified ψ0 (σ α pα ) = 0.
(3.13)
If the σ s are interpreted as matrices, remembering that p is null, the only way the equation 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 and ψ0 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 consideration the previous assignments ψ = A(σ4 m + p ∓ σ0 E)eu(±Et+p·x+mτ+δ ) ;
(3.14)
where A is the amplitude and x = σm xm 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 matrices are called idempotents because they reproduce themselves when squared. These idempotents 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 =ψ
I + σ4 I − σ4 +ψ = ψ+ + ψ− . 2 2
(3.15)
This apparently trivial decomposition produces some surprising results due to the following relations eiθ (I + σ4 ) = (cos θ + i sin θ )(I + σ4 ) = (I cos θ + iσ4 sin θ )(I + σ4 ) = e
iσ4 θ
(3.16)
(I + σ4 ).
and similarly eiθ (I − σ4 ) = e−iσ4 θ (I − σ4 ).
(3.17)
If we had chosen a different idempotent the result would have been similar; we will see how the various idempotents are arranged in a symmetry group and it has been argued that they may be related to elementary particles.[15] 4. Relativistic dynamics When working in curved spaces the monogenic condition is naturally modified, replacing the vector derivative ∇ with the covariant derivative D. A generalized monogenic function 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 monogenic function must verify the second order differential equation D2 ψ = 0.
(4.2)
It is possible to write a general expression for the covariant Laplacian in terms of the metric tensor components (see [16, Section 2.11]) but we will consider only situations where that complete general expression is not needed. When Eq. (4.1) is multiplied on the left by D, we are applying second derivatives to the function, but we are simultaneously applying first order derivatives to the reciprocal frame vectors present in the definition of D itself. We can simplify the calculations if the 2 Matrix
σ 4 is the same as matrix γ 5 = iγ 0 γ 1 γ 2 γ 3 .
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variations of the frame vectors are taken to be much slower than those of function ψ so that frame vector derivatives can be neglected. With this approximation, the covariant Laplacian 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 derivatives of pα are neglected. This approximation is usually of the same order as the former one and should not be seen as a second restriction. Inserting Eq. (3.3) one sees that it is a solution if gαβ pα pβ = 0. (4.4) This equation is the curved space equivalent to Eq. (3.4) and it means that the square of vector p = gα pα is zero, that is, p is a vector of zero length; for this reason it is called a null vector or nilpotent. Vector p is the momentum vector and should not be confused with 4-dimensional conjugate momentum vectors defined below. We arrive again at Eq. (3.5) and the condition describing 4D wave motion can be expanded as gαβ dxα dxβ = 0. (4.5) This condition effectively reduces the spatial dimension to four but the resulting space is non-metric because all displacements have zero length. We will remove this difficulty by considering two special cases. First let us assume that vector g0 is normal to the other frame vectors so that all g0i factors are zeroed; condition (4.5) becomes g00 (dx0 )2 + gi j dxi dx j = 0.
(4.6)
All the terms in this equation are scalars and we are allowed to rewrite it with (dx0 )2 in the lhs gi j i j (dx0 )2 = − dx dx . (4.7) g00 We could have arrived at the same result by defining a 4-dimensional displacement vector −1 dx0 v = √ gi dxi ; g00
(4.8)
and then squaring it to evaluate its length; v is a unit vector called velocity because its definition is similar to the usual definition of 3-dimensional velocity; its components are dxi . dx0 Being unitary, the velocity can be obtained by a rotation of the σ4 frame vector vi =
˜ 4 R. v = Rσ
446
(4.9)
(4.10)
The rotation angle is a measure of the 3-dimensional velocity component. A null angle corresponds to v directed along σ4 and null 3D component, while a π/2 angle corresponds to the maximum possible 3D component. The idea that physical velocity can be seen as the 3D component of a unitary 4D vector has been explored in several papers but see [17]. Equation (4.8) projects the original 5-dimensional space into an Euclidean signature 4 dimensional space, where an elementary displacement is given by the variation of coordinate x0 . In the particular case where g0 = σ0 the displacement vector simplifies to dx0 v = gi dxi 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 of the 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 other frame vectors in order to isolate (dx4 )2 instead of (dx0 )2 , as we did before; (dx4 )2 = −
gµν µ ν dx dx . g44
(4.11)
We have now projected onto 4-dimensional space with signature (+ − −−), known as Minkowski signature. In order to check this consider again the special case with g0 = σ0 and the equation becomes (dx4 )2 =
1 gmn m n (dx0 )2 − dx dx ; g44 g44
(4.12)
the diagonal elements gii are necessarily positive, which allows a verification of Minkowski signature. 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 1
dx4 ν = p
g44 g44
gµ g4 dxµ .
(4.13)
All the products gµ g4 are bivectors because we imposed g4 to be normal to the other frame vectors. When (dx4 )2 is evaluated by an inner product we notice that g0 g4 has positive square while the three gm g4 have negative square, ensuring that a Minkowski signature is obtained. Naturally we have to impose the condition that none of the frame vectors depends on x4 . Bivector ν is such that ν 2 = νν = 1 and it can be obtained by a Lorentz transformation of bivector σ04 . ν = T˜ σ04 T,
(4.14)
where T is of the form T = exp(B) and B is a bivector whose plane is normal to σ4 . Note that 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 frame of vectors γµ such that (γ0 )2 = 1 and (γm )2 = −1, producing the desired Minkowski signature [8]. The geometric algebra of this space is isomorphic to the even sub-algebra of
447
G4,1 and so the area element dx4 ν (4.13) can be reformulated as a vector called relativistic 4-velocity. The four γ bivectors are defined in a similar way to the γ matrices used in Eq. (3.12), which is to be expected from the isomorphism between geometric and matrix algebras already mentioned. Equations (4.7) and (4.11) define two alternative 4-dimensional spaces, those of 4dimensional 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 parameter while in the latter it is x4 that takes such role. In fact Eq. (4.11) only covers the spacelike part of GTR space, because (dx4 )2 is necessarily non-negative. Naturally there is the limitation that the frame vectors are independent of both x0 and x4 , equivalent to imposing a static metric, and also that g0i = gµ4 = 0. Provided the metric is static, the geodesics of 4DO can be mapped one-to-one with spacelike geodesics of GTR and we can choose to work on the space that best suits us for free fall dynamics. For a physical interpretation of geometric relations it will frequently be convenient to assign new designations to the 5D coordinates that acquire the role of affine parameter in the null subspace. We recall the assignments x0 ≡ t and x4 ≡ τ; total derivatives with respect to these coordinates will receive a special notation: d f /dt = f˙ and d f /dτ = fˇ. Unless otherwise specified, we will assume that the frame vector associated with coordinate x0 is unitary and normal to all the others, that is g0 = σ0 and g0i = 0. Recalling from Eq. (4.7), these conditions allow the definition of 4DO space with metric tensor gi j . Although we could try a more general approach, we would loose the possibility of interpreting time as a line element and this, as we shall see, provides very interesting and novel interpretations of physics’ equations. In many cases it is also true that g4 is normal to the other frame vectors and we have seen that in those cases we can make metric conversions between GTR and 4DO; as we shall see, electromagnetism requires a non-normal g4 and so we leave this possibility open. For the moment we will concentrate on isotropic space, characterized by orthogonal refractive index vectors gi whose norm can change with coordinates but is the same for all vectors. Normally we relax this condition by accepting that the three gm must have equal norm but g4 can be different. The reason for this relaxed isotropy is found in the 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 ignores what happens with dimension 4. We will therefore characterize an isotropic space by the refractive index frame g0 = σ0 , gm = nr σm , g4 = n4 σ4 . Indeed we could also accept a non-orthogonal g4 within the relaxed isotropy concept but we will not do so for the moment. Equation (4.7) can now be written in terms of the isotropic refractive indices as dt 2 = (nr )2 ∑(dxm )2 + (n4 dτ)2 .
(4.15)
m
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 previous equation then becomes dt 2 = (nr )2 dr2 + r2 (dθ 2 + sin2 θ dϕ 2 ) + (n4 dτ)2 . (4.16) Since we have g4 normal to the other vectors we can apply metric conversion and write the equivalent quadratic form for GTR
2
dτ =
dt n4
2
nr − n4
2
2 dr + r2 (dθ 2 + sin2 θ dϕ 2 ) .
(4.17)
In the case of a central mass, we can examine how the Schwarzschild metric in GTR can be transposed to 4DO. The usual form of the metric is dτ
2
2M 2M −1 2 2 = 1− dχ − dt − 1 − χ χ −χ 2 dθ 2 + sin2 θ dϕ 2 ;
(4.18)
where M is the spherical mass and χ is the radial coordinate, not the distance to the centre of the mass. This form is non-isotropic but a change of coordinates can be made that returns the expression to isotropic form (see D’Inverno [18], section 14.7): p r = χ − M + χ 2 − 2Mχ /2; (4.19) and the new form of the metric is M 2 1− M 4 2 2 2 2r dτ = dt − 1 + dr − r2 dθ 2 + sin2 θ dϕ 2 . M 2r 1+ 2r
(4.20)
From this equation we immediately define two coefficients, which are called refractive index coefficients, M 3 M 1+ 1+ 2r 2r n4 = , nr = . (4.21) M M 1− 1− 2r 2r These refractive indices provide a 4DO Euclidean space equivalent to Schwarzschild metric, allowing 4DO to be used as an alternative to GTR. Recalling that we derived trajectories from solutions (3.3) of a 4-dimensional wave equation (4.3), it becomes clear that orbits can also be seen as 4-dimensional guided waves by what could be described as a 4-dimensional optical fibre. Modes are to be expected in these waveguides and we shall 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 the one that makes the travel time an extremum; usually it is the path that minimizes the time but in some cases a ray can follow a path of maximum or stationary time. These solutions are usually unstable, so one takes the view that light must follow the quickest path. In Eq. (4.7) we have defined a time interval associated with a 4-dimensional elementary displacement, which allows us to determine, by integration, a travel time associated with displacements of any size along a given 4-dimensional path. We can then extend Fermat’s principle to 4D and impose an extremum requirement in order to select a privileged path between any two 4D points. Taking the square root to Eq. (4.7) r gi j i j dt = − dx dx . (5.1) g00 Integrating between two points P1 and P2 Z P2 r Z P2 r gi j i j gi j i j − − t= dx dx = x˙ x˙ dt. g00 g00 P1 P1
(5.2)
In order to evaluate the previous integral one must know the particular path linking the points by defining functions xi (t), allowing the replacement dxi = x˙i dt. At this stage it is useful to define a Lagrangian gi j i j L=− x˙ x˙ . (5.3) 2g00 The time integral can then be written Z P2 √
t=
2L dt.
(5.4)
P1
Time has to remain stationary against any small change of path; therefore we envisage a slightly distorted path defined by functions xi (t) + ε χ i (t), where ε is arbitrarily small and χ i (t) are functions that specify distortion. Since the distortion must not affect the end points, the distortion functions must vanish at those points. The time integral will now be a function of ε and we require that dt(ε) = 0. (5.5) dε ε=0 Now, the Lagrangian (5.3) is a function of xi , through gαβ and also an explicit function of x˙i . Allowing for a path change, through ε makes t in Eq. (5.4) a function of ε t(ε) =
Z P2 q
2L(xi + ε χ i + x˙i + ε χ˙ i ) dt.
P1
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(5.6)
This can now be derived with respect to ε Z P 2 1 ∂L i ∂L i dt(ε) √ = χ˙ + i χ dt . dε ε=0 ∂x P1 2L ∂ x˙i ε=0
(5.7)
Note that the first term on the rhs can be written Z P2 P1
1 ∂L i √ χ˙ dt = 2L ∂ x˙i
Z P2 √ ∂ ( 2L) i χ˙ dt. P1
∂ x˙i
(5.8)
This can be integrated by parts Z P2 √ ∂ ( 2L) i χ˙ dt = P1
∂ x˙i
"
#P2 Z √ P2 d ∂ ( 2L) i − χ P1 dt ∂ x˙i P1
! √ ∂ ( 2L) χ i dt. ∂ x˙i
(5.9)
The first term on the second member is zero because χ i vanishes for the end points; replacing in Eq. (5.7) Z P2 dt(ε) d 1 ∂L i 1 1 ∂L +√ χ dt. (5.10) =√ −√ dε ε=0 L ∂ x˙i L ∂ xi 2 P1 dt The rhs must be zero for arbitrary distortion functions χ i , so we conclude that the following set of four simultaneous equations must be verified d 1 ∂L 1 ∂L √ =√ ; (5.11) i dt L ∂ x˙ L ∂ xi these are called the Euler-Lagrange equations. Consideration of Eqs. (4.8) and (4.11) allows us to conclude that the Lagrangian defined by (5.3) can also be written as L = v2 /2 and must always equal 1/2. From the Lagrangian one defines immediately the conjugate momenta vi =
−gi j j ∂L = x˙ . i ∂ x˙ g00
(5.12)
Notice the use of the lower index (vi ) to represent momenta while velocity components have an upper index (vi ). The conjugate momenta are the components of the conjugate momentum vector gi vi v= √ (5.13) −g00 and from Eq. (2.18)
√ −g00 v = gi vi = gi gi j x˙ j = g j x˙ j .
451
(5.14)
The conjugate momentum and velocity are the same but their components are referred to the reciprocal and refractive index frames, respectively.3 Notice also that by virtue of Eq. (3.4) it is also pi vi = . (5.15) p0 The Euler-Lagrange equations (5.11) can now be given a simpler form v˙i = ∂i L.
(5.16)
This set of four equations defines trajectories of minimum time in 4DO space as long as the frame vectors gα are known everywhere, independently of the fact that they may or may not be referred to the orthonormed frame via a refractive index. By definition these √ trajectories 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 now defined by 1 gµν µ ν L =− xˇ xˇ . (5.17) 2 g44 Consequently the conjugate momenta are νµ =
−gµν ν ∂L = xˇ . µ ∂ xˇ g44
(5.18)
From Eq. (3.4) we have νµ = pµ /p4 ; the associated Euler-Lagrange equations are νˇ µ = ∂µ L .
(5.19)
"These are, by definition, spacelike geodesics of GTR with metric tensor −gµν /g44 and we have thus defined a method for one-to-one geodesic mapping between 4DO and spacelike GTR. Recalling the conditions for this mapping to be valid, all the frame vectors must be independent of both t and τ and g0 and g4 must be normal to the other 3 frame vectors. In tensor terms, all the gαβ must be independent from t and τ and g0i = gµ4 = 0." 6. The sources of refractive index The set of 4 equations (5.16) defines the geodesics of 4DO space; particularly in cases where there is a refractive index, it defines trajectories of minimum time but does not tell us anything about what produces the refractive index in the first place. Similarly the set 3 In
most cases g00 = −1, the velocity can be conveniently written v = gi x˙i and conjugate momenta vi = gi j x˙ j .
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of equations (5.19) defines the geodesics of GTR space without telling us what shapes space. In order to analyse this question we must return to the general case of a refractive frame gα without other impositions besides the existence of a refractive index. Considering the momentum vector p = pα gα = pα nβ α σ β ,
(6.1)
β
with nα γ nβ γ = δα , 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 = σ 0 . We can then invoke the fact that for an elementary particle in flat space the momentum vector components can be associated with the concepts of energy, 3D momentum and rest mass as p = Eσ 0 + p + mσ 4 (see Sec. 3.) If this consequence is extended to curved space and to mass distributions, we write p = Eσ 0 + p + mg4 , where now E is energy density, p = pm gm is 3D momentum density and m is mass density. The previous equation then becomes ˙ 0 + p˙ + mg˙4 = x˙ · G. Eσ
(6.3)
When the Laplacian is applied to the momentum vector the result is still necessarily a vector D2 p = S. (6.4) Vector S is called the sources vector and can be expanded into 25 terms as S = (D2 nβ α )σβ pα = Sβ α σβ pα ;
(6.5)
where pα = gαβ pβ . Tensor Sα β contains the coefficients of the sources vector and we call it the sources tensor. The sources tensor influences the shape of geodesics as we shall see in one particularly important situation. One important consequence that we don’t pursue here is that by zeroing the sources vector one obtains the wave equation D2 p = 0, which accepts gravitational wave solutions. If σ 0 is normal to the other frame vectors we can write p = E(σ 0 + v) in the reciprocal frame, with v a unit vector or p = E(−σ0 + v) in the direct frame. Equation (6.2) can then be given the form ˙ 0 + v) + E v˙ = σ0 + v · G. E(σ (6.6) Since G can have scalar and bivector components, the scalar part must be responsible for the energy change, while the bivector part rotates the velocity v. The bivector part of G is generated by D ∧ p, which allows a simplification of the previous equation to v˙ = v · (D ∧ v),
453
(6.7)
if the frame vectors are independent of t. This equation is exactly equivalent to the set of Euler-Lagrange equations (5.16) but it was derived in a way which tells us when to expect geodesic movement or free fall. We will now investigate spherically symmetric solutions in isotropic conditions defined by Eq. (4.16); this means that the refractive index can be expressed as functions of r. The vector derivative in spherical coordinates is of course 1 1 1 1 D= σr ∂r + σθ ∂θ + σϕ ∂ϕ − σt ∂t + στ ∂τ . (6.8) nr r r sin θ n4 The Laplacian is the inner product of D with itself but the frame vectors’ derivatives must be considered; all the derivatives with respect to t, r and τ are zero and the non-zero ones are ∂θ σr = σθ , ∂ϕ σr = sin θ σϕ , ∂θ σθ = −σr , ∂ϕ σθ = cos θ σϕ , (6.9) ∂θ σϕ = 0, ∂ϕ σϕ = − sin θ σr − cos θ σθ . After evaluation the curved Laplacian becomes 1 2 n0r 1 2 D = ∂ + ∂ − ∂r + 2 ∂θ θ + rr r 2 (nr ) r nr r 2 cot θ csc θ 1 + 2 ∂θ + 2 ∂ϕϕ − ∂tt + ∂ττ . r r (n4 )2
(6.10)
The search for solutions of Eq. (6.4) must necessarily start with vanishing second member, a zero sources situation, which one would implicitly assign to vacuum; this is a wrong assumption as we will show. Zeroing the second member implies that the Laplacian of both nr and n4 must be zero; considering that they are functions of r we get the following equation for nr 00
nr +
2n0r (n0r )2 − = 0, r nr
(6.11)
with general solution nr = b exp(a/r). It is legitimate to make b = 1 because the refractive index must be unity at infinity. Using this solution in Eq. (6.10) the Laplacian becomes 2 a 1 2 −a/r D = e ∂rr + ∂r + 2 ∂r + 2 ∂θ θ + r r r 2 cot θ csc θ 1 + 2 ∂θ + 2 ∂ϕϕ − ∂tt + ∂ττ ; (6.12) r r (n4 )2 which produces the solution n4 = nr . So space must be truly isotropic and not relaxed isotropic as we had allowed. The solution we have found for the refractive index components 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 produces a GTR metric which is verifiably in disagreement with observations; consequently it has purely geometric significance. The inadequacy of the isotropic solution found above for relativistic predictions deserves some thought, so that we can search for solutions guided by the results that are expected to have physical significance. In the physical world we are never in a situation of zero sources because the shape of space or the existence of a refractive index must always be tested with a test particle. A test particle is an abstraction corresponding to a point mass considered so small as to have no influence on the shape of space; in reality a point particle is a black hole in GTR, although this fact is always overlooked; one wonders how a black hole is postulated not to influence space geometry. A test particle must be seen as source of refractive index itself and its influence on the shape of space should not be neglected in any circumstances. If this is the case the solutions for vanishing sources vector may have only geometric meaning, with no connection to physical reality. The question is then what should go into the second member of Eq. (6.4) in order to find physically meaningful solutions. If we are testing gravity we must assume some mass density to suffer gravitational influence; this is what is usually designated as noninteracting dust, meaning that some continuous distribution of non-interacting particles follows the geodesics of space. Mass density is expected to be associated with S4 4 ; on the other hand we are assuming that this mass density is very small and so we use flat space Laplacian to evaluate it. We consequently make an ad hoc proposal for the sources vector in the second member of Eq. (6.4) S = −∇2 n4 σ4 .
(6.13)
D2 x˙ = −∇2 n4 σ4 ;
(6.14)
Equation (6.4) becomes as a result the equation for nr remains unchanged but the equation for n4 becomes 00
n4 +
00 2n04 n0r n04 2n0 − = −n4 + 4 . r nr r
(6.15)
√ When nr is given the exponential form found above, the solution is n4 = nr . This can now be entered into Eq. (4.11) and the coefficients can be expanded in series and compared 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 dτ 2 = e−2M/r dt 2 − e2M/r ∑(dxm )2 . m
455
(6.17)
Expanding in series of M/r the coefficients of this metric one would find that the lower order terms are exactly the same as for Schwarzschild’s and so the predictions of the metrics are indistinguishable for small values of the expansion variable. Montanus [19] arrives at the same solutions with a different reasoning; Yilmaz was probably the first author to propose this metric [20, 21, 22]. Equation (6.14) can be interpreted in physical terms as containing the essence of gravitation. When solved for spherically symmetric solutions, as we have done, the first member provides the definition of a stationary gravitational mass as the factor M appearing in the exponent and the second member defines inertial mass as ∇2 n4 . Gravitational mass is defined with recourse to some particle which undergoes gravitational influence and is animated with velocity v and inertial mass cannot be defined without some field n4 acting upon it. Complete investigation of the sources tensor elements and their relation to physical quantities is not yet done; it is believed that 16 terms of this tensor have strong links with homologous elements of stress tensor in GTR, while the others are related to electromagnetic field. 7. Electromagnetism in 5D spacetime Maxwell’s equations can easily be written in the form of Eq. (6.4) if we don’t impose the condition that g4 should remain normal the other frame vectors; as we have seen in section 3 this has the consequence that there will be no GTR equivalent to the equations formulated in 4DO. We will consider the non-orthonormed reciprocal frame defined by q gµ = σ µ , g4 = Aµ σ µ + σ 4 ; (7.1) m where q and m are charge and mass densities, respectively, and A = Aµ σ µ is the electromagnetic vector potential, assumed to be a function of coordinates t and xm but independent of τ. The associated direct frame has vectors q gµ = σµ − Aµ σ4 , g4 = σ4 ; (7.2) m and one can easily verify that Eq. (2.18) is obeyed. The momentum vector in the reciprocal frame is p = Eσ 0 + pm σ m + qAµ σ µ + mσ 4 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 is convenient 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)
˙ p˙ = qx˙ · F − qA.
(7.4)
and rearranging
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The first term in the second member is the Lorentz force and the second term is due to the radiation of an accelerated charge. Recalling the wave displacement vector Eq. (B.1) we have now dx = σα dxα −
q Aµ σ4 dxµ . m
(7.5)
This corresponds to a refractive index tensor whose non-zero terms are n4 µ = −
nα α = 1,
q Aµ . m
(7.6)
According to Eq. (6.5) the sources tensor has all terms null except for the following S4 µ = −
q 2 D Aµ ; m
(7.7)
where D is the covariant derivative given by D = gα ∂α = σ µ ∂µ + (σ 4 +
q Aµ σ µ )∂4 . m
(7.8)
We can then define the current vector J verifying µ 2
∇ A = µ∇F = J,
where J=−
m 4 µ S µσ . q
(7.9)
(7.10)
Please refer to [8, Chap. 7] or to [7, Part 2] to see how these equations generate classical electromagnetism. In free space we make J = 0 and Eq. (7.9) accepts plane wave solutions for F which are of course electromagnetic waves. Notice that these solutions propagate in directions normal to proper time, which is perfectly consistent with the classical relativistic formulation. The Dirac equation for a free particle has been derived from the 5-dimensional monogenic 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 inherit 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 translated into the more usual Dirac matrix formalism. Electrodynamics can now be implemented in the the same way used in Sec. 7 to implement classical electromagnetism. The monogenic condition must now be established with the covariant derivative given by Eq. (7.8) q σ µ ∂µ ψ + σ 4 + Aµ σ µ ∂4 ψ = 0. (7.11) m
457
Multiplying on the left by σ 4 and taking ∂4 ψ = imψ µ γ (∂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-dimensional waves, that is, they have 3-dimensional wavefronts normal to the direction of propagation. Whenever the refractive index distribution traps one of these waves a 4-dimensional waveguide is produced, which has its own allowed propagating modes. In the particular case of a central potential, be it an atom’s or a galaxy’s nucleus, we expect spherical harmonic modes, which produce the well known electron orbitals in the atom and have unknown manifestations in a galaxy. 8. Hyperspherical coordinates Deriving physical equations and predictions from purely geometrical equations is an exercise whose success depends on the correct assignment of coordinates to physical entities; the same space will produce different predictions if different options are taken for coordinate assignment. In the previous sections we assumed that empty space could be modelled by an assignment of time, three spatial directions and proper time to five orthogonal directions in 5D spacetime. We are now going to experiment with a different assignment of flat space coordinates, which will explore the possibility that physics and the Universe have an inbuilt hyperspherical symmetry. The exercise consists on assigning coordinate x4 = τ to the radius of an hypersphere and the three xm coordinates to distances measured on the hypersphere surface; time, x0 , will still be measured along a direction normal to all others. If the hypersphere radius is very large we will not be able to notice the curvature on everyday phenomena, in the same way as everyday displacements on Earth don’t seem curved to us. The Universe as a whole will manifest the consequences of its hyperspherical symmetry; using the Earth as a 3-dimensional analogue of an hyperspherical Universe, although our everyday life is greatly unaffected by Earth’s curvature the atmosphere senses this curvature and shows manifestations of it in winds and climate. What we propose here is an exercise consisting of an arbitrary assignment between coordinates and physical entities; the validity of such exercise can only be judged by the predictions it allows and how well they conform with observations. Hyperspherical coordinates are characterized by one distance coordinate, τ and three angles ρ, θ , ϕ; following the usual procedure we will associate with these coordinates the frame vectors {στ , σρ , σθ , σϕ }. The position vector for one point in 5D space is quite simply x = tσt + τστ . (8.1)
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In order to write an elementary displacement dx we must consider the rotation of frame vectors, but we don’t need to think hard about it because we can extend what is known from ordinary spherical coordinates. dx = σ0 dt + σ4 dτ + τσρ dρ + τ sin ρσθ dθ + τ sin ρ sin θ σϕ dϕ.
(8.2)
Just as before, we consider only null displacements to obtain time intervals; dt 2 = dτ 2 + τ 2 dρ 2 + sin2 ρ dθ 2 + sin2 θ dϕ 2 .
(8.3)
The velocity vector, v = x˙ − σ0 , can be immediately obtained from the displacement vector dividing by dt ˙ v = σ0 τ˙ + τσρ ρ˙ + τ sin ρσθ θ˙ + τ sin ρ sin θ σϕ ϕ.
(8.4)
Geodesics of flat space are naturally straight lines, no matter which coordinate system we 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 = τ˙ 2 + τ 2 ρ˙ 2 + sin2 ρ θ˙ 2 + sin2 θ ϕ˙ 2 . (8.5) Because de Lagrangian is independent of ϕ we can establish a conserved quantity ˙ Jϕ = τ 2 sin2 ρ sin2 θ ϕ.
(8.6)
It may seem strange that any physically meaningful relation can be derived from the simple coordinate assignment that we have made, that is, proper time is associated with hypersphere radius and the three usual space coordinates are assigned to distances on the hypersphere radius. This unexpected fact results from the possibility offered by hyperspherical coordinates to explore a symmetry in the Universe that becomes hidden when we use Cartesian coordinates. In the real world we measure distances between objects, namely cosmological objects, rather than angles; we have therefore to define a distance coordinate, which is obviously r = τρ. It does not matter where in the Universe we 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 the Hubble relation. Applying the chain rule and then replacing ρ ˙ = r˙ = ρ τ˙ + ρτ
τ˙ ˙ r + ρτ. τ
(8.7)
We expect objects that have not suffered any interaction to move along στ ; from (8.4) we see that this implies ρ˙ = θ˙ = ϕ˙ = 0 and then τ˙ becomes unity. Replacing in the equation above and rearranging r˙ 1 = . (8.8) r τ
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What this equation tells us is exactly what is expressed by the Hubble relation. The value of τ can be taken as constant for any given observation because the distance information is carried by photons and these preserve proper time, as we have seen in our discussion about electromagnetic waves.4 The first member of the equation is the definition of the Hubble parameter and we can then write H = 1/τ. In this way we find the physical meaning of coordinate τ as being the Universe’s age. Underlying the present discussion there is an assumption a preferred frame where stillness 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 Universe which are galaxy clusters, as much as we can tell. This is far from the orthodox point of view, because galaxy clusters are seen as moving relative to each other and so cannot possible define a fixed frame. But in our formulation still objects move in straight lines along the proper time direction and keep their angular separations constant; this is naturally perceived as increasing mutual distances. If there is any relation between our formulation and an ether it must be found in the fact that movement has an absolute meaning, so it is defined relative to something that is fixed; we call the fixed reference a preferred frame while other authors call it ether. How does the use of hyperspherical coordinates affect dynamics in our laboratory experiments? We would like to know if these coordinates need only be considered in problems of cosmological scale or, on the contrary, there are implications for everyday experiments. The answer implies rewriting (8.2) with distance rather than angle coordinates; replacing ρ, r dx = σ0 dt + σ4 − σρ dτ + σρ dr + r(σθ dθ + sin θ σϕ dϕ). (8.9) τ Evaluating time intervals from the null displacement condition, as before r 2 r dτ 2 − 2 dτdr + dr2 + r2 (dθ 2 + sin2 θ dϕ 2 ). dt 2 = 1 + τ τ
(8.10)
This would be a version of (3.8) in spherical coordinates, were it not for the extra terms with powers of r/τ in the second member. The coefficient r/τ implies a comparison between the distance from the object to the observer and the size of the Universe; remember that τ is both time and distance in non-dimensional units. We can say that ordinary special relativity will apply for objects which are near us, but distant objects will 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 for the dynamics near a massive sphere using Cartesian coordinates; since this is frequently applied on a cosmological scale, we must find out how the dynamics is modified by 4 In
order to preserve proper time photons must travel on the hypersphere surface and thus don’t follow geodesics.
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the use of hyperspherical coordinates. Using the refractive indices and hyperspherical coordinates, noting that nr = n24 , Eq. (4.7) becomes dt 2 = n24 dτ 2 + n44 τ 2 dρ 2 .
(8.11)
Dividing both members by dt 2 and reversing the equation n24 τ˙ 2 + n44 τ 2 ρ˙ 2 = 1;
(8.12)
˙ and replacing τ ρ˙ by r˙ − rτ/τ " n24 τ˙ 2 + n44
# 2 r τ˙ 2 r˙ + r − 2τ˙ r˙ = 1. τ τ 2
(8.13)
Dividing both members by n44 r2 and rearranging results in the equation 2 2 τ˙ r˙ r˙ 1 τ˙ 2 1 τ˙ − 2 = − +2 . 4 2 r τ τr n4 n4 r
(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 in series of M/r taking only the two first terms. 2 2 r˙ 1 − τ˙ 2 (2τ˙ 2 − 4)M τ˙ τ˙ r˙ + − ≈ +2 . 2 3 r r r τ τr
(8.15)
The previous equation applies to bodies moving radially under the influence of mass M located at the origin which is, remember, the observer’s position. For comparison we derive the corresponding equation in Cartesian coordinates; starting with (8.12) it is now n24 τ˙ 2 + n44 r˙2 = 1;
(8.16)
2 r˙ 1 − τ˙ 2 (2τ˙ 2 − 4)M 1 τ˙ 2 1 − ≈ + . = r r2 r3 n44 n24 r2
(8.17)
dividing by n44 r2 and rearranging
If we want to apply these equations to cosmology it is easiest to follow the approach of Newtonian cosmology, which produces basically the same results as the relativistic approach but presumes that the observer is at the centre of the Universe [18, 23]. In order to adopt a relativistic approach we need equations that replace Einstein’s in 4DO. A set of such was proposed above Eq. (6.4) but their application in cosmology has not yet been tested, so we will have to defer this more correct approach to future work. The strategy we will follow here is to consider a general object at distance r from the observer, moving
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away from the latter under the gravitational influence of the mass included in a sphere of radius r. If we designate by µ the average mass density in the Universe, then mass M in (8.15) is 4π µr3 /3; this will have to be considered further down. Friedman equation governs standard cosmology and can be derived both from Newtonian and relativistic dynamics, with different consequences in terms of the overall size of the Universe and the observer’s privileged position. From the cited references we write Friedman equation as 2 r˙ 8π Λ k = µ + − 2; (8.18) r 3 3 r with Λ a cosmological constant and k the curvature constant; the gravitational constant was not included because it is unity in non-dimensional units and the equation is written in real, not comoving, coordinates. In order to compare (8.15) with Friedman equation there is a problem with the last term because the Hubble parameter r˙/r does not appear isolated in the first member; we will find a way to circumvent the problem later on but first let us look at what (8.15) tells us when the mass density is zeroed. In this case n4 = 1 and we find from (8.12) that τ˙ is unity, unless ρ˙ is non-zero, for which we can find no reasonable 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 immediately obvious that µc = 3/(8πτ 2 ). Let us not overlook the importance of this conclusion because 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 is just a consequence of space symmetry and not of dynamics. Let us now see what happens when we consider a small mass density; here we are talking 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 accepted value. It is therefore just a perturbation of the flat solution that we described above and the fact that we are presuming a privileged observer has to be taken just for this perturbation. The first thing we note when we consider matter density is that τ˙ < 1, because there is now a component of the velocity vector along σρ . Ideally we should solve the Euler-Lagrange equations resulting from (8.12) in order to find τ˙ and ρ˙ but this is a difficult process and we shall carry on with just a qualitative discussion. Considering that we ˙ and the two last terms in are discussing a perturbation it is legitimate to make r˙/r ≈ τ/τ ˙ 2 , the same as the second member of (8.15) can be combined into one single term (τ/τ) we 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 equation that this corresponds to a negative curvature constant, k, and to an open Universe. Lastly the second term includes the mass M of a sphere with radius r and can be simplified to 8π µ(τ˙ 2 − 2)/3; this has the effect of a negative cosmological constant; the combined ef-
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fect of the two terms is expected to close the Universe [23, 24]. The previous discussion was done in qualitative terms, making use of several approximations, for which reason we must question some of the findings and expect that after more detailed examination they may not be quite as anticipated; in particular there is concern about the refractive indices used, which were derived in Cartesian coordinates both by the author and those that preceded him in using an exponential metric; it may happen that the transposition to hyperspherical coordinates has not been properly made, with consequences in the perturbative analysis that was superimposed on the flat solution. The latter, however, is totally independent of such concerns and allows us to state that the assumption of hyperspherical symmetry for the Universe dispenses with dark matter in accounting for the gross of observed expansion. Dark matter is also called in cosmology to account for the extremely high rotation velocities found in spiral galaxies [25, 26] and we will now take a brief look at how hyperspherical symmetry can help explain this phenomenon. Galaxy dynamics is an extremely complex subject, which we do not intend to explore here due to lack of space but most of all due to lack of author’s competence to approach it with any rigour; we will just have a very brief outlook at the equation for flat orbits, to notice that an effect similar to the familiar Coriollis effect on Earth can arise in an expanding hyperspherical Universe and this could explain most of the observed velocities on the periphery of galaxies. Let us recall (8.9), divide by dt and invoke null displacement to obtain the velocity r ˙ v = σ4 − σρ τ˙ + σρ r˙ + r(σθ θ˙ + sin θ σϕ ϕ). (8.19) τ If orbits are flat we can make θ = π/2 and the equation simplifies to rτ˙ ˙ 4 + r˙ − ˙ ϕ. v = τσ σρ + rϕσ (8.20) τ Suppose now that something in the galaxy is pushing outwards slightly, so that the paren˙ and can be caused by a pressure gradient, for thesis is zero; this happens if r˙/r = τ/τ ˙ which is exinstance. The result is that (8.20) now accepts solutions with constant rϕ, actly what is observed in many cases; swirls will be maintained by a radial expansion rate ˙ which exactly matches the quotient τ/τ. In any practical situation τ˙ will be very near unity and the quotient will be virtually equal to the Hubble parameter; thus the expansion rate 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, the expansion velocity is about 2.43 km s−1 ; this is to be compared with the orbital velocity of near 300 km s−1 and probably within the error margins. An expansion of this sort could be present in many galaxies and go undetected because it needs only be of the order of 1% the orbital velocity.
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9. Symmetries of G4,1 algebra In this algebra it is possible to find a maximum of four mutually annihilating idempotents, which generate with 0 an additive group of order 16; for a demonstration see Lounesto [14], section 17.5. Those idempotents can be generated by a choice of two commuting basis elements which square to unity; for the moment we will use σ023 and σ014 . The set of 4 idempotents is then given by (1 + σ023 )(1 + σ014 ) (1 + σ023 )(1 − σ014 ) , f2 = , 4 4 (1 − σ023 )(1 + σ014 ) (1 − σ023 )(1 − σ014 ) , f4 = . f3 = 4 4
f1 =
(9.1)
Using the matrices of Sec. 3 to make matrix replacements of σ023 and σ014 one can find matrix equivalents to these idempotents; those are matrices which have only one non-zero element, located on the diagonal and with unit value. SU(3) symmetry can now be demonstrated by construction of the 8 generators σ3 + σ02 , 2 −σ2 + σ03 λ2 = σ03 ( f1 + f2 ) = , 2 σ014 − σ1234 , λ3 = f1 − f2 = 2 −σ1 − σ04 λ4 = −σ1 ( f2 + f3 ) = , 2 −σ4 + σ01 , λ5 = −σ4 ( f2 + f3 ) = 2 σ012 + σ034 λ6 = σ012 ( f1 + f3 ) = , 2 σ013 − σ024 λ7 = −σ024 ( f1 + f3 ) = , 2 f1 + f2 − 2 f3 2σ023 + σ014 + σ1234 √ √ λ8 = = . 3 2 3
λ1 = σ02 ( f1 + f2 ) =
464
(9.2)
These have the following matrix equivalents 0 −j 0 0 0 1 0 0 j 0 0 0 1 0 0 0 λ1 ≡ 0 0 0 0 , λ2 ≡ 0 0 0 0 , 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 −j 0 λ4 ≡ 0 1 0 0 , λ5 ≡ 0 j 0 0 , 0 0 0 0 0 0 0 0 0 0 −j 0 1 √ 0 0 0 0 0 λ7 ≡ 0 j 0 0 , λ8 ≡ 1/ 3 0 0 0 0 0 0
1 0 λ3 ≡ 0 0 0 0 λ6 ≡ 1 0
0 −1 0 0
0 1 0 0 0 0 0 0 0 0 0 1 0 0 , 0 −2 0 0 0 0
0 0 0 0
0 0 , 0 0 0 0 0 0
(9.3)
which reproduce Gell-Mann matrices in the upper-left 3 ∗ 3 corner [15, 27, 28]. Since the algebra is isomorphic to complex 4 ∗ 4 matrix algebra, one expects to find higher order symmetries; Greiner and Müller [27] show how one can add 7 additional generators to those of SU(3) in order to obtain SU(4) and the same procedure can be adopted in geometric algebra. We then define the following additional SU(4) generators σ1 − σ04 , 2 σ4 + σ01 λ10 = σ4 ( f1 + f4 ) = , 2 −σ012 − σ034 λ11 = −σ012 ( f2 + f4 ) = , 2 σ013 + σ024 λ12 = σ024 ( f2 + f4 ) = , 2 σ3 − σ02 , λ13 = σ3 ( f3 + f4 ) = 2 σ2 + σ03 λ14 = σ2 ( f3 + f4 ) = , 2 f1 + f2 + f3 − 3 f4 σ023 − σ014 − σ1234 √ √ λ15 = = . 6 6 λ9 = σ1 ( f1 + f4 ) =
(9.4)
Once again, making the replacements with Eq. (3.9) produces the matrix equivalent gen-
465
erators
0 0 λ9 ≡ 0 1 0 0 λ12 ≡ 0 0
0 0 0 0
0 0 0 0 0 0 0 j
0 0 0 0
1 0 0 0 , λ10 ≡ 0 0 0 j 0 0 0 −j , λ13 ≡ 0 0 0 0
0 0 −j 0 0 0 , λ11 ≡ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 , λ14 ≡ 0 0 0 1 0 0 1 0 1 0 0 0 √ 0 1 0 0 λ15 ≡ 1/ 6 0 0 1 0 . 0 0 0 −3 0 0 0 0
0 1 , 0 0 0 0 0 0 0 0 , 0 0 −j 0 j 0 0 0 0 1
0 0 0 0
(9.5)
The standard model involves the consideration of two independent SU(3) groups, one for colour and the other one for isospin and strangeness; if generators λ1 to λ8 apply to one of the SU(3) groups we can produce the generators of the second group by resorting to the basis elements σ3 and σ04 . The new set of 4 idempotents is then given by (1 + σ3 )(1 + σ04 ) (1 + σ3 )(1 − σ04 ) , f2 = , 4 4 (1 − σ3 )(1 − σ04 ) (1 − σ3 )(1 + σ04 ) f3 = , f4 = . 4 4
f1 =
(9.6)
Again a set of SU(3) generators can be constructed following a procedure similar to the previous one σ02 + σ023 , 2 σ01 + σ013 , α2 = σ01 ( f1 + f2 ) = 2 σ04 − σ034 α3 = f1 − f2 = , 2 σ2 + σ024 α4 = σ2 ( f2 + f3 ) = , 2 −σ1 − σ014 , α5 = −σ1 ( f2 + f3 ) = 2 σ4 − σ03 α6 = σ4 ( f1 + f3 ) = , 2 σ012 + σ1234 , α7 = σ012 ( f1 + f3 ) = 2 f1 + f2 − 2 f3 2σ3 + σ04 + σ034 √ √ α8 = = . 3 2 3 α1 = σ02 ( f1 + f2 ) =
466
(9.7)
This new SU(3) group is necessarily independent from the first one because its matrix representation involves matrices with all non-zero rows/columns, while the group generated by λ1 to λ8 uses matrices with zero fourth row/column. In the following section we 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 into left and right spinors and here we discuss how the different idempotents are related to the symmetries discussed above, suggesting a relation between idempotents and the different elementary particles. We have already established that each set of 4 idempotents is generated by a pair of commuting unitary basis elements. Let any two such basis elements be denoted as h1 and h2 ; then the product h3 = h1 h2 is itself a third commuting basis element. For consistence we choose, as before, h1 ≡ σ023 ,
h2 ≡ σ014 ;
(9.8)
to get h3 ≡ σ1234 ,
(9.9)
which commutes with the other two as can be easily verified. The result of this exercise is the existence of triads of commuting unitary basis elements but no tetrads of such elements. We are led to state that a general unitary element is a linear combination of unity and the three elements of one triad h = a0 + a1 h1 + a2 h2 + a3 h3 . Since h is unitary and the three hm commute we can write h2 = (a0 )2 + (a1 )2 + (a2 )2 + (a3 )2 + 2(a0 a1 − a2 a3 )h1 + + 2(a0 a2 − a1 a3 )h2 + 2(a0 a3 − a1 a2 )h3 = 1
(9.10)
(9.11)
The only form this equation can be verified is if the term in square brackets is unity while all the others are zero. We then get a set of four simultaneous equations with a total of sixteen solutions, as follows: 8 solutions with one of the aµ equal to ±1 and all the others zero, 6 solutions with two of the aµ equal to −1/2 and the other two equal to 1/2 and 2 solutions with all the aµ simultaneously ±1/2. The aµ coefficients play the role of quantum numbers which determine the particular idempotent that goes into 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’t propose here any direct relationship between the various idempotents and the known elementary particles, although the fact that the standard model gauge symmetry group is found as direct consequence of the monogenic condition which itself generates the Dirac equation is rather intriguing.
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Table 1: Coefficients for the various unitary elements. 1 (a0 ) 1 0 0 0 −1 0 0 0 −1/2 −1/2 −1/2 1/2 1/2 1/2 1/2 −1/2
σ023 (a1 ) 0 1 0 0 0 −1 0 0 −1/2 1/2 1/2 −1/2 −1/2 1/2 1/2 −1/2
σ014 (a2 ) 0 0 1 0 0 0 −1 0 1/2 −1/2 1/2 −1/2 1/2 −1/2 1/2 −1/2
σ1234 (a3 ) 0 0 0 1 0 0 0 −1 1/2 1/2 −1/2 1/2 −1/2 −1/2 1/2 −1/2
λ3
λ8
λ15
0 0 1 −1 0 0 −1 1 0 −1 1 −1 1 0 0 0
√0 2/√3 1/√3 1/ 3 √0 −2/√3 −1/√3 −1/ 3 √0 1/√3 1/√3 −1/√3 −1/ 3 √0 2/√3 −2/ 3
p 0 p2/3 −p2/3 − 2/3 p 0 −p2/3 p2/3 p2/3 − 3/2 √ 1/√6 1/√6 −1/√6 −1/ p 6 3/2 √ −1/√6 1/ 6
10. Conclusion and future work Monogenic functions applied in the algebra of 5-dimensional spacetime have been shown to originate laws of fundamental physics in such diverse areas as relativistic dynamics, quantum mechanics and electromagnetism, with possible, still unclear, consequences for cosmology and particle physics. To say that those functions provide us with a theory of everything is certainly unwarranted at this stage but it is clear that there is a case for much greater effort being invested in their study. There are unanswered questions in the present work. For instance, how can we avoid an ad hoc definition of inertial mass or what is the true relation between the symmetries generated by monogenic functions and elementary particles? In spite of its various loose ends, the formalism is perfectly capable of unifying relativistic dynamics, quantum mechanics and electromagnetism, which in itself is no small achievement. Certain developments seem relatively straightforward but they must be made, even if no new predictions are expected. Applying monogenic functions to the Hydrogen atom should not be difficult because the form of the Dirac equation we arrived at is perfectly equivalent to the standard one; one should then find the same solutions but in a GA formalism.
468
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 in planetary mechanics or galaxy dynamics. Gravitational waves are predicted by the monogenic function formalism as we pointed out but did not investigate. How important are they and what chance is there of them being detected by experiment? We don’t know the answer and we don’t know what difficulties lie on the path of those who try to solve the equations; this is an open area. The sources’ tensor must be clearly understood and directly related to geometry; at the moment all densities, mass, electromagnetic energy, etc. must be inserted in the equations but 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 answer to the problem but the concept has not yet been formalized and there are no clear ideas for achieving such goal. In conclusion, the present work opens the gate of a path that will possibly lead to an entirely new formulation and understanding of physics but this path is very likely to have many 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 5dimensional space but we are also interested in two of its 4-dimensional subspaces and one 3-dimensional subspace; ideally our choice of indices should clearly identify their ranges in order to avoid the need to specify the latter in every equation. The diagram in 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. Indices in the range {0, 3} will also receive Greek letters but chosen from µ, ν, ξ . For indices in the range {1, 4} we will use Latin letters i, j, k and finally for indices in 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 = gα dxα = nβ α σβ dxα .
(B.1)
Because this vector is nilpotent, by virtue of Eq. (4.6), the five coordinates are not independent and we can divide both members by dx0 = dt defining the nilpotent vector x˙ = g0 + gi x˙i = nα 0 σα + nβ i σβ x˙i .
(B.2)
Suppose we have a 5D vector a = σα aα and we want to find its time derivative along a path parameterized by t, that is all the xi are functions of t. We can write a˙ = ∂β aα x˙β σα ;
(B.3)
where naturally x˙0 = 1. Remembering the definition of covariant derivative (2.25) and Eq. (B.2) we can modify this equation to a˙ = x˙β gβ · gβ ∂β aα σα = x˙ · (Da).
(B.4)
We have expressed vector a in terms of the orthonormed frame in order to avoid vector derivatives but the result must be independent of the chosen frame. This procedure has an obvious dual, which we arrive at by defining xˇ = gµ xˇµ + g4 .
(B.5)
The proper time derivative of vector a is then aˇ = xˇ · (Da).
(B.6)
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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 nonclassical 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 quasiclassical 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 quasilocal 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 TroutonNoble 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,
[email protected] University
of
Cambridge,
CB2
1PZ
E-mail:
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 http://www.physicsfoundations.org/Ether_spacetime/book.htm
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.
http://www.physicsfoundations.org/Ether_spacetime/book.htm * Contributions to future volumes should be sent to: Dr J Levy
[email protected] Dr M C Duffy
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