2595 lines
213 KiB
Plaintext
2595 lines
213 KiB
Plaintext
Old Physics
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for New
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–a worldview alternative to
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Einstein’s relativity theory
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Foreword by D.F. Roscoe
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Thomas E. Phipps, Jr.
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Old Physics for New:
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a worldview alternative to
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Einstein’s relativity theory
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Thomas E. Phipps, Jr.
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Apeiron
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Montreal
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Published by C. Roy Keys Inc. 4405, rue St-Dominique Montreal, Quebec H2W 2B2 Canada http://redshift.vif.com
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© C. Roy Keys Inc. 2006
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First Published 2006
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Library and Archives Canada Cataloguing in Publication
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Phipps, Thomas E., 1925Old physics for new : a worldview alternative to Einstein's relativity theory / Thomas E. Phipps.
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Includes bibliographical references and index. ISBN 0-9732911-4-1
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1. Special relativity (Physics). 2. Electromagnetic theory. I. Title.
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QC173.585.P52 2006 530.11 C2006-905979-9
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Appendix quoted from: A Threefold Cord: Philosophy, Science, Religion, Viscount Samuel and Herbert Dingle, © 1942 George Allen and Unwin. Reproduced by permission of Taylor and Francis Books UK.
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Front cover: A blue Fu dog (at Allerton Park, near Monticello, Illinois), symbolizing Nature or Old Physics, regards with dismay the glittering New Physics of the “black Garuda” (a mask crafted by Balinese artist Nyoman Setiawan), one of various incarnations of Vishnu, in the present context seen as the incarnation of Albert Einstein’s special relativity theory.
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Table of Contents
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Foreword ..............................................................................................................i Author’s Preface ...............................................................................................vii
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Chapter 1
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What’s Wrong with Maxwell’s Equations? .......................................................1 1.1 Problems of first-order description ........................................................1 1.2 The under-parameterization of Maxwell’s equations ....................................................................................................9 1.3 The problem about Faraday’s observations: d/dt................................10 1.4 Justification for a Hertzian form of Faraday’s law .............................13 1.5 Other problems of Maxwell’s equations..............................................15 1.6 Chapter summary ...................................................................................16
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Chapter 2
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What to Do About It ... (the Hertzian Alternative).......................................17 2.1 First-order invariant field equations ....................................................17 2.2 History: Why did Hertz fail?.................................................................23 2.3 Invariance vs. covariance: The physics of it.........................................26 2.4 Invariance or covariance: Which is physics?.......................................28 2.5 Hertzian wave equation.........................................................................30 2.6 Potier’s principle .....................................................................................34 2.7 Sagnac effect and ring laser ...................................................................37 2.8 A bit of GPS evidence.............................................................................42 2.9 Chapter summary ...................................................................................43
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Chapter 3
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Higher-order Electrodynamics ... (the neo-Hertzian Alternative).................. 45 3.1 The higher-order kinematic invariants ................................................45 3.2 Neo-Hertzian field equations ................................................................53 3.3 Neo-Hertzian wave equation ................................................................57 3.4 Phase invariance......................................................................................64 3.5 Doppler effect ..........................................................................................65 3.6 Chapter summary ...................................................................................66
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Chapter 4
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Stellar Aberration .............................................................................................69 4.1 Appreciation of the phenomenon .........................................................69 4.2 SA according to SRT ...............................................................................71 4.3 SA according to neo-Hertzian theory...................................................77 4.4 SRT’s unrecognized conceptual difficulties with SA .....................................................................................................84 4.5 Einstein’s state of mind: a speculation .................................................86 4.6 A rebuttal .................................................................................................87 4.7 Another “first test” failure of dσ: the rigid body................................89 4.8 Newtonian point particle mechanics....................................................93 4.9 Chapter summary ...................................................................................95
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Chapter 5
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Electrodynamic Force Laws ..............................................................................99 5.1 Electromagnetic force in SRT.................................................................99 5.2 Neo-Hertzian force law ........................................................................101 5.3 Evidence of the Marinov motor ..........................................................108 5.4 Other electrodynamic force laws ........................................................109 5.5 Sick of field theory? ... (the Weber alternative) ................................114 5.6 Chapter Summary.................................................................................118
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Chapter 6
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Clock Rate Asymmetry ...................................................................................123 6.1 Distant simultaneity, acausality..........................................................123 6.2 Einstein’s train on a different track.....................................................127 6.3 Clock slowing: actual or symmetrical? (The twin paradox) ...............................................................................132 6.4 GPS evidence for clock rate asymmetry.............................................139 6.5 Clock rates, free-falling vs. supported in a gravity field ....................................................................................147 6.6 Platonic time and simultaneity ...........................................................150 6.7 Length Invariance .................................................................................152 6.8 Clock rate as an energy state function................................................155 6.9 Reversible work.....................................................................................159 6.10 Atomic clocks: prospects for their improvement.............................160 6.11 Chapter summary ................................................................................161
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Chapter 7
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Collective Time ...............................................................................................165 7.1 Principles governing proper time.......................................................165
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7.2 Collective time and relativity principles............................................167 7.3 Related observations.............................................................................176 7.4 Philosophical context............................................................................180 7.5 Particle mechanics, again .....................................................................182 7.6 Field theory revisited............................................................................185 7.7 The light clock in orbit..........................................................................189 7.8 Two more forms of the relativity principle (distinguishable by a crucial experiment) .........................................193 7.9 Kinematics for uncompensated clocks...............................................194 7.10 The need for more facts.......................................................................197 7.11 Chapter summary ................................................................................199
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Chapter 8
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Linkages of Time, Energy, Geometry.............................................................. 203 8.1 Connection of time and action (and the effect of gravity).....................................................................203 8.2 An effect of gravity on mass in equations of motion?.......................................................................209 8.3 Kinematics for compensated clocks ...................................................211 8.4 Velocity composition: More than you wanted to know .........................................................217 8.5 The many-body problem: γ as integrating factor.............................222 8.6 A fable.....................................................................................................226 8.7 The demo problem problem................................................................230 8.8 Collective time in a nutshell ................................................................233 8.9 Chapter summary .................................................................................235
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Appendix
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Dingle’s “Proof that Einstein’s Special Theory cannot correspond with fact” ..................................................................241 Remark on the psychology of scientific revolutions ............................247
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Index ..............................................................................................................249
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Old Physics for New: a worldview alternative to Einstein’s relativity theory i Thomas E. Phipps, Jr. (Montreal: Apeiron 2006)
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Foreword
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hese few words of introduction are primarily directed at those readers who are not familiar either with Tom Phipps’ singular style of scientific prose or, more importantly, with his rigorously applied view that, when theorizing about the world around us, we must pay absolute attention to the practicalities of the measurement processes by which the quantities involved in this theorizing are measured. Let me talk about the Phippsian prose style first: the common experience upon reading a scientific text is to be confronted by a finished article—that is, by a text from which all sense of intellectual journeying has been exorcised, cleansed, deleted. The experience may be necessary but it is rarely exciting and never invigorating—it becomes merely a job that must be done, a dusty dry road along which weary feet must be dragged. But Phipps refutes this puritanical model; he is renaissance man—the man who glories in the splendour of the written word and its capacity to illuminate the obscure, and to decorate the plain. And so the experience of reading Phippsian scientific prose is not unlike that of reading a good detective novel—the dim detective, the obvious clues overlooked, the false trail followed, the unsolved crime written up as solved so that the bureaucrat can sleep his dreamless sleep and, finally, Sherlock Holmes with his pipe and Dr. Watson ... Now let me consider the (for me) perfectly commonsensical view that the practicalities of the measurement process must play an unambiguously prominent role in the theorizing process: As an example of a theory where this was not done (with hugely significant consequences), we need look no further than classical Maxwell electrodynamics. In this case, the formalism absolutely requires that the detectors used by (inertial) observers to measure field quantities be at rest in the observer’s frame. Thus, if we have two observers, each in his own inertial frame, then, since their instruments are physical objects and unable to occupy the same place at the same time, it is absolutely impossible for these two observers to make simultaneous measurements of the same field
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T
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ii D.F. Roscoe
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point. In other words, certain choices made at the theorizing level have rendered impossible a perfectly reasonable thing—that distinct observers can have direct knowledge of conditions occurring at a particular place at a given time. Phipps’ answer to this conundrum is simple: there is no reason on Earth why the detector measuring field quantities should be fixed in the (inertial) observer’s frame. After all, the source currents which generate the field are not, so why should the test-particles (which comprise the detectors) be? And since the detector need not be fixed in one observer’s inertial frame, why should it be fixed in any inertial frame? Following this logic, if we allow the detector to have free motion, then the formalism of electrodynamics which follows must somehow allow for the parameterization of the detector’s motion. A natural candidate for this formalism already exists in the equations of Hertz’s electromagnetic theory (the known failure of his theory was the fault not of his equations but of his physical interpretation) and these are easily written down: just take Maxwell’s equations and replace all appearances of ∂ ∂t by d dt . This replacement introduces a convective velocity which must be interpreted, and Phipps’ solution is to use this convective velocity to describe the motion of the free detector. A simple and elegant idea, don’t you think? ... but now comes the crux: by this simple process, which is driven by the idea that there is no reason on God’s Earth why an observer cannot use a freely moving detector, the equations of electromagnetism become Galilean invariant; thus, at a stroke, solving one of the great conundrums of 19th century physics and, in removing the primary raison d’être of Special Relativity (SRT), putting a huge question mark over a large chunk of 20th century theoretical physics. Now Phipps is a realistic and honest man and each of these traits has its consequence on the way his thinking proceeds. Realism first: the story outlined above makes plain that SRT, and all that has flowed from it, is an unfortunate accident of history for some and an incredible stroke of good fortune for others—and it is the ‘others’ who are in the driving seat here. What is required, Phipps realizes, is an example of some physical circumstance in which SRT can be shown to have failed ... unambiguously. One does exist, although careful reading of the standard texts (when one is wide awake and on top of one’s game) is required to spot it, otherwise the cardsharp cleans you out: stellar aberration is the bone in the fish pie. Briefly, and as Phipps points out in entertain
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Foreword iii
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ing detail, SRT claims to provide the complete explanation for the Doppler shift and for stellar aberration—both phenomena affecting light that comes from stars. To see the problem immediately, it is sufficient to observe that in order to explain the Doppler shift, the velocity used by SRT is defined as the relative velocity between emitter and detector (v = ve – vd) which, of course, is perfectly consistent with SRT’s own internal logic. However, in order to explain stellar aberration, the velocity used by SRT is defined as the Earth’s orbital velocity in the solar frame (v = vorb) ... stellar velocities are nowhere to be seen ... and there is no source-sink relativity whatsoever! So, in order to ‘explain’ two different aspects of the same starlight, SRT must submit to two different interpretations, one consistent with its own internal logic and one inconsistent with that logic. If you work in a University physics department, try putting that position to any of your colleagues. Honesty second: there are several good reasons for being extremely sceptical about SRT—Phipps is eloquent on them allbut he knows that the clock cannot be turned back to 1894 (the year Hertz died). Physics has moved on since then (and I do not mean merely theoretical physics); in particular, although we can with reason reject SRT, the time dilation prediction of SRT has been verified to high accuracy many times over. Indeed, without using the time-dilation formula of SRT to calibrate the relative clock-rates of the Earth-based clocks and orbiting clocks, the GPS system could never work as well as it does. So, Phipps accepts that time dilation is a fact of physics and that the time-dilation formula of SRT is verified and must therefore be properly built into theory. So, how does Phipps respond to this state of affairs? Well, close analysis is hardly required ... for the Emperor is clearly naked to the innocent eye ... SRT makes two independent statements, of which we are all aware: firstly, there is the statement about time dilation (with a formula which works in well-defined situations) and, secondly, there is a statement about length contraction ... which Phipps correctly points out is a prediction of an effect which (a) has never been observed and (b) creates all kinds of difficulties, not least of which is making it impossible to consider SRT as a generalization (or covering theory in Phippsian lingo) of Newtonian Mechanics. It is the length contraction prediction, for example, that makes the science of rigid body mechanics impossible for the “relativist.” For Phipps, and for any right-thinking per
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iv D.F. Roscoe
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son in my view, the notion of length-contraction is a metaphysical fantasy that can have no place in a theory of physics. And because length contraction and time dilation are independent statements then—as Phipps points out—we can cherry-pick. We can have a theory which assumes the reality of time dilation whilst denying that of length contraction. The way forward is formally trivialjust replace ordinary clock time, t, in the Hertz formalism by the proper time parameter, τ, defined in the usual SRT way where the velocity parameter, vd, is the velocity of the detector in the (inertial) observer’s frame. The result is the Neo-Hertzian formalism, the ramifications of which Phipps works through in great detailbut I shall stick with the big canvas: in denying the existence of length contraction but accepting the existence of time dilation Phipps is, in fact, denying spacetime symmetry; but, in doing so, is regaining the possibility of rigid-body mechanics and, through the neo-Hertzian formalism, is finding mutually consistent treatments for the Doppler shift and stellar aberration. This is already a huge bonus. This Phippsian saga closes with a couple of chapters devoted to discourse on the nature of timekeeping (rather than on the nature of time). As I see it, this section is driven by three circumstances: firstly, there is no identifiable causal mechanism within SRT for the “predicted” physical effect of clock retardation. If there were, the twin-paradox would never have arisen in the first place. Secondly, there is the (almost) self-evident fact that any man (or, in this politically correct world, person) engaged in theorizing about the world armed with a sensibly constructed clock which furnishes a time t, can either choose to use t directly as his measure of time or choose to use an arbitrarily defined strictly monotonically increasing function T = g(t) as his measure of time. The only consequence is that there will be some choice of g which provides maximal simplicity to his theorizing—but all choices are equally valid. Thirdly, there is the empirical fact of the engineers’ experience about how to make the GPS system work in practicethe fact of an Earth-bound Master Clock against which all the tobe-launched satellite clocks are calibrated so that once they are in orbit they keep synchronous time with the Earth-bound Master Clock. This calibration process amounts to the choice of a set of gfunctions –g1,g2,... say—each one tailored separately to account for the distinct operating conditions of its associated clock. In effect, Phipps argues that there are no reasons whatsoever
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Foreword v
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beyond vain prejudice and ideology—for believing that there exists for any system an inherently fundamental measure of time (or “proper time” in the sense intended within SRT and GR). And, upon reflection, I find myself agreeing with him. In which case, he argues, the most simple system of time-keeping is the one pioneered by the GPS engineers—that of an agreed (almost inertial) Master Clock against which all other clocks placed wheresoever are synchronized by a g-transformation chosen according to the operating conditions of the clock concerned (gravitational potential, relative velocity, etc., accounted for). Thus, the vision spawned by SRT & GR according to which there are as many different “proper” clocks as there are particles in the universe is replaced by one in which there is a single (arbitrarily chosen) inertial Master Clock against which all other clocks are synchronized. As always, Phipps provides an exhaustive analysis of the ramifications of this timekeeping methodology—but two can be mentioned in a single breath: the absoluteness of the here and now is restored to the discourse of physics—with the corresponding consignment of the relativity of simultaneity to the proverbial dustbin; and the resurrection of the distinct possibility of a realistic theory of many-particle physics. I shall finish my few words in praise of this lovely, lovely book by remarking briefly on that aspect of the neo-Hertzian formalism which I find to be most remarkable: as a student (forty years ago) I struggled with Maxwellian electrodynamics, and part of my problem was that I always found two things rather odd: firstly was the fact that here we had a theory in which the (supposedly) most important parts were the fields, E and B, which were unashamedly defined in terms of Newtonian forces—and yet this very same theory was proclaimed the fountain-head of all that was non-Newtonian in the whole world; secondly was the fact that, although ideas of force were hard-wired into the definitions of the field quantities, the theory still required an additional postulate (the Lorentz force law) to make it into a useful theory of electrodynamics. One can accept such things in an entirely mechanical way, of course. But they left me feeling perpetually slightly disconnected from any claim to a real understanding of the Maxwellian picture. At a stroke, Phipps has removed all such impedimenta to clear sight: no longer is electrodynamics claimed as the portal to a shining new world, quite different from the old; instead, it sits firmly and squarely as an integral part of
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vi D.F. Roscoe
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that old world. And, almost by magic—yet not really—Phipps shows us that, in its neo-Hertzian reincarnation, electromagnetism is already electrodynamics; there is no need to postulate force laws additional to those inherent in the basic definitions of the field quantities ... read, marvel and enjoy!
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David Roscoe Sheffield, October 2006
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Old Physics for New: a worldview alternative to Einstein’s relativity theory vii Thomas E. Phipps, Jr. (Montreal: Apeiron 2006)
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The great discoveries of science often consist ... in the uncovering of a truth buried under the rubble of traditional prejudice, in getting out of cul-de-sacs into which formal reasoning divorced from reality leads; in liberating the mind trapped between the iron teeth of dogma.
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—Arthur Koestler, The Sleepwalkers.
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Author’s Preface
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cience can be seen as the fitting to nature of a mask—a mask beneath which the probings of human curiosity may be able to discover only other masks. For nature, as the ancients knew, is something else. Some spectators today, sufficiently concerned to qualify as “dissidents”—of whom I happen to be one—view the mask currently popular among professional physicists as especially and unnecessarily hideous. Beauty is in the eye of the beholder ... and ugliness, too. A keen sense of the ugly is as essential to the success of a would-be contributor to science as any of the more obvious talents. For the points of special ugliness of a theory are the loci of its vulnerability, at which a sharply-directed attack is most likely to lead to progress. In the human genetic mix the lovers of ugliness seem to be a majority dominant in every era. By them, ugliness is perceived as beauty ... and there is little arguing with them, since they command, as needed, all the machinery of political dominance, including both democracy (the voice of consensus) and autocracy (the voice of authority). Consequently, identifiable “progress” is a matter of accident, a statistical fluctuation against odds. Old ideas pass out of fashion, old beauties succumb to new. In the seething ferment of transiently fashionable ideas that is the frontier of today’s theoretical physics, the only trend visible to the eye of the detached, not to say bemused, beholder is a secular increase of ugliness—as of entropy. Heading the parade of modern physics ugliness is Special Relativity Theory (SRT), an icon now so sacred that to breathe a word of negative criticism is to be automatically awarded the
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viii Old Physics for New: a worldview alternative
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jester’s bells and mantle of “crackpot.” Many critics (misled by the tale of “The Emperor’s New Clothes”) have tried to laugh, expecting to evoke a chorus ... and all such have left their bones to whiten the Juggernaut’s path. I anticipate no different fate, and am not concerned except to leave one more set of footprints on the path. A hallmark of ugliness is pretentiousness, and few ideas in the history of physics have been more overweeningly pretentious than the one underlying SRT, that of Universal Covariance. Maxwell’s equations are ugly enough, being invariant not even at first order ... but Einstein and Minkowski had the inspiration to universalize this form of ugliness and make it beautiful by making it transcendent—over mechanics and all the rest. The present investigation is concerned with showing ways in which that particular mask fails to fit nature. Though a goodly portion of this book will be devoted to destructive criticism—for ugliness deserves no less—most of it will be concerned with an attempt to rebuild, to offer constructive alternatives based on the pioneering electromagnetic theory of Heinrich Hertz and on an approach, suggested by the success of the Global Positioning System (GPS), to establishing a consistent way of telling a kind of “time” divorced from environment—very like Newton’s original conception. These alternatives may strike many viewers as themselves uglyfor that is an understandable response to any ringing in of the Old to replace the New. The nice thing about science, its redeeming feature, is that human aesthetic preferences and value judgments, differences of opinion regarding ugliness and beauty, conceptions of old and new, make no lasting difference. What matters, what lasts, is what works. What works is ultimately decided by experiment. This book will not have been written in vain if it succeeds in calling attention to two particular experiments that it claims to have crucial impact. If they are done, it is just possible that what they reveal about the shape of nature will enable (and motivate) the artisans who follow to craft a better-fitting mask. The writing of this book has been an educational experience for me. When I began it I was long aware of the shortcomings of Maxwell’s equations and the superiority of Hertz’s approach. But I had not conceived of the extent to which “time” needed to be reappraised, and the far-reaching consequences throughout theoretical physics of the needed reappraisal. Later, I became aware of
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Author’s Preface ix
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others who preceded me in recognizing important aspects of the “alternative physics” problem. These include Charles M. Hill, Al G. Kelly, Curt Renshaw, and of course that dean of dissidents, John Paul Wesley. Unfortunately, most of these are believers in some form of fundamental substrate in our universe possessing a determinate state of motion, the equivalent of Maxwell’s “luminiferous ether.” In contrast, I claim reconcilability of the facts of observation with a form of relativity principle. My book has been written with an absence of scholarly trappings such as footnotes, obligatory scientific “if’s,” “but’s,” and “on-the-other-hand’s,” and with a regrettable appearance of assurance on my part that I have got things right. Indeed, that is to a very limited degree my current illusion, and I have chosen my didactic mode of exposition in order to leave everywhere as little uncertainty as possible about my meaning ... but my true feelings lie closer to the quotation from Xenophanes that heads Chapter 5. In the final analysis, the book amounts to propaganda motivating a couple of simple experiments ... and the reader who knows how to get those done would save time to lay down this wordy book and at the first opportunity get busy doing them. Colleagues and friends whose advice and moral support have eased my task include Michael H. Brill, Dennis P. Allen, Jr., Peter and Neal Graneau, J. Guala-Valverde, David F. Roscoe, Ronald G. Newburgh, C. J. Carpenter, Paul B. Coggins, Ruth R. Rains, and Kathleen Leahr. Particular recognition is due the publisher, C. Roy Keys, who—with a few others, such as Petr Beckmann, Howard C. Hayden, Harold W. Milnes, Eugene Mallove, and Cynthia K. Whitney, none of whom history (which is written by the winners) is ever likely to honor adequately—has taken a crucial leadership role among dissidents in physics and astronomy by giving them the rarest and most essential gift for any would-be contributor to science, a way to communicate. In effect these have provided the outcast, the rejected, the up-staged, the downtrodden, the politically incorrect, the wrongthinkers of physics, with a Samizdat. Finally, a word of special thanks to David F. Roscoe for setting aside his academic obligations to read my text and provide a much appreciated Foreword.
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T. E. Phipps, Jr. Urbana, Illinois August, 2006
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x Old Physics for New: a worldview alternative
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Special notice regarding length invariance: The assumption or postulate of length invariance is built into all aspects of the alternative theory developed in this book. Therefore if the reader is inherently intolerant of this idea—that is, if his mind is permanently closed against it, whether through the revelations of inner voices or through private knowledge of incontrovertible contrary experimental evidence—he should read no farther. To go on would be a waste of time, as this book is for him the purest form of crackpot literature. Those who recognize that the Lorentz contraction could be a myth are encouraged to go on. And if there are some on the edge, curious as to why anyone in the twenty-first century would propose such a bizarre idea as length invariance, full understanding requires reading the text, but a shortcut can be found in Chapter 6, Section 7.
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Old Physics for New: a worldview alternative to Einstein’s relativity theory 1 Thomas E. Phipps, Jr. (Montreal: Apeiron 2006)
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In view of all our present difficulties it would seem that we ought at least to try to start over again from the beginning and devise concepts ... which come closer to physical reality ... If we are ever successful in carrying through such a modified treatment, it is evident that not only will the structure of most of our physics be altered, but in particular the formal approach to those phenomena now treated by relativity theory must be changed, and therefore the appearance of the entire theory altered. I believe that it is a very serious question whether we shall not ultimately see such a change, and whether Einstein’s whole formal structure is not a more or less temporary affair.
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—P. W. Bridgman, The Logic of Modern Physics
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Chapter 1
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What’s Wrong with
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Maxwell’s Equations?
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1.1 Problems of first-order description
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irtually the whole of “established” modern fundamental theoretical physics (quantum mechanics aside) is based upon two sacred cows, Einstein’s special relativity theory (SRT) and Maxwell’s equations of electromagnetism, the latter being postulationally supplemented by a Lorentz force law. Of these two, Maxwell’s equations are clearly the more fundamental, in that they came first chronologically—thus forming a formally prerequisite basis for SRT—and also in that they constitute the original prototype of field theory, the basis for all later “elementary particle” physics developments acceptable to modern authorities. Field theory has taken theoretical physics by storm. For most of the past century theoretical physicists have been convinced that field theory (the continuum mode of description) is the “natural language” of physics on both large and small physical scales. (I happen to disagree profoundly, but that is another
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2 Old Physics for New: a worldview alternative
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story I shall not even try to tell in the present book.) For this reason it seems particularly important to get field theory right on its own terms ... a process that can be begun only by doing the same for Maxwell’s equations themselves. That will be our task in Chapters 2 and 3. In this chapter we briefly prepare the way by noting a few of the manifold shortcomings of Maxwell’s equations in their accepted form (and, by incidental implication, of SRT). Before anything else, let us set down those equations for the simplest case of free space, which suffices for present purposes:
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14
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s
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E
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Bj
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ct c
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∂π
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∇× − =
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∂
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GG
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G
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G (1.1a)
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1B
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E ct
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∂
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∇× =− ∂
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G
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G
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G (1.1b)
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∇ ⋅ BG = 0
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G (1.1c)
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∇ ⋅ EG = 4πρ
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G , (1.1d)
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ρ and jGs being the Maxwellian charge and current densities, respectively. These four field equations, together with suitable physical boundary conditions, define and determine the electric
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field EG and magnetic field BG at the “field point” in space and time specified by the arguments ( )
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x, y, z,t of those two vector field quantities. {I use Gaussian units (a) for the sake of history, (b) to declare my independence from dicta of international committees, (c) to commemorate the brightest feature of my personal higher education, that I did not go to MIT. Those wishing to use other units can consult Table 2, p. 618 of Jackson[1.1] or equivalent.} On the broadest plane of philosophical generality, the field continuum idea is subject to the type of objection applicable to all mathematical idealizations parading as physical descriptions: The mathematical continuum, like the mathematical point, is something dwelling in the head of the mathematician. If it happens to find occasional usefulness outside that locus, this is more plausibly viewed as a happenstance, a bit of good luck not to be pressed too hard, than as a basis for religious experience. Yet we find in history the temporarily successful idealization, put forward tentatively by one generation, promoted into the next generation’s eternal truth ... and the third generation so worshipful of it that consensus greets further experimental testing, if unsuccessful, with suspicion suited to the Anti-Science. The real sin in
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1. What’s Wrong with Maxwell’s Equations? 3
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all this is not in over-estimating the descriptive scope of an idealization; that is mere rashness or bad judgment. Rather, it is in the arrogance of self-assurance-via-mutual-assurance through which crowds of scholars commit cumulative, massive follies that would be tolerated by no single scholar under restraints of individual responsibility or personal prudence. The scientific mistakes of individuals are healed and forgiven by history; the scientific arrogance of academic mobs is not. That is my comment on today’s final arbiter of scientific tastes, “consensus.” But if the greatest crime of modern physics against humanity were a crime against humility, we could deal with that in a paragraph and be left with no material for a book. No, it is in the details (where God is said to reside) that we shall discover more pressing problems with existing field theory. So, nothing more will be said here against the continuum as a descriptive abstraction. The first preliminary to be noted about the above or any other field continuum equations is that they offer no hint of that fundamental aspect of nature known as the wave-particle dualism. For that, of course, Maxwell is not to blame. It is strictly in hindsight that one recognizes all “fields” to be classical surrogates for spatially extended states (the wave aspect) of what quantum theory terms virtual particles. These can manifest themselves locally (the observable particle aspect) only through “process completions” enabled by interaction with material objects categorized by physicists (with typical professional parochialism) as “detectors.” For a field to make an observable appearance, there must always be a detector present. From this more modern perspective, it seems that the detector is central to all observability, physics being the description of what is observable. In that sense Maxwell’s equations are not about physics, because they concern what is supposed to be abstractly true at a disembodied mathematical “field point,” not what happens circumstantially at a detector—the Maxwell field being commonly conceived as a separately “existing” Ding an sich (whenever it isn’t claimed to be a “physical vector,” whatever that may be, besides an oxymoron). Only by imagining the detector to occupy the field point can the two views be reconciled ... but that exacts a serious penalty, to be appraised presently, in that the physical degrees of freedom of any actual (or idealized point) detector are suppressed through such stipulated superposition upon a mathematical “field point,” which by definition is entirely lacking in
|
||
|
||
|
||
4 Old Physics for New: a worldview alternative
|
||
motional freedoms—to the lasting and significant detriment (as it happens) of the scope of all field theory. The most prominent deficiency to be noted about the above specific field equations is that they are not invariant under firstorder (Galilean) inertial transformations. This is an extremely serious matter. It implies that in electromagnetism there exists an order of physical description—indeed, a dominantly important order, the first—at which the relativity principle (valid since Newton) does not hold. That is, if we are limited in our accuracy of physical measurement in such a way that we can observe only effects of first order in velocity, we ought seemingly to be able to observe actual violations of the relativity principle ... such as firstorder fringe shifts when our inertial system moves with respect to some “fundamental” system. This putative non-invariance can be seen from the fact that an operator such as ∂ ∂t , appearing in Eq. (1.1), is non-invariant under the Galilean transformation. The latter asserts that
|
||
′= −
|
||
G GG
|
||
r r vt , t t
|
||
′ = (1.2) for a first-order inertial transformation between primed and unprimed systems. (That is, ′ ′
|
||
rG ,t and rG,t specify coordinates of the same event point in the two uniformly-moving “inertial” frames,
|
||
vG being the constant velocity of the primed frame with respect to the unprimed one, and “first order” implying that considerations of order v2 or higher are ignored.) From this Galilean transformation we find[1.2] the operator relations
|
||
∇′ = ∇
|
||
G G, v
|
||
tt
|
||
∂∂
|
||
= + ⋅∇
|
||
∂′ ∂
|
||
G
|
||
G . (1.3a,b)
|
||
The first of these may be derived by applying (1.2), remembering
|
||
that vG is constant and recognizing that in partial differential operator actions upon field quantities [traditionally considered functions of ( )
|
||
x, y, z,t ] the chain rule applies: y
|
||
x zt x xx xy xz xt x
|
||
′
|
||
′ ′′
|
||
∂
|
||
∂ ∂ ∂ ∂ ∂ ∂ ∂∂ ∂
|
||
= + + +=
|
||
′ ′ ′ ′′
|
||
∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ , etc. ′
|
||
→∇ =∇
|
||
G G.
|
||
Similarly, since x
|
||
x x vt
|
||
′= − , ( ) x
|
||
xt v
|
||
′
|
||
∂ ∂ = − , etc., we have
|
||
.
|
||
xyz
|
||
y
|
||
x zt t tx ty tz tt
|
||
vvv v v
|
||
t x y zt t
|
||
′
|
||
′ ′′
|
||
∂
|
||
∂ ∂ ∂ ∂ ∂ ∂ ∂∂
|
||
= + ++
|
||
′ ′ ′′
|
||
∂ ∂∂ ∂∂ ∂∂ ∂∂
|
||
⎛⎞
|
||
∂ ∂ ∂ ∂∂ ∂
|
||
′
|
||
= − + + = − ⋅∇ = − ⋅∇
|
||
⎜⎟
|
||
′ ′ ′ ′′ ′
|
||
∂ ∂ ∂ ∂∂ ∂
|
||
⎝⎠
|
||
GG
|
||
GG
|
||
|
||
|
||
1. What’s Wrong with Maxwell’s Equations? 5
|
||
(Essential to this derivation is the constancy of vG implied by the stipulation of inertiality.) So, although the space derivative operator is invariant under first-order inertial transformations, the par
|
||
tial time derivative operator is non-invariant ( )
|
||
tt
|
||
′
|
||
∂ ∂ ≠ ∂ ∂ —a feature that prevents invariance of the field equations. Maxwell himself (who, by the way, never wrote nor saw “Maxwell’s equations”) was not disturbed by such non-invariance, since he did not subscribe to motional relativity and thought of electromagnetic description as subject to simplification in a fundamental system of ether “at rest.” In the nineteenth century this feature of non-invariance was taken seriously. Maxwell’s predicted fringe shifts were looked for experimentally and not found. Relativity at first order was thus discovered (by Mascart and others) to be an empirical fact. That forced the conclusion that Maxwell’s equations were wrong, or that something else was wrong. A “solution” was offered by Lorentz and subsequently reinforced by Einstein (in 1905). This was that “inertial” motions are to be described not by the Galilean transformation, Eq. (1.2), but by a more complicated set of equations known as “Lorentz transformations.” These introduced second-order fiddlings with both space and time variables (with “Lorentz covariance” substituted for invariance) that allowed the retention of Maxwell’s equations unscathed. This makes eminently good sense provided one is convinced a priori that Maxwell’s equations are the be-all and end-all of electromagnetic physics—or that wishing and diddling can make it so. Such a conviction stands permanently in the way of any progressive evolution in the foundations of field theory and has petrified itself into a doctrine of universal covariance. Over the years the latter has become to theoretical physics what Virgin Birth is to Christianity. How legitimate is it to treat first-order invariance problems by second-order solutions? In my (idiosyncratic) opinion, not at all. The Lorentz transformations for motion parallel to the x-axis are
|
||
22
|
||
1
|
||
x vt
|
||
x vc
|
||
−
|
||
′ = − (1.4a)
|
||
yy
|
||
′ = (1.4b)
|
||
zz
|
||
′ = (1.4c)
|
||
|
||
|
||
6 Old Physics for New: a worldview alternative
|
||
2
|
||
22
|
||
1
|
||
v
|
||
tx
|
||
c
|
||
t vc
|
||
−
|
||
′ = − . (1.4d)
|
||
It will be observed that in (1.4d) there is a first-order term in v that must survive even if all terms in v2 are discarded. Thus at first order these transformations take the form x x vt
|
||
′ = − (1.5a)
|
||
yy
|
||
′ = (1.5b)
|
||
zz
|
||
′ = (1.5c)
|
||
( 2)
|
||
t t vc x
|
||
′ = − . (1.5d)
|
||
Evidently at sufficiently long distances (large enough x-values) there will be a first-order departure (1.5d) from the Galilean transformation (1.2), affecting the time coordinate. (In this connection, see Appendix.) Consequently on a sufficiently large scale of distance there is no limiting conformity between the Lorentz and Galilean transformations. “Inertiality”—a physical property that both common sense and Einstein tell us should not depend on choice of distance scale—is not described by the Lorentz and Galilean transformations in a mutually consistent way at first order. To believe that at first-order the Lorentz way is right and the Galilean way is wrong requires us to accept a never-verified proposition—that at great distances time and synchronized clocks behave in a different way from their counterparts in our near vicinity. This in turn means that Newtonian physics is not right on a large physical scale, even as a first-order approximation. The reader can believe this proposition as an act of faith, if he wants to; but there is no objective basis for belief. Nor will he find it acknowledged up front in any of the SRT texts—which prefer to represent Einstein’s mechanics as a covering theory of Newton’s mechanics, in order to claim all the credit of the latter, preparatory to collecting extra dividends. (A “covering theory” is one that replicates all predictions of its covered theory in some parametric limiting case, but makes different predictions when that case is not realized.) Because of failure to make a first-order connection with the Galilean transformation, no Lorentz covariant mechanical theory can be a covering theory of Newtonian mechanics. The traditional method by which higher accuracy is attained in physics—namely, through orderly progression from lower to
|
||
|
||
|
||
1. What’s Wrong with Maxwell’s Equations? 7
|
||
higher orders of approximation, each successive order being so contrived as to be a covering theory of the lower orders—is certainly violated by such an expression of faith. Let that belief (that on a large scale the physics of the first order is different from what it is locally) be yours, dear relativist, not mine. I choose to view this as a signal, a clue, that all is not well in the foundations. In fact I cannot make it fit very well with ordinary formulations of the relativity principle, much less with assumptions of cosmological homogeneity rife among today’s general relativists. The relativity principle, as normally phrased, says that the laws of nature are the same in different inertial systems. It says nothing about their being different even within one inertial system. An effect of distance on clock settings in a given inertial system might be thought not to be the same as a change of “laws of nature.” Yet any first-order changing of clock settings as a function of distance, departing from the Galilean t t
|
||
′ = , will certainly affect at first order that law of nature known as Newton’s second law of
|
||
motion, FG = maG , which is founded unequivocally on t t
|
||
′= .
|
||
On the other hand, if we accept that locally the Lorentz transformations reduce at first order to the Galilean ones, then the “relativistic” theory seemingly inherits at first order the difficulty of Maxwell’s predicted first-order fringe shifts under the Galilean transformation, discussed above—the shift-disproving experiments being of local character and at first order. As it happens, there is a save based on Potier’s principle (a deduction from Fermat’s principle), whereby the putative fringe shifts prove to be theoretically unobservable after all. This will be one of our topics in Chapter 2. The matter is a bit subtle, and is never discussed nor alluded to in modern texts of electrodynamics. This is part of the general conspiracy of silence concerning first-order electromagnetic physics. As nearly as I can tell, the claim that at first order a physical
|
||
inertial transformation is described on the timelike side by
|
||
( 2)
|
||
t t vc x
|
||
′ = − is without a shred of observational support. Rather the contrary would seem to be the case: Since the Earth reverses its velocity every six months, it switches inertial systems with the corresponding frequency, so all its co-moving clocks throughout space should require continual resetting in those ever-changing states of motion, in such a manner that the apparent timings of astronomical events at great distances (large x-values) should vary with an annual period, and should vary (from correspond
|
||
|
||
|
||
8 Old Physics for New: a worldview alternative
|
||
ing event timings in our immediate spatial vicinity) proportionally to x. The first-order annual velocity change effect resulting from Eq. (1.5d) would thus imply a first-order annual change of clock readings and apparent dynamical evolutions at the locations of distant galaxies. There is no way clock readings (settings) at a given place can change periodically without associated clock rates appearing to change (as an artifact of the continual clock resettings associated with frame changes, about which the reader can learn from standard SRT texts such as Taylor and Wheeler[1.3]—see the twin paradox discussion there). In other words the earth’s circling should periodically affect the apparent (measured) rates of remotely distant physical processes—although the processes themselves do not factually vary in rate. But in actuality no such distance-dependent appearances appear. Note that at first order there is no space-side fiddle (contraction) to “compensate” the claimed time-side fiddle (clock phase changes)—as there is at second order—since the space coordinates at first order transform in Galilean-Euclidean fashion, Eqs. (1.5a-c). The Lorentzian “first-order world” defined by Eq. (1.5) thus seems decidedly unworldly. If our argument about distant astronomical process appearances is correct, it would seem that SRT and its mathematics of “Lorentz covariance” (too often sloppily referred to as “Lorentz invariance” or, heaven help us, simply “invariance”) get by on the strength of an intergalactic gentlemen’s agreement to ignore the observational physics of the first order. This is such an obvious and trivial criticism that it is a waste of time to make it, since minds are closed. Dingle,[1.4] who was put through the wringer by the physics establishment and accused of “dementia” for his alleged crime of failing to “understand” SRT, remarked, “It is ironical that, in the very field in which Science has claimed superiority to Theology, for example—in the abandoning of dogma and the granting of absolute freedom to criticism—the positions are now reversed. Science will not tolerate criticism of special relativity, while Theology talks freely about the death of God, religionless Christianity, and so on.” He found out the hard way that when all gentlemen subscribe to an agreement, all non-subscribers are by definition non-gentlemen. It is to this pass that SRT has brought the physics of the twenty-first century. And there, for all I can tell, it will stick—perhaps for millennia. One can only hope that a single millennium will suffice to eradicate the Einsteinian folly, as it did
|
||
|
||
|
||
1. What’s Wrong with Maxwell’s Equations? 9
|
||
the Ptolemaic. But the durability of a myth grows nonlinearly with its perceived beauty—which is far more a matter of selfconsistency than of external-world-consistency.
|
||
1.2 The under-parameterization of Maxwell’s equations
|
||
Much lip service is paid by Maxwell’s followers to “source-sink reciprocity.” Oddly, however, in the equations themselves there is no reciprocity of parameterization between source and sink motions. Look closely at Maxwell’s equations, (1.1). You will see that source motions are parameterized by s s
|
||
jG = ρvG , where vGs is indeed the velocity in the observer’s inertial frame of the particle that is the source of the field (or a corresponding charge density). But look as you may, you will find no parameter describing velocity of the field detector, absorber, or sink. Now, pause and reflect ... What is going on? After all, the sink is physically as important to the radiation process as the source (no photon being able to land without a landing-place), and far more so for purposes of observation or “measurement”—recognized in quantum theory as the quintessence of any micro process. In field theory what has become of this quintessence? What is going on is that the detector must be pictured as permanently at rest at the field point. Remember the observer’s “field point”? Well, that’s where the sink is. And nowhere else. Ever. Think of that! In this fickle world of motion, flux, and relativity, here is something eternal, fixed, immobile ... something you can count on. Dare I say it ... ? Something absolute. The source, in contrast, is not absolute. Like everything else in nature it can move freely, knows no home, has its velocity and degrees of freedom parameterized, etc. But the sink by definition sticks like glue to “the observer,” as a sort of extension of his personality. Since the observer is always by definition inertial, so is the sink. Here is a composition of matter (try making a detector without matter) that is by definition always inertial in its motions. Strive as it may to break that bond with the inertial observer, the poor thing just cannot do so consistently with Maxwell’s field theory. The sink lacks parameters that would grant it in theory those physical degrees of freedom which it clearly possesses in nature. It is always dangerous to build theory that departs from nature in ways so elementary you can count them on one hand, even with a thumb and finger missing.
|
||
|
||
|
||
10 Old Physics for New: a worldview alternative
|
||
Looking at it another way, we can view the observer as tied to a particular composition of matter (the field detector). He thus becomes a “preferred observer” with respect to that bit of matter. The state of motion of the matter in question then defines a fundamental physical inertial system that is preferred (by that observer) above all others. Clearly, in either way of looking at it, an element of absolutism contrary to the relativity idea has entered in a basic way—an element ineradicable as long as we refuse (by declining to parameterize sink motions) to allow arbitrary relative motions between every observer and every sink. The motional relativity idea, in contrast, properly requires that all observers be free to move with respect to all compositions of matter. There must be no preferred observer or bit of matter, ever, anywhere in the universe. Pause for a moment to reflect on the present claim: Here in the heart of Maxwell’s equations is a completely anti-relativistic structural element—an omission of parameterization that prevents relative motion between the observer and a corresponding favored composition of matter. And Maxwell’s equations lie at the heart (formally and structurally) of SRT, which is billed as the purest expression of the relativity idea. In other words, in the heart of the heart of “relativity,” as we know it today, dwells a heartworm.
|
||
1.3 The problem about Faraday’s observations: d/dt
|
||
A directly related difficulty evidenced by the Maxwell magnetic induction equation, (1.1b), is that it misrepresents the Faraday observations on which it is allegedly based. According to all the texts, Faraday’s observations are described by dd
|
||
E d B dS
|
||
dt dt
|
||
Φ
|
||
⋅ =− =− ⋅
|
||
∫ ∫∫ G
|
||
GG G
|
||
A
|
||
v , (1.6)
|
||
where Φ is the BG -field flux through the surface bounded by a closed electrical circuit. Note particularly the appearance of the total time derivative d dt here. The line integral on the left represents the electromotive force (emf) generated in the fluxpenetrated circuit when any of various experimental parameters are changed. Among those changed by Faraday was the shape of his circuit. That is, he moved part of the circuit in a magnetic field and observed that this produced an emf. It is this shape-changing aspect that necessitates using a total time derivative operator d dt in (1.6), rather than the partial derivative ∂ ∂t native to tra
|
||
|
||
|
||
1. What’s Wrong with Maxwell’s Equations? 11
|
||
ditional field theory. There is no escape from d dt , because a shape change cannot occur without accelerated relative motions of circuit parts. Such different motions in different places require for their localized (differential) description different values of a local velocity parameter ( )
|
||
vd t
|
||
G , of the sort that is present in d dt but not in ∂ ∂t . There being no inertial system in which the circuit as a whole can be considered even approximately “at rest,” there is no applicability of SRT, which by its basic terms of reference is limited to inertial motions. According to the ideology of relativists, such problems of mixed local accelerations and non-accelerations should rigorously be handled by general relativity theory (GRT). As far as I know, the doctrine to that effect has never been put into practice in this case, perhaps because even relativists—though a rigorously sober lot, as befits those teetering on the very tippy-top rung of the intellectual ladder—have a sufficient sense of humor to smile at any claim of logical necessity to treat Faraday’s elementary observations via four-index tensor symbols (or to drag in equivalence of acceleration to “gravity”), when a simple total time derivative will manifestly do the job with elegance and precision. So, let us not argue with the electrodynamics texts, but take their word for (1.6) as a statement of empirical fact. How, then, do relativists and field theorists get from (1.6), with its empirically correct d dt , to Maxwell’s (1.1b), with its politically correct ∂ ∂t ? Ah, now that is a tale that would interest psychologists. Some authorities, such as Panofsky and Phillips[1.5] just brutally set down a howler,
|
||
dB
|
||
E d B dS dS
|
||
dt t
|
||
∂
|
||
⋅ =− ⋅ =− ⋅
|
||
∂
|
||
∫ ∫∫ ∫∫
|
||
G
|
||
GG
|
||
GG G
|
||
A
|
||
v . (?)
|
||
If you can square that with your mathematical conscience you are off to the races in “deriving” the Maxwell (1.1b), involving a partial time derivative. These authors stifle their scruples by fast talk about “a differential expression valid for free space or a stationary medium.” Indeed, this might be a legitimate representation of Faraday’s observations if they had been confined to free space or a stationary medium. However, they involved changing the
|
||
shape of a circuit—altering the path of dGA in flux-penetrated space. That’s a different matter, which mathematically requires retention of the d dt operator.
|
||
|
||
|
||
12 Old Physics for New: a worldview alternative
|
||
Or, take an alternative dodge perpetrated by J.D. Jackson[1.1] and reflecting equivalent desperation to get to the known right answer. He says, “Faraday’s law can be put in differential form by use of Stokes’s theorem, provided the circuit is held fixed in the chosen reference frame ... ” So, his draconian choice is simply to ignore the experiment and pretend that Faraday held his circuit fixed in an inertial frame. That gets us to Maxwell’s (1.1b), all right ... but at what a cost! Lorrain and Corson[1.6] carry out a variant of this reasoning even more conscientiously. If I understand their analysis, they allow non-inertial motions, but require the circuit at all times to move as a rigid whole—still in deliberate disregard of what Faraday did and saw. Wangsness[1.7] claims to allow shape changes of the Faraday circuit, but when his formulas are examined they are found to contain only a single velocity
|
||
dimensioned parameter vG , our old SRT friend, the relative velocity of two inertial systems ... so, although the shape change was talked about, by what was it parameterized? When you change the shape of a circuit you impart different velocities to different portions of it. If your theory lacks a velocity parameter that can take different numerical values on different circuit portions, your mathematics is plainly inadequate to the physics ... though it may be adequate to the political demands of your day. From the variety of such swindles by modern authoritiesall of whom insist on making bricks without straw (reducing global circuit descriptions to differential form without benefit of local shape-change parameterization)—and the unanimity of critical silence by which they are greeted, the degree of degeneracy of modern physical judgment and ethics could be judged ... and mathematical integrity as well ... if anyone were judging. But physics has shown itself a profession (like others) mightily resistant to whistle-blowing. Some more sentient observers of the electromagnetic scene have noticed that information is lost in passing from the integral formulation (1.6) to the differential formulation (1.1b). Consequently they insist on integral formulations of field theory as more “fundamental.” [Thus (1.6) allows for continuous deformations of the global integration contour, whereas (1.1b) contains no way of describing the resulting continuous local departures from inertial motion.] None of these experts has recognized that the simple way to recover all the lost information is by changing the differential formulation to make it include an extra velocity parame
|
||
|
||
|
||
1. What’s Wrong with Maxwell’s Equations? 13
|
||
ter ( )
|
||
vd t
|
||
G capable of describing the said local departures by taking on different local values ... and the simple way to do that is to avoid the strenuous mental gymnastics involved in replacing d dt by ∂ ∂t , and instead to relax and peacefully leave d dt in the differential formulation. This is the sort of thing readily grasped by freshmen but utterly hidden from the ken of academic savants (unless their peers are telling them about it, in which case they invented it back in ’01).
|
||
1.4 Justification for a Hertzian form of Faraday’s law
|
||
Why this concerted professional willingness or compulsion to make obvious elementary mathematical mistakes in the interest of getting to Eq. (1.1b), as if at all costs? Simply because the stakes are tremendous. The costs are indeed “all.” The reputation of every physicist of the modern era, dead or alive, depends on that little ∂ ∂t . Empiricism calls for the field equation (1.1b) to be replaced by the Hertzian invariant form
|
||
1 dB
|
||
E c dt
|
||
∇× =−
|
||
G
|
||
G
|
||
G , (1.7)
|
||
but that would destroy spacetime symmetry, which arises from and depends critically upon the appearance of the time variable t in the field equations (1.1) of electromagnetism solely in a balanced form
|
||
,,,
|
||
x y z ict
|
||
⎛⎞
|
||
∂∂∂ ∂
|
||
⎜⎟
|
||
∂∂∂∂
|
||
⎝⎠
|
||
that manifests a fundamental symmetry of space and time partial derivative operators. This is the mother lode—where spacetime symmetry originates. The total time derivative, in contrast, is traditionally expressed as
|
||
()
|
||
d
|
||
dv
|
||
dt t
|
||
∂
|
||
= + ⋅∇
|
||
∂
|
||
G
|
||
G , (1.8)
|
||
where vGd may be treated as constant or more generally as an ar
|
||
bitrary function of t (at any rate constant under the action of ∇G ). To use d dt is to upset the balance of space and time:
|
||
()
|
||
, , , vd
|
||
x y z ict
|
||
⎛⎞
|
||
∂ ∂ ∂ ∂ + ⋅∇
|
||
⎜⎟
|
||
∂∂∂∂
|
||
⎝⎠
|
||
G
|
||
G.
|
||
So, it’s hello d dt , goodbye spacetime symmetry. Note that this
|
||
vGd is a local velocity parameter entirely different from the global
|
||
|
||
|
||
14 Old Physics for New: a worldview alternative
|
||
parameter vG descriptive of an inertial frame transformation. The proof of Eq. (1.8) for the general case in which the field detector is treated as a point particle in arbitrary (possibly accelerated) motion, ( )
|
||
dd
|
||
v =v t
|
||
G G , is immediate from the chain rule:
|
||
d
|
||
dy
|
||
d dx dz dt v
|
||
dt dt x dt y dt z dt t t
|
||
∂ ∂ ∂ ∂∂
|
||
= + + + = + ⋅∇
|
||
∂ ∂ ∂ ∂∂
|
||
G
|
||
G . (1.9)
|
||
It is important to note, as an implicit feature of this derivation and conception, that the field quantities or other operands of d dt are not viewed as arbitrary functions of ( )
|
||
x, y, z,t , as is standard in traditional field theory, but are instead viewed as functions of ( ) ( ) ( )
|
||
()
|
||
x t , y t , z t ,t . This means that one maintains always an inflexible regard for the fact that all field values are measured quantities—measured on the trajectory of the point-like detector ... that trajectory being described by () () () ()
|
||
()
|
||
r t = x t ,y t ,z t
|
||
G at any locus in space where the “field” is to be interrogated, or at each place along a conducting circuit sub
|
||
ject to deformations. Consequently, our vGd is not a “velocity field.” The introduction of the d dt operator completely spoils the formal symmetry of space and time differentiations and thus destroys the basis (in electromagnetism) for SRT and for all modern physics built upon it. And it leaves no justification for “universal covariance,” the mathematical expression of spacetime symmetry that is the touchstone or shibboleth of our scientific age. The ap
|
||
pearance here of an extra velocity-dimensioned parameter vGd fits hand-in-glove precisely to compensate the Maxwellian underparameterization mentioned in Section 1.2. Exploitation of this lucky fit will be our main objective in the next chapter. The upshot of the present discussion is that the physicist faces a choice between respecting empirical fact, as embodied in Faraday’s observations (demanding total time differentiation), and imposing mathematical beauty, as embodied in covariant formalisms (for which it is essential that all time differentiation operators be of the partial type). Among theorists of our era there has never been a moment’s hesitation about that choice. They have stopped at nothing to legislate their foreknown truth. This is to say that the people who currently call themselves, and are paid to be, physicists are flying under false colors. They are ideologues, playthings of their collective infatuation with beauty. The world as it is interests them less than the world as it ought to be. They
|
||
|
||
|
||
1. What’s Wrong with Maxwell’s Equations? 15
|
||
are what happens when physicists morph into mathematicians (of sorts). But since they constitute an intellectual monopoly that controls all media of communication in their field, whistleblowing by individual out-groupies like myself is a vain endeavor. Those who control today’s scientific communication exert absolute power to allow or prevent progress, and by that power have been corrupted absolutely.
|
||
1.5 Other problems of Maxwell’s equations
|
||
In conclusion regarding the deficiencies of Maxwell’s equations, we have barely scratched the surface of this bountiful topic. Plenty more such are to be seen by any beholder not blinded by science. A major portion of this book will, in one way or another, be concerned with exploring the ramifications of this rich subject. I have thus far omitted criticism, for example, of sins against logical economy, such as that entailed in defining a “field” as force on unit charge and then postulating a separate force law for charges. And I have omitted criticism of internally contradictory aspects of field theory itself. For instance, Eq. (1.6) defines “flux” as an integral. This implies that any circuit senses instantly via its emfe.g., by a set of voltmeters placed everywhere around the perhaps infinitely spatially extended circuit—any change of a global (integral) property. (If not, please tell me which voltmeter measures the flux change first. And feel free to place yourself in any inertial system!) This can only betoken instant and simultaneous actions at-a-distance—supposedly forbidden by the very terms of reference of field theory, not to mention SRT. Yet here we perceive field theory (always presented as the bastion of causal thinking) to be founded on a conception that is manifestly acausal and contrary to the relativity of simultaneity. The bones of quantum mechanics move perceptibly beneath the skin of the Maxwell field[1.8]—instant action-at-a-distance being an integral aspect of quantum theory. The only known exception in the entire range of physical experience to the rule of instant action is radiation (locally completed quantum processes)—the tail that hitherto has wagged the dog. Suffice it to say that Maxwell was a genius. He earned that status by eclipsing the original geniuses of electrodynamics, André-Marie Ampère and Wilhelm Weber. It is the fate of genius to be eclipsed. Pace ... let genius be sufficient unto the day
|
||
|
||
|
||
16 Old Physics for New: a worldview alternative
|
||
thereof. The whole concept of the field-continuum mode of description is over-ripe for reappraisal, if our era could move beyond its complacently dismissive view of electrodynamics as a closed book. About this subject there is only one thing trivial, in the sense of being true clearly to any child—namely, that it is barely (and thus far poorly) begun.
|
||
1.6 Chapter summary
|
||
In this chapter we have concentrated on three major deficiencies of Maxwell’s equations:
|
||
• They are not invariant at the lowest (first) order of approximation under first-order inertial (Galilean) transformations. • They are under-parameterized, in that source motions are described but not sink motions. Such an implicit promotion of the sink to a preferred motional status flouts any conceivable form of relativity principle. (This is why invariance fails.) • They misrepresent Faraday’s observations of induction through the false implication that ∂ ∂t can treat that for which d dt is mathematically necessary: viz., description of the emf generated by changing the shape of a conducting circuit penetrated by magnetic flux.
|
||
Other shortcomings are readily cited, but these will suffice to give us the clues needed in order to commence a plausible reconstruction of electromagnetic theory in the next chapter.
|
||
References for Chapter 1
|
||
[1.1] J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1962). [1.2] M. Jammer and J. Stachel, Am. J. Phys. 48, 5 (1980). [1.3] E. F. Taylor and J. A. Wheeler, Spacetime Physics (Freeman, San Francisco, 1966). [1.4] H. Dingle, Science at the Crossroads (Martin, Brian, and O’Keefe, London, 1972). [1.5] W. K. H. Panofsky and M. Phillips, Classical Electricity and Magnetism (Addison-Wesley, Reading, MA, 1962), 2nd ed. [1.6] P. Lorrain, D. R. Corson, and F. Lorrain, Electromagnetic Fields and Waves: Including Electric Circuits (Freeman, New York, 1988), 3rd ed. [1.7] R. K. Wangsness, Electromagnetic Fields (Wiley, New York, 1979). [1.8] C. A. Mead, Collective Electrodynamics (MIT, Cambridge, MA, 2001).
|
||
|
||
|
||
Old Physics for New: a worldview alternative to Einstein’s relativity theory 17 Thomas E. Phipps, Jr. (Montreal: Apeiron 2006)
|
||
There is no physical phenomenon whatever by which light may be detected apart from the phenomena of the source and the sink ... Hence from the point of view of operations it is meaningless or trivial to ascribe physical reality to light in intermediate space, and light as a thing travelling must be recognized to be a pure invention.
|
||
—P. W. Bridgman, The Logic of Modern Physics
|
||
Chapter 2
|
||
What to Do About It ...
|
||
(the Hertzian Alternative)
|
||
2.1 First-order invariant field equations
|
||
he criticisms of Maxwell’s field equations in the preceding chapter were made in order to gain constructive clues to the building of alternative electrodynamic theory. In that we have been successful: We saw that those equations were under-parameterized (lacking a velocity-dimensioned parameter to describe sink motions), and also that they misrepresented Faraday’s observations of induction by using the non-invariant operator ∂ ∂t instead of the first-order invariant operator d dt (which
|
||
contains an extra velocity-dimensioned parameter vGd needed to describe local departures of his circuit from inertiality). Putting these clues together—matching parameter deficit with parameter surplus—we have the inference forced upon us that it would be worthwhile to aim for an invariant (instead of covariant) formulation of electromagnetism in terms of total time derivatives,
|
||
wherein vGd is interpreted as describing sink motion. (As a confirmation and dividend for using an invariant time derivative operator, we shall discover that the resulting field equations are themselves invariant at first order.) No inspiration can be claimed
|
||
T
|
||
|
||
|
||
18 Old Physics for New: a worldview alternative
|
||
for such an idea, or confluence of ideas, since the facts make it inescapable. The idea, over-ripe, has fallen from the tree and has only to be picked up. We know that motional relativity is an empirical fact, and that invariance is the natural mathematical expression of a relativity principle. Covariance is supposed to do as well, but certainly not better! So, first things being first, let us get busy and get first-order electromagnetic field theory right for a change. That will be our aim in this chapter. Then, in the next chapter, we shall see whether higher-order description can be induced to follow suit (as a covering theory) in the course of nature ... as nature, so to speak, intended. To prepare the way, we gather up a loose end by proving our assertion of the invariance of d dt under the Galilean (first-order) inertial transformation, Eq. (1.2). Applying the Galilean velocity addition law,
|
||
′= −
|
||
GGG
|
||
dd
|
||
v v v , (2.1)
|
||
where vGd is sink or detector velocity measured in the unprimed
|
||
inertial frame, G′
|
||
vd is the same measured in the primed frame, and
|
||
vG is velocity of the primed relative to the unprimed inertial frame, together with Eqs. (1.3) and (1.8), we find
|
||
()
|
||
′′
|
||
∂∂
|
||
⎛⎞⎛ ⎞ ′ ′
|
||
= + ⋅∇ = + ⋅∇
|
||
⎜⎟
|
||
⎜⎟ ∂ ∂
|
||
⎝⎠
|
||
⎝⎠
|
||
∂ ∂ ⎛⎞
|
||
⎛⎞
|
||
= + ⋅∇ + − ⋅∇ = + ⋅∇ =
|
||
⎜ ⎟ ⎜⎟
|
||
∂∂
|
||
⎝ ⎠ ⎝⎠
|
||
GG
|
||
GG
|
||
G GG
|
||
G GG G
|
||
'
|
||
,
|
||
dd
|
||
dd
|
||
dv v
|
||
dt t t
|
||
d
|
||
v vv v
|
||
t t dt
|
||
(2.2)
|
||
which verifies the first-order invariance of d dt . Note that in this proof it is essential to distinguish between the two velocity
|
||
dimensioned parameters, vG and vGd . They are physically and
|
||
mathematically unrelated, vG being always necessarily constant
|
||
for inertial motions, and vGd being more generally descriptive of arbitrary non-inertial motions via an arbitrary Lagrangian particle trajectory, ( )
|
||
dd
|
||
v =v t
|
||
G G —but not via an Eulerian “velocity field”
|
||
()
|
||
, ,,
|
||
dd
|
||
v =v x y z t
|
||
G G . The latter requires a more elaborate analysis, patterned on that of Helmholtz,[2.1] who, according to Miller,[2.2] derived the result
|
||
() ( )
|
||
dU U V U V U
|
||
dt t
|
||
= ∂ + ∇⋅ −∇× ×
|
||
∂
|
||
G G G G GG
|
||
G.
|
||
If we apply this to arbitrary vector fields, U,V
|
||
G G , then a standard vector identity[2.3] converts it to
|
||
|
||
|
||
2. What to Do About It ... (the Hertzian Alternative) 19
|
||
( ) ( )( )
|
||
dU U V U U V U V
|
||
dt t
|
||
= ∂ + ⋅∇ + ∇⋅ − ⋅∇
|
||
∂
|
||
G G G GG G G G
|
||
G G G . (2.3)
|
||
If ∇G operating on VG yields zero, then (2.3) reduces to (1.8). Such results have been given also by Abraham-Becker[2.4] and more recently by Dunning-Davies,[2.5] as well as by other authors. They provide a generalization of the traditional total or convective derivative (1.8) from the Lagrangian specialized case ( )
|
||
dd
|
||
v =v t
|
||
G G , under which (1.8) and (2.2) are valid, to the general field-theoretical case, ( )
|
||
VG = VG x, y, z,t . However, I see no use for such generality in the present context. Physically (setting aside entirely all those hardy perennial interpretations, of the Lorentz ether pedigree, that postulate an electromagnetic “medium” in an identifiable state of motion), they correspond to the case of an extended “mollusk” detector (to borrow a figure of speech from Einstein), or electric eel—which could be expanding and contracting, with
|
||
non-vanishing VG -divergence or gradient. Prosaically, attention will be confined in this book to idealized inanimate point detectors—hence to the Lagrangian ( )
|
||
dd
|
||
v =v t
|
||
GG .
|
||
We proceed to postulate the first-order invariant electromagnetic field equations, here called Hertzian because—though often (and still continually being) re-invented—they were first proposed by Heinrich R. Hertz.[2.6] Getting rid of the non-invariant ∂ ∂t everywhere in Maxwell’s equations in favor of the invariant d dt , we have for free space, to which attention will be confined here:
|
||
14
|
||
m
|
||
dE
|
||
Bj
|
||
c dt c
|
||
π
|
||
∇× − =
|
||
GG
|
||
G
|
||
G (2.4a)
|
||
1 dB
|
||
E c dt
|
||
∇× =−
|
||
G
|
||
G
|
||
G (2.4b)
|
||
∇ ⋅ BG = 0
|
||
G (2.4c)
|
||
∇ ⋅ EG = 4πρ
|
||
G . (2.4d)
|
||
These equations are identical to the Maxwell Eqs. (1.1) except that d dt replaces ∂ ∂t and the source terms are interpreted differ
|
||
ently. In (2.4a) the modified source term jGm is interpreted as the current density that is measured by a suitable point detector moving at velocity ( )
|
||
vd t
|
||
G in the observer’s inertial frame. It is evi
|
||
dently related to the Maxwellian source current density jGs
|
||
|
||
|
||
20 Old Physics for New: a worldview alternative
|
||
(measured at the stationary field point by a detector at rest there) by the first-order vector summation,
|
||
ms d
|
||
j = j − ρv
|
||
G G G . (2.5)
|
||
The −ρvGd term here represents an equivalent current (of reversed direction, signified by the minus sign) due to detector motion at
|
||
velocity vGd . In effect, a current-density detector is pictured as
|
||
passing with velocity vGd through the observer’s “stationary” field point and as having a snapshot taken of it there at the moment of detection. The snapshot shows it to be displaying on its read-out
|
||
a measured value jGm , which is the vector sum of the Maxwell source current and the convection current produced by its own motion. It will be observed that (2.4b) reproduces the empirical result (1.7), which we decided in Chapter 1 was needed to describe Faraday’s observations. And since there is no virtue in going half-way with invariance, the invariant total time derivative has been used in (2.4a) as well. By the nature of invariance, the Hertzian invariant formulation of field theory obeys a first-order relativity principle and gets rid automatically of the spurious first-order fringe shifts predicted by Maxwell’s non-invariant equations when subject to the Galilean transformation. We stipulate that various field-related quantities such as source charge and current are measured (simultaneously) at the
|
||
field point, as are the field quantities EG and BG themselves. Moreover, the various physical detectors (as of electric and magnetic fields, charge densities, etc.) that make such measurements
|
||
all share a common state of motion parameterized by vGd . Because of this shared-motion assumption we may picture all these quantities as measured by a single multi-purpose “detector.” This we idealize as a mathematical point (but one sufficiently massive to possess a classical trajectory), which moves through the observer’s idealized stationary “field point” and coincides with it at the instant of measurement. This constitutes a minor extension of the (seldom-discussed) idealizations employed by Maxwell and field theorists generally—who tacitly assume immobility of the detector at the field point. The term “point detector,” as used here, will always refer to a physical detector, usually of several field-related quantities, idealized as a mathematical point, yet recognizable as surrogate for an actual macro instrument through the fact of its possessing a well-defined trajectory. Such preliminaries need not be dwelt on further. They are usually skipped
|
||
|
||
|
||
2. What to Do About It ... (the Hertzian Alternative) 21
|
||
over almost entirely by textbook expositors, who break a leg to get to the mathematics. The more the textbook-writing professors cripple themselves in this way, the more their students do the same when they in turn grow up to be textbook-writing professors. Since the Maxwell case of a stationary detector represents the special case 0
|
||
vd =
|
||
G , d dt → ∂ ∂t [as shown by Eq. (1.8)], we see that Hertz’s electromagnetism, Eq. (2.4), constitutes a “covering theory” of Maxwell’s equations, (1.1). That is, all predictions of Maxwell’s theory are reproduced by Hertz’s theory in the special case that the field detector is at rest, 0
|
||
vd =
|
||
G , at the field point in the observer’s inertial system. If the detector moves with respect to the field point ( 0)
|
||
vd ≠
|
||
G , new effects are predicted by Hertz ... and the prediction is invariant at first order, so that all inertial observers agree on it. The electromagnetic “laws of nature” are the same for all of them, so a first-order relativity principle is automatically obeyed in conformity with the mathematical invariance (under Galilean transformations). Einstein or Maxwell might handle this physical situation of detector motion through the field point by transferring observer and measuring apparatus bodily into another frame, also required to be inertial ... and then forever after would be limited to strictly inertial motion of a detector fixed at the observer’s new field point. The Hertzian analysis, not being limited to inertial motions of the sink, allows arbitrary non-inertial ( )
|
||
dd
|
||
v =v t
|
||
G G . It is thus a “covering theory” in more ways than one.
|
||
Since the operators ∇G and d dt appearing in the Hertzian field equations (2.4) are Galilean invariant [per Eqs. (1.3) and
|
||
(2.2)], and the source terms ρ and jGm are measured quantities on which all observers must agree, it follows that the field equations must be Galilean invariant; consequently that the field transformation law at first order is
|
||
EE
|
||
′=
|
||
G G, B B
|
||
′=
|
||
G G ; (2.6) that is, the field quantities themselves are invariant. Let us pause to prove our claimed field equation invariance in detail. First we note the Galilean source transformation equations ( ) ()
|
||
ρ r ,t ρ r,t
|
||
′ ′ ′=
|
||
G G (2.7)
|
||
and
|
||
( ) () ()
|
||
ρ
|
||
′ ′ ′= −
|
||
GG
|
||
G G GG
|
||
, ,,
|
||
ss
|
||
j r t j r t r t v , (2.8)
|
||
|
||
|
||
22 Old Physics for New: a worldview alternative
|
||
this vG being the velocity parameter appearing in the Galilean transformation, Eq. (1.2). Eq. (2.7) follows from the facts (a) that
|
||
rG′ and rG refer to the same field point in space, viewed in primed and unprimed inertial systems, (b) that t t
|
||
′ = , and (c) that charge density is measured by counting charges in the given small (infinitesimal) detector volume, which has the same size and shape as viewed in each system and regardless of its relative motion (because we postulate length invariance, as will be discussed more fully in due course—cf. Chapter 6, Section 7). Such a charge count, a pure number, must be invariant under changes of the viewing system. Similarly (2.8) holds for the Maxwellian current density, with inclusion of a convective current-density term due to the inertial relative motion. These are well-known results from Maxwell theory, to be found for instance in Jammer and Stachel.[1.2] From this we prove the invariance of the Hertzian measured current density,
|
||
( ) ( )( )
|
||
ρ ρ ρρ
|
||
ρ
|
||
′
|
||
′ ′ ′′
|
||
= − =− = − − −
|
||
=− =
|
||
GG G G
|
||
G G G GG
|
||
GG
|
||
G,
|
||
ms d s ds d
|
||
s dm
|
||
j j v j v j v vv
|
||
j vj
|
||
(2.9)
|
||
where use has been made of Eqs. (2.1), (2.5), (2.7), and (2.8). With these preparations, proof of the Hertzian field equation invariances becomes a matter of inspection. Thus invariance of (2.4c),
|
||
( B) B B 0
|
||
′ ′′
|
||
∇⋅ =∇ ⋅ =∇⋅ =
|
||
G GG
|
||
G G G , (2.10)
|
||
follows from (2.6) and (1.3a). Invariance of (2.4d),
|
||
()
|
||
E 4πρ E 4πρ E 4πρ 0
|
||
′ ′′ ′
|
||
∇⋅ − =∇ ⋅ − =∇⋅ − =
|
||
G GG
|
||
G G G , (2.11)
|
||
follows from (2.6), (1.3a), and (2.7). That of (2.4b),
|
||
1 1 10
|
||
dB d dB
|
||
E E BE
|
||
c dt c dt c dt
|
||
′′
|
||
⎛ ⎞ ⎛⎞
|
||
′′ ′
|
||
∇× + =∇ × + =∇× + =
|
||
⎜ ⎟ ⎜⎟
|
||
′⎝ ⎠
|
||
⎝⎠
|
||
GG
|
||
G G GG
|
||
G G G , (2.12)
|
||
follows from (2.6), (1.3a), (2.2), and the assumed first-order constancy of the units ratio,
|
||
cc
|
||
′ = . (2.13) Finally, the invariance of (2.3a),
|
||
|
||
|
||
2. What to Do About It ... (the Hertzian Alternative) 23
|
||
ππ
|
||
π
|
||
′′
|
||
⎛ ⎞ ⎛⎞
|
||
′′ ′ ′
|
||
∇× − − =∇ × − −
|
||
⎜ ⎟ ⎜⎟
|
||
′′
|
||
⎝⎠
|
||
⎝⎠
|
||
=∇× − − =
|
||
GG G
|
||
G GG
|
||
GG
|
||
GG
|
||
G
|
||
G
|
||
14 1 4
|
||
1 4 0,
|
||
mm
|
||
m
|
||
dE d
|
||
B j B Ej
|
||
c dt c c dt c
|
||
dE
|
||
Bj
|
||
c dt c
|
||
(2.14)
|
||
follows from the above together with (2.9). In the same way we may identify a first-order invariant continuity equation, which generalizes the customary one involving ∂ ∂t :
|
||
ρρ
|
||
ρ
|
||
′′
|
||
⎛ ⎞ ⎛⎞
|
||
′′ ′
|
||
∇⋅ + =∇ ⋅ + =∇⋅ + =
|
||
⎜ ⎟ ⎜⎟
|
||
⎝ ⎠ ⎝⎠
|
||
GG G
|
||
G G G0
|
||
mm m
|
||
d dd
|
||
jj j
|
||
dt dt dt . (2.15)
|
||
The generalization resides, of course, in the circumstance that the
|
||
detector measuring total current jGm can move with respect to the field point; whereas such motion is (without physical justification) forbidden in the traditional ∂ ∂t formulation. The above invariances all hold at first order under the Galilean transformation, Eq. (1.2). We shall have more to say presently about the physical meaning of a field invariance such as (2.6), which is entirely different from the corresponding “covariance.” Finally, we note in passing that Eq. (2.1), ′ = −
|
||
GGG
|
||
dd
|
||
v v v , implies various expressions of the reciprocity idea. Thus, if we formally put primes on both sides of this equation we get ′′ ′ ′
|
||
=−
|
||
GGG
|
||
dd
|
||
v v v . Interpreting one prime as effecting a switch from one inertial frame to another, and two primes as effecting a switch back, we have the requirement ′′ =
|
||
GG
|
||
dd
|
||
v v , which implies ′′ ′ ′
|
||
==−
|
||
GGGG
|
||
ddd
|
||
v v v v , or ′′
|
||
=+
|
||
GGG
|
||
dd
|
||
v v v . Comparing with (2.1), we see that vv
|
||
′=−
|
||
G G , (2.16) which is the customary expression of Galilean velocity reciprocity between inertial frames. If the latter is taken as given, the argument can be worked backwards to prove ′′ =
|
||
GG
|
||
dd
|
||
v v.
|
||
2.2 History: Why did Hertz fail?
|
||
It is of interest to digress for a moment to see how history explains why the reader has never heard of Hertz’s first-order invariant version of electromagnetic theory. First, let it be noted that Heinrich Hertz, although generally credited only as the experimentalist who confirmed in his laboratory Maxwell’s ideas about the wave propagation of light, was also a powerful theorist in his own right. In the last chapter of his book Electric Waves,[2.6] Hertz developed a formally (Galilean) invariant generalization of Max
|
||
|
||
|
||
24 Old Physics for New: a worldview alternative
|
||
well’s theory, involving a new velocity-dimensioned parameter with components ( )
|
||
α,β,γ . He conceived of his theory (formally the same as that developed above) as describing an electrodynamics of “moving media,” and interpreted his new velocity parameter as ether velocity. This was a serious mistake, a false interpretation. He compounded that error by postulating a Stokesian ether 100% convected by ponderable matter. This made his theory testable, because it reified the ether—giving it “hooks” to observable matter. (Maxwell, too, assumed an ether, but cleverly avoided giving it hooks to anything observable!) Soon after Hertz’s death an experimentalist, Eichenwald,[2.7] went into his laboratory and disconfirmed Hertz’s predictions. The invariant theory was thus discredited and relegated to history’s trash bin. One important lesson to be learned: It is never theory alone that is proven or disproven in the laboratory; it is theory plus interpretation of the symbols employed in it. We have seen that by the simple expedient of interpreting Hertz’s ( )
|
||
α,β,γ as our vGd , with no reference to “ether”—specifically, by re-interpreting Hertz’s “ether velocity” as field-detector velocity relative to the laboratory inertial system—all conflicts with observation are eliminated through the obtaining of a covering theory of Maxwell’s electromagnetism. In fact Hertz possessed such a formal covering theory, but spoiled it as physics by his bad guess about symbol interpretation. The irony of history is that Maxwell’s (ether) interpretation, too, was discarded, leaving to posterity only his formalism. That residual (once and future) formalism was inferior to the discarded Hertzian formalism, in exactly the sense that any non-invariant formulation is inferior to its corresponding invariant covering theory. The final irony is that if there were justice based on chronological priorities Maxwell’s theory would have been discarded even before Hertz’s was discardedand on identical grounds, namely, that of hair-trigger observational “disproof.” For we mentioned that Mascart[2.8] (in 1872) and others had done experiments looking for Maxwell’s predicted first-order fringe shifts and not finding them. So, Maxwell’s theory was observationally “discredited” long before Hertz’s, fully as firmly as Hertz’s, and on identical grounds, viz., a bad “ether” interpretation ruined it as physics. Still, as a fact of history, Maxwell’s inferior formalism was retained and Hertz’s superior formalism was discarded and utterly forgotten. (If extra irony is needed on top of irony, Hertz himself was quoted as doing Max
|
||
|
||
|
||
2. What to Do About It ... (the Hertzian Alternative) 25
|
||
well the favor of saying “Maxwell’s theory is Maxwell’s equations”—meaning, forget Maxwell’s ether interpretation—yet no physicist ever did Hertz the favor of saying “Hertz’s theory is Hertz’s equations”—meaning, forget Hertz’s ether interpretation!) That’s the way history of science works, and you won’t find it in history of science books. Most of the latter are written to specs of political correctness of their day. They themselves are part of the history of science, and not the most admirable part. Still, it would be difficult to get along without them. For instance, the work of A. I. Miller,[2.2] an historian of science who yields place to nobody in his abject fealty to Einstein, is invaluable for giving a rare glimpse of the electromagnetic science that preceded Einstein. Thus, Miller’s Eq. (1.8), attributed to Hertz, is identical to our Eq. (1.7), which we asserted to be demanded by Faraday’s empirical evidence. So, although Miller does not make the point, Hertz’s grasp of Faraday’s physics—of the empirical basis of the budding electromagnetic science—was superior to that of the Maxwellians. However, Miller’s ideological bias in favor of his hero is so strong that he cannot resist making serious expository mistakes. He says of Hertz that “his axiomatic assertion of the form invariance of the electromagnetic field equations [or “covariance” as Hermann Minkowski (1908a) described this mathematical property] led Hertz to predict new effects whose empirical confirmation could in turn serve to confirm his axiom of covariance.” This short quotation is riddled with misinformation. Hertz did not make an “axiomatic assertion.” He simply and explicitly alleged the invariance of his equations (an allegation factually correct at first order, as we have seen). Unfortunately he left it at that, not bothering to prove his allegation—probably because he considered the reader intelligent enough to provide his own proof. Hertz rightly said nothing about covariance, because it had not been invented, was irrelevant, and was inapplicable to his equations. Those equations were, exactly as he said, invariant, not and never covariant. Concerning the vast physical difference between these two kinds of form preservation we shall have more to say below. The mathematical difference is equally vast, as should be evident to any reader who has had the mathematics of covariance drilled into him in the course of a modern higher education.
|
||
|
||
|
||
26 Old Physics for New: a worldview alternative
|
||
2.3 Invariance vs. covariance: The physics of it
|
||
It is worth pausing to say a few words about the neglected physical aspect of the distinction between covariance and invariance. It might be thought that there is no physics involved and that the distinction is purely mathematical—and not much of a distinction at that, since authorities as distinguished as Dirac slur over the difference by alluding to “Lorentz invariance.” This writer, in fact, has never seen a clear, precise, mathematically kosher statement of the difference between invariance and covariance. So, I shall slur a bit myself, simply saying that under an invariant transformation a mathematical relationship (equation) preserves each symbol in place, from its untransformed to its transformed interpretation, without altering the formal relationships among symbols and without redefining individual symbol meanings. A covariant transformation, in contrast, allows similar form preservation of the relationships among symbols, but does so by introducing separate explicit relations of dependence (linear, in the case of Lorentz covariance) among transformed and untransformed symbols, amounting to symbol redefinitions. Symbol redefinition may be said to be the hallmark of “covariance.” Examples: The Maxwell field equations (1.1) are covariant under a Lorentz transformation, Eq. (1.4). (The explicit relations of linear dependence among field components are given in any of the texts, such as Jackson[1.1] or Panofsky and Phillips.[1.5] In effect there is a “scrambling” of electric and magnetic field components, such that electric field components in one frame are redefined as a linear combination of both electric and magnetic components in the other, etc.) The Hertz field equations (2.4) are invariant under the Galilean transformation (1.2), as we showed. Their symbols transform in place without redefinitions. E, B
|
||
G G in one frame mean exactly the same thing in any other, as per Eq. (2.6). It is my bias to characterize covariance as a clever contrivance, invariance as mathematically straightforward. (The symbol-redefinition approach succeeds through achieving self-consistency—and that in itself seems to me potentially dubious, in the sense that it would be better not to have to rely on it as physics. Mathematicians, in contrast, view self-consistency as self-justifying. Mary’s little lamb, physics, is supposed to follow. Like the French king who said “L’état, c’est moi,” or the American Admiral who calls himself “Sixth Fleet,” the mathematician declares, “Physics, it is my sym
|
||
|
||
|
||
2. What to Do About It ... (the Hertzian Alternative) 27
|
||
bols ... and my symbols are what I define them to be.”) Both invariance and covariance exploit the theme of “form preservation,” and on this basis both can lay formal claim to expressing a physical relativity principle. But physically the two describe entirely different sorts of “fields,” and no confusion should be allowed to exist between the Maxwellian and Hertzian types. The easiest way to recognize the difference is to contrast the ways in which the two are measured. In Einstein-Maxwell theory each of an infinite ensemble of inertial observers has his personal point-like measuring instrument (a multi-purpose type capable of measuring field components, charge and current densities, etc., in the manner indicated above) at rest at his own field point, which is fixed in his system. At the instant these field points coincide (the points being located on lines in space parallel to the relative motions, all intersecting to allow such coincidence), each of the measuring instruments (“detectors” brought into coincidence) is read by its respective observer and the various sets of field-component numbers so recorded are found on comparison to satisfy the relations of linear dependence mentioned above and specified by the Lorentz transformation (with group parameter v descriptive of the relative motion of any pair of these inertial systems). Thus operational procedures or “measurements” are replicated in the various inertial systems, with the unspoken assumption (basic to all classical physics) of physical replicability of experiments.
|
||
Notice that this entails a collision of the various detectors, if those are constructed of matter. To avoid this, one could picture “near misses” of the detectors. But that is a compromise with operationalism. It would be more instructive to recognize that Maxwell’s conception of “field” as Ding an sich comes into play here, so that Maxwellianism, sang pur, does not base its conceptualizations upon measurement at all (or upon the pesky material mechanisms of measurement), but simply upon abstract numbers floating in space (attached to an independently existing “field” reality) that are revealed by the equations without need for measurement, even in thought. Hence in inertial transformations the “collision” (harmless and non-destructive) is of mathematical field points only, not of actual material (or notional) field detectors ... so there is no need to arrange near misses. This of course does not fit with Einstein’s working version of operationalism, whereby everything hangs on Gedanken measurements, so it is
|
||
|
||
|
||
28 Old Physics for New: a worldview alternative
|
||
anomalous that Einstein took unquestioning satisfaction in exactly preserving Maxwell’s formalism, root and branch. One wonders if he understood the full philosophical implications of what he was preserving. So much for the Maxwell-Einstein form of electromagnetic field theory. Let us turn next to the Hertzian version. Instead of a raft, passel, or pride of inertial observers, each equipped with his own “personal” point detector permanently fixed at his field point, there is only a single point detector present—and this can be unequivocally, without apologies or evasions, a genuine composition of matter. It is public property. Differently-moving inertial observers, of which there may be any number present, are all provided, not with any tangible apparatus, but with a (unique)
|
||
numerical parameter value vGd descriptive of the velocity of this single “public” detector relative to their own inertial system. There is never any collision of real or notional detectors, because there is only one detector present in the whole universe of discourse ... and each observer is at rest with respect to his own field point, as in Maxwell’s theory. Each field point is in motion aligned to pass through the location of the point detector. At the moment when all field points coincide with the point detector’s location, all observers consult the read-out of that single (public) instrument. Since they all read the same numbers from the same instrument at the same instant, it follows that numerical invariance and form invariance are trivially satisfied. There is only one physically unique point event of detection. That is what all observers describe. Eq. (2.6), E E
|
||
′=
|
||
G G, B B
|
||
′=
|
||
G G , just expresses this necessary identity of numbers read from the same instrument at the same time by different observers. So, genuine invariance is a Kindergarten matter—as distinguished from covariance, which is for fast-track college sophomores and other candidates for initiation into the Orphic and Eleusinian mysteries of higher mathematics, who—trudging valiantly toward general relativity theory—bear ’mid snow and ice their banner with a strange device, “Excelsior!”
|
||
2.4 Invariance or covariance: Which is physics?
|
||
Is it a Mexican standoff, then, between invariant and covariant formulations of electromagnetism? Is there no way to decide which is physics? Will either describe all observable aspects of nature? At the level of classical description, in what may be called
|
||
|
||
|
||
2. What to Do About It ... (the Hertzian Alternative) 29
|
||
the strong-field approximation (many photons present to represent the “field”) this is undeniably the case. Maxwell’s theory and its Hertzian covering theory do entirely equivalent jobs. Until about 1925 physicists had no reason to think in other terms. But then came quantum mechanics to overturn all ideas of the replicability of experiments on the micro scale. Recall that the Maxwell-Einstein model requires each member of an ensemble of inertial observers to be equipped with his own macro fielddetection instrument, at rest at his field point, and all to replicate experimental procedures. This, as we noted, causes a conceptual problem of “collisions,” when the covariant measurement is made at the instant of coincidence of the field points. To dodge that, we babbled of “near misses,” but in the weak-field case all such gestures of accommodation fail. For, consider the single-photon limit: In the simplest case we have two Maxwell-Einstein inertial observers with their macroinstruments coming together to make a simultaneous “measurement” of this photon. Do they both succeed? Of course not. The photon, by the basic nature of any “quantum” process, can be absorbed in at most one macro-detector. On a random basis, one of the two detectors present, let us say, “wins” and makes a successful measurement of the photon’s “field.” The other then necessarily “loses” and measures zero—a big goose-egg! Experimental replicability thus fails catastrophically. Turn and twist as he may, there is no way any self-respecting mathematician, not to mention physicist, can represent the one observer’s “zero” as related by the Lorentz transformation to the other observer’s finite measured values of the field components. Hence covariance clearly and manifestly fails irreparably in the weak-field limit. The idea of universal covariance is a joke, a propagandist’s mantra, an unchecked flowering of New York Times science. It does not work even within all electromagnetism, much less within all nature. Does invariance succeed in this limit? Of course it does, trivially so—because there is only a single Hertzian macro-detector present. That either detects the photon or it doesn’t, on a random basis. The fraction of trials on which it succeeds is related to the Born (probabilistic) interpretation of the photon’s wave function. All very neat—no problems, because no “interference” or “competition” among macro detection devices for the weak-field quantum. The upshot, then, is that Hertzian invariance works in both
|
||
|
||
|
||
30 Old Physics for New: a worldview alternative
|
||
strong- and weak-field cases, whereas Maxwell-Einstein covariance works only for strong (or “classical”) fields. If physics wants to progress smoothly, with integrity and consistency, into the quantum descriptive domain, it must drop covariance at the threshold and switch to invariance. (Having done that, it might as well drop covariance, period ... and all the elaborate cloud-castle of twaddle with which it is laden.) This is such a simple and irrefutable argument that I have found it to beget an equally simple and irrefutable scientific response: universal ignoration. The Big Yawn. Such a response may be bolstered by the modern trend of philosophers and philosophically inclined physicists to view operationalism as discredited. The urge to see things as either credited or discredited for good and all is one of the subliminal influences toward simplism that prevent much net progress of scientists en masse toward adulthood. If physics is to be about what is “measurable,” it needs to be able to give at least a conceptual or Gedanken account of how the necessary measurements are made. That is what I have insisted upon here. I call this “instrumentalism” or “operationalism.” Natural philosophers and scholars of science are notable for sicklying o’er with the pale cast of thought whatever they view as fundamental in the description of nature. I decline to chop logic with them about the sicklying job they did in allegedly “discrediting” operationalism. But I do want to know: Where have they been all these years since that hatchet job? Have they mislaid their sicklying gear? Why no sicklying o’er of the problem of electromagnetic measurement? How has field theory escaped so long any mobilization of scholarly critical apparatus capable of probing just how (by what instruments in what states of motion) the “field” is measured? In other words, how has covariance, in its imperial nakedness, so long escaped being laughed at? And for how much longer? Surely modern scholarship is itself some kind of joke—another weapon in the arsenal of political correctness. We seem to have built the kind of world in which only cynics know how or when to laugh.
|
||
2.5 Hertzian wave equation
|
||
The first-order invariant wave equation is derived in the customary way: Taking the curl of Eq. (2.4a), we have in the source-free
|
||
case ( 0)
|
||
jm =
|
||
G
|
||
|
||
|
||
2. What to Do About It ... (the Hertzian Alternative) 31
|
||
() 2
|
||
22
|
||
1 dE 1 d 1 d B
|
||
BE
|
||
c dt c dt c dt
|
||
∇× ∇× = ∇× = ∇× =−
|
||
GG
|
||
GG
|
||
GG G G ,
|
||
this last equality following from taking the total time derivative of Eq. (2.4b). Applying a standard vector identity,[2.3] we have
|
||
( )( ) 2
|
||
2
|
||
22
|
||
1dB
|
||
B B B c dt
|
||
∇× ∇× =∇ ∇⋅ −∇ =−
|
||
G
|
||
G GG
|
||
G G GG ,
|
||
where ∇2 = ∇ ⋅ ∇
|
||
G G G ; and, since ∇ ⋅ BG = 0
|
||
G from (2.4c), the Hertzian wave equation—invariant at first order under Galilean transformations—is
|
||
2 2
|
||
22
|
||
10
|
||
dB
|
||
B c dt
|
||
∇− =
|
||
G
|
||
G . (2.17)
|
||
Similarly, taking the curl of Eq. (2.4b) and the total time derivative of (2.4a) for 0
|
||
jm =
|
||
G , we derive
|
||
2 2
|
||
22
|
||
10
|
||
dE
|
||
E c dt
|
||
∇− =
|
||
G
|
||
G . (2.18)
|
||
Taken together, (2.17) and (2.18) describe electromagnetic wave propagation in free space (vacuum). These wave equations are invariant at first order under the Galilean transformation (because derived from field equations already shown to be invariant). They possess this property by virtue of our having substituted the first-order invariant operator d dt for the traditional Maxwellian non-invariant ∂ ∂t . From this substitution follow interesting physical consequences, several of which we shall presently explore. Although the quantity c appearing in (2.17) and (2.18) is invariant, in agreement with (2.13), this does not imply that the physical speed of light is invariant. To explore this, we must solve the Hertzian wave equation. Applying standard field theoretical analytic techniques, let us look for a solution of (2.18) of d’Alembertian type. (For simplicity we may confine attention to
|
||
the electric field EG , since the results for BG are formally identical.) The solution we seek is of the form ( )
|
||
EG = EG p , where
|
||
x yz
|
||
p = k ⋅ r −ωt = xk + yk + zk −ωt
|
||
G G . (2.19)
|
||
We find that
|
||
() ()
|
||
( ) () ()
|
||
222 2
|
||
2 22
|
||
222 2 ,
|
||
xyz
|
||
Ep Ep
|
||
xyz
|
||
k k k E p kE p
|
||
⎛⎞
|
||
∂∂∂
|
||
∇ = ++
|
||
⎜⎟
|
||
∂∂∂
|
||
⎝⎠
|
||
′′ ′′
|
||
= ++ =
|
||
GG
|
||
G G (2.20)
|
||
|
||
|
||
32 Old Physics for New: a worldview alternative
|
||
where double prime denotes two differentiations with respect to p. Similarly, using Eq. (1.8), we have
|
||
() ()
|
||
( ) ( ) ()
|
||
( )( )
|
||
( ) ()
|
||
()
|
||
2
|
||
2
|
||
2
|
||
22
|
||
2
|
||
2
|
||
2
|
||
2
|
||
2
|
||
2.
|
||
d
|
||
dd
|
||
dd d
|
||
d Ep v Ep
|
||
dt t
|
||
v v Ep
|
||
tt
|
||
ω ωv k v k E ω v k E
|
||
⎛∂ ⎞
|
||
= + ⋅∇
|
||
⎜⎟
|
||
⎝∂ ⎠
|
||
⎛⎞
|
||
∂∂
|
||
= + ⋅∇ + ⋅∇
|
||
⎜⎟
|
||
∂∂
|
||
⎝⎠
|
||
′′ ′′
|
||
= − ⋅+ ⋅ = − ⋅
|
||
GG
|
||
G
|
||
G
|
||
G
|
||
GG
|
||
GG
|
||
GG G
|
||
GG
|
||
GG G
|
||
(2.21)
|
||
Eqs. (2.18), (2.20), (2.21) imply
|
||
()
|
||
( )2
|
||
2 2
|
||
10
|
||
d
|
||
k vk E
|
||
cω
|
||
⎡ ⎤ ′′
|
||
−+ − ⋅ =
|
||
⎢⎥
|
||
⎣⎦
|
||
GG
|
||
G . (2.22)
|
||
From the vanishing of the coefficient of EG′′ it follows that
|
||
d
|
||
ck = ω− v ⋅ kG
|
||
G or d
|
||
k
|
||
cv
|
||
kk
|
||
ω= ± + ⋅
|
||
G G , (2.23)
|
||
where k = kG ⋅ kG . It is useful to define a wave phase propagation speed (“phase velocity”) u relative to the observer’s inertial system, as
|
||
d
|
||
k
|
||
u cv
|
||
kk
|
||
=ω= ± + ⋅
|
||
G G . (2.24)
|
||
The corresponding result in Maxwell’s theory is u = ±c . Consequently we see that the universal constancy of light speed (Einstein’ second postulate) is not satisfied in Hertzian theory—so a new kinematics will have to be devised to describe high-speed motions. That we leave for later chapters. Evidently, the most general d’Alembertian solution of Eq. (2.18) can be written, from (2.19) and (2.24), as
|
||
() ()
|
||
1 d2 d
|
||
E E k r ck k v t E k r ck k v t
|
||
⎡ ⎤⎡ ⎤
|
||
= ⋅+ −⋅ + ⋅− +⋅
|
||
⎣ ⎦⎣ ⎦
|
||
GG GG
|
||
GG G
|
||
G G G G , (2.25)
|
||
where 1 2
|
||
E ,E
|
||
G G are arbitrary vector functions. Invariance of the field solutions, Eq. (2.6), implies that ( ) ( )
|
||
E p E′ p′
|
||
=
|
||
G G . This in turn implies a condition of phase stability, p p′
|
||
= , or, from (2.19),
|
||
k r ωt k r ωt
|
||
′ ′ ′′
|
||
⋅− = ⋅ −
|
||
GG
|
||
G G , (2.26) which describes a constant value of the phase of the propagating wave field as measured in the primed and unprimed inertial systems. Suppose we limit attention to the case of uniform motion of the field detector. Then we can specialize by considering the detector at rest in the primed inertial system, so that vd = v
|
||
G G , where
|
||
vG is the (constant) velocity of the primed system with respect to
|
||
|
||
|
||
2. What to Do About It ... (the Hertzian Alternative) 33
|
||
the unprimed one. In this case, eliminating r′,t′
|
||
G from (2.26) by means of the Galilean transformation, Eq. (1.2), we find after rearrangement that
|
||
( ) ( d)
|
||
k k r tω ω k v
|
||
′ ′′
|
||
− ⋅= − − ⋅
|
||
GG G
|
||
G G . (2.27)
|
||
Since rG,t are arbitrary and independently variable, their coefficients must vanish. Consequently,
|
||
kk
|
||
G′ = G (2.28) and
|
||
d
|
||
ω ω kv
|
||
′ = − G ⋅ G . (2.29)
|
||
The first of these results describes first-order aberration, the second the first-order Doppler frequency effect of detector motion. Since stellar (Bradley) aberration is ordinarily described as
|
||
an apparent small turning of the kG -vector of starlight propagation, we see that no such apparent turning is predicted by Eq.
|
||
(2.28), which asserts first-order invariance of the kG -vector. Starlight from the pole of the ecliptic [for which 0
|
||
d
|
||
k⋅v =
|
||
G G , hence u = c from (2.24)], cannot depart from constancy in either speed or direction. A direct way to describe stellar aberration will therefore involve a higher-order refinement of the Hertzian theory, which will be our topic in Chapter 3. Another way will be dis
|
||
cussed in Chapter 7. This kG -constancy consideration alone is sufficient to show that unadorned Hertzian theory, as most simply interpreted, is inadequate to the physics and needs higher-order refinement. We have worked it out here in detail to show how a self-consistent first-order invariant treatment can be developed formally—as well as to provide a model for the higher-order calculation. The reader should not be disturbed that first-order electromagnetic theory is unable to handle stellar aberration. [Maxwell’s theory does no better, since it shares Eq. (2.28).] SRT claims to handle it (about that claim we shall have more to say in Chapter 4), but it is a second-order theory. To give invariance an equal chance to do the job, we owe it a second-order development. The phenomenon appears to be a first-order effect because of proportionality of the observed aberration angle to a velocity. But stellar aberration is in fact a much subtler phenomenon than it seems or was thought to be classically. We shall argue in Chapter 4 that
|
||
what it involves is not a directional turning of the kG -vector but a second-order effect of detector motion on light speed. Because of
|
||
|
||
|
||
34 Old Physics for New: a worldview alternative
|
||
the one-way light propagation involved, stellar aberration also represents a much more important and revealing aspect of the physics of light than is generally recognized.
|
||
2.6 Potier’s principle
|
||
Before leaving the first-order development of invariant field theory, it is necessary to revert to some nineteenth-century lore that has a direct bearing on why the bizarre and counter-intuitive “convection of light by the absorber,” apparently predicted by our wave equation solution (2.24), is not in general observable. As we study Eq. (2.24), we see that it suggests a very counterintuitive thing—that motion of the field detector or photon absorber with respect to the observer changes the speed of light, by pulling the photons along with it—as it were, convecting them. If
|
||
the direction of light propagation, specified by kG k , is parallel to
|
||
detector motion vGd , then light speed (“phase velocity”) according to (2.24) is d
|
||
c + v , and if anti-parallel it is d
|
||
c − v . [This tacitly assumes the choice of plus sign for ±c in (2.24).] It might be thought that such a grossly acausal notion (after all, how does the photon know about its future absorber’s motion before it has even reached that absorber?) is directly counter to experience, so the whole Hertzian fabrication can be dismissed. However, phase velocities (like photon trajectories and mental “pictures” of light propagation or other quantum processes generally) are without directly observable counterpart in nature. Moreover—and this is the point of immediate consequence—there is a theoretical result of nineteenth-century physics, known as Potier’s principle, which makes it understandable that the simplest forms of first-order observations, such as ordinary laboratory fringe shifts, can reveal no
|
||
observable effect on light speed of any additive term of type kG ⋅ vG entering the expression for phase velocity. Actually, to be true to history, we should say that Potier’s principle was aimed at ether theories. It showed that fringes would not be shifted by any additive phase velocity term of the
|
||
form kG ⋅ VG , if VG were interpreted as ether velocity. But the
|
||
mathematics of Potier’s proof applies to any “ VG , ” regardless of
|
||
how interpreted physically. So, it applies here with VG interpreted
|
||
as detector velocity vGd . The principle is discussed in several modern references, for instance.[2.9-2.11] It will be derived here for the convenience of readers to whom such references are not avail
|
||
|
||
|
||
2. What to Do About It ... (the Hertzian Alternative) 35
|
||
able. Nineteenth century geometrical optics is assumed, but presumably a modernized wave formulation would not alter the gist of the treatment or result. Potier’s principle:[2.12] Let two fixed points in an inertial system, P1 and P2, of emission and absorption, respectively, be joined by a light path consisting of n0 connected straight-line segments, the ith one of which may be denoted vectorially as i = iεi
|
||
GG
|
||
A A , where
|
||
i ii
|
||
ε = kG k
|
||
G is a unit vector of light propagation on the ith segment,
|
||
0
|
||
i = 1,2,...,n . Let the index of refraction in the vicinity of the ith
|
||
path segment be given by a law of the form
|
||
()
|
||
()
|
||
11
|
||
ii
|
||
n = + aε ⋅ V + O V c
|
||
G
|
||
G , where a is a scalar constant and VG is a
|
||
constant vector. [Here ( )
|
||
O V c denotes any quantity of the order of V c in magnitude, considered small.] Then
|
||
(a) The total light transit time from P1 to P2 is increased (com
|
||
pared to the case V = 0 ) by an amount
|
||
() ( )
|
||
()
|
||
a c L⋅V 1+O V c
|
||
G G , where 0
|
||
1
|
||
n i
|
||
i
|
||
L=
|
||
=∑
|
||
G GA is the vector (straight-line) distance from P1 to P2 . (b) The spatial path taken by physical light in passing from P1
|
||
to P2 is unaffected by arbitrary changes in VG (for V c ).
|
||
Proof: The time required for light to transit the ith path segment is
|
||
() ()
|
||
()
|
||
()
|
||
( ) ()
|
||
()
|
||
11
|
||
1 1.
|
||
iii i i ii ii
|
||
ii ii i
|
||
t n a V OVc
|
||
u cn c c
|
||
Va
|
||
al O V c V O V c
|
||
c c cc
|
||
ε
|
||
ε
|
||
⎡⎤
|
||
== = = + ⋅ +
|
||
⎣⎦
|
||
=+ ⋅ + =+ ⋅ +
|
||
G
|
||
AAA A G
|
||
G GG
|
||
AA
|
||
GA
|
||
Hence the total transit time from P1 to P2 is
|
||
() ( )
|
||
()
|
||
() ( )
|
||
()
|
||
00 0
|
||
11 1
|
||
11
|
||
01 ,
|
||
nn n
|
||
ii i ii i
|
||
a
|
||
TV t V OVc
|
||
cc
|
||
a
|
||
T LV OVc
|
||
c
|
||
== =
|
||
⎛⎞
|
||
= = + ⋅+
|
||
⎜⎟
|
||
⎝⎠
|
||
= +⋅ +
|
||
∑∑ ∑
|
||
G GG
|
||
AA
|
||
G G (2.30)
|
||
where T (0) is the transit time for VG = 0 . This proves part (a) of the principle. To establish part (b) we invoke Fermat’s principle, which states that for arbitrary path variations in close proximity to the physical light path P, connecting fixed end points P1 and P2, the time of light transit is an extremum (least) for the actual physical path P. We have just shown that total light transit time (the
|
||
only observable) is determined by the constant vector LG between fixed endpoints P1 and P2 , and is independent of the index i, hence of the particular values of the i
|
||
GA ; that is, independent of the
|
||
|
||
|
||
36 Old Physics for New: a worldview alternative
|
||
detailed geometry of the path between those endpoints. Hence
|
||
for an arbitrary constant value of VG
|
||
T(V ) =T(0)+
|
||
G (Constant for all varied paths). (2.31)
|
||
Since T (0) does depend on path and is a minimum on path P, it
|
||
follows that ( )
|
||
T VG is also a minimum on P for arbitrary VG . Therefore the physical path of light—that which takes the least
|
||
time—is the same for VG ≠ 0 as for VG = 0 , q.e.d. It will be understood that the statement here of Potier’s principle in terms of an “index of refraction” is arbitrary and does not follow history. (Potier’s original formulation treated “ether” motion in the context of Fresnel’s theory.) The above form of the principle can be connected to the present work by reference to Eq. (2.24). Choosing the plus sign for c in that equation and defining an index of refraction of free space by
|
||
()
|
||
()
|
||
1 2
|
||
1
|
||
11
|
||
d dd
|
||
d
|
||
cc k k
|
||
n v v Ovc
|
||
u kc c k
|
||
k
|
||
cv
|
||
k
|
||
−
|
||
⎛⎞
|
||
= = = + ⋅ =− ⋅ +
|
||
⎜⎟
|
||
⎜⎟
|
||
⎝⎠
|
||
+⋅
|
||
GG
|
||
GG
|
||
G G , (2.32)
|
||
we see that this satisfies the conditions of the theorem, given the
|
||
identifications ( )
|
||
a = − 1 c , εi = kG k
|
||
G for all i, and d
|
||
VG = vG . So, just as ether wind is unobservable in the spatial domain at first order, the same is true of our apparent phenomenon “convection of light by the absorber.” (Formally, our identification d
|
||
VG = vG implies that we are thinking of the “ether” as convected by the absorber. That is, the light detector is always at rest in the luminiferous medium, or carries the medium with it.) The index of refraction of free space is indistinguishable from unity (the value for 0
|
||
vd =
|
||
G ) by any experiments in the spatial domain (interferometry, diffraction, etc., related to “path”). This conclusion is as valid as, and is founded upon, Fermat’s principle. Historically, Potier’s principle established the futility of all experimental attempts to challenge the relativity principle at first order. It was doubtless a major motivator of Michelson-Morley’s investigation at second order. To interject a personal note, I may remark that I wasted several years of experimental effort looking for first-order effects of “light convection by the absorber,” using interferometry, diffraction gratings, etc., all with null results. These trials are summarized in Chapter 7 of Heretical Verities.[2.11] At the time I knew of Potier’s principle, but failed to make the simple connection from
|
||
|
||
|
||
2. What to Do About It ... (the Hertzian Alternative) 37
|
||
ether convection to absorber-motion convection. It is sobering to find years of laboratory labor obviated by a few minutes of thought. The same might prove true of theoretical labor, if this lesson were to be applied more generally. Does this mean there is no hope of observing any distinction between invariant Hertzian and covariant Maxwellian theories of electromagnetism? No, it merely means that effects of the d
|
||
kG ⋅ vG phase term will not be directly observable as a basis for the distinction. Furthermore, part (a) of Potier’s principle leaves open the possibility of experiments in the time domain. In analyzing stellar aberration in Chapter 4, we shall find that the Hertzian second-order phase term has a first-order effect on the observable phenomenon. Consequently, stellar aberration will play a stellar role in deciding empirically between the Hertzian and MaxwellEinstein views of field theory.
|
||
2.7 Sagnac effect and ring laser
|
||
In the Sagnac experiment the detector moves circularly in an inertial system, somewhat as in stellar aberration, so one might hope from the foregoing that such an experiment would cast light on the issue of Hertzian vs. Maxwellian electromagnetism. Unfortunately, Potier’s principle spoils this hope, as we shall see. In the Sagnac apparatus monochromatic light circulates around a planar area A of any shape in opposite directions on a platform, establishing interference fringes. The platform rotates, along with A, at angular velocity ω about an axis normal to A located anywhere (inside or outside A). As a result of the rotation either of two results is observed in practice,
|
||
(a) zero fringe shift relative to the case of no rotation, (b) a first-order fringe shift of magnitude consistent with a circulation time discrepancy between the two beams of
|
||
2
|
||
Δt = 4Aω c , this latter result being evidence of the “Sagnac effect.”
|
||
What makes the difference between these two cases? Detailed experimental conditions, seldom discussed. If, as in the original experiments of Sagnac, there is a good deal of vibration, or if there is deliberate “jitter” introduced into the light propagation process, result (b) is observed. If the apparatus is comparatively steady and no dither is employed, result (a) is apt to be obtained.
|
||
|
||
|
||
38 Old Physics for New: a worldview alternative
|
||
A jargon has been developed to describe the physics of the latter case; viz., it is called “frequency pulling,” resulting in “mode locking.” Frequency pulling refers to an empirically observed tendency of the light frequency to shift spontaneously to the nearest eigen-frequency such that an integral number of wavelengths fits the cavity length (or light path optical length). The cavity then behaves as if “sticky” and carries the standing wave pattern along with it without a Sagnac shift, as if the apparatus were not rotating. To cause this phenomenon of mode locking, some interaction between the two counter-propagating modes must be present, known as mode coupling, and indications are that the coupling agency is generally the backscattering of light from imperfectly reflecting mirrors. Although mirrors of reflectance as high as 99.9999% have been used, there is always some mode locking. Kelly[2.13] (page 52 of his book) comments as follows:
|
||
There was a considerable element of luck in the original Sagnac experiment. It was later found that the light signals can lock onto the circuit and mirrors unless there is considerable vibration; such vibration was present in the Sagnac experiment. In later designs a dither is introduced to ensure that locking does not occur. History would, no doubt, have taken a different turn had the Sagnac test given a zero result, which would have been the case had the equipment been rock steady ...
|
||
The dithering introduced to combat mode locking may involve a small superposed twisting oscillation about the rotation axis, or some other ingenious form of perturbation. Physicists view the phenomenon as not of fundamental physical interest, but merely as an apparatus effect to be overcome. For purposes of present discussion, I am willing to go along with this and to accept that the “real physics” is result (b), above. Let us turn to theory, then, and see how it explains that outcome. First, what does Maxwell-Einstein theory predict about the Sagnac experiment? Identifying the laboratory as close enough to an inertial system, and declaring light always to have speed c in inertial systems, they note that light circulating at that speed relative to the laboratory in the direction of platform rotation will have to go an extra increment of laboratory travel distance Δs in order to “catch up” in completing the light circuit, whereas light circulating counter to the sense of rotation will travel a shortened
|
||
|
||
|
||
2. What to Do About It ... (the Hertzian Alternative) 39
|
||
distance Δs in returning to its starting point. [It is easily shown for a circular path rotating about its center that Δs bears to path
|
||
length the ratio ( )2
|
||
v c . However, Δt = Δs v , the time (or phase) discrepancy, is of first order in v.] When the beams reunite and interfere to form fringes there will be a travel time discrepancy Δt between them, and the first-order fringe shift of outcome (b) is expected to be observed. This is the prediction of SRT—and of course of classical first-order kinematic theory as well. All that’s really involved is a small “travel distance” discrepancy in the lab for the counter-circulating beams. The physics behind this seems to be that optical phenomena, like mechanical ones, need to be analyzed in inertial systems to yield simplest descriptions. This necessity appears to depend only on the Newtonian theoretical framework. The fact that there is a first-order travel time discrepancy Δt observed by the rotating platform rider R as well as by the lab observer—both observing the same factual phase discrepancy or fringe shift—suggests that R must attribute different speeds to light going in the two directions, since the two beams, starting simultaneously and traveling equal total first-order distances in the rotating system, come back to the same place in that system at different times ... but many Einsteinians cannot bear this implication, because it asserts a first-order non-constancy of light speed in a non-inertial system. Other Einsteinians take it in stride, accepting (Einstein, The Meaning of Relativity, 1922) that “vacuum light is propagated with the velocity c, at least with respect to a definite inertial system.” Now let’s look at the same picture from the standpoint of Hertzian theory. As before, our observer is at rest in the inertial laboratory, but we note that the light path contains a number of mirrors, optical fibers, light pipes, or other tangible objects in rotary motion that may be viewed as “detectors” or absorber-re
|
||
emitters of the light. A typical one of these, moving at velocity vGd in the laboratory, will formally convect the light traveling with it, in accordance with Eq. (2.24)—imparting relative to the labora
|
||
tory a phase speed ( ) d
|
||
u = ±c + k k ⋅ v
|
||
G G . However, Potier’s principle applies (given that we conceive of this as a fixed endpoint problem, owing to the fact that the inertial observer’s field points, source and sink, are indeed fixed in the lab, regardless of detector motion) and we are forced to conclude that only the u = ±c part of this is observable, as in Maxwell’s theory. Consequently, with ef
|
||
|
||
|
||
40 Old Physics for New: a worldview alternative
|
||
fective speed c in the laboratory, we are back to the classical travel distance Δs effect, previously discussed, as the only first-order observable influence on the fringes. Such reasoning predicts outcome (b), the Sagnac effect. The latter, then, is disqualified as a means of deciding between Maxwell and Hertz. If the rotating platform rider R attempts to use Hertzian theory he is also no better off than the Maxwellian observer, since for both the detector is at relative rest ( 0)
|
||
vd =
|
||
G and the two theories become equivalent. Thus both teach the lesson mentioned above, that optical problems, like mechanical ones, require analysis with respect to an inertial system for simplest description. The ring laser (often used as a frictionless gyroscope) differs in significant details from the Sagnac interferometer. In most cases no dithering is employed, and the observable quantity is not a fringe shift but the frequency of passage past a fixed observation point of moving “beats” of a standing-wave pattern set up in a fixed optical fiber, light pipe, or other light-conducting circuit, the whole together rotating as a rigid unit at angular frequency ω in an inertial system. The formula describing this phenomenon is df = 4AωλP , where A is the area of the optical circuit, P the optical (not physical) path length of its perimeter, λ the light wavelength, and df the frequency at which nodes of the moving “standing wave” beat pattern pass the observation station at any fixed point along the circuit. Because of the appearance of a geometry-dependent factor A/P in the formula, the observed df depends on the shape of the circuit ... but for all shapes and sizes the observed df is proportional to ω. This method has been used to improve (vastly) on the accuracy of the Michelson-Gale[2.15] measurement of the spin angular velocity of the earth (which employed a large-scale Sagnac interferometer). In the active form of the ring laser the cavity, generally a lowpressure gas-containing Pyrex tube, square or triangular in planar configuration, with high-reflectance mirrors at the corners, is caused to “lase” by electromagnetic excitation. Beats of the resulting standing wave pattern move at the frequency df past a detection station fixed with respect to the tube. A frequency-pulling phenomenon in such geometry is observed only for small enough df values, below a certain “lock-in threshold.” Generally, no attempt is made to combat this; but the threshold is made very low, even into the micro-Hertz range, by using nearly perfect mirrors to reduce backscatter. Although the experimental details differ
|
||
|
||
|
||
2. What to Do About It ... (the Hertzian Alternative) 41
|
||
between the ring laser and the classic Sagnac interferometer, the gist of the analysis is the same. Hertzian theory, as above, is predictively equivalent to Maxwellian theory. Different versions of the Sagnac experiment have been carried out in which the light source rotates with the rotating platform or rests in the laboratory. Similar fringe shifts are observed in the two cases. This confirms that source motion is irrelevant, just as the relativists maintain. Further variants of the Sagnac idea include fiber-optic gyroscopes, fiber-optic conveyors, etc. A wealth of reference material is accessible on the Internet. The use of fiber optics allows easy exploitation of Sagnac’s recognition of the arbitrariness of optical circuit shape. Thus Wang et al.[2.14] have distorted the circuit shape into linearity on most of the circuit, so that the fiber motion can be made mainly inertial, parallel and anti-parallel to light propagation direction. These workers have demonstrated interference fringe shifts of counter-propagating light beams in such fiber optic circuits, major portions of which move uniformly and rectilinearly in the laboratory. Experiments of this kind are in a way analogous to the Faraday experiment in which a portion of an electrical circuit was moved in a magnetic field. As in that case, they defy conventional SRT analysis because of lack of an inertial frame in which the circuit as a whole can be considered to be at rest. Wang’s reported
|
||
empirical finding was an optical fringe shift Δφ= 4πv ⋅ dG cλ
|
||
GA , which is a linear expression of the Sagnac formula in differential form. Wang was under the impression that his observations (of fringe shifts in optical circuits consisting mainly of inertial portions, with translatory motions instead of rotary) refuted Einstein’s second postulate of light-speed constancy, inasmuch as a shift must correspond to equal travel distances of countercirculating light beams in unequal times (implying light speed non-constancy in inertial systems). But this can doubtless be countered by the observation that fringe formation requires circuit completion, and the circuit as a whole is not at rest in any single inertial system. When the Lorentz transformation event calculus is applied to counter-moving inertial systems, it will doubtless turn out that the first-order situation is saved for SRT by what amounts to a disguised version of the classical “travel distance” argument already rehearsed. Still, although SRT may not be refuted, the physical motions in Wang’s experiment are essentially inertial, and one becomes painfully aware of the possi
|
||
|
||
|
||
42 Old Physics for New: a worldview alternative
|
||
bility of other, less contrived, explanations of the observations. Indeed, as Stedman[2.16] remarks with a straight face, “it is now generally recognized that the prediction of [the Sagnac formula] is remarkably robust to the assumed theoretical framework .... ”
|
||
2.8 A bit of GPS evidence
|
||
The Global Positioning System (GPS) provides further evidence related to the Sagnac effect and to other topics that will interest us. We shall have more to say about it in subsequent chapters. Dr. Ron R. Hatch, a GPS expert, sent me the following e-mail on 24 August, 2005:
|
||
Using the GPS system and the clocks set to run at a common earth surface time ... The range to the GPS satellites is computed from the transit time from satellite to ground. The transit time is multiplied by the speed of light to get the range. But the speed of light has to be adjusted by the component of the receiver velocity (including the earth spin) away or toward the satellite. Thus, the speed used is (c + v) and (c—v). They call it a Sagnac effect even though the deviation from a straight line during the transit time is on the order of 10–11 meters. I have argued the circular path has nothing to do with it and in the latest GPS ICD they admit that any receiver motion in addition to the rotation must be used in the adjustment.
|
||
It is to be noted that the velocity “v” used in the GPS calculations just mentioned is referred to an inertial system S moving with the center of a non-rotating earth. In that system the firstorder Hertzian phase velocity of light, Eq. (2.24), is
|
||
( )d d
|
||
u = ±c + k k ⋅ v → c ± v
|
||
G G , where vGd is the velocity of a detector at rest or in motion on or near the surface of the spinning earth. The field point is at rest in S, so Potier’s principle (applicable alike to one-way and two-way paths with fixed endpoints) again rules out any observable distinction between Hertz’s and Maxwell’s theories. In both theories we are left with a distance effect Δs , as in the Sagnac effect ... although in this case because of linearity it might more aptly be termed a Wang effect. Again, the main lesson is that simplest descriptions require reference to a “sufficiently inertial” system. In all these applications, what shall be meant by the assertion that a reference system is sufficiently inertial? The only possible answer is that circumstances alter cases. In the original Sagnac
|
||
|
||
|
||
2. What to Do About It ... (the Hertzian Alternative) 43
|
||
experiment, to good enough approximation, the inertial reference system was the laboratory. In the Michelson-Gale version the sufficiently inertial system was one moving with the center of a nonrotating earth. The same is true for the GPS. In the case of stellar aberration (cf. Chapter 4), since the tiny aberration phenomenon exhibits an annual period, a more rigorous criterion of “inertiality” applies, requiring reference to the barycenter of the solar system or equivalent. For very long-term observations, wherein the solar system deviates appreciably from linear motion, the barycenter of the galaxy might be needed.
|
||
2.9 Chapter summary
|
||
Hertz’s version of first-order electromagnetic field theory, an invariant (under the Galilean transformation) covering theory of Maxwell’s non-invariant version, has been sketched here. It failed historically because of a false “ether” interpretation, but when that is corrected very few observable departures from Maxwell are predicted. Appearances indicative of a gross Hertzian “convection of light by the absorber” prove deceptive, in that such putatively acausal effects are shown by Potier’s principle (a consequence of Fermat’s principle) to be unobservable at first order in somewhat the same sense that “ether wind” is unobservable. (That is, unobservable in the space domain, as by fringe shifts; the time domain being another matter, as yet unprobed in the laboratory.) The Sagnac effect, in its various guises—because of Potier’s principle—provides no evidence to decide between the Maxwell and Hertz pictures of light propagation. The issue of covariant (Maxwell) vs. invariant (Hertz) formulations of field theory, while not resolvable by ordinary laboratory observations, is settled on the side of theory through the intervention of quantum considerations: Invariant (single detector) theory handles both weak-field and strong-field cases, whereas covariant (multi-detector) theory works only in the strong-field limit. Covariance is thus discredited on its home ground, electromagnetism. Consequently, the historic favor of “discreditation” is returned, a century later, from the Hertz (German) to the Maxwell (English) political camp. We find, however, that the Hertzian wave equation, taken in its original context, fails to describe stellar aberration. For that, the inadequacy of a first-order formulation is responsible. A “neo-Hertzian” higher-order ap
|
||
|
||
|
||
44 Old Physics for New: a worldview alternative
|
||
proximation, to be developed in the next chapter, will be found to do better.
|
||
References for Chapter 2
|
||
[2.1] H. von Helmholtz, Borchart’s J. Math. 78, 273-324 (1874).
|
||
[2.2] A. I. Miller, Albert Einstein’s Special Theory of Relativity Emergence (1905) and Early Interpretation (1905-1911) (Addison-Wesley, Reading, MA, 1981). [2.3] J. A. Stratton, Electromagnetic Theory (McGraw-Hill, NY, 1941), App. II. [2.4] M. Abraham and R. Becker, The Classical Theory of Electricity and Magnetism (Blackie and Son, London, 1932), Vol. 1. [2.5] J. Dunning-Davies, Progress in Phys. 3, 48-50 (2005).
|
||
[2.6] H. R. Hertz, Electric Waves, translated by D. E. Jones (Dover, NY, 1962), Chap. 14. [2.7] A. A. Eichenwald, Ann. Phys. (Leipzig) 11, 1 (1903); ibid., 421. [2.8] E. E. N. Mascart, Ann. École Norm. 1, 157-214 (1872); ibid., 3, 157-214 (1874). [2.9] R. G. Newburgh, Isis 65, 379 (1974). [2.10] R. G. Newburgh and O. Costa de Beauregard, Am. J. Phys. 43, 528 (1975). [2.11] T. E. Phipps, Jr., Heretical Verities: Mathematical Themes in Physical Description (Classic Non-fiction Library, Urbana, 1986). [2.12] A. Potier, J. de Physique (Paris) 3, 201 (1874).
|
||
[2.13] A. G. Kelly, Challenging Modern Physics—Questioning Einstein’s Relativity Theories (BrownWalker, Boca Raton, 2005). Also A. G. Kelly, “Synchronization of Clock Stations and the Sagnac effect,” in Open Questions in Relativistic Physics, F. Selleri, ed. (Apeiron, Montreal, 1998). [2.14] R. Wang, Y. Zheng, and A. Yao, Phys. Rev. Ltrs. 93, No. 14, 143901 (2004); also R. Wang, Y. Zheng, A. Yao, and D. Langley, Phys. Ltrs. A, 312, 7-10 (2003) and R. Wang, Galilean Electrodynamics 16, No. 2, 23-30 (2005). [2.15] A. A. Michelson and H. G. Gale, Astroph. J. 61, 137-145 (1925). [2.16] G. E. Stedman, Rep. Prog. Phys. 60, 615-688 (1997).
|
||
|
||
|
||
Old Physics for New: a worldview alternative to Einstein’s relativity theory 45 Thomas E. Phipps, Jr. (Montreal: Apeiron 2006)
|
||
... the supreme goal of all theory is to make the irreducible basical elements as simple and as few as possible without having to surrender the adequate representation of a single datum of experience.
|
||
—Albert Einstein, On the Method of Theoretical Physics (1933)
|
||
Chapter 3
|
||
Higher-order Electrodynamics
|
||
... (the neo-Hertzian Alterna
|
||
tive)
|
||
3.1 The higher-order kinematic invariants
|
||
n the last chapter it seemed that an operationalist or instrumentalist philosophy could contribute to clear thinking by forcing us to examine the details of our measurement procedures—if only in Gedanken terms. We found that first-order electromagnetic theory could benefit immensely from applying such intellectual discipline. For instance, the reasoned contemplation of how measurements would have to be made ruled out spacetime symmetry and covariance (through consideration in the ultimate weak-field limit of the single quantum competed-for by a plurality of macro-instruments) in favor of true invariance. Let us see if a similar philosophy can guide us in the labyrinth of higher-order approximation. The leitmotif of instrumentalism is the question, how do we measure whatever it is we recklessly conceptualize in our theorizing? Pursuit of that inquiry proves to be a relatively painless way of disciplining our idealizations. The young Einstein took the lead in asking that question about one of the most fundamental of physical descriptors, time. His answer hinged upon recognition of a crucial distinction between (a) inertial “frame time” t, measured
|
||
I
|
||
|
||
|
||
46 Old Physics for New: a worldview alternative
|
||
by a spatially extended set of clocks at rest in that inertial frame and subject to a specified (“Einstein”) synchronization convention, and (b) “proper time” τ of a particle in an arbitrary state of motion, measured by a single co-moving clock. In my opinion Einstein’s most enduring and important contribution to relativistic physics was his definition relating the differentials of these two types of “time;” namely,
|
||
2 2 22
|
||
2 2 2 2.
|
||
d dt dr c invariant
|
||
dr dx dy dz
|
||
τ≡ − =
|
||
= + + (3.1)
|
||
It is understood that all differentials here refer to a pair of events occurring successively on the trajectory of a single (classical or macro) particle, as measured by instruments at rest in an inertial frame in the case of dr,dt , and by a clock at rest with respect to the particle in the case of dτ. It is also understood that on this trajectory “other things” (such as gravity) are “equal.” This definition was actually a leap of inspiration, not a strict deduction from observation. It was, however, a most grand and far-reaching leap—and one for which I can only express admiration ... for it has been copiously confirmed in its implications by experiments of the sort for which I profess esteem. Indeed, I find it tempting to make a further leap and promote (3.1) to the status of fact. This is not because I think such a course logically defensible, nor acceptable as methodology, but simply because adopting it here proves to be a great expository convenience, and I am a lazy expositor. So I hope for the reader’s indulgence; given which, (3.1) provides us with half of the “facts” we shall need to get started in developing higher-order electromagnetic theory. In other words, we define or postulate the higher-order timelike invariant of physics to be τ and seek no further justification for this choice. Why should one have any confidence in this? The instrumentalist answer is: because we have an instrument that can measure this “proper time” τ. The instrument is, of course, the single clock we have alluded to, the one in an arbitrary state of motion and present at each of the two events bounding the interval dτ. Its readings are irrefragable fact, agreed on by all observers (hence “invariant” under aliasing transformations)unlike the non-invariant t, which depends on the state of motion of a plurality of co-moving clocks. Still, once we adopt the Einstein convention (discussed in all the relativity texts, equivalent to setting distant co-moving clocks by light signals of speed ex
|
||
|
||
|
||
3. Higher-order Electrodynamics ... (the neo-Hertzian Alternative) 47
|
||
trapolated to infinity—see Appendix), t becomes a useful parameter for linking up any subsequent higher-order refinements of our first-order physics—for, as will be discussed later in more detail, its differential (unlike that of proper time) is exact, hence best suited to “coordinate” representations. In a given inertial system such a t specifies the distant simultaneity on which depends Newtonian physics, including Newton’s third law, etc.; and t is also the timelike parameter appearing in all classical theories of electromagnetism. The last three chapters of this book will be devoted to developing a slightly different kind of “t”—a “collective” variant of frame time that will benefit from the virtues of frame time for many-body description while eliminating many of its restrictions and allowing it to acquire invariance properties of its own. Without apology, then, we shall identify the t of Eq. (3.1) with the t of our previous chapters, and shall assume that τ in (3.1) refers to the invariant proper time of the (point) field detector. That is, in the context of higher-order electromagnetic field theory, we consider d
|
||
τ =τ , but here and elsewhere may sometimes omit the subscript “d” denoting the detector. The latter is located at position ( )
|
||
rG = rG t , measured with respect to a given inertial system S, and is in an arbitrary state of motion relative to S specified by
|
||
()
|
||
dd
|
||
dr d dr
|
||
v v t dt dt d
|
||
τ τ
|
||
= ==
|
||
GG
|
||
G G . (3.2)
|
||
From (3.1) we have in general
|
||
( )2 2 2 2 1
|
||
11
|
||
d dr dt c v c
|
||
dt
|
||
τ γ
|
||
= − = − ≡ , (3.3a)
|
||
or in particular for the proper time of the field detector clockparticle
|
||
22
|
||
1
|
||
1
|
||
d
|
||
dd
|
||
dt
|
||
d vc
|
||
γ
|
||
τ = = − . (3.3b)
|
||
One can also define a “proper velocity” of the field detector,
|
||
d dd dd
|
||
dr dt dr
|
||
Vv
|
||
d d dt γ
|
||
ττ
|
||
≡= =
|
||
GG
|
||
G G , (3.4)
|
||
relative to S, and can obtain from (3.1) a useful general operator relationship
|
||
d dt d d d d dt dt
|
||
γ
|
||
ττ
|
||
= = . (3.5)
|
||
|
||
|
||
48 Old Physics for New: a worldview alternative
|
||
The proper velocity in (3.4) is “invariant” in the sense that all observers in whatever state of motion will agree on its numerical
|
||
value, provided they also measure the event separation drG in
|
||
some agreed inertial system S. Unless drG is attached to a material object, it does not, of course, possess invariance in the sense that different inertial observers assign it equal numerical values with
|
||
reference to their own frames ( vGd and VGd both being frame dependent for the description of pairs of independent physical events). Pitfalls of analogy: We pause to remark in connection with (3.1) that Einstein took another closely-related leap of genius, with disastrous consequences. He and Minkowski jumped to the conclusion, on the basis of “spacetime symmetry,” that not only was dτ the timelike invariant of physics, but that the analogous space-dimensioned interval,
|
||
dσ = −c2dτ2 = icdτ = invariant (?) , (3.6)
|
||
was the spacelike invariant of physics. To an instrumentalist, this has no plausibility whatever, because there is no invariant way to measure such a quantity, no “space clock” to give it operational meaning. Its invention (out of whole cloth of the mind, without encouragement from any objective aspect of the physical world) is strictly an artefact of spacetime symmetry—an alleged “symmetry” itself created out of nothing but a parametric deficiency of Maxwell’s equations—a symmetry which we have seen in Chapter 2 to be physically untenable where adequate account is taken of Faraday’s observations of induction. Intelligent beings from outer space, unacquainted with Maxwell’s equations, would be baffled by the concept of spacetime symmetry except as a possible aspect of religion, like the three-fold symmetry of the Trinity. In short, Eq. (3.6) asserts a physical symmetry that is not merely imaginary but in all likelihood counterfactual. Let us try to specify the trouble a little more closely. Operationalist critique: Although I am no philosopher, let me speak here for the “discredited” viewpoint associated with what has been called operationalism. I have been able to discern one and only one way, consistently with the ideology of SRT, to give an operational definition of the spacelike interval defined by Eq. (3.6), and this is to make a Lorentz transformation to the inertial system in which δt = 0 ; for in that uniquely preferred inertial sys
|
||
tem 2 2 2
|
||
dσ = δr − c δt becomes equal to δr . That can be measured,
|
||
|
||
|
||
3. Higher-order Electrodynamics ... (the neo-Hertzian Alternative) 49
|
||
using as “instrument” a meter stick, so in this case dσ is not only a real number but a measurable quantity. However, by this very token the operationalist must disavow dσ as a physical invariant—for the fact that only a single preferred reference system exists in which dσ can be operationally defined disqualifies that particular mathematical applicant for candidacy as physics. That is, the operations involved in the definition of any physical invariant should conform to a relativity principle: they should themselves be invariant, not dependent on frame of reference. That criterion is met by the on-trajectory invariant dτ, but not by the would-be trajectory-linking “invariant” dσ . The single clock’s usefulness for measuring the true physical invariant dτ holds consistently in every inertial system. The meter stick’s usefulness for measuring the alleged “invariant” dσ holds only in that preferred system in which it is at rest. The operationalist sees such an alleged dσ -invariance as nonsense. Physical descriptive primacy belongs not to the mental creation dσ but to the meter stick itself. That is ultimately what “invariance,” as well as measurement (which itself needs to be an invariant concept in order to effectuate a relativity principle), is about. This suggests that there is no viable alternative to using δr itself as the spacelike invariant of physics. It is spacelike and it is measurable. Would not the existence of another spacelike, but non-measurable, invariant be something considerably worse than a redundancy? If the much put-upon Herrgott made δr , why in the name of all vain labors would He also make dσ ? The idealized meter stick (in differing states of motion) exists invariantly in every frame. What it measures, length, exists (why not invariantly?) in every frame. Length is the very embodiment—the sine qua non—of the “spacelike.” Try thinking of space without it. Go on—boggle your mind ... That’s precisely what Einstein is asking you to do. The Einsteinians have been down in that boggle so long it looks like up to them. Defence of operationalism: Why should one employ operational criteria in winnowing physical theories? Consider the case of
|
||
dσ = 2 2 2
|
||
δr − c δt for δr > cδt , so that dσ is real. Given input numerical values of δr,δt , SRT predicts by paper and pencil calculation some real number dσ . In principle the input numerical values are obtained by measurement. These number inputs to the theory represent the outcomes of measurement operations we can perform with real instruments. The operationalist viewpoint is
|
||
|
||
|
||
50 Old Physics for New: a worldview alternative
|
||
that the corresponding output dσ , being also a real number, should also have an operational counterpart, a measurement conceivably performed with some instrument that might be quantified by that number. In other words, a sort of logical economy is implicit: It is displeasing to have real numbers floating around in a theory with nothing measurable “out there” to attach them to. Imaginary or complex numbers are another matter, but real numbers need real referents. Any real number input to or output from a physical theory ought to pay its freight by possessing a referent in the realm of real-number measurement. Bridgman’s operationalism was allegedly defeated by its inability to find such a referent for the quantum-mechanical wave function. That, as I understand it, is the cause and full extent of its “discrediting.” But the wave function Ψ is not a real number. When you connect it to reality via Ψ * Ψ , it submits to a probability interpretation, hence becomes measurable. Real numbers are c-numbers in Dirac’s terminology. We need to develop greater respect for c-numbers. Both their presence and their absence in a theory are crucial regulators of the theory’s relationship to reality. They are in fact its only link to reality. If I may differ from Bridgman on one point, “paper and pencil operations” detached from observability (which he introduced in desperation occasioned by the wave function issue) are exactly what must be avoided. They are the whole trouble. Uncontrolled paper and pencil operations permit the building of unlimited cloud castles of inference. Operationalism serves as a sobering antidote to this potential frivolity and its attendant trivializing of physics. (I judge febrile, overimaginative mathematizing to be trivializing ... and the instinctive ability to recognize this to be one of the marks of a physicist.) The “invariant” dσ exists only as a paper and pencil operation. That’s what’s wrong with it. Correspondingly, the strength of dτ lies in its measurability: there is an instrument that will do the job without reliance on chains of inference. That guarantees the existence of an aspect of reality corresponding to dτ and legitimizes the claim of dτ to “invariance.” dσ conspicuously lacks such legitimacy. In summary: concerning the invariants of kinematics, the physical distinction between (3.1) and (3.6) is that (3.1) asserts an instrumentally-measurable, objectively real relationship between two event points on the trajectory of a single particle, whereas (3.6) asserts a corresponding not-instrumentally-measurable rela
|
||
|
||
|
||
3. Higher-order Electrodynamics ... (the neo-Hertzian Alternative) 51
|
||
tionship between event points on two different particle trajectories, such that a light signal cannot connect those two events. When you think about the latter “world structural” proposition, you will see that it is truly spooky: Here is an alleged descriptor of “reality,” a number inaccessible to instrumental measurement and accessible only to calculation, such that even a light signal cannot check on it; yet we hold it, as an article of faith, to be more real than our yardstick, more real than the stone that Samuel Johnson kicked ... because the yardstick and the stone (saith Einstein, forsooth) are non-invariant, whereas the calculated interval (3.6) is claimed to be invariant. The claim rests on no firmer basis than beauty (a certain imagined symmetry—and an erroneous one at that ... ). A calculated number is thus more real than the stone that bruises our foot. And Einstein in later life disapproved of spooky things! Remember: Invariance is not something to kid about ... it is to be used sparingly, with reverence, to describe what is really “out there” in nature. Should not “really” mean verifiably? Did not Dr. Johnson have the right idea? So, we are free (and well-advised) to disregard (3.6) and all the evil empire built upon it, and to postulate a different higherorder spacelike invariant of physics. What shall we choose? All the world is at our disposal. The simplest possibility, already adduced, is that object length (measured as Euclidean separation distance of endpoints) is the higher-order invariant. Here is the object. It is part of our experience—that which it is theory’s job to describe. The object is thus as “real” as our experience, which is to say as real as we are. If it is real, there should be in our theory an invariant to match it. Unless there is good reason to seek something more sophisticated, it makes sense to try that first. Loose cannons, loose realities adrift without invariant theoretical counterparts—like loose invariants without counterparts in reality—are surely to be avoided in sound theory. The operationalist or instrumentalist will point to a meter stick as a perfectly good instrument for measuring length, and I know of nothing better, although a yardstick is just as good. We have educated our youth to higher things, to be sure, but the scorn of youth is no harder to bear than that of age and century-long tradition. (All right ... meter sticks are affected by temperature and light waves aren’t, so no modern experimentalist would dream of using a meter stick for accurate measurement of object length. But he would use some instrument ... and my “meter stick” is in any case a Ge
|
||
|
||
|
||
52 Old Physics for New: a worldview alternative
|
||
danken one immune to mundane perturbations. The secret of doing physics lies in the finding of harmless idealizations—those that reveal more than they conceal. There is no formula for it. It is an art ... but also a matter of taste, guided by experience. For instance, I happen to like rigid bodies, as will be discussed in Chapter 4—they suit my taste. Relativists abhor them and cannot coexist in the same “world” with them.) Why, indeed, should we seek anything better than Euclidean length for our invariant? At this point vague rumblings are apt to be heard from some quasi-physicists about the Michelson-Morley experiment[3.1]—to the effect that length has to contract to prevent fringes from shifting, so length cannot be a higher-order invariant. The M-M experiment, to which we have already alluded, merely extends to higher order the first-order observations of Mascart[2.6] and others of the nineteenth century showing the physical validity of a relativity principle. Any mathematics that supports a relativity principle is supported by M-M. And we saw that Hertz’s invariant mathematics did a perfectly good job of that at first order. In order to accommodate M-M automatically at higher order, all we have to do is find a higher-order invariant counterpart of Hertz’s mathematics. Indeed, we can accomplish that with almost any choice of spacelike invariant ... but the choice of object length is surely the first one to try, and we shall find no need to go beyond it. The proof will be in the pudding. In summary, then, the ratiocinations of this section have led us to postulate the invariants of kinematics to be:
|
||
2 22
|
||
:
|
||
:.
|
||
d dt dr c invariant
|
||
r invariant
|
||
τ
|
||
δ
|
||
=− =
|
||
=
|
||
Timelike
|
||
Spacelike (3.7)
|
||
These will be our postulates throughout this book—which means we accord them in effect a factlike status. In (3.7) we use a notation dr to denote spatial separation of two successive events (of timelike separation) on the trajectory of a single particle, and distinguish this from δr , which denotes spacelike separation of events marking, e.g., the two ends of an extended structure, a meter stick. [To be entirely consistent, we should be able to speak of the “distant simultaneity” of such events. The reader must take this on trust for now, as we defer to Chapter 6 our investigation of the absoluteness of simultaneity. Here it will suffice that δr = invariant be understood to mean that extended structures undergo no metric (measurable dimensional) changes, such as
|
||
|
||
|
||
3. Higher-order Electrodynamics ... (the neo-Hertzian Alternative) 53
|
||
Lorentz contraction, as a result of relative motion or of changes of environment, such as gravity potential. In the relativity language, which we use sparingly in this book, d in (3.7) denotes an onworldline event separation, whereas δ denotes a separation of events on two different worldlines.] Of course we have not said anything yet about what the physical “inertial transformations” are, mathematically, at higher order. (We consider them to be Galilean at first order.) Nor have we examined higher-order kinematics. For the moment we take a carefree attitude toward those problem areas, to be treated later, since electromagnetism is our immediate concern. Einstein put electromagnetism ahead of mechanics—making the latter conform to the former. This is a distorted judgment, but let us accept it for the moment and proceed.
|
||
3.2 Neo-Hertzian field equations
|
||
The first-order Hertzian field equations, (2.4), were patterned faithfully on the corresponding Maxwell equations, (1.1). The only difference, apart from source term adjustments (resulting from detector motion), was that Maxwell’s ∂ ∂t was replaced everywhere by d dt , where (in both cases) t is the frame time of an inertial system. To proceed beyond this first-order approximation, it is evident that we must retain the “total” aspect of Hertz’s total time derivative and make a further replacement in his firstorder equations of non-invariant frame time t by the higher-order invariant proper time τ of something. Proper time of what? Of the field detector, of course. We keep coming back to that ... “fields” being what field theory is about, and “fields” being what field detectors detect. Recall that we idealize the field detector as a mathematical point sufficiently massive to possess a trajectory, so there is no ambiguity about measurement of this τ. It is the time displayed by a co-moving idealized classical “clockparticle.” Such a clock or pocket-watch is just one more of the comoving instruments incorporated in the multi-purpose instrument composite we have idealized as the “point detector.” The motion need not be inertial but can be represented by any function of frame time t , which by (3.7) is related to τ by
|
||
22
|
||
τ = 1 − vd c dt
|
||
∫ , (3.8a)
|
||
|
||
|
||
54 Old Physics for New: a worldview alternative
|
||
where ( ) ( )
|
||
dd
|
||
v = v t = dr t dt is an arbitrary smooth function descriptive of detector velocity measured in an inertial frame. For .
|
||
d
|
||
v = v = const , this reduces to
|
||
t
|
||
τ = γ , (3.8b)
|
||
where γ is given by (3.3a). As just indicated, we carry forward to higher approximations the covering theory theme by formally replacing the non-invariant t wherever it appears in the Hertzian field equations (2.4) by the higher-order invariant parameter d
|
||
τ =τ , the proper time of the field detector. Thus the higher-order invariant field equations, here termed neo-Hertzian, are postulated to be
|
||
14
|
||
m
|
||
dE
|
||
Bj
|
||
cd c
|
||
π τ
|
||
∇× − =
|
||
GG
|
||
G
|
||
G (3.9a)
|
||
1 dB
|
||
E c dτ
|
||
∇× =−
|
||
G
|
||
G
|
||
G (3.9b)
|
||
∇ ⋅ BG = 0
|
||
G (3.9c)
|
||
∇ ⋅ EG = 4πρ
|
||
G . (3.9d)
|
||
This is a covering theory of the Hertzian first-order equations (2.4), because τ differs from t only at second order. Hence, as detector velocity slows relative to our inertial system S, τ becomes indistinguishable from t and (3.9) goes identically into (2.4) as a “covered” case. The latter is itself a covering theory of Maxwell’s equations, so all the observational-agreement credits of the accepted school-taught electromagnetic theory automatically accrue to the neo-Hertzian variant. The length-invariance Ansatz, Eq. (3.7), carried over from first
|
||
order, ensures that the grad operator ∇G remains invariant at higher orders, as at first order [Eq. (1.3a)]. It also ensures that our small detector volume does not change its dimensions in any observer’s view; hence, the enclosed-charge count stays the same and Eq. (2.7) remains true at higher orders: ( ) ()
|
||
ρ r ,τ ρ r ,τ .
|
||
′ ′ ′=
|
||
G G (3.10)
|
||
Similarly for Eq. (2.8),
|
||
( ) () ()
|
||
, ,,
|
||
ss
|
||
j r τ j rτ ρrτV
|
||
′ ′= −
|
||
GG G
|
||
G G G (3.11)
|
||
holds, where in formal analogy to Eq. (3.4) we have from (3.1) a proper relative velocity of inertial frame S′ with respect to S,
|
||
|
||
|
||
3. Higher-order Electrodynamics ... (the neo-Hertzian Alternative) 55
|
||
( )2
|
||
1
|
||
dr dt dr v
|
||
Vv
|
||
d d dt v c
|
||
γ
|
||
ττ
|
||
== = = −
|
||
GG G
|
||
G G . (3.12)
|
||
Here the proper time τ is that of any point at rest in inertial frame S′ . Evidently in this special case τ can be identified with
|
||
the frame-time t′ of S′ , and vG (a constant) is the frame-time velocity of S′ measured by instruments at rest in S. In order to proceed, we need to sketch a higher-order analog of the Galilean velocity composition law, Eq. (2.1). Consider our
|
||
point detector to be instantaneously located at position rGd′ in iner
|
||
tial frame S′ , which moves at uniform velocity vG relative to S. At all orders, its rigorously exact location in S is ′
|
||
=+
|
||
GGG
|
||
dd
|
||
r r vt , (3.13)
|
||
where t is the frame-time of S. Transposing and differentiating with respect to t, we find
|
||
′′
|
||
= − → =−
|
||
G GG G GG
|
||
dd dd
|
||
dr dr v v v v
|
||
dt dt , (3.14)
|
||
the implication being valid only at first order, where t t
|
||
′ = , in agreement with the Galilean transformation (1.2). Thus we confirm our first-order velocity addition result, (2.1). However, if instead we differentiate (3.13) with respect to detector proper time τ, we get
|
||
γ
|
||
τ τ ττ
|
||
′= − = −
|
||
GG G
|
||
GG
|
||
dd d
|
||
dr dr dr
|
||
dt
|
||
vv
|
||
d d d d , (3.15)
|
||
or, with (3.4) and (3.12),
|
||
′= −
|
||
GGG
|
||
dd
|
||
V V V , (3.16)
|
||
which is the proper-time form of our desired higher-order velocity composition law. In the above calculations we have dropped the notational distinction between δr and dr, since it is evident that spatial-increment lengths are meant in each case with reference to a particular inertial system. Finally, nothing in our discussion limits the detector to immobility in S′ . The result (3.16) therefore remains valid if arbitrary detector motion ( )
|
||
Vd τ
|
||
G in S is described in S′ by ( )
|
||
′ τ′
|
||
G
|
||
Vd for instantaneous velocities. (Keep in
|
||
mind that VG is a constant for inertial motion.) Applying (3.12) in S and in S′ we can write out (3.16) in terms of frame-time velocities:
|
||
|
||
|
||
56 Old Physics for New: a worldview alternative
|
||
( ) ( ) ()
|
||
′= − ′
|
||
−−−
|
||
G GG
|
||
2 22
|
||
111
|
||
dd
|
||
dd
|
||
v vv
|
||
v c v c vc
|
||
. (3.17)
|
||
In the simplest case of collinear velocities, say, a particle moving with speed d
|
||
w = v c with respect to S′ , speed d
|
||
u = v c with respect to S, and S′ moving with speed z = v c with respect to S, all moving along the same line, (3.17) becomes
|
||
2 22
|
||
1 11
|
||
wuz
|
||
wuz
|
||
=−
|
||
− − − (3.18)
|
||
for w, u, z each restricted to the open interval between –1 and 1, to avoid non-physical singularities. [This means, as in SRT, that all frame-time velocities are limited in magnitude to c as an unreachable upper limit (pace tachyons).] The quadratic (3.18) can be solved for w to yield
|
||
w = ± N D , (3.19)
|
||
where
|
||
2 2 22 2 2
|
||
2 2 22
|
||
21 1 2
|
||
2 1 1 1.
|
||
N uz u z u z u z
|
||
D uz u z u z
|
||
= − −+ −−
|
||
= − −+ −
|
||
When this w-function is plotted, the choice of roots becomes obvious. Since the non-collinear case is still more complicated, it is apparent that this law of velocity composition, based on proper times, is not simply expressible in terms of frame-time velocities. Moreover, (3.17) does not describe the most general case of relative velocity of two arbitrarily-moving particles viewed in different inertial frames. It treats only the case of a single particle as viewed in two different inertial frames ... but it does so by means of two proper times—that of the particle moving arbitrarily with instantaneous speed u in S and that of any point at rest in S′ (the latter proper time being equivalent to the frame time t′ in S′ ). We defer further discussion of topics related to kinematics to later chapters, as this would distract us here. For the present all we need to retain is that analysis in terms of proper times entails great complexity. Because our velocity composition laws, (2.1) at first order and (3.16) at higher orders (actually, simple “addition” laws), conveniently have identically the same form, all formal aspects of the invariance proof for Hertzian field equations hold for the neoHertzian ones. The invariance proof for the higher-order case therefore need not be repeated. We need only pause to note that
|
||
|
||
|
||
3. Higher-order Electrodynamics ... (the neo-Hertzian Alternative) 57
|
||
the higher-order “inertial” transformations under which this invariance holds, which might be termed “neo-Galilean,” are
|
||
r r Vτ, τ' τ
|
||
′= − =
|
||
G
|
||
G G . (3.20)
|
||
Here τ τ
|
||
′ = has the trivial meaning that the same clock or particle shows the same elapsed times as viewed by differently-moving observers. The general relation
|
||
()
|
||
Vdτ =γv dt γ = vdt
|
||
G G G (3.21a)
|
||
[which follows from (3.12) and (3.3a)] can be integrated in the
|
||
case of constant vG (implying constant VG ) to
|
||
Vτ = vt
|
||
G G . (3.21b) Hence the spatial part of the neo-Galilean transformation (3.20) can be written in traditional Galilean form, r r vt
|
||
′= −
|
||
G G G , (3.22) while the time part can be written [by identifying τ′ as t′ for the case of the detector at rest in S′ and applying (3.3a) with the rec
|
||
ognition that vG is constant] as
|
||
t t /γ
|
||
′ = (3.23)
|
||
in frame-time (not explicitly invariant) form. This expresses “relativistic” time dilation without the clock-phase dependence on distance claimed by SRT. Needless to say, it is the time dilation aspect that is confirmed by experiment. (It may be added that Franco Selleri[3.2] has argued in numerous publications for (3.23)which allows for absolute simultaneity—as the physically correct representation of time dilation. However, he also insists on the Lorentz contraction of spatial quantities ... and on this we disagree.)
|
||
3.3 Neo-Hertzian wave equation
|
||
The formal similarity of the higher-order field equations (3.9) to the Hertzian ones, (2.4), allows us to set down immediately the neo-Hertzian wave equation by analogy with (2.18),
|
||
2 2
|
||
22
|
||
10
|
||
dE
|
||
E c dτ
|
||
∇− =
|
||
G
|
||
G . (3.24)
|
||
This equation has been treated previously.[2.11, 3.3] Considering first
|
||
the case that vGd (hence γd ) is constant, we shall give two different derivations of its solution.
|
||
Solution using invariant forms. The invariances E E
|
||
′=
|
||
G G of Eq. (2.6) and c c
|
||
′ = of Eq. (2.13) may be assumed valid at higher or
|
||
|
||
|
||
58 Old Physics for New: a worldview alternative
|
||
ders. Hence the wave equation (3.24) is of manifestly invariant form. It should therefore be capable of solution by means employing only invariant quantities. Calling to mind the first-order wave equation (2.18), namely,
|
||
2 2
|
||
22
|
||
10
|
||
dE
|
||
E c dt
|
||
∇− =
|
||
G
|
||
G,
|
||
and its solution (2.24), namely,
|
||
d
|
||
k
|
||
u cv
|
||
kk
|
||
=ω= ± + ⋅
|
||
G G,
|
||
we have only to introduce higher-order counterparts u*,ω* of u,ω and to recognize [for example by comparing (3.12) with
|
||
(2.8)] that VG is the higher-order counterpart of vG , to set down at once by formal similarity the solution of (3.24) as
|
||
*
|
||
*d
|
||
k
|
||
u cV
|
||
kk
|
||
=ω =± + ⋅
|
||
G G . (3.25)
|
||
Here we have taken account of k* = k
|
||
G G , which reflects the fact that
|
||
spatial vector quantities such as kG behave in a classical way (in consequence of length invariance, implying Euclidean geometry). In order to “translate” (3.25) into a more useful frame-dependent form, we recognize that ω*τ =ωt ; (3.26) that is, this dimensionless product represents the same pure number, whether expressed in terms of invariant or framedependent quantities (there being no juncture in the smooth transition from lower- to higher-order description at which a numerical discontinuity can occur). Then *
|
||
d d
|
||
t dt d
|
||
ωγ
|
||
ωτ τ
|
||
= = = (3.27)
|
||
from (3.3b). Hence
|
||
*
|
||
*d
|
||
d
|
||
uu
|
||
kk
|
||
ωγ
|
||
= ω = = γ , (3.28)
|
||
and Eq. (3.25) can be written with the aid of (3.4) as
|
||
*d
|
||
d dd
|
||
k
|
||
uu c v
|
||
kk
|
||
ωγ
|
||
= γ = = ± + ⋅γ
|
||
G G . (3.29)
|
||
Finally, dividing through by γd , we arrive at
|
||
22
|
||
dd
|
||
k
|
||
u cv v
|
||
kk
|
||
=ω= ± − + ⋅
|
||
G G , (3.30)
|
||
|
||
|
||
3. Higher-order Electrodynamics ... (the neo-Hertzian Alternative) 59
|
||
which is our higher-order wave equation solution expressed in frame-measurable form, for easiest comparison with observation. We shall make much use of this result in what follows.
|
||
Solution using frame-dependent forms. This treatment is more tedious but also more straightforward. As above, we simplify by
|
||
considering vGd (hence γd ) to be a constant. Following our method of solving the Hertzian equation, we seek a solution of the form ( )
|
||
EG = EG p , where
|
||
p = k ⋅ r −ωt
|
||
G G , (3.31)
|
||
the same as (2.19). Then, as in Eq. (2.20), we have
|
||
() () ()
|
||
222 22 2 22
|
||
Ep Ep kE p
|
||
xyz
|
||
⎛⎞
|
||
∂ ∂ ∂ ′′
|
||
∇ = ++ =
|
||
⎜⎟
|
||
∂∂∂
|
||
⎝⎠
|
||
G G G . (3.32)
|
||
Applying (3.5) with d
|
||
τ =τ and (1.9), we find for constant γd
|
||
() ()
|
||
( ) ( ) ()
|
||
( )( ) ( )
|
||
2
|
||
2 2 2
|
||
22
|
||
2 2
|
||
2
|
||
2
|
||
22 2
|
||
2
|
||
2.
|
||
dd
|
||
dd
|
||
d d d dd
|
||
d Ep v Ep
|
||
dt
|
||
v v Ep
|
||
tt
|
||
vk vk E vk E
|
||
γ
|
||
τ
|
||
γ
|
||
γω ω γω
|
||
⎛∂ ⎞
|
||
= + ⋅∇
|
||
⎜⎟
|
||
⎝∂ ⎠
|
||
⎡⎤
|
||
∂∂
|
||
= + ⋅∇ + ⋅∇
|
||
⎢⎥
|
||
∂∂
|
||
⎣⎦
|
||
⎡ ⎤⎡ ⎤
|
||
′′ ′′
|
||
= − ⋅+ ⋅ = − ⋅
|
||
⎢ ⎥⎣ ⎦
|
||
⎣⎦
|
||
GG
|
||
G
|
||
G
|
||
G
|
||
GG
|
||
GG
|
||
GG G
|
||
GG
|
||
GG
|
||
(3.33)
|
||
From (3.24), (3.32) and (3.33) it follows that
|
||
( ) ()
|
||
22
|
||
2
|
||
20
|
||
d d
|
||
k vk Ep
|
||
c
|
||
γω
|
||
⎧⎫
|
||
⎡ ⎤ ′′
|
||
− −⋅ =
|
||
⎨⎬
|
||
⎣⎦
|
||
⎩⎭
|
||
GG
|
||
G . (3.34)
|
||
From the vanishing of the coefficient of EG′′ we have
|
||
dd
|
||
ck = γ ω− v ⋅ kG
|
||
G , (3.35)
|
||
or, dividing by d
|
||
kγ and defining a frame-time phase velocity u as before, we get
|
||
22
|
||
dd
|
||
k
|
||
u cv v
|
||
kk
|
||
=ω= ± − + ⋅
|
||
G G , (3.36)
|
||
in agreement with our previous solution (3.30). Finally, it is of possible interest to remove the above restric
|
||
tion to constant vGd .
|
||
Solution for arbitrary ( )
|
||
dd
|
||
v =v t
|
||
G G . In this more general case[3.4] we look for a d’Alembertian solution of the form ( )
|
||
EG = EG p where ()
|
||
p= k⋅r − f t
|
||
G G , (3.37)
|
||
|
||
|
||
60 Old Physics for New: a worldview alternative
|
||
kG being constant as before and f an arbitrary function that generalizes the previous ωt . The spatial part, (3.32), holds as before. The time part is
|
||
() ()
|
||
2
|
||
2 dd d
|
||
d dd
|
||
Ep Ep
|
||
d dt dt
|
||
γγ
|
||
τ=
|
||
G G . (3.38)
|
||
In order to evaluate this, we need to know dp dt . From (1.9)
|
||
()
|
||
()
|
||
,
|
||
d x yz
|
||
dx x dy y dz z d
|
||
dp p v xk yk zk f t
|
||
dt t
|
||
f vk vk vk f v k
|
||
⎛∂ ⎞
|
||
= = + ⋅∇ + + −
|
||
⎜⎟
|
||
⎝∂ ⎠
|
||
=− + + + =− + ⋅
|
||
G
|
||
G
|
||
|
||
G
|
||
G
|
||
(3.39)
|
||
where f ≡ df dt . From (3.38) we obtain
|
||
() 2 22
|
||
d d dd d d
|
||
d d E p pE pE p E
|
||
dt dt
|
||
γ γ γγ γ γ
|
||
⎛ ⎞ ′ ′ ′′
|
||
= ++
|
||
⎜⎟
|
||
⎝⎠
|
||
G GG G
|
||
, (3.40)
|
||
with d d
|
||
γ ≡ dγ dt
|
||
. Use of this and (3.32) in the wave equation (3.24) yields
|
||
22 2 2
|
||
22 0
|
||
dd
|
||
d
|
||
d
|
||
pp
|
||
k EpE
|
||
cc
|
||
γγ
|
||
γ
|
||
γ
|
||
⎛⎞
|
||
⎛ ⎞ ′′ ′
|
||
− −+ =
|
||
⎜⎟
|
||
⎜⎟
|
||
⎝ ⎠⎝⎠
|
||
GG
|
||
|
||
. (3.41)
|
||
In order for a d’Alembertian solution to exist it is necessary that
|
||
the coefficients of both EG′ and EG′′ vanish. For EG′ this means
|
||
2
|
||
2
|
||
1d
|
||
d
|
||
d p dp
|
||
d
|
||
dt dt dt
|
||
γ
|
||
γ
|
||
⎛⎞
|
||
=− ⎜ ⎟
|
||
⎝ ⎠.
|
||
Let y ≡ dp dt . Then
|
||
1d d
|
||
dd
|
||
dy dy
|
||
dd
|
||
y
|
||
dt dt y
|
||
γγ
|
||
γγ
|
||
⎛⎞
|
||
=− → =−
|
||
⎜⎟
|
||
⎝ ⎠ . (3.42)
|
||
The solution of this is d
|
||
y = b γ , with b an integration constant. With (3.39) this yields
|
||
d d
|
||
dp b
|
||
y f vk
|
||
dt γ
|
||
= =− + ⋅G=
|
||
G
|
||
. (3.43)
|
||
The vanishing of the coefficient of EG′′ in (3.41) implies that
|
||
d d
|
||
dp
|
||
ck f v k
|
||
γ = dt = − + ⋅ G
|
||
G
|
||
. (3.44)
|
||
Taking the absolute value of (3.43) and comparing with (3.44), we see that b = ck , i.e., b = ±ck . (3.45)
|
||
|
||
|
||
3. Higher-order Electrodynamics ... (the neo-Hertzian Alternative) 61
|
||
We assume this condition, which assures simultaneous vanishing of the coefficients of E′, E′′
|
||
G G in (3.41) to be satisfied. Eq. (3.43) then implies
|
||
d d
|
||
ck
|
||
f vk
|
||
γ
|
||
=± + ⋅G
|
||
G
|
||
, (3.46)
|
||
whence
|
||
() 0 0
|
||
tt
|
||
d d
|
||
ck
|
||
f t fdt v k dt
|
||
γ
|
||
⎛⎞
|
||
= = ± +⋅
|
||
⎜⎟
|
||
⎝⎠
|
||
∫∫ G
|
||
G
|
||
. (3.47)
|
||
(Here it will be recalled that vGd and γd are functions of t.) The d’Alembertian wave function argument p can always be written in the form k ⋅ r −ωt = k ⋅ r − ukt
|
||
GG
|
||
G G , which may be considered to define the phase velocity u. Comparing with (3.37), we see that ()
|
||
ukt = f t ; so
|
||
() 2 2
|
||
00
|
||
22
|
||
11
|
||
,
|
||
tt dd
|
||
dd
|
||
av av
|
||
ft
|
||
u c v dt v kdt
|
||
kt t kt
|
||
k
|
||
c v vk
|
||
= =± − + ⋅
|
||
=± − + ⋅
|
||
∫ ∫G
|
||
G
|
||
G
|
||
G (3.48)
|
||
where av denotes a time average over the interval zero to t.
|
||
Eq. (3.48) is our desired generalization of the previous results
|
||
(3.30) and (3.36), which were obtained for the special case of vGd constant. It will be observed that the form of the solution is not affected, beyond introduction of a time-averaging process over an interval of length t. What is this t? Here we need help from the physics. Two likely candidates suggest themselves: (1) It is the propagation time of the photon from emission to absorption. (2) It is the absorption time interval. The first of these seems ruled out by common sense, as some starlight (presumably, on the customary causal model) has been propagating since long before the Earth came into existence, and it seems infeasible to “average” over the motions of an absorber not yet in existence during part of the alleged averaging interval. More plausible is alternative (2), since it asks us to average only during an absorption process we know must be occurring because it constitutes the “field detection” our field theory is concerned with. Adopting this view, we can say that as a practical matter the interval zero to t can be considered so short (of the order of 18
|
||
10− second for single-photon absorption, or flash duration for a group of photons) as to be practically instantaneous. Thus the “averaging” is in general an
|
||
|
||
|
||
62 Old Physics for New: a worldview alternative
|
||
unnecessary refinement, and in treating problems of wave (radia
|
||
tion) detection we can view vGd in our formulas as having a constant instantaneous value associated with the moment of detection. However, this question should not be considered settled on the basis of mere ratiocination. It deserves to be kept open. The upshot is that all three of the above methods of solving the neo-Hertzian wave equation tentatively agree on a solution having phase velocity (propagation speed)
|
||
22
|
||
dd
|
||
k
|
||
u cv v
|
||
kk
|
||
=ω= ± − + ⋅
|
||
G G . (3.49)
|
||
Referring to Eq. (3.31), we see that the most general d’Alembertian solution of that equation is consequently
|
||
()
|
||
() ()
|
||
()
|
||
22 1
|
||
22
|
||
2,
|
||
dd
|
||
dd
|
||
E E kr k c v kv t
|
||
E kr k c v kv t
|
||
= ⋅+ − −⋅
|
||
+ ⋅− − +⋅
|
||
GG
|
||
GG G G
|
||
GG
|
||
G G G (3.50)
|
||
where 1 2
|
||
E ,E
|
||
G G are arbitrary vector functions. Thus the neoHertzian light speed u is not a constant except in the special (Maxwellian) case of a stationary detector, 0
|
||
vd = .
|
||
Our discussion of Potier’s principle in Chapter 2, establishing the unobservability (by ordinary laboratory techniques of interferometry, etc.) of an additive term of type d
|
||
kG ⋅ vG in the phase velocity, of course applies here as well. Because of this unobservability, little attention need be paid to the apparently “acausal” aspects of the description of light “propagation” given by (3.50). Still, there are many who will be disturbed by the implication that the emitter “knows” enough about the future absorption event to regulate photon speed. It is worth pausing to note once again that this confronts us directly with the quantum nature of basic electromagnetic processes—at least those observed in the far zone (“radiative”). Quantum mechanics is fundamentally acausal by the very nature of its pedigree, being based upon a formal Correspondence with a classical instantaction-at-a-distance form of mechanics. Moreover “propagation,” as we have repeatedly noted, is Everyman’s paradise of inference—being as far from direct operational verification as ever an “ether wind” was. Einstein chose to think about propagation in exactly the same plodding, causal way Maxwell did, but without the concrete justification of Maxwell’s picture of contiguous contacts within an ether to flesh-out a
|
||
|
||
|
||
3. Higher-order Electrodynamics ... (the neo-Hertzian Alternative) 63
|
||
physical picture of what was doing the propagating. So, Einstein has left us with mathematical vectors “propagating” (straight out of our minds) through physical space. Why such fancy-free mathematical creations should be fettered by further imaginings of etherless contiguous causal evolution, as if successive ghostly “contacts” operated upon “vectors” across space, is hard to trace to anything but cultural inertia of physical theorists or lack of roughage in their mental diet. In fact, trying to picture what is going on during propagation is identically the same thing as trying to picture what is going on in a quantum pure state—for the simple reason that the photon propagates in a pure state. The dangers of pictorialization are notorious, since they make up one of the central discovery themes of early twentieth-century physics. Of course, Einstein didn’t know about that in 1905, and never wanted to know about it. Can you blame him? He was the last causal thinker about the quantum world, except for ten million successors who carry on his tradition to this day by thinking causally about light propagation ... and being cocksure about it, at that, by virtue of their unwavering allegiance to Maxwell’s equations. Meanwhile, are there published premonitions of acausal formulations of electromagnetic theory, including non-standard views of “propagation”? Indeed, many such. Right off the bat we notice that Maxwell’s equations themselves specify no inherent preference between advanced and retarded descriptions of “propagation.” It is we, the users, who superpose that gratuitously by selecting retarded and discarding advanced solutions of the wave equation. That’s our privilege, to be sure, but let’s not be cocksure about it! As Oliver Cromwell wrote in 1650 to the General Assembly of the Church of Scotland, “I beseech you, in the bowels of Christ, think it possible you may be mistaken.” A lightsphere shrinking in acausal response to a detection event is operationally indistinguishable from a light-sphere expanding in causal response to an emission event. And, of course, lightspheres have little to do with the quantum reality, anyway—for all that Huyghens was a great man. The ambiguity inherent in this situation allowed WheelerFeynman to propound their theory[3.5] of “the absorber as the mechanism of radiation.” This employed a balanced combination of half-advanced and half-retarded potentials, with an assumption of perfect absorption, such that no photon sets out on its
|
||
|
||
|
||
64 Old Physics for New: a worldview alternative
|
||
journey without being assured of an absorber to arrive at. This is another way of edging up on our present Hertzian suggestion that absorber motion can influence one-way light “propagation” speed. And, to assert the (causal) hypothesis that the universe is guaranteed to be so full of absorbers that no photon is ever orphaned (and energy is thus conserved), is surely a postulationally spendthrift way of achieving what is more efficiently accomplished by the (acausal) hypothesis that no photon is ever emitted without its absorption being pre-arranged. The latter requires us to feign no hypotheses about the “universe.” Fokker (I lack the reference, but it can be found in Wheeler and Feynman[3.5]) said, “The sun would not radiate if it were alone in space.” He had the right idea, perhaps. Another unconventional thinker about photons and their absorption was G. N. Lewis (who first proposed the name “photon” in 1926). Finally, it might be noted that SRT itself provides one prominent opening for unconventional views of light. The timelike interval of that theory, dτ, vanishes for any two event points connected by a light signal. This could be interpreted to mean that the emitting and absorbing atoms are in “virtual contact.” (If I am not mistaken, this, too, was G. N. Lewis’s idea and terminology.) That is, the photon ages by an amount zero during its “journey;” so the journey is in some sense of zero spatial length (if we take the photon’s word for it). If we don’t understand that, it may be merely because we are not photons. These matters are mentioned here only to bring out the point that spookiness in the description of light “propagation” is not a private innovation of this writer’s, but is in fact something of a hardy-perennial cottage industry ... or once was, before the great SRT mental ice age set in hard.
|
||
3.4 Phase invariance
|
||
Another implication of the neo-Hertzian wave equation concerns the transformation properties of phase: The first-order invariant relation (2.6) applies also at higher orders,
|
||
( ) ()
|
||
E p Ep
|
||
′ ′=
|
||
G G . (3.51)
|
||
This implies numerical phase equality, p p
|
||
′ = ; hence from (3.31)
|
||
k r ωt k r ωt
|
||
′ ′ ′′
|
||
⋅− = ⋅ −
|
||
GG
|
||
G G . (3.52)
|
||
|
||
|
||
3. Higher-order Electrodynamics ... (the neo-Hertzian Alternative) 65
|
||
This expresses the classical principle of phase invariance. For simplicity, consider the detector at rest in inertial system S′ ,
|
||
which moves with velocity vG with respect to S . Then vd = v
|
||
G G and the proper time τd of the detector is equal to the S′ frame time
|
||
d
|
||
t tγ
|
||
′ = , according to Eq. (3.23). Using this and Eq. (3.22) to eliminate the ( )
|
||
r′,t′
|
||
G variables, we obtain
|
||
( d) d
|
||
k r ωt k r v t ω t γ
|
||
′′
|
||
⋅− = ⋅ − −
|
||
GG
|
||
G GG
|
||
or
|
||
( )( )
|
||
dd
|
||
k k r k v ω ωγ t
|
||
′′ ′
|
||
− ⋅ =− ⋅ + −
|
||
GG G
|
||
G G . (3.53)
|
||
Since rG and t may vary independently, their coefficients must vanish. Hence we obtain the higher-order generalizations of (2.28) and (2.29):
|
||
Coeff. of rG : k k
|
||
G′ = G (3.54)
|
||
Coeff. of t: ( )
|
||
dd
|
||
ω γω kv
|
||
′ = − G ⋅ G . (3.55)
|
||
The second of these results describes the Doppler effect for source stationary in S and detector at rest in S′ . The first seems to describe aberration—and to do so incorrectly, by denying that light propagation vectors are directionally affected by inertial
|
||
transformations. Since such an effect of light kG -vector turning is a core teaching of SRT, and since the subject will turn out to have some subtleties, as well as considerable interest in its own right, we shall devote the next chapter to the topic of stellar aberration, basing our discussion there on our principal neo-Hertzian results, (3.49) and (3.54). For now, let us examine the Doppler effect.
|
||
3.5 Doppler effect
|
||
Recall that in deriving our Doppler result, Eq. (3.55), we assumed the detector to be at rest in S′ . Hence the S′ observer sees a motionless detector, ′ = 0
|
||
vd , and we have the Maxwellian case u c
|
||
′= and ω′ k′c kc
|
||
= = , in view of k k
|
||
′ = from (3.54). The “transmitted” frequency of the light source, 0
|
||
ω=ω , measured in the rest system S of the source, then obeys
|
||
1
|
||
0d d
|
||
ω ω′γ− k v
|
||
= + G ⋅ G , (3.56)
|
||
from (3.55); whence
|
||
12 2 00
|
||
1
|
||
1 12
|
||
d dd dd
|
||
v vv c cc
|
||
ω ωγ γ ω
|
||
−⎡ ⎤
|
||
⎛ ⎞ ⎛⎞
|
||
⎛⎞
|
||
′= + ≈ − + + +
|
||
⎢⎥
|
||
⎜⎟
|
||
⎜ ⎟ ⎜⎟
|
||
⎝⎠
|
||
⎝ ⎠ ⎝⎠
|
||
⎣⎦
|
||
A A A " , (3.57)
|
||
|
||
|
||
66 Old Physics for New: a worldview alternative
|
||
where ( ) ( )
|
||
dd
|
||
= kk⋅v v
|
||
GG
|
||
A is the cosine of the angle between the light propagation direction and the detector velocity measured in the source system S , and we have used d d d
|
||
k v kv v ω′ c
|
||
⋅= =
|
||
G G A A in (3.56). Clearly, to describe the stellar Doppler effect, it is necessary
|
||
to take d source sink
|
||
vv −
|
||
=
|
||
G G , the velocity of the telescope relative to the star—which is of course different for each stellar source. Eq. (3.57) is to be compared with a standard result of SRT for the Doppler effect [see Chapter 4, Eq. (4.2d)]; viz.,
|
||
2
|
||
00
|
||
1
|
||
1 12
|
||
d dd SRT d
|
||
v vv c cc
|
||
ω ωγ ω ⎡ ⎤
|
||
⎛ ⎞ ⎛⎞
|
||
′= − ≈ − + +
|
||
⎢⎥
|
||
⎜ ⎟ ⎜⎟
|
||
⎝ ⎠ ⎝⎠
|
||
⎣⎦
|
||
A A " . (3.58)
|
||
This has been discussed in Heretical Verities.[2.11] [Since vGd is detector velocity relative to source ( S′ relative to S ), the way of looking at it that employs source velocity relative to the detector must replace vd by source
|
||
−v in these formulas.] The neo-Hertzian result (3.57) agrees at first order with the SRT result (3.58), but disagrees at second order. Unfortunately, the disagreement is not easily tested. For the case A = 0 of transverse Doppler, the two formulas agree with each other and with the observationally confirmed time dilation, to second order, which is all that can be observed by methods such as those employed by Ives-Stillwell.[3.6] For the case A ≠ 0 the first-order term is non-vanishing and dominates over the very small second-order term. So the older experimental methods will not serve and no existing data can settle the issue. However, frequency measurements by modern techniques have become extremely precise, so it is not out of the question that a crucial experiment could be devised to allow a choice between (3.57) and (3.58). This has not been investigated ... but it offers the possibility of an independent empirical check on other claims in this book
|
||
3.6 Chapter summary
|
||
In this chapter we have set up the most obvious form of field theory based on Hertz’s theme of invariance, and have carried this one step beyond the first-order stage treated in Chapter 2 by substituting the invariant proper time of the field detector for noninvariant frame time. The resulting higher-order invariant electromagnetic theory, which includes the effects of time dilation, we have termed neo-Hertzian. On the spatial side we elected to leave well enough alone and try the simplest thing, length invariance
|
||
|
||
|
||
3. Higher-order Electrodynamics ... (the neo-Hertzian Alternative) 67
|
||
and Euclidean geometry. In the next chapter we shall see that this succeeds in describing stellar aberration where SRT (as its supporters are blissfully unaware) fails irremediably. In this connection we shall be led to propose an astronomical observation both feasible and crucial, in the sense that it should (if there is such a thing as a crucial experiment) settle for all time the issue between invariance and covariance. Here we have prepared the way by working out in full detail, by three different (but mutually consistent) methods of calculation, the solution of the neo-Hertzian wave equation. This tells us everything that classical field theory, amended to make it invariant under inertial transformations, has to say about the propagation of light.
|
||
References for Chapter 3
|
||
[3.1] A. A. Michelson and E. W. Morley, Am. J. Science 31, 377-386 (1886); ibid., 34, 333-345 (1887). [3.2] F. Selleri, Found. Phys. 26, 641-664 (1996); “Sagnac effect: end of the mystery,” in Relativity in Rotating Frames, G. Rizzi and M. L. Ruggiero, eds. (Kluwer, Dordrecht, 2004), pp. 57-77. [3.3] T. E. Phipps, Jr., “Hertzian Invariant Forms of Electromagnetism,” in Advanced Electromagnetism Foundations, Theory and Applications, T. W. Barrett and D. M. Grimes, eds. (World Scientific, Singapore, 1995). [3.4] T. E. Phipps, Jr., Apeiron 7, 76-82 (2000). [3.5] J. A. Wheeler and R. P. Feynman, Rev. Mod. Phys. 17, 157 (1945) and 21, 425 (1949). [3.6] H. E. Ives and G. R. Stilwell, J. Opt. Soc. Am. 28, 215 (1938); 31, 368 (1941).
|
||
|
||
|
||
|
||
|
||
Old Physics for New: a worldview alternative to Einstein’s relativity theory 69 Thomas E. Phipps, Jr. (Montreal: Apeiron 2006)
|
||
Having set ourselves the task to prove that the apparent irregularities of the five planets, the sun and moon can all be represented by means of uniform circular motions, because only such motions are appropriate to their divine nature ... we are entitled to regard the accomplishment of this task as the ultimate aim of mathematical science based on philosophy.
|
||
—Claudius Ptolemy, Almagest I
|
||
Chapter 4
|
||
Stellar Aberration
|
||
4.1 Appreciation of the phenomenon
|
||
t may seem uncultured to interrupt the oleaginous flow of theory by obtruding an aspect of the real world ... but physics is about such interruptions and even physicists cannot indefinitely postpone occasional contacts with physics. I adopt a disrespectful attitude toward the efforts of my fellow laborers in the vineyard of physical description because most have a side of them of which they are unaware—a side in urgent need of a thorn. And the thorn is precisely the subject matter of this chapter, stellar aberration (SA). Why should a physicist care about SA? (The great majority don’t.) First, because it is one of the few examples of genuine oneway propagation of light, so it just might have something to teach us about that interesting subject. More importantly, the facts of SA turn out to be essentially irreconcilable with special relativity theory (SRT), the rock on which the church of modern physics is founded. This is such a darkly guarded secret that you may be reading of it here for the first time. I wonder if I have earned enough credit with you to hope for your attention while I outline the facts and prove the irreconcilability. That will be one of the main goals of this chapter. As a reason for investigating SA more
|
||
I
|
||
|
||
|
||
70 Old Physics for New: a worldview alternative
|
||
cogent to our present studies, it will be recalled from Chapter 2 that Hertzian (first-order) electromagnetism failed to describe the phenomenon—and our claim that neo-Hertzian (higher-order) theory could do better was only promissory. So, it remains to deliver on the promise. Let us begin with the basics. Stellar aberration is a phenome
|
||
non first observed by James Bradley[4.1] (in 1728) at first order in
|
||
()
|
||
v c . Hence it is generally treated as “classical” in nature. This view is epitomized in Arthur Eddington’s explanation of the phenomenon by his famous “umbrella” analogy: a man with an umbrella, running at speed v through raindrops falling vertically at speed c, must tilt his umbrella forward through an angle whose tangent is v c in order to stay as dry as possible. This deduction is facilitated by consulting a vector triangle whose perpendicular sides are v and c, and whose hypotenuse is therefore of length
|
||
22
|
||
c + v . The result is correct for rain, and the model rightly suggests that forward in the sense of the Earth’s motion in its orbit is the proper direction to tilt the telescope. Moreover, to first order (given proper interpretation of v), the suggested angle of tilt is correct. However, the fact that the hypotenuse of the vector triangle, representing the relative speed of light traveling down the telescope tube, exceeds c should alert us that something is amiss. In fact, if that leg of the triangle is shortened to match the presumed speed limit c, this closes up the aberration angle to zero. Thus the phenomenon is not explained at all, compatibly with the existence of a limit on the speed of light relative to the telescope tube. So, Eddington’s model does not generalize from rain to photons, and is in fact quite misleading in its application to starlight. Another attempt to concoct a first-order model of the phenomenon was made by Bergmann[4.2] (in 1946) as follows: “ ... when a light ray enters the telescope, let us say from straight above, the telescope must be inclined ... so that the lower end will have arrived straight below the former position of the upper end by the time that the light ray has arrived at the lower end.” If that were true, then filling the telescope with water, which has a refractive index greater than unity and hence slows the light passing through the tube to speed c c
|
||
′ < , should observably
|
||
change the SA angle from ( )
|
||
1
|
||
tan v c
|
||
− to ( )
|
||
1
|
||
tan v c
|
||
− ′ . The experiment was done by Airy[4.3] in 1871, with no such result: the water was found to have no effect whatever on SA. It seems that the harder people try to over-simplify the phenomenon by classical
|
||
|
||
|
||
4. Stellar Aberration 71
|
||
models, the less they understand it. (We may add that Bergmann’s book[4.2] had Einstein’s approval, and that throughout his life Einstein appears to have over-simplified the SA phenomenon ... and this applies to almost all modern physicists as well.) The fact is that starlight gives us a tantalizing glimpse into the “most quantum” phenomenon in nature, the large-scale nonlocal action of the quantum of light. We shall never get closer to a personal apperception of the quantum world, and of the meaning of quantum non-locality, as well as of localized process completion, than we do through the act of viewing starlight. The physicist, if he is to enter into the quantum world with any degree of understanding, must bare his head and acknowledge that he stands in the presence of something recognizable (for his professional purposes) as holy. In that way he can get over his temptation to trivialize the phenomenon by classical models—including SRT (which in its handling and entire conceptualization of light is purely classical). A first step toward grasping the cogency of this last observation is to review what SRT has to say on the subject. We shall find that SRT, in its own mathematically impeccable way, goes every bit as wrong physically as either of the two classical attempts mentioned above.
|
||
4.2 SA according to SRT
|
||
In this section we refresh the reader’s memory regarding wellknown textbook material. Our treatment, following Aharoni,[4.4] is ponderous but systematic and rigorous. SRT describes a one-way propagating plane-wave or ray of starlight in an inertial system S
|
||
by a four-vector ( )
|
||
k k,i c
|
||
μ = G ω , μ= 1,",4 , where kG is a threevector of magnitude k = 2π λ= 2πν c =ω c , directed along the wave normal, and ν =ω 2π is the light frequency (as affected by Doppler effect). In studying SA we may focus attention on monochromatic radiation idealized as plane waves, in recognition of the great distances of all stellar sources from Earth. For simplicity we treat our telescope (at rest in S) as situated above the atmosphere of a uniformly-moving non-rotating Earth, to avoid all concerns with non-inertiality, or with refraction and dispersion, consequently with distinctions among phase velocity, group velocity, energy velocity, etc. (The SRT stipulation of strict inertiality is misleading, because the physics demands non-inertiality of detector motion in order for SA to become a measurable phenomenon.)
|
||
|
||
|
||
72 Old Physics for New: a worldview alternative
|
||
As with all four-vectors, the sum of the squares of the four components is an invariant, in this case zero. Thus 2
|
||
kk k
|
||
μμ μ
|
||
μ=
|
||
∑
|
||
= k ⋅ k + (iω c)2 = k2 −ω2 c2 = 0
|
||
G G is a null vector. This merely expresses the fact that ck =ω or λν = c in vacuum. Propagation in vacuum takes the photon from its source atom in the star to its earthly detector, which we suppose to be another atom in the photodetector of a telescope sharing the state of motion of the Earth. Our given wave-normal or ray of starlight is described by
|
||
()
|
||
k ,iω c
|
||
G in an inertial system S and by ( )
|
||
′ ω′
|
||
kG ,i c in another inertial system S′ . For the moment we refrain from identifying these systems physically, but shall merely turn the crank of the formalism. According to SRT, the two four-vectors have components that can be related by the special Lorentz transformation or boost,
|
||
()
|
||
()
|
||
11 4
|
||
22
|
||
33
|
||
4 4 1,
|
||
k k i vc k
|
||
kk
|
||
kk
|
||
k k i vc k
|
||
γ
|
||
γ
|
||
′⎡ ⎤
|
||
=+
|
||
⎣⎦ ′=
|
||
′=
|
||
′⎡ ⎤
|
||
=−
|
||
⎣⎦
|
||
(4.1)
|
||
where 1 2 3
|
||
kG = k , k , k , k4 = iω c , 2 2
|
||
γ = 1 1 − v c , etc., v is the velocity of S′ relative to S , and the relative motion of S and S′ is directed along their shared x-axes. Invariance of the null four
|
||
vector is preserved in S′ , ( )
|
||
ωω
|
||
′′ ′ ′ ′
|
||
⋅+ = − =
|
||
kG kG i c 2 k 2 2 c2 0 .
|
||
We follow SRT textbook procedures[4.4] to learn how the kμ four-vector transforms: It is convenient to introduce direction cosines A = cosα, m = cosβ, n = cosγ in S , so that 1 2 3
|
||
kG = k , k , k = kA, km, kn , where A2 + m2 + n2 = 1. Since k =ω c , we can write our four-vector in S as ck , m, n,i
|
||
μ =ωA ω ω ω. In S′ the same light ray is described by μ ω ω ω ω
|
||
′ ′′ ′ ′ ′ ′
|
||
ck = A , m , n,i . The Lorentz transformation (4.1) then yields
|
||
( )( ) ( )
|
||
ω γ ω ω γω
|
||
′′ ⎡ ⎤
|
||
= + =−
|
||
⎣⎦
|
||
A A iv c i A v c (4.2a)
|
||
ωω
|
||
′m′ = m (4.2b)
|
||
ωω
|
||
′n′ = n (4.2c)
|
||
()
|
||
ω γω
|
||
′ = 1 − Av c . (4.2d)
|
||
Eq. (4.2d) expresses the SRT relativistic Doppler effect, which was discussed in Chapter 3, and provides the derivation of Eq. (3.58) used there. From Eqs. (4.2) we obtain the transformation formulas for the direction cosines,
|
||
|
||
|
||
4. Stellar Aberration 73
|
||
()
|
||
γωω
|
||
−
|
||
′= − =
|
||
′−
|
||
A
|
||
A A 1A
|
||
vc
|
||
v c v c (4.3a)
|
||
()
|
||
1
|
||
m
|
||
m m vc
|
||
ω
|
||
ωγ
|
||
′= =
|
||
′ − A (4.3b)
|
||
()
|
||
1
|
||
n
|
||
n n vc
|
||
ω
|
||
ωγ
|
||
′= =
|
||
′ − A . (4.3c)
|
||
It is a simple algebraic exercise to verify that ′ ′′
|
||
A 2 + m 2 + n 2 = 1 , so the transformed quantities are indeed direction cosines in S′ . Also, we note that when multiplied by c these relations reduce to the light velocity composition laws obtained by differentiating the Lorentz transformation equations (Møller[4.5]). From these general relations we can determine at once how much the light ray or telescope axis changes its direction in 3-space in the change from S′ to S . By vector analysis, or from the geometry of direction cosines, we know that the cosine
|
||
of the angle α between the two unit vectors kG k and ′ ′
|
||
kG k (i.e., between the wave normals in S and S′ ) is the scalar product ′ ′′
|
||
AA + mm + nn . This “classical” relation holds because under the special Lorentz transformation the spatial coordinate axes in S and S′ retain their parallelism, so directions and changes of direction in the two spaces are unambiguously defined by projections onto the coordinate axes. Using 2 2 2
|
||
m + n = 1 − A , we find from (4.3)
|
||
() ( )( )
|
||
()
|
||
()
|
||
22 1
|
||
2
|
||
1 22
|
||
22
|
||
23
|
||
2 2
|
||
cos 1 1 1
|
||
1
|
||
cos 1 1 1
|
||
1
|
||
1 12
|
||
1
|
||
12 6
|
||
SRT
|
||
vc m n
|
||
vc vc vc
|
||
vc
|
||
vc
|
||
vv v cc c
|
||
α γγ
|
||
−
|
||
−
|
||
⎡⎤
|
||
−
|
||
= ++
|
||
⎢⎥
|
||
− −−
|
||
⎢⎥
|
||
⎣⎦
|
||
⎡⎤
|
||
−
|
||
= − −− ≈
|
||
⎢⎥
|
||
−
|
||
⎣⎦
|
||
−+
|
||
−
|
||
⎛⎞ ⎛⎞ ⎛⎞
|
||
−+ + +
|
||
⎜⎟ ⎜⎟ ⎜⎟
|
||
⎝⎠ ⎝⎠ ⎝⎠
|
||
AA A AA
|
||
A A
|
||
AA
|
||
AA
|
||
A"
|
||
(4.4)
|
||
The expansion of arc cosine in powers of v c here is a bit tricky and rather interesting, but since the interest is purely mathematical we shall take the result as given. For historical interest, it may be worth a brief digression to compare with Einstein’s original result. In his 1905 paper[4.6] he gave the formula, essentially a velocity composition law,
|
||
|
||
|
||
74 Old Physics for New: a worldview alternative
|
||
φβ
|
||
φβ
|
||
βφ
|
||
−
|
||
′= =
|
||
−
|
||
cos
|
||
cos ,
|
||
1 cos
|
||
v
|
||
c,
|
||
which is very elegant but not very useful. We need to know the telescope tilt angle α φ φ
|
||
′
|
||
=−
|
||
SRT . The following strategy is as good as any for worming out this information. Let φ φ
|
||
′′
|
||
q = cos , q = cos . A Pythagorean right triangle with small vertex angle φ′ can be drawn, with hypotenuse 1 − βq and long side q − β. The third (short) side is then of length 2 2
|
||
1 − β 1 − q , so
|
||
β
|
||
φβ
|
||
−−
|
||
′= −
|
||
22
|
||
11
|
||
tan q
|
||
q.
|
||
Einstein’s formula shows the symmetry φ φ β β
|
||
′′
|
||
↔ , q ↔ q, ↔ − , so
|
||
β
|
||
φβ
|
||
′
|
||
−−
|
||
= ′+
|
||
22
|
||
11
|
||
tan q
|
||
q.
|
||
Then a standard trigonometric identity[4.7] states that ()
|
||
φφ
|
||
φ φ φφ
|
||
′−
|
||
′− = ′
|
||
sin
|
||
tan tan cos cos ,
|
||
whence
|
||
() β β
|
||
φφ α β β
|
||
⎛⎞
|
||
′
|
||
−− −−
|
||
⎜⎟
|
||
′′
|
||
−= = −
|
||
⎜⎟
|
||
′
|
||
−+
|
||
⎝⎠
|
||
22 2 2
|
||
11 11
|
||
sin sin SRT
|
||
qq
|
||
qq q q .
|
||
On eliminating q′ by means of Einstein’s composition relation above, viz.,
|
||
β
|
||
β
|
||
−
|
||
′= 1−
|
||
q
|
||
q q,
|
||
we get an expression for SRT
|
||
α as an arc sine equivalent to the arc cosine expression (4.4). Both show that α φ φ
|
||
′
|
||
=−
|
||
SRT can be represented as the small vertex angle of a Pythagorean right triangle with hypotenuse 1 − βq , long side
|
||
( )( )
|
||
22
|
||
1 − βq − 1 − q 1 − 1 − β ,
|
||
and short side
|
||
()
|
||
22
|
||
1 q β q1 1 β
|
||
⎡⎤
|
||
− − −−
|
||
⎢⎥
|
||
⎣ ⎦.
|
||
Einstein’s formulas are consistent with these results when q is replaced by his cosφ. When q is replaced by A = − sinθcosφ they
|
||
|
||
|
||
4. Stellar Aberration 75
|
||
are consistent with angles more generally defined as in Fig. 4.1, so that A,m,n = − sinθcosφ,sinθsinφ,− cosθ, (4.5)
|
||
the signs signifying downward propagation of the starlight. Opposite signs could equally well be used, signifying upward pointing of the telescope. Although only one parameter, A = − sinθcosφ, appears in the expansion (4.4), it depends on two angles, the polar angle θ and the azimuthal or orbital angle φ, the latter varying with the time of year. It is well known (since Bradley) that over the course of a year any given star traces out on the celestial sphere a small ellipse of semi-major axis a and semi-minor axis b, known as the “figure of aberration.” This ellipse is roughly the projection along the mean direction to the star of the Earth’s (approximately circular) orbit onto the celestial sphere. Thus for a star near the zenith (θ small) the figure of aberration is almost a circle, whereas toward the horizon (θ near 90° ) it becomes flattened into a line. Holding θ constant at some intermediate value, we note that as the Earth progresses counter-clockwise around its orbit (φ varying) the star’s position on the projected elliptical figure of aberration advances synchronously, but with a 90° phase advance—i.e., in phase quadrature. When φ= 0° we see from Fig. 4.1 that the Earth is advancing directly toward the star, and that the aberra
|
||
Fig. 4.1. Earth’s orbit in plane iˆ1,iˆ2 of the ecliptic, showing Earth’s posi
|
||
tion at azimuth φ and orbital velocity vGorb ; also fixed kG -vector of starlight propagation at polar angle θ, lying in fixed plane 2 3
|
||
iˆ ,iˆ normal to the ecliptic.
|
||
|
||
|
||
76 Old Physics for New: a worldview alternative
|
||
tion angle α is a minimum, corresponding to the b-value (semiminor axis) of the projected ellipse, αmin = b . Three months later, when φ= 90° , α is a maximum, corresponding to the semi-major axis αmax = a . These extremal properties can be verified directly from the coefficient 2
|
||
1− A of the dominant term in (4.4): For φ= 0° , 2
|
||
1− A = ( )2
|
||
1 − − sinθ = cosθ, which is minimal for the given θ-value; and when φ= 90° we have A = 0 and 2
|
||
1−A =1, which is maximal. A convenient gauge of stellar aberration that should be readily measurable is the ratio of semi-axes, as predicted by (4.4); viz., () ()
|
||
min
|
||
max
|
||
2
|
||
2
|
||
23
|
||
2 24
|
||
sin 0
|
||
cos sin cos sin
|
||
cos 2 3
|
||
1
|
||
cos sin sin
|
||
6 4 2
|
||
cos sin sin
|
||
9 5
|
||
b a
|
||
vv cc
|
||
v c
|
||
v c
|
||
αθ
|
||
α
|
||
αα
|
||
θθ θ θ
|
||
θ
|
||
θθ θ
|
||
θθ θ
|
||
=−
|
||
== =
|
||
⎛⎞ ⎛⎞
|
||
≈− +
|
||
⎜⎟ ⎜⎟
|
||
⎝⎠ ⎝⎠
|
||
⎛⎞
|
||
+
|
||
⎜ ⎟⎛ ⎞
|
||
⎝⎠
|
||
− ⎜⎟
|
||
⎝⎠
|
||
⎛⎞
|
||
+
|
||
⎜ ⎟⎛ ⎞
|
||
⎝⎠
|
||
++
|
||
⎜⎟
|
||
⎝⎠
|
||
A A
|
||
"
|
||
(4.6)
|
||
The leading term here, cosθ, corresponds to the projection of the Earth’s “circular” orbit on the celestial sphere. Owing to the ellipticity of the actual orbit, these calculations would have to be modified in application (and small diurnal variations allowed for), but the circular-orbit approximation, used throughout this chapter, will illustrate the features we wish to emphasize. In particular, the crucial point is that SRT predicts a firstorder departure of the shape of the figure of aberration from the simple classical Earth’s-orbit projection recipe ( )
|
||
cosθ , corresponding to Bradley aberration. This departure term has a maximum magnitude at θ= 45° above the plane of the ecliptic. Since the angular figure of aberration itself is of the order (v/c) radian [cf. Eq. (4.4)], to verify alteration of its shape requires measure
|
||
ments of order ( )2
|
||
v c . Writing in 1946, Bergmann[4.2] stated that “the relativistic second-order effect is far below the attainable accuracy of observation.” But since then the observational situation has changed dramatically. With the advent of the Very Long Baseline Interferometry (VLBI) system, it is claimed that angles can be resolved to around 4
|
||
10− arc sec, or roughly 10
|
||
5 10−
|
||
× radian. The
|
||
|
||
|
||
4. Stellar Aberration 77
|
||
value of (v/c) is about 4
|
||
10− radian for the earth’s orbital motion.
|
||
Therefore ( )2
|
||
v c effects, of order 8
|
||
10− radian, should be measurable. An opportunity thus arises for astronomers to perform a crucial test of SRT in an area (special relativistic shape modification of the classical Bradley figure of aberration) not hitherto probed, so far as this writer knows. To do this would be an important contribution to physics. However, some form of the VLBI system has been in existence since the late 1960’s, and nobody has suggested using it to check the second-order term in Einstein’s prediction, Eq. (4.4). Why should they? Einstein was right about everything else, how could he be wrong about that? If you offered ten randomly-selected physics professors the chance to use the VLBI system for such a purpose, none would be interested—no career advancement opportunities there. Why face the possibility of failure to obtain the politically correct answer—when failure means instant pariah-hood? Besides, they know the answer from a theory they trust completely. What interests them is the unknown, the frontier. It is not unlike the case of the savants who declined to look through Galileo’s telescope. I have not checked, but feel sure they were academicians to a man. We note an ageless pattern: Known theory trumps empirical inquiry every time. The above derivation is conducted in typical textbook style, in that it leaves the impression that whatever a bunch of mathematics has to say about the phenomenon is all that needs to be said. In this case that is far from the truth. Much more needs to be said, as we shall soon see.
|
||
4.3 SA according to neo-Hertzian theory
|
||
To review the preceding chapters in one sentence: Starting from Maxwell’s field equations as a covered theory, successive improvements in the scope of “invariance” can be achieved by the formal replacements
|
||
d dd d d
|
||
d dd
|
||
vv
|
||
t dt t d dt t
|
||
γγ
|
||
τ
|
||
∂ ∂ ⎛∂ ⎞
|
||
→ = + ⋅∇ → = = + ⋅∇
|
||
⎜⎟
|
||
∂ ∂ ⎝∂ ⎠
|
||
GG
|
||
G G,
|
||
the arrows representing progression from Maxwellian to Hertzian to neo-Hertzian formulations of field theory. In the discussion that follows we shall be concerned with neo-Hertzian predictions alone. The central result from Chapter 3, Eq. (3.49),
|
||
|
||
|
||
78 Old Physics for New: a worldview alternative
|
||
was that the neo-Hertzian wave equation has a d’Alembertian solution for which
|
||
()
|
||
22
|
||
dd
|
||
u=± c −v + k k ⋅v
|
||
G G , (4.7)
|
||
where u is light (in vacuo) propagation phase velocity and vGd is detector velocity with respect to the observer’s inertial frame. Maxwell’s result, u = ±c , corresponds to the particular case in which the light detector is at rest in the observer’s frame ( 0
|
||
vd =
|
||
G ). To get a feeling for the situation, let us first apply (4.7) to the special case of aberration of light from a star at the zenith. The present discussion is based on previous work by the author.[4.8]
|
||
First off, we have the problem, in defining vGd , of where to place the “observer” in order to specify “the observer’s inertial frame.” For most purposes of ordinary observation the earthly lab or observatory suffices as sufficiently “inertial;” but in the case of stellar aberration, as we have already noted, it happens that the phenomenon achieves observability only through the slow annual changes resulting from non-inertiality of Earth motion. So we must place our observer in a reference system more nearly inertial than that of the orbiting Earth. Referring things, then, to the more nearly inertial system of the Sun (or the solar system barycenter), we have
|
||
d orb
|
||
v =v
|
||
G G , (4.8)
|
||
where vGd is the velocity of our detector (telescope) at rest on the Earth’s surface and orb
|
||
vG is the tangential velocity in the Sun’s inertial system of the Earth (neglecting diurnal effects) in its orbit, approximated as circular. This orbit defines the plane of the eclip
|
||
tic, to which our ray of starlight kG from the zenith is normal. Thus 0
|
||
d
|
||
k⋅v =
|
||
G G in (4.7), with choice of the plus root, yields
|
||
22 d
|
||
u = c − v . (4.9)
|
||
We see that the light, propagating straight down from the zenith, is slightly slowed by a second-order amount. We know also that
|
||
for an observer at rest with respect to the telescope tube vGd vanishes and u = c , in agreement with Maxwell. What can this mean? Only one possible thing: for the light to have speed c relative to the telescope tube and speed 2 2
|
||
d
|
||
c − v relative to the vertical (zenith) direction, the telescope tube has to be tilted away from the vertical. This tilting is evidence of “stellar aberration.” It is a phenomenon whose existence is clearly predicted ab initio by the line
|
||
|
||
|
||
4. Stellar Aberration 79
|
||
of reasoning given here. The predicted angle of tilt for starlight from the zenith, as shown by the vector triangle of Fig. 4.2, is
|
||
() ( ) 3
|
||
1 1 22 1
|
||
sin tan 6
|
||
dd neo Hz d d d
|
||
vv
|
||
vc v c v c c
|
||
α− −
|
||
−
|
||
⎛⎞
|
||
= = − ≈+ +
|
||
⎜⎟
|
||
⎝ ⎠ " , (4.10)
|
||
and we may take the sense of tilt to be “forward” in the direction of Earth’s orbital velocity, to agree with Bradley’s observations. (The present argument fails to specify the sense of the telescope tilt, which can nevertheless be deduced from elementary consid
|
||
erations.) Since the wave-normal kG -vector is vertical [and is not
|
||
affected by inertial transformations, as the invariance G′ = G
|
||
k k of Eq. (3.54) shows], filling the telescope tube with water does not alter the horizontal orientation of the wave-fronts. It does slow the propagation along the inclined tube direction in the same ratio as the vertical descent, so the Pythagorean diagram, Fig 4.2, is not changed, and the tilt angle is unaffected, in agreement with the observations of Airy.[4.3] We recall that the classical diagram corresponding to Fig. 4.2, based on the Eddington “umbrella” model mentioned above, is also Pythagorean, but with a vertical side of length c and an hypotenuse of (impossible) length 2 2
|
||
d
|
||
c + v , so that
|
||
Fig. 4.2. Vector diagram (Pythagorean triangle) showing telescope tilt angle α for starlight from the zenith, as described by neo-Hertzian electromagnetism.
|
||
|
||
|
||
80 Old Physics for New: a worldview alternative
|
||
()
|
||
3
|
||
11
|
||
tan 3
|
||
dd classical d
|
||
vv
|
||
vc c c
|
||
α − ⎛⎞
|
||
= ≈− +
|
||
⎜⎟
|
||
⎝ ⎠ " (4.11)
|
||
All theories agree as to the first-order term. Note that neoHertzian theory, according to Eq. (4.9) or (4.10), specifically identifies detector velocity as the parameter appropriate to the description of SA. Neither light source speed nor source-sink relative speed enters. Some other theories, such as SRT, are less clear about identifying what “relative velocity” is involved. We shall say more about this presently. By coincidence it happens for our special case of starlight from the zenith that SRT agrees to third order with the neo-Hertzian result (4.10)—as follows from Eq. (4.4) with θ= 0 , A = 0 . However, such agreement does not hold in general. Our use of the Sun as a referent in the above discussion was merely for simplicity of presentation. More generally, we may consider an initial (pseudo-) inertial system S that co-moves with the telescope at time t0 and a second similar system S′ that co
|
||
moves with it at later time t1 . Then if vGd is interpreted as the velocity of S′ relative to S , the formula (4.10) and analysis remain valid with this new meaning of the symbols. The “tilt angle” so described is the angle of tilt change between changes of inertial system. Thus we are freed from concern about the Sun and can recognize that only detector motions need be considered in giving a complete empirical description of the SA phenomenon. Such a Sun-free description has been recommended by Synge.[4.9] A particular virtue of this perception is that it reemphasizes and forces us to recognize that the observability of SA arises purely through non-inertiality of the detector’s motion. If our telescope remained at rest permanently in an inertial system, or if any integral multiple of 12 months marked the time interval
|
||
10
|
||
t − t , SA would be unobservable. Thus, if S and S′ are the same inertial system, 0
|
||
vd =
|
||
G and from (4.10) 0
|
||
neo Hz
|
||
α − = ; so there is no observable aberration (i.e., no change of aberration angle). One could write Δα instead of α, to be more explicit about this. [Here, though, we may view α alternatively as the “constant of aberration,” or angular radius of the near-circular figure of aberration traced out on the celestial sphere over a year’s time by the image of a star at the zenith. For this purpose Eq. (4.8) applies.]
|
||
|
||
|
||
4. Stellar Aberration 81
|
||
Let us now work out the neo-Hertzian analysis for the general case of a star not at the zenith. From the geometry of Fig. 4.1 we see that
|
||
23
|
||
ˆˆ
|
||
k k = −i sinθ− i cosθ
|
||
G (4.12)
|
||
and
|
||
12
|
||
ˆˆ
|
||
sin cos
|
||
dd d
|
||
v = −i v φ+ i v φ
|
||
G . (4.13)
|
||
Hence
|
||
( ) sin cos
|
||
dd
|
||
k k ⋅ v = −v θ φ
|
||
G G , (4.14)
|
||
so that Eq. (4.7) yields
|
||
2 2 sin cos
|
||
dd
|
||
u = c − v − v θ φ. (4.15)
|
||
We may consider the telescope axis to be pointed along the direc
|
||
tion of a “telescope vector” TG , which is the negative of the vector
|
||
sum ( )
|
||
kku
|
||
G and −vGd , just as in the velocity composition diagram of Fig. 4.2. Then
|
||
()
|
||
()
|
||
()
|
||
23 1 2
|
||
12 3
|
||
ˆˆ ˆ ˆ
|
||
sin cos sin cos
|
||
ˆˆ ˆ
|
||
sin sin cos cos
|
||
d
|
||
dd
|
||
dd
|
||
T kku v
|
||
u i i iv iv
|
||
iv i u v iu
|
||
θθ φ φ
|
||
φ θφ θ
|
||
⎡⎤
|
||
=− −
|
||
⎣⎦
|
||
= +− +
|
||
=− + + +
|
||
G
|
||
GG
|
||
(4.16)
|
||
where u is given by (4.15). The magnitude of TG is
|
||
()
|
||
22
|
||
222 2
|
||
2
|
||
sin cos .
|
||
dd
|
||
d
|
||
T TT u ukk v v
|
||
cv θ φ
|
||
= ⋅= − ⋅ +
|
||
=−
|
||
G
|
||
GG G
|
||
(4.17)
|
||
The scalar product of unit vectors TG T and kG k is the cosine of the angle between these vectors, which is our telescope tilt angle (or rather, tilt angle change associated with change of velocity
|
||
vector −vGd , as discussed above),
|
||
()
|
||
1
|
||
cos
|
||
αneo Hz − T k Tk
|
||
−⎡⎤
|
||
=⋅
|
||
⎣⎦
|
||
G
|
||
G . (4.18)
|
||
Since, with the help of (4.15), we have
|
||
() ( )
|
||
12 3
|
||
22 23
|
||
ˆˆ ˆ
|
||
sin sin cos cos
|
||
ˆˆ
|
||
sin cos sin cos .
|
||
dd
|
||
dd
|
||
T k k iv i u v iu
|
||
i i uv c v
|
||
φ θφ θ
|
||
θ θ θφ
|
||
⎡⎤
|
||
⋅ =− + + +
|
||
⎣⎦
|
||
⎡⎤
|
||
⋅ − − =− − =− −
|
||
⎣⎦
|
||
G
|
||
G
|
||
(4.19)
|
||
Eq. (4.18) with (4.17) yields [exploiting the sign ambiguity of the radical in (4.7)]
|
||
|
||
|
||
82 Old Physics for New: a worldview alternative
|
||
22 1
|
||
222 2
|
||
cos sin cos
|
||
d neo Hz
|
||
d
|
||
cv cv
|
||
α θφ
|
||
− −
|
||
−
|
||
= − . (4.20)
|
||
For purposes of easiest expansion, this is advantageously reexpressed as
|
||
()
|
||
()
|
||
()
|
||
2 1 2
|
||
2 23 2
|
||
2 24 5
|
||
1
|
||
tan 1
|
||
1 12
|
||
16
|
||
1 34 8
|
||
40
|
||
d neo Hz
|
||
d
|
||
dd
|
||
d
|
||
v
|
||
c vc
|
||
vv cc
|
||
v c
|
||
α−
|
||
−
|
||
⎛⎞
|
||
−
|
||
⎜⎟
|
||
=⎜ ⎟
|
||
−
|
||
⎝⎠
|
||
−+
|
||
⎛⎞ ⎛⎞
|
||
≈− +
|
||
⎜⎟ ⎜⎟
|
||
⎝⎠ ⎝⎠
|
||
− + + ⎛⎞
|
||
++
|
||
⎜⎟
|
||
⎝⎠
|
||
A
|
||
AA
|
||
A
|
||
A AA "
|
||
(4.21)
|
||
where A = − sinθcosφ, as in Eq. (4.5), and the angles are those of Fig. 4.1. The simple trigonometry governing passage from (4.20) to (4.21) is that of the Pythagorean right triangle shown in Fig. 4.3. Study of (4.21) reveals the same phase effect as was noted for SRT, such that the motion of the stellar image along the annual figure of aberration on the celestial sphere is in phase quadrature with the Earth’s position in its orbit (leading by 90° ). Further discussion is given in earlier work by the author.[4.8] We record the neo-Hertzian result analogous to the SRT prediction (4.6), viz.,
|
||
Fig. 4.3. Pythagorean triangle for telescope tilt angle α in the neoHertzian general case of starlight propagating with direction cosine A = − sinθcosφ.
|
||
|
||
|
||
4. Stellar Aberration 83
|
||
() ()
|
||
1
|
||
22 min
|
||
max 1
|
||
22
|
||
22
|
||
24
|
||
2
|
||
cos
|
||
tan sin
|
||
0 tan
|
||
2
|
||
cos sin sin
|
||
cos sin 9
|
||
cos 3 5
|
||
d
|
||
d
|
||
d
|
||
d
|
||
dd
|
||
v
|
||
cv
|
||
b
|
||
av
|
||
cv
|
||
vv cc
|
||
θ
|
||
αθ
|
||
α
|
||
αα
|
||
θθ θ
|
||
θθ
|
||
θ
|
||
−
|
||
−
|
||
⎛⎞
|
||
⎜⎟
|
||
⎜⎟
|
||
−
|
||
= ⎝⎠
|
||
== =
|
||
= ⎛⎞
|
||
⎜⎟
|
||
⎜⎟
|
||
−
|
||
⎝⎠
|
||
⎛⎞
|
||
+
|
||
⎜⎟
|
||
⎛⎞ ⎛⎞
|
||
⎝⎠
|
||
≈+ + +
|
||
⎜⎟ ⎜⎟
|
||
⎝⎠ ⎝⎠
|
||
A A
|
||
"
|
||
(4.22)
|
||
The absence of a second-order term in the neo-Hertzian result, (4.21), and the presence of such in the SRT result, Eq. (4.4) [or the
|
||
absence of a first-order term in ( )
|
||
vd c in (4.22) and its presence in (4.6)], is especially to be noted. It provides the basis for a crucial test to decide between the two theories. We pointed out above that, following development of the VLBI system in the late 1960’s, it appears to have become technically feasible to make secondorder astrometric observations of sufficient refinement to do this. All that has been lacking is the will. No new apparatus is needed, just observation time and some peanut bars to defray the labor of graduate students. Finally, it may be wondered why Potier’s principle, which denied the observability of any distinction between SRT and Hertzian theory in the case of the Sagnac effect, does not do the same for SA. I believe the reason is that Potier’s principle refers to a single light path with fixed endpoints; whereas here we may consider that we have either two light paths (at different times) or a single light path with an accelerated endpoint. It is worth mentioning that both Potier’s principle and Fermat’s principle from which it derives (which states that physical light takes the “least time” in going between two points) are cast into limbo by the advent of SRT. The latter introduces two types of “time,” proper and frame. Fermat’s principle certainly cannot refer to an interval of proper time, since that is zero on all light paths. But if it refers to frame time, SRT informs us that frame time is “meaningless,” nonexistent because non-invariant. Only some Raffiniert thing of beauty called “spacetime” is meaningful. So, Fermat’s principle seemingly dies without burial—another victim of the great destroyer. In Chapter 6 we shall introduce a type of “collective time” that would allow its reincarnation. I admit I have not given this subject area much thought. The reader may find it worthwhile to do better. |