zotero/storage/A37VSCN4/.zotero-ft-cache

Einstein Plus Two
By
Petr Beckmann
Professor Emeritus o f Electrical Engineering, University o f Colorado Fellow,
Institute o f Electrical and Electronic Engineers
THE GOLEM PRESS Boulder, Colorado 1987

Library o f Congress Catalog Card No.: 85-82516 ISB N 0-911762-39-6
C opyright © 1987 by T he G olem Press All rights reserved Printed in the U .S.A .
THE GOLEM PRESS Box 1342 Boulder, Colorado 80306

To the two great physicists o f our time,
Edward Teller
and
Andrei Dmitriyevich Sakharov

Preface
W hen I run, I feel a wind; but not one that will m ake a windmill turn. As long as an observer is at rest on the ground, it does not m atter whether the velocity o f the wind is referred to the observer or the windmill. A physicist who falsely assumes that the effect-producing velocity (that makes the windmill turn) is th at with respect to the observer, but correctly applies the relativity principle, will expect the windmill to turn when he is running. The experimental evidence will contradict his expectation, and he can then either abandon his false premise, or he can so distort space and time that the observer’s m otion produces two exactly equal and opposite forces on the windmill, keeping the mill motionless as observed. The Einstein theory, in effect, takes the latter road; but I believe the laws o f physics, including the relativity principle, must hold regardless o f any observer, who should do nothing but observe. A n electric or magnetic field will accelerate an electron. Its magnetic field will therefore increase, which causes the induced electric field to decelerate it. T hat will decrease the magnetic field and the induced electric field will accelerate the electron again. The resulting oscillations are derived from the Maxwell equations in Part Two o f this book. They explain the quantization of electron orbits, the de Broglie relation and the Schrodinger equation simply and without further assumptions. The natural frequency o f these oscillations depends on the velocity o f the elec­ tron; but the velocity with respect to what? The velocity that will make the Lorentz force and the Maxwell equations valid, claims the Einstein theory, is the velocity with respect to the observer. But if so, does the electron oscillate for me because I am moving past it, but not for you because it lies still in your rest frame? T o answer yes is to kill the relativity principle. As I will attem pt to show, the velocity that makes the Maxwell-Lorentz elec­ trodynamics valid is that o f charges with respect to the local fields they traverse. That squares with the experimental evidence in electromagnetics and optics, and it leads to the derivation of two phenomena for which no explanation other than ad hoc postulates has hitherto been available: the quantization of electron orbits and in the realm o f gravity, the Titius series. Why, then, has the Einstein theory celebrated an uninterrupted series of brilliant successes for more than 80 years?

6

Because in all past experiments the observing instruments have always been nailed to the local field, so that they could not reveal whether the observed effect was associated with an observer-referred or a field-referred velocity. The technology for testing that difference may not be available for some time.
But if it is field-referred velocities that are the effect-producing ones, then the Maxwell equations automatically become invariant to the Galileian transform a­ tion; the undisputed fact that the Lorentz force and the Maxwell equations with o fo m w -re fe rre d velocities are Lorentz-invariant is one that becomes both trivial and irrelevant.
I am not so naive as to think that the first attem pt to move the entire Einstein theory en bloc onto classical ground will turn out to be perfectly correct. W hat 1 do hope is that the approach will provide a stimulus for the return o f physics from description to comprehension. Attempting to redefine the ultimate founda­ tion pillars of physics, space and time, from what they have been understood to mean through the ages is to move the entire building from its well-established and clearly visible foundations into a domain o f unreal acrobatics where the observer becomes m ore im portant than the nature he is supposed to observe, where space and time become toys in abstract mathematical formalisms, and where, to quote a recent paper on modern approaches to gravitation theory, “the distinctions between future and past become blurred.”
This book is for those who do not wish to blur such distinctions (“H e will com m it posthum ous suicide yesterday”?). It is for those who seek to understand rather than merely to describe; for those who will accept the Einstein theory as a brilliant, powerful and productive equivalence, but not as a physical reality.
It is for those who are prepared to sacrifice a lifetime’s investment in learning; and perhaps more importantly, for the young students who have not yet made such an investment.

Boulder, Colorado
1983-1987

P.B.

Contents

PREFACE

5

INTRODUCTION: TRUTH AND EQUIVALENCE

11

1. EINSTEIN MINUS ZERO

1.1. The Static Inverse Square Law

25

1.2. The Velocity of Light: W ith Respect to W hat?

27

1.3. The “Purely Optical” Evidence

1.3.1. A berration

30

1.3.2. Fresnel’s Coefficient o f D rag

33

1.3.3. Fizeau’s and Airy’s Experim ents

35

1.3.4. Double Stars and Other Objections

to the Ballistic Theory

37

1.3.5. The Michelson-Morley Experiment

39

1.3.6. Moving Mirrors

40

1.3.7. The Michelson-Gale Experiment

42

1.4. Magnetic Force and the Gravitational Field

46

1.5. Electromagnetic M omentum

5 3

1.6 . The Field o f a Moving Charge

57

1.7. Mass and Energy

62

1.8. The Modified Newton-Colulomb Law

66

1.9. The Electromagnetic Evidence

1.8.1. Mass, M omentum and Energy

73

1.9.2. C ham pion’s Experim ent

75

1.9.3. Time Dilation: Ives-Stillwell, Mesons,

and Clocks Around the Globe

77

1.10. Galileian Electrodynamics

1.10.1. The Maxwell Equations and the Lorentz Force

82

1.10.2. Electromagnetics of Moving Media

85

1.10.3. Invariance o f Relative Velocities

88

1.10.4. Invariance of the Maxwell Equations

92

1.11. Mercury, Mesons, Mossbauer, and Miscellaneous

98

.2 EINSTEIN PLUS ONE

2.1. Strictly Central M otion

103

2.2. Self-Induced Oscillations of an Accelerated Charge

108

2.3. The Faraday Field and Electron Oscillations

112

2.4. Slightly Off-Central Motion

118

2.5. The Quantization of Electron Orbits

124

2.6. Electromagnetic Mass

128

2.7. Electromagnetic Mass and Acceleration

13 7

2.8. Energy Balance

140

2.9. Planck’s C onstant

143

2.10. The Root Problem

149

2.11. The Schrodinger Equation

152

2.12. Radiation and Some Other Matters

156

3. EINSTEIN PLUS TWO

3.1. Gravitation

165

3.2. Mercury

170

3.3. T heTitius Series

176

3.4. The Stable Planetary Orbits

179

3.5. Siblings, Twins, or One Identical Child?

184

3.6. Inertia

188

EPILOGUE

193

Appendix: The Devil’s Advocates

195

References

203

Index

207

Introduction:
Truth and
Equivalence

Introduction:
Truth and Equivalence

T r u t h , som e say, is w hat agrees with experiment.

Necessary, but not sufficient: fata m organas can be photographed, and all

astronomic measurements on earth record the same position of a star that may not

exist: next week’s observations o f the star’s light may bring the news that it blew up

in the 14th century. The m irror image o f a candle behaves as if it were emitting

light, and a body immersed in water behaves as if it had lost mass.

To develop a workable guideline for what is true and what is equivalent, con­

sider first some uncontroversial cases o f equivalence.

A good example is provided by the ionospheric equivalence theorems (there are

two, but for our purposes they can be merged into one). W hen radio waves are

returned by the ionized layers in the upper atmosphere, they are not reflected by

them like a tennis ball is bounced off a wall. Even using the geometric optics

simplification, a radio pulse travels with variable speed along a path similar to the

one sketched in the figure below.

On entering the ionized layer at

A , the pulse slows and the path

BI

curves (for reasons given in any

textbook of ionospheric propaga­

tion) until at the point B it becomes

horizontal and the pulse comes to a

standstill — in the geometric optics

approximation, anyway. The pro­

cess then reverses itself symmetri­

cally, and the pulse leaves the

ionized layer with the velocity of light at the point C.
It is not a simple process, and the ionospheric equivalence theo­

The ionospheric equivalence theorem (true and ef­ fective height). A rad io wave pulse is slowed along the segment A B C in the ionosphere, but the transit tim e is the sam e as if it ran the p ath A B 'C w ith constant velocity c.

rems provide welcome relief: as proved in any textbook on ionospheric radio wave

propagation, the time taken by the pulse to m ake it from transm itter T to receiver

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IN T R O D U C T IO N

R by the true, curved, slow path via B is exactly equal to the time that would be taken by a fictitious pulse traveling with constant free-space velocity from T to R via the straight sides of the triangle with apex at B'.
Thus the true height o f the reflection point OB is replaced by the effective height (the actual term used in ionospheric research) OB' o f the reflection point, which is the height that the pulse would reach in the same time if it propagated with the velocity of light throughout the trip. Since an ionospheric station, like any other radar, measures the time elapsed between transmission and reception, the two are equivalent. The real height is true, but involves bothersom e calculations; the effective height is fictitious — a “just as if” equivalent height — but much simpler to use. (The two heights are related by a Volterra integral equation.)
Now here it is quite uncontroversial which o f the two heights is true as a physical reality, and which is merely equivalent in producing the same effect on the measuring instrum ent. The obvious criterion for distinguishing between the two is that the effective height has limited validity: it will work when we measure the time for the echo to return, but not otherwise. A satellite measuring ionization directly will agree only with the real height o f the layer, as will any other independent method.
Limited validity is, in fact, the first o f my two proposed guidelines o f how to separate truth from equivalence.

C /o n s id e r two more examples of the limited-validity guideline. A real image o f an object is one whose points are sources o f optical rays, just as they are on the original object. A real image is, for example, produced by an object located beyond the focal distance of a concave mirror. But the plane bathroom m irror will produce only a virtual image — it is ju st as i f the rays em anated from points on the image behind the wall, but in reality they do not. Limited-validity guideline: a real image behaves optically like a real object under all conditions; a virtual image only under some. Intercept the rays from a real image at any point between the image and the eye, and the image will disappear from sight just as an original object would. But if the equivalent rays are intercepted by an obstacle between image and eye just behind the bathroom wall, the virtual image stays in the mirror. A second example is provided by Thevenin’s Theorem , which permits the simplification of complicated electrical circuits. It states that in any linear circuit the voltage between any two points, such as A and B on p. 13, is the same as if it were caused by a single source in series with a single impedance (with values also given by the theorem). Truth and equivalence are sharply separated here by the limited-validity guideline: Let us assume that figure (a) represents a real circuit, and (b) is the equivalent circuit calculated by Thevenin’s Theorem . Then in figure (b) what is to the right o f A -B corresponds to voltages, currents and

IN T R O D U C T IO N

13

T h iv e n in ’s T heorem
circuit elements in the real world; but what is to the left o f it is a “just as i f ’ m athematical equivalence which is fictitious, notw ithstanding the fact that such a circuit could, if we so desired, be very easily realized as a physical reality.
T jt't us now apply the limited-validity guideline to the Einstein Theory (there is
a good reason why I am reluctant to call it the Theory “o f Relativity”). Is it limited or universally valid?
It is certainly universally valid in its claims, and there is no experimental evidence to contradict it. However, such evidence can be obtained only when sources o f light or elementary particles move with a velocity com parable with the velocity o f light, and this, at present, restricts the verified results to a surprisingly narrow field: a handful o f optical experiments (which are also supported by an alternative hypothesis), and electromagnetics — and please hold back your protest until I fully explain what I mean.
f irst, 1 have singled out the optical experiments because they make no use of the electromagnetic nature o f light. They use light simply as something that has the capacity to interfere and that travels from here to there with velocity c.
The rest o f the acceptable evidence virtually always relies directly or indirectly on electromagnetic theory, as will be shown in Part One. In particular, the velocity o f elementary particles is rarely measured directly (as, say, the ratio o f distance covered to time elapsed), but is usually inferred from the directly measured voltage and the Lorentz force, which is assumed to remain valid at high velocities, which are defined to be velocities with respect to the observer. Similarly, the decrease in the ratio o f charge to mass o f elementary particles at high velocities is always attributed to the increase in inertial mass, for the invariance o f electric charge has simply been postulated. M ore examples will be given in Part One, where these points will be discussed m ore fully.
This faith in the extrapolated validity o f our presently accepted electromagnetics at high velocities makes the Einstein theory very different from other universal principles in physics. The law o f the conservation o f energy, for example, has been

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IN T R O D U C T IO N

verified in all branches o f physics and beyond — biology and chemistry, for example. If the kinetic theory of gases or even all of therm odynam ics were to collapse tom orrow , the energy conservation law would not budge, for it would continue to be supported by the orbits of the planets, the tides of the ocean, and the nitrogen-fixing bacteria in the soil. But if electromagnetics for high velocities were to be refuted tom orrow (and let me recall that historically, the Maxwell equa­ tions and the Lorcntz force grew out of a belief in an elastic, all-pervasive ether), the first thing they would take with them is the experimental evidence for the Ein­ stein theory.
Note that I am not complaining about the amount of supportive evidence for the Einstein theory; only a crank (and there seem to be plenty) would go to war against Einstein on that account. W hat I am complaining about is the narrow field from which this plentiful evidence is gleaned.
No length contraction has ever been shown on a well-defined, charged or uncharged body with well-defined dimensions and a velocity measured by several independent methods, if not directly; no time dilation experiment has ever pro­ vided proof that the changed rate o f the clock is only perceived by the moving observer and has not taken place in the clock itself.
The Einstein theory has never proved its two tacit postulates: that the MaxwellLorcntz electrodynamics, remain valid at high observer-referred velocities; and that the m otion o f m atter through a force field does not inherently — in­ dependently o f any observer — change its own force field.

^ ^ t this very objection also shows that the limited-validity criterion is not (or not yet) usable on the Einstein theory. W ithout an experimental refutation o f the theory, we do not know whether its limited validity is inherent, as it is in a virtual image, or whether it is merely due to our technological limitations in being unable to impart a sufficiently high velocity to anything but elementary particles.
Let us then examine another possibility for distinguishing between truth and equivalence when the difference cannot be established by full vs. limited experimental confirmation. For this purpose I have thought up the Grandiose Theory of the Railroad Track.
The rails of a railroad track appear to converge as they recede into the distance, as we have all seen with our own eyes; yet we all know that in reality they are reasonably parallel. The reason why nobody considers that a paradox, I suppose, is that we have learned from childhood to trust our mind and experience when our eyes deceive us — for railroad tracks if not for TV documentaries.
The explanation is “perspective” — the way in which images are projected onto the retina or onto the cam era’s focal plane. It is not terribly complicated, but it is not the simplest thing in the world, either: most o f us would rather pay for readyto-use perspective software than go through the chore of writing it ourselves.

IN T R O D U C T IO N

15

But my Grandiose Theory of the Railroad Track offers an alternative explana­ tion: lengths shrink with distance from the observer.
Now you and 1 know that this is an absurdity, but imagine some M artian Mole, who is intelligent, logical and erudite, but has no means o f remote sensing. Suppose he visits us and wants to know why humans perceive a railroad track as converging, and is given the two theories: perspective and distance-shrink. “1 use Ockham ’ s razor,” he might say, “and I buy the shrinkage theory. O f the two, it is by far the simpler.”
Don’t try to use measuring rods; they contract as they are carried away from you along the track; and don’t go there with the measuring rod yourself, because the track will shrink behind you.
A closed loop with an interferometer? No: the wavelength shrinks with distance from the observer — that’s why railroad tracks are notorious for the absence of fringe shifts.
But if the wavelength changes without producing a Doppler effect, the fre­ quency o f the light must have changed, you say. O f course it has; have you never heard of time dilation?
You install a second beamsplitter and interferometer (plus observer) at the far end o f the loop, proving that the distance between the rails is the same at both ends of the track at the same time.
But you have proved no such thing. The wavelength is shortened away from the observer: it shrinks for one this way, and for the other that way, and each observer observes, from his own point o f view, the same outcom e o f a different process. T hat’s what modern physics understands by “relativity;” and whatever measuring instrument we may use is subject to the same perversion as the quantity it attem pts to measure.
Now suppose the theory could not be disproved experimentally; how would we know it is absurd?
To some extent, o f course, the flaw in the G randiose Railroad Theory lies in the fact that, like the Einstein theory, it is not tied to nature itself, but to the observer or instrum ent that measures it. If 1 had tied the contracting distances to Grand Central Station, you would not need an interferom eter to disprove it; you could go uptown and jum p across the tracks.
However, observer-dependence in itself need not be flawed. Velocity is observer-dependent; it has no meaning unless we specify with respect to what stan­ dard of rest we measure it. Some functions o f velocity — such as the Doppler effect — must necessarily be observer-dependent, too.
In the Dialogues on Two World Systems, Salviati, fronting for Galileo, took great pains to persuade Simplicio, representing Aristotle and the Church, that the path of a stone dropped from the mast of a moving ship would appear oblique to a stationary observer on shore, though it would hit the deck at the same distance from the mast as when it was dropped ([Galileo 1630], pp. 142-144 o f the Eng­

16

IN T R O D U C T IO N

lish translation). There is no flaw per se in certain quantities being observerdependent.
But not everything is observer-dependent. Note that in Galileo’s example the velocity vector is observer-dependent, but the distance (from the mast) is not. Surely space and time must be the same for all objects dwelling in them if any con­ sistency is to be preserved; and our measuring standards, if they are to be stan­ dards, must not be subject to the fluctuations of the quantities of which they are supposed to be standards.
Both the Einstein and the railroad track theories break that rule, and they do so in the particularly critical case o f space and time, which are something special in that together with mass, they are used to define velocity, m omentum, acceleration, force, and progressively higher concepts. But space and time themselves cannot be defined; if they could, any non-circular definition would have to involve a more primitive concept still. When a philosopher says that time is “that which flows from future into the past” he is using descriptive lyrics, not a one-to-one mathematical definition.
Assuming that the railroad track theory could not be disproved by direct experiment, it could be recognized as (at best) an equivalence by its tampering with the fundam ental, and hence undefinable concepts on which everything else is built; and this tam pering with the primitive fundam ents is what I propose as a second guideline for discerning truth from equivalence. M athematics is perfectly free and unfettered by experimental observation to define its axioms from which it deduces their consequences; physics, if it is to understand the real world, must build on the two primitive and undefinable pillars. It must not tam per with them in order to accom m odate higher concepts. It must not redefine the undefinable; more par­ ticularly, it must not make the primitive pillars observer-dependent.
Note that this proposal has nothing whatever to do with “absolute” space or “absolute” time. We are still free to choose the origin o f our coordinate system in both space and time where we please. Tor there is no evidence o f any system being more privileged (though it may often be much m ore convenient) than any other. And most certainly we need not give up the Principle of Relativity.

T h e Einstein theory, then, may not turn out as general as experiments relying on presently accepted electromagnetics m ake it appear; and it defines the undefinable primitives space and time via the higher-order concept of velocity, arguably making the definition circular, and certainly making the two primitives observer-dependent.
But there is a third point that makes it highly suspicious: One o f its two postulates may be inherently irrefutable.
A theory may be irrefutable because it is true; or it may be irrefutable because it is inherently protected against refutation, even though it may be false. A crude

IN T R O D U C T IO N

17

example o f a theory that is close to irrefutable, but patently untrue, would be the claim that the earth has a second moon, made of a material that becomes perfectly transparent when illuminated.
The Einstein theory rests on two postulates. The first is the Principle o f Rela­ tivity, known for more than three centuries, with which few reasonable men will quarrel. But the other, known as the Second Postulate, postulates a constant velocity of light independent o f the state o f motion o f the emitting source (and therefore, by the relativity principle, also independent of the state of motion of the receiver).
With respect to what? In the Einstein theory, with respect to the observer: if two observers move with different velocities with respect to the same source, each measures the same velocity o f its light. This is not only sharply different from what we are used to with low velocities, but plays havoc with space, time and simultaneity. The usual explanation for this bizarre postulate is that there is no reason why we should expect high velocities to add in a m anner linearly extrapolated from our experience with low velocities.
But the Second Postulate violates a lot more than unimaginative thinking; in­ deed, it violates a lot m ore even than the tim e-honored concepts o f space and time.
Imagine that the Second Postulate were valid, on some planet in a distant galaxy, not for light, but for water squirted from a fountain in periodic pulses act­ ing as time signals. No m atter whether you stood still, ran with the water or against it, you would always measure the same velocity of the water with respect to yourself.
W ould this have to be a planet where space and time are something quite different from what we are used to? Not at all: it would have to be no more than a planet on which nothing moves faster than the water squirted by the fountain (with standardized velocity and pulse frequency). You would then set c in the Lorentz transform ation equal to the velocity o f the water and proclaim it a uni­ versal constant — and the Lorentz transform ation will do the rest, for it will so distort space and time that it will force the w ater postulate to be “true,” i.e., agree with measurement.
All measurements would keep confirming the water postulate beautifully due to Einstein’s theorem for the addition o f velocities as long as only velocities slower than that o f the water are used. Suppose, for example, that this imagined planet is inhabited by highly intelligent beings who are, in our vocabulary, deaf and blind, and the water squirted from the fountain in their National Bureau o f Standards is the fastest thing they know. The theory would be much acclaimed, because it predicts everything correctly in spite o f its bizarre water postulate.
But there is a flaw: the theory is revealed as incorrect one day when a scientist discovers the microphone and makes sound dectectable by his people’s senses. He uses sound signals to measure distances, time intervals and velocities, and the sham-theory will now predict imaginary velocities.

IN T R O D U C T IO N
But not all is lost. The physicists o f the planet simply amend the theory and set c in the Lorentz transform ation equal to the velocity o f sound (in air at 0 ° C and 1000 m bar pressure). The Second Postulate now checks out beautifully for every velocity up to that of sound; but one day a scientist discovers the photocell and the existence of light, and the amended theory is refuted by the velocity o f light signals.
So they amend the sham -theory once more and set c in the Lorentz transform a­ tion equal to the velocity of light, and what do they get?
The Einstein theory in its full glory. Perhaps you can now see what 1 am getting at. If we define space and time to cater to a constant velocity o f water, the theory is refuted by sound signals; if we define space and time to cater to a constant velocity o f sound, the theory is refuted by light signals; and if we define space and time to cater to the constant velocity of lig h t.. . but there is nothing faster than light. This implies that the Second Postulate may well be something that is not inherently true, but that is merely protected from refutation by the lack o f a “messenger” velocity faster than that o f light. This possibility — and with it the possibility that the Einstein Theory is merely an equivalence — gains weight when it is realized that the Second Postulate (from which the Lorentz transform ation immediately follows) has never been dem onstrated by direct experiment.
'T h e r e is an o th e r point o f interest associated w ith the logical flaw, alleged or genuine, o f tam pering with the fundam ental concepts and in effect defining them by highcr-order concepts — not to m ention points o f built-in irrefutability. As I will point out below , there is, 80 years after the E instein th eo ry m ade its appearance, a sizable com m unity o f scientists w ho have not accepted it. A nd there is a far larger gro u p o f scientists w ho feel a p ronoun ced distaste fo r it, th o u g h they shrug it oil' and accept the theory because there is no viable alternative. (M ost scientists, o f course, are in a third group: they never get deeply into the Einstein theory and “accept” it as 1 accept the theory o f the genetic code an d o th e r theories o utside my expertise.) It is my belief that this distaste stem s from the o p position, conscious or not, to tam pering with fundam ental concepts such as time and sim ultaneity.
But no, we are told, the reason why people have difficulty with the Second Postulate, and hence with the rest o f the Einstein theory, is quite simple. W hat prevents a few cranks, mavericks and flat-earthers from accepting such an unorthodox view o f space and time is their inability to accept anything that is different from the world they are used to.
Then why are there no cranks rejecting the existence of atoms that nobody has ever seen? Why are there no “underground” scientific journals doubting the vali­ dity of thermodynamics? (There are several doubting the Einstein theory.) Why

IN T R O D U C T IO N

19

does the quantization of energy raise no hackles in a world in which all energy varies smoothly from a fly’s sneeze to a 100,000 megaton-equivalent volcanic eruption?
It is, o f course, the exact opposite that is true: not only physicists, but people in general love phenomena that are quite different from the world they are used to. They spend hard-earned money for a toy gyroscope just to see it balance on a piece o f string when it really “ought to” fall off, and they are doubly fascinated when they see that it is no swindle. W hat they do not like is being asked to abandon reason: they grow wary when they sense a logical flaw.
They would be offended by a theory defining a straight line in terms o f a rect­ angle, especially if its area is dependent on the state o f the student contemplating it.

T h e r e is a counterexample to people’s wariness o f logical flaws: the Principle of Relativity itself, which never has any trouble being accepted. It is quite misleading to call the Einstein theory “the” theory o f relativity, a name that I will not use. Ein­ stein did not discover the principle, which was known to Galileo, though he did not
explicitly state it. It was explicitly stated, though not under that name, by Newton in the Principia: Corollary V, Book 1, says Corporum data spatio inclusorum iidem sunt motus inter se, sive spatium illud quiescat, sive moveatur idem uniJ'ormiter in directum sine molu circulari — “The m om enta o f the bodies included in a given space are the same, whether that space is at rest or whether it moves uni­
formly in a straight line without rotation.” There was no electromagnetics then; all of physics (then called “natural
philosophy”) consisted o f mechanics and optics, the latter — in either the corpuscle or wave theory — considered to obey mechanical laws. Since all o f mechanics can be reduced to m om enta o f bodies, Newton’s statem ent surely is an explicit 17th cen­ tury form ulation o f the relativity principle, which is today often stated as “the laws
of physics hold equally well in all inertial fram es.” Newton’s belief in a system o f absolute rest, based on considerations of
accelerated (rotational) motion may have been unnecessary, but it did not contradict the principle of relativity valid for uniformly moving systems (inertial
frames) which he had thus stated.
Let me take this opportunity to dispel another myth, namely that Einstein’s theory con­ tradicts Newton’s Laws. The statement that force equals mass times acceleration was put in Newton’s mouth posthumously: there is no place in the Principia where Newton makes such a statement. He always writes about the rate o f change o f momentum (mutatio motus, or “change o f m otion,” the latter defined as the product quantitatis materiae et velocitatis). In present notation — the Principia make their case by geometry — Newton never took the m out o f the parentheses in d(m v)/dt, for he was too careful a man to ignore the possibility that inertial mass might be variable. When Einstein introduced velocity-dependent mass explicitly, he did not have to change one iota in Newton’s Laws o f Motion for any part o f his theory; that he developed it in contradiction to them is one o f the numerous fables surround­ ing the Einstein theory. (Newton’s law o f gravitation is not, o f course, one o f the three Laws o f Motion, nor does it have their generality and fundamental significance.)

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But how relativistic is the Einstein theory, “the” theory o f relativity?
If the laws o f physics are conserved in all inertial frames, one would expect that it makes no difference whether an electric charge moves through a stationary magnetic field or a magnetic field sweeps past a stationary charge. The reason for this expectation, I submit, is our unperverted subconscious which says that magnetic fields and charges interact all by themselves, without the benefit o f observers. But that is not what the Einstein theory says. A charge moving through a uniform magnetic field is acted on by a force; but a moving uniform magnetic field (which has no space or time derivatives, and therefore cannot induce an elec­ tric field) does not affect a stationary charge, for “moving” and “stationary” is defined with respect to the observer, not with respect to the field.
Even stranger, in the Einstein theory a moving charge does not act with the same force on a stationary charge as the stationary charge acts on the moving one. (The observer “sees” the moving charge with its electric field intensifiied in the direction perpendicular to the velocity, but the field o f the stationary charge is unmodified.) Action and reaction are therefore no longer equal and opposite when the charges are interacting at a distance and not actually colliding at one point in “space-time.” Only erudite Einsteinians are aware o f this, and their answer is “So w hat?”
So what we have, if we believe in an objective reality unchanged by observers’ perceptions, is a theory that fulfills the principle o f relativity by distorting space and time in order to enforce the validity o f laws expressed in terms o f observerreferred velocities.

\ ^ ^ i y , then, have scientists universally accepted the Einstein theory?
They haven’t. Most scientists have not studied it beyond a freshm an course. Among those who have, most do accept it without reservations. But some turn away in queasiness — and in silence, for they have nothing better to turn to.
It is true that am ong the heretics there is a sizable percentage o f cranks and simpletons; but there are others. There are those who lack mathematical training and simply feel that the Second Postulate does not square with com m on sense. And there are also, to this day, some rebels of academic standing whose grumblings can occasionally be heard in public.
Louis Essen, director (now retired) of the Time and Frequency Division of Bri­ tain’s prestigious N ational Physical Laboratory, and a physicist o f international renown, writes “A com m on reaction o f experimental physicists to the theory is that although they do not understand it themselves, it is so widely accepted that it must be correct. 1 must confess that until recent years this was my own attitude.” His analysis [Essen 1971] finds the theory self-contradictory.

IN T R O D U C T IO N

21

Prof. Thomas G. Barnes, Professor Emeritus of Physics at the University of Texas, writes “It is time to return physics to a philosophy that puts physical reasoning ahead of blind faith in relativistic concepts that lead to nonsensical contradictions.” [Barnes 1983.]
The late Herbert Dingle, Professor of the History and Philosophy of Science at University College, London, was originally an enthusiastic supporter of the Ein­ stein theory, but in his study o f the theory he found flaws and turned against it in numerous articles and a book [Dingle 1972].
Burniston Brown, retired Reader (Associate Professor) in Physics at University College, London, is the author o f a recent book [Brown 1982], which makes the case for retarded action at a distance as an alternative to the Einstein theory. (So does the present book, but giving more emphasis to the effects of the aberrational component of the retarded force.)
These are but four o f a sizable list o f contem porary or recently deceased Ein­ stein critics, and no offense is intended to those not listed here.
But perhaps no less impressive are the names o f some o f Einstein’s opponents in his own time, and I do not mean the “natural” enemies o f any new theory — the mediocre fossils who are threatened with having to unlearn a lifetime’s investment in the old theories. I mean the names o f those whose work is closely associated with the theoretical basis or experimental verification of the Einstein theory, but who — and this may come as a surprise to m any — vigorously opposed it.
Hendrik Antoon Lorentz, author of the Lorentz Transformation, would have nothing to do with the Einstein theory and opposed it until his death in 1928.
Herbert E. Ives o f the Ives-Stilwell experiment not only seethed in his personal correspondence over Einstein’s contradictions, “guesses” and “hunches,” [Hazelett and Turner, 1979], but also had the stature to be given space for his heretic attacks on the Einstein theory in established scientific journals as late as 1953, the year of Ives’ death. A nd the incom parable A lbert A. Michelson o f the Michelson-Morely experiment remained doggedly faithful, until his death in 1931, to the “entrained ether” theory (with which, indeed, that experiment was perfectly compatible) .1

\ ^ i y , then, can objections to the Einstein theory be published only in the “underground” scientific press?
Because they merely show that there may be something radically wrong with the theory; but they have no full substitute to offer.
1 R eferring to the experim ent, M ichelson is said to have quipped “ I created a m o n ster.” I have found no confirm ation o f this in the tw o biographies o f A .A . Michelson that I have read, one of them by his daughter [Livingston 1973]. O n the other hand, both books are som ewhat apologetic about his refusal to accept the Einstein theory, and it could be that they did not w ant to throw even m ore “b ad ” light o n him . But if the rem ark is apocryphal, it is well invented.

22

IN T R O D U C T IO N

In a world where every possible experimental verification has shown uncanny agreement with the results predicted by the Einstein theory, such approaches will only get you a smile and a shoulder shrug. To beat the Einstein theory, it is not good enough to provide an alternative that does equally well; you have to show that it can do better.
Can it be done?

Part One
Einstein Minus Zero

Sec. 1.1

25

1.1 The Static Inverse Square Law

Consider Newton’s Law o f G ravitation. For low relative velocities o f the two interacting bodies (“low” velocity meaning here, and in the rest of the book, negligible com pared with the velocity o f light) it is quite uncontroversial; in polar coordinates, with the origin at the center o f mass o f one o f the two bodies, and r0 the unit radial vector, it says that the force between the two masses is

„

r m i rri2

Fg=

— r„

(1)

where T = 6.67 x 10' 11 N m V kg2 is the gravitational constant, and the rest o f the form ula, in fact the rest o f this book, is also in SI units; the minus sign says that the force is directed against the unit vector r0, i.e. attractive.
The inverse square o f a distance from a point is indicative o f something — a force — em anating from a source at that point. We will assume that it propagates with the velocity o f light c. We know from experience that in the electric analogy o f ( 1) this is the case: for example, if we remove (discharge) the charge, the removal o f the force at a distance r is delayed by a time r/c. We assume (with Einstein and practically every other gravity theoretician) that the same holds for gravity: that if we were able to “dismass” a mass as we are able to discharge a charge, then the result o f this (or any other) modification would reach the field at a distance r only after a delay o f r/c, the disturbance o f the field traveling out­ wards with a velocity c.
This is quite a conventional assum ption. It not only emerges from the Einstein theory, but it was also made by the late 19th century classics; in fact, it was made even earlier by Pierre Simon Laplace himself in Book 10, C hapter 8, o f his Mecanique celeste (publ. 1799-1825). W ith no electromagnetism to go on, Laplace could not have foreseen that the velocity of propagation was that of light, but he explicitly worked with a velocity of propagation of the gravitational force.
The velocity c with which the force propagates from its source is measured with respect to the source, and this again is uncontroversial, for there are only two static bodies, and the interaction is that o f one body in the field o f the other. For

26

EINSTEIN M INUS ZERO

Sec. 1.1

this static case all theories — propagation, emission, ether, and the Einstein theory
(with the observer located on one of the two masses) — yield the same result, and
there is nothing substantial to determine the deeper nature o f the mechanism that
transmits the force with velocity c.
This deeper nature will not be needed in the following; nevertheless, it is intriguing to contemplate the product of the masses (charges) in the Newton-Coulomb Law. This implies that gravitational or electric attraction is a force quite unlike, say, the force stretching the rope in a tug-of-war, where the tension is proportional to the sum of athletes on each side. Masses and charges are evidently not team players: they interact individually, each particle of one body with each particle of the other.
A possible interaction that exhibits such uncollectivist behavior is a wave emitted by one source and interacting with all similar sources struck by its wave fronts. If such waves are the solution of the wave equation for the force field, they need no ether or particle flow or medium in which to propagate (though they do not contradict any of them); they are simply a wave motion of force in unspecified form.
Analogously, the Coulom b Law for the force between two electric charges is

F, =

(2)

47r<or i

where e0= 8.854 x 10" '2 F /m is the free-space permittivity; if the charges are ele­ mentary (electronic, q= 1.6 x 10*19 C) but have opposite signs, then the constant K = 2.32 x 10~28 N m 2.
This force again propagates outw ard from its two sources with velocity c , and this time we may regard the hypothesis as experimentally verified, at least to the extent that the force ceases to act with a corresponding delay when its source is removed.

Sec. 1.2

27

1.2. The Velocity of Light: With Respect to What?
W hen light is emitted by a >ource moving uniform ly through a vacuum, its velocity is constant; but with respect to what?
W ith respect to all observers, regardless of their velocities relative to the source, says the Second Postulate o f the Einstein Theory. This is today the generally accepted answer despite the absence o f a direct proof and despite the objections pointed out in the Introduction.
Before the advent o f the Einstein theory, it was generally believed that light pro­ pagated in an all-pervading “luminoferous” medium, the ether. The velocity of light was constant with respect to the ether, just as the velocity o f sound is con­ stant with respect to the air in which it propagates, even though the source and the observer might be moving with different velocities with respect to the air.
There were, however, two varieties o f the ether theory. In the first, the earth and other objects moved through the ether without affecting it, so that the velocity of light with respect to an observer moving through the ether was c - v, where v was the velocity o f the observer, and both velocities were measured with respect to the ether.
In the “entrained” ether theory, the earth dragged the ether in its neighborhood along as it moved round the sun. The velocity o f light would therefore be constant in all terrestrial laboratory experiments (including those made with starlight), since the ether was at rest with respect to the laboratory.
In the “ballistic” theory o f light, whose main exponent was the brilliant young Swiss physicist W alther Ritz (1878-1909, died at age 31), it was assumed that the velocity o f light is constant with respect to its source, like bullets from a machine gun on a moving train. It did not need an ether.
The alternative to the Second Postulate that I will work with is that the velocity o f light is constant with respect to the local gravitational field through which it propagates.
The reason for this assum ption is the reason for all assum ptions in physics: it is supported by all the available experimental evidence and contradicted by none — as I hope to show in the following sections.
Let me first explain w hat is meant by “with respect to the local gravitational field.” As in any other conservative vector field, any point o f a gravitational force field is defined by the line o f force and the equipotential passing through it; its coordi­ nates can therefore serve as a standard o f rest. This approach will yield the correct result, though it throws no light on the physical mechanism involving it. Alternatively, we may think o f light as a disturbance o f the gravitational field itself (something like sound, which is a disturbance o f a pressure field); this will again yield the correct result, but there is no evidence whether this is a physical reality.

28

EINSTEIN MINUS ZERO

Sec. 1.2

The “local” refers to light propagating through gravitational fields moving with respect to each other, as is the case for the planets, the sun and the stars. If the sun is the rest-frame, light from a terrestrial source would first move with a velocity c + v (where v is the orbital velocity o f the earth, about 30 km /sec) in the dom i­ nant terrestrial gravitational field, and then with velocity c in the rest frame. In the transitional region there would be a transitional velocity, marked by the pro­ perties of most transients: difficult and of secondary importance.
Beyond this simple consequence o f Galileian relativity, the experimental evidence (bend­ ing o f light rays in a gravitational field) suggests that the velocity o f light varies with the intensity o f a gravitational field; this is not surprising, since all cases o f wave motion show a velocity dependence on the properties o f their environment (the index o f refraction). It is, however, a minor point to which we will not return until Sec. 1.11.
There is also hard experimental evidence that the velocity o f light remains constant with respect to the earth’s gravitational field, but not with respect to the earth rotating in it; this will be discussed in Sec. 1.3.7.
The assum ption that the velocity o f light is constant with respect to the local gravitational field is one that may raise many hackles as a conceptual form ulation, but as an experimental fact it is not at all absurd:
hirst, it satisfies the relativity principle without attem pting to redefine space and time. Like waves on the water of a stream flowing into a river and into the sea, light travels with different relative velocities through a vacuum in the terrestrial field, through that in the solar field, and through that o f the fields that lie beyond; none o f them is privileged or at absolute rest. If inertial frames are related to each other by the Galileian transform ation, and time flows at the same rate in all of them , the laws o f physics will hold equally well in all o f them, as will be shown for optics and electromagnetics in the following, and as is surely obvious for the velocity o f light by itself, without regard to its electromagnetic nature.
Second, there is a rarely noted, but nevertheless firm, precedent o f an electro­ magnetic quantity that depends on a velocity with respect to uncharged matter (the source o f gravitation). It is the magnetic field, not as it appears in thought experiments by this or that theory, but as it is measured in the macroscopic world. It is too weak to be measured unless the electric field o f the moving charges is first neutralized, as is the case when a current flows in an overall neutral conductor. This is no new assum ption, but a consequence o f perfectly orthodox (including Einstcinian) electromagnetics, as will be pointed out in more detail in Sec. 1.4.
Third, this assumption cannot experimentally contradict the Einstein theory, for no observer or measuring instrument has ever traveled through a gravitational field with a velocity com parable to that o f light — certainly not in uniform , recti­ linear m otion. In the cases where the motion was rotational, i.e. in Sagnac-type experiments (rotation of a double interference loop), the evidence supports both the present assum ption and Einstein’s general theory for rotating systems.
Thus, in all optical experiments supporting the Einstein theory, the observer was always nailed to the gravitational field of the earth; on the other hand, the

Sec. 1.2

VELOCITY OF LIG H T

29

Michelson-Gale experiment utilizing the earth’s rotational velocity (Sec. 1.3.7), which did register a fringe shift, is explained by the Einstein theory as a Sagnactype experiment (an argument that can also be used for satellites). In these cases using the earth’s rotational velocity, both assum ptions lead to the observed result; however, Einstein’s general theory, valid for accelerated frames, is mathematically so complicated and physically so opaque that only a comparatively small circle of specialists has mastered it.
Let us then quickly run through the crucial experiments of a purely optical character, that is, those that m ake no use o f the electromagnetic nature o f light, but treat it simply as something that moves with a m easurable velocity and that is capable o f interfering with itself. These are the experiments that do not in any way rely on electromagnetic inferences — such as those based on the tacit assum ption that the expression for the Lorentz force remains valid at high, observer-referred velocities. This group of purely optical experiments thus excludes those involving charged particles.
The second, electromagnetic type o f experiment can also be characterized by
another property: it always involves the square o f the quantity 13 —v/ c, whereas
the purely optical experiments are most often limited to first-order observations, i.e. to observing quantities depending linearly on /3.
This makes the purely optical evidence not only more easily obtainable, but also less dependent on possibly flawed conclusions, and we shall examine it first.

30

Sec. 1.3.1

1.3. The “Purely Optical” Evidence
Among the experiments that treat light simply as something propagating with a m easurable velocity w ithout reference to its electromagnetic character, we will examine the crucial ones performed with moving sources (including moving m irrors and moving media o f transmission). By “crucial” I mean those helping to support or reject one of the four competing theories — ether, ballistic, gravita­ tional, or Einstein’s Second Postulate.
1.3.1. Aberration
In 1728, Jam es Bradley (1692-1762), then Savilian Professor o f A stronom y at O xford, sent the A stronom er Royal (N ewton’s good friend Hailey) an Account o f u new discovered motion o f the Fix’d Stars, noting that a star in the constellation of the Dragon crossed the meridian more to the south in the winter of 1725-26 than in the preceding and following summers, an effect that could not be explained by parallax. 1 The effect, called aberration, is reminiscent o f vertical rain leaving slanted tracks on the side window of a traveling car: while the star light travels through the telescope with velocity c, the telescope moves forward with the earth’s orbital velocity u (about 30 km /sec), so that the ray passing through the telescope makes an angle o f aberration e with the true direction of the star.

O O/

S' S

f a)

(b)

A berration: (a) general geometry, (b) wave theory (see also p. 201)
Bradley’s discovery was erroneously interpreted as a victory o f the ballistic theory o f light over the wave theory, probably for two reasons: the explanation by the ballistic theory (corresponding to the rain on the moving window) is much simpler; and Newton’s criticism o f the wave theory was misinterpreted as approval
' Phil. Trans, vol. 35, p.637 (1728). W hittaker [1910/62, p.94] notes that Roemer (the first to m easure the velocity o f light, using Ju p iter’s m oons), in a letter to H uygens dated 30 D ecem ber 1677, suspected the apparent displacement of a star and gave the correct explanation, thus preceding Bradley by half a century.

Sec. 1.3.1

ABERRATIO N

31

of the corpuscular theory. In fact, as those who have read the Opticks know, Newton refrained from endorsing either.
In reality, Bradley’s discovery was not an experimentum crucis, for it can be explained satisfactorily by any one of the four theories. However, aberration plays a significant role in the theory to be proposed, so we will review it for later reference.

If c and v are, respectively, the velocities o f light and o f the object on which it is incident, both referred to the fram e in which the source o f light (or force!) is at rest, then by the ballistic theory, which treats light as it would machine gun bullets, we find the aberration angle e by resolving the velocity o f light in the telescope system into the direction o f v in the star (E) system and into the direction perpen­ dicular to it:

c. s in a

sin a

tan(a + r) = —v + c cos o = T[1T+ cos o

(*)

where 3 - v/c. After elementary manipulations this yields

B sin o

t a i u = _ 71 T+ 7/1i co s o

-

(2 )

or neglecting second-order terms in j3 and e we have approximately

( % —ft sin o

(3)

The negative sign means that the aberration subtracts from the angle a and therefore deviates toward the direction o f the velocity. However, this is true only o f light or other agents that are emitted from a source. We shall soon have occasion to consider the aberra­ tion o f an attractive force, such as Coulomb’s or that o f gravitation, that is directed toward the source (which might more accurately be called a sink). In that case the aberration is positive, so that it deviates away from the direction o f thevelocity. This is immediately apparent by notingthat the attraction by the sun S in thefigure is,as far as the geometry of aberration is concerned, equivalent to the emission o f light by the fictitious star £ .
The wave theory can, o f course, do equally well, for the phase fronts or planes
o f constant phase in a system in which the star is at rest are given by

$ = u ( - k ■r

(4)

where r is the position vector based at an origin fixed somewhere on the earth’s orbit, u is the angular frequency o f the light, and k is the propagation constant with scalar value

k — —Ld = —2n

(5)

c A

If k is oriented as in the figure, we have

<f> = cut + kx cos a 4- ky sin n

(6)

32

EINSTEIN MINUS ZERO

Sec. 1.3.1

To find the direction o f the phase fronts, we set <F- oof = const (that is, we look for the locus o f a constant phase 4> at a fixed time t ), yielding the family o f planes

y = —x co t a + c o n st

(7)

which is not surprising, since the phase fronts are perpendicular to the direction of propagation _y= x ta n a .
But substituting t = x / v in (6), we find the family o f phase fronts as

u<P/

= x(l + fl cos a ) + fly sin a = const

(8)

U)

or

X + f l COS ft

y =

f—l Sill ft 1- const

(9)

Comparing (7) and (9), we see the factor multiplying x in (9) plays the role of the cotangent of the same angle, namely the aberration angle by which the planes of equal phases shift when t = x /v instead of t= const. Hence

1 + f l COS f t

cotf =

f~lfTS~'ill ft

( 10)

which is identical with (2). The same derivation is valid for the gravitational field theory. There is an aber­
ration as the ray from a star enters the gravitational field o f the sun if it has a velocity with respect to the star; and there is a further aberration as the ray passes from the gravitational field o f the sun to that o f the earth — for simplicity 1 am replacing a continuous transition from the dom inance o f one field to the other by a sharp discontinuity, assuming that a more careful treatment would introduce time-consuming details, but no substantial modifications. We are then back to the preceding derivation: we simply take the entire blob representing the earth’s gravitational field instead o f considering it a point as above. This will be discussed in m ore detail in connection with Airy’s experiment below.
The aberration formula derived by the Einstein theory agrees with (1) to first order in fl. (For the orbital motion o f the earth, f l - 10'4 ; second-order verifica­ tion would thus require the measurement o f an angle with a precision o f 1 in 100 million — supporting the assertion in the introduction that the crucial verifications o f the Einstein theory always rely on electromagnetics, the only field where signifi­ cant values o f fl2 are achievable.) It is derived directly from the Lorentz transfor­ mation and is a property o f the different values observed in the space-times o f the two inertial frames.
However, here and in the following I will not reproduce the Einsteinian deri­ vations. There is an abundance o f available books making the case for the Einstein theory, and the observant reader may have detected that this is not one o f them.

Sec. 1.3.2

F R E S N E L ’S C O E F F IC IE N T

33

1.3.2. Fresnel’s Coefficient of Drag
Augustin Jean Fresnel (1788-1827) was a rare genius, whose work should fill one with humility not only because o f its volume and significance, but because of the pause-giving thought that virtually all the results derived by him remain valid to this day even though they were derived from the concept of an elastic, compressible ether.
This circumstance is striking enough in such cases as the reflection and transmis­ sion coefficients, diffraction formulas, and path clearance criteria without which contem porary microwave relay lines could not be designed. But it is almost uncanny in the case of the dragging coefficient for moving media: Fresnel derived it, without experimental evidence, from the idea o f a compressible ether being par­ tially dragged along by a body moving through it. Yet the form ula he thus derived in the “wrong” way has not only stood the test o f time, but it also had a significant effect on the acceptance of the Einstein theory: the only competing theory at the time, Ritz’s ballistic theory, was unable to provide an explanation for the experimentally confirmed dragging coefficient, whereas Einstein obtained it as the leading term o f the series resulting from his velocity-addition theorem. (The double-star argument against Ritz, apart from being flawed, had not yet appeared. On the other hand, Ritz apparently did not know about the work of Hoek, half a century earlier, whose “etherless” derivation of the drag coefficient he might have used as a defense.)
The Fresnel coefficient o f drag describes the velocity o f light in a moving material medium.
The refractive index n o f a medium is defined by

n ——

( 1)

Cm

where cm is the velocity of light in that medium when it is at rest. If now a transparent medium moveswith velocity v with respect toa fixed system of coordinates, thenthe velocityo f light in that medium,which is c /n at rest, is increased by an am ount 8v when measured with respect to the same system:

< " ' = —+ bv

(2)

n

where

,

is Fresnel’s coefficient of drag. This expression will be derived in Sec. 1.10.2 from purely electromagnetic con­
siderations using the Galileian transform ation; but in this non-electromagnetic survey it is im portant to note that neither ether nor Einstein are needed to obtain

34

EINSTEIN M INUS ZERO

Sec. 1.3.2

it; it was derived by Dutch astronom er M. Hoek on using the result o f a little known experiment that he performed more than a century ago [Hoek 1868]1
Hoek compared the velocity of light in glass and vacuum (air), orienting a glass rod east-west, so that one o f the two counter-running interference loops (see figure) passed through the glass in the direction o f the rotating earth (west-east), and the other against it. He found no fringe shift when he reversed the direction of the entire apparatus.
I repeated the experiment with laser light in 1970 at the Engineering Center of the University o f Colorado in Boulder, about 800 m north o f the 40th parallel, where the rotational velocity o f the earth is u = 355 m /sec, so that 0 = 1.18 x 10'6. Since laser light allows an optical path difference of thousands of wavelengths, this set-up allows the paths in air and glass to be com pared directly without a double

r

screen

H oek's experim ent 11868]

R epetition by Beckm ann in 1970

loop; moreover, the whole arrangem ent was m ounted on a styrofoam float in a tank of water, so that the fringes could be observed as the interferometer was rotated through 180°. The fringe shift after reversal was the same as H oek’s: none.
The experiments themselves are o f no great importance; H oek’s is not widely known, and 1 never published mine.
W hat is of fundam ental im portance, however, is the way in which Hoek [1868] derived the Fresnel drag coefficient without recourse to an ether. Most textbooks never mention Hoek, and imply that there are only two ways to derive the coeffi­ cient o f drag: Fresnel’s using an elastic ether, and Einstein’s using the first term in the power expansion of the velocity addition theorem.

1 A paper published by an obscure D utch jo u rn al in 1868 is not the easiest to ob tain , and 1 am greatly indebted to my friend and countrym an lr. Pavel Dolan, now of Rijswijk, Netherlands, for sending me a photostat.

Sec. 1.3.3.

F IZ E A U 'S A N D A I R Y ’S E X P E R IM E N T S

35

The essence o f H oek’s derivation (applied to my simpler experiment) is the following. If / is the length o f the glass rod (the other parts o f the path cancel without affecting the result) with refractive index n, and the rotational velocity of the earth is v, then the time difference o f the two phase fronts, with the light going east to west against the rotation o f the earth, is by Galilean relativity

I

I

AT =

—

(4)

c/n —ov —v c + v

and when the apparatus is reversed for the light to travel east to west, the time difference is given by the same expression with the sign o f p reversed. Since there is no fringe shift, the two A7s must be equal, and on neglecting second order terms in 0, we obtain (3).
Hoek’s double loop is subject to the same procedure, and again yields the Fresnel coefficient with nothing but classical relativity — without ether or Einstein.

1.3.3 Fizeau’s and Airy’s Experiments

Fresnel had derived (2) and (3) o f the

preceding section without experimental

evidence, and his hour o f trium ph came in

1851, 24 years after his death in 1817,

when Fizeau confirmed the formula by an

interference loop immersed in pipes run­

ning water with and against the loop

branches at a velocity of v - 1 m/sec. The

experiment is too well known to be de­

scribed in detail, especially since the dis­

cussion o f H oek’s derivation above shows that Fizeau’s experiment is no m ore crucial to support of the Einstein theory than sup­ port o f the ether. (Fizeau’s experiment preceded H oek’s, but because o f H oek’s derivation, 1 listed it first.) However, we

Fizeau’s experim ent (1851) Ssource of light, / interference loop running with the water, / / interference loop running against it. The length / was 1.5 m , the velocity o f the water 7 m/sec.

can well imagine how Fresnel’s confirmed prediction caused the scientific com ­

munity to be unshakably convinced of the physical reality of an elastic, partially

entrained ether that only cranky mavericks could doubt — just as today it is con­

vinced of the physical reality o f Einstein’s theory for exactly the same reason: have

its predictions not been vindicated by experiment?

Yet Fresnel’s ultim ate trium ph did not come until 1871, when Sir George Airy

(1801-92) performed an experiment that Fresnel, by then 44 years dead, had pro­

posed: repeat Bradley’s experiment using a telescope filled with water. By the

36

EINSTEIN M INUS ZERO

Sec. 1.3.3

ballistic theory, this would result in replacing c by c /n in (1), Sec. 1.3.1, and (3

by p / n ; when the effect o f refraction is considered together with that o f aberra­

tion, the aberration angle is found proportional to n2, which (for water) would

increase the aberration angle by 7.6°.

Fresnel, however, concluded that the aberration angle would not change: the

velocity of light would slow in the water, but the coefficient of drag in the water

moving through a stationary ether would increase the velocity, and in calculating

the aberration angle, the two effects would exactly cancel, so that the angle of

aberration would be independent o f the index o f refraction. He wrote to A rago in

1818, “Although this experiment has never been perform ed, I do not doubt that it

will confirm my conclusion.”1

Airy’s experiment found no change in aberration and once m ore fully confirmed

Fresnel’s concept o f an elastic, partially entrained, compressible ether. , a A'

To see that Airy’s experiment does

not contradict the gravitational-field

assumption, consider the analogy of

a water-filled submarine moving

through stationary water, represent­

ing the earth’s gravitational field mov­

ing through the sun’s. As before, we

substitute a sharp boundary (the walls of the submarine) for a gradual trans­ ition. Now consider the aberration of a sound signal. The calculation of Sec. 1.3.1 will show that there is in­

A nalogy o f A iry’s experim ent: in a water-filled subm arine moving through the water, the same ab erratio n will set in when sound waves are observed at the end C of a tube, no matter w hether the tube is filled with air or w ater.

deed an aberration if the submarine is adopted as a rest frame, since the walls of

the subm arine will reradiate the sound wave in the direction o f arrival from the

outside water. (If there are no walls marking an abrupt transition, the change in

direction comes about in a curve rather than a “corner” at B but makes no

difference to the end effect.) If we now slow the sound with an air bubble

representing the telescope (remember that unlike light, sound travels faster in the

denser medium), what will happen to the direction o f the sound ray in the bubble

inside the submarine?

Nothing whatsoever: the aberration has already changed the direction at the

interface of moving and stationary medium, and there can be no additional aber­

rational change inside the submarine. Indeed, if we could detect a change in direc­

tion, we would be able to detect uniform motion without looking out of the win­

dow or otherwise referring to a rest standard: we would resurrect absolute rest and

kill the relativity principle.

' Fresnel's derivation is now only o f historic interest, an d I do not w ant to waste space on it. Readers will find it well sum m arized in [W hittaker 1910], an d those w ho read R ussian will find a m uch sim pler description o f Fresnel’s argum ent in G .S . L andsberg’s O ptika (M oscow 1952), pp. 363-4, showing how Fresnel worked with a compressible ether.

Sec. 1.3.4

DOUBLE STARS

37

But would that not also protect the entrained ether theory from Airy’s experiment?
Yes, it would — if one were to introduce the entrained ether as a confusing and unnecessary synonym for “gravitational field.”

1.3.4 Double Stars and Other Objections to the Ballistic Theory

Consider the binary stars A and B revolving about a common center of mass and emitting light, as assumed by the ballistic theory, traveling at

h k— — -----I

velocity c with respect to its source. Then in a

system at rest with respect to (say) the common

center o f mass, the light emitted in the direction

m arked c by the receding star B is slower than that

of the advancing star A , but the latter has a handi­

cap h o f up to the m ajor axis o f its orbit. Its faster

light should, therefore, eventually catch up with its

V v

slower brother. This, it was claimed, would lead to several effects when these two stars are observed on

“Ballistic” light from star A in the direction c would

the earth: the orbits viewed in such light should deviate from Kepler’s laws [De Sitter, 1913], and

eventually catch up with the slower light from star B.

the Doppler-shifted spectral lines of their light should double and triple. But no

such effect has ever been observed, and this was used to reject the ballistic theory.

The evidence from laboratory experiments does indeed refute the ballistic

theory, as we shall see; however, the double-star argum ent is flawed for several

reasons, o f which only one is o f interest here, because it illustrates the

gravitational-field hypothesis. The alleged refutationtacitly assumes that thelight

emitted by double stars will remain constant with respect to the center o f the

double star from the moment of emission through the years or centuries of travel

until it arrives at the terrestrial spectroscope. But from the point o f view o f the

gravitational hypothesis this tacit assum ption is false: the two light rays will indeed

at first travel with different velocities, constant with respect to either star; but they

will soon stabilize at a com m on velocity as the gravitational fields o f the two merge

into one; the velocity will change to a different, but again com m on, value as the

light enters the next dom inant field on its journey, for it is still c, but now with

respect to a different field; and so forth until it enters the telescope o f a terrestrial

observatory, with no reason for any special effects.

There is, however, convincing laboratory evidence against the ballistic theory.

There are several experiments disproving the theory if it is assumed that the velo­

city o f light is not modified as it passes through lenses and is reflected by moving

mirrors like a tennis ball bounces off a racket — with twice the velocity o f the

moving m irror added to the incident velocity. Among the more convincing

38

EINSTEIN M INUS ZERO

Sec. 1.3.4

experiments are the one by Tolm an [1912] and the two by M ajorana [1917, 1918a and 1918b, 1919],
Tolm an [1910] observed the limbs o f the sun with a Lloyd interferom eter (a plane mirror causing interference with the direct ray near grazing incidence). The sun, at its equator, rotates with a circumferential velocity o f about 2 km /sec, so that by the ballistic theory the light emitted by the receding limb should have a lower velocity than that emitted by the advancing limb, and a fringe shift should therefore be observed on pointing the interferometer first at one limb, then at the other; but none was observed. The experiment was repeated with lavish equipment by Bonch-Bruyevich and Molchanov [1956], who were apparently unaware of T olm an’s experiment 44 years earlier, for they m ake no reference to it.
M ajorana [1917, 1918a] m ounted 10 m irrors on a rotating wheel with a circum­ ferential velocity o f up to 70 m /sec and let the light, after traversing the indicated path, pass into a Michelson interferometer with unequal arms. The simple ballistic theory should have shown a Doppler shift in addition to the one predicted by either the ether or the Einstein theory, and the “tennis ball” version should have shown a further shift due to the stationary mirrors. But only the normal Doppler shift was observed. The same result was obtained when the mirrors were replaced with active sources (mercury lamps) [M ajorana 1918b, 1919].
There is, however, one intriguing variant o f the ballistic theory that is less easily refuted: the “reradiation” version, by which mirrors and lenses reradiate the received (ballistic) light with a velocity c with respect to themselves, so that if they are stationary, the light transm itted or reflected by them propagates with the same velocity as predicted by either the ether or the Einstein theory. Since most inter­ ferometers, such as Tolm an’s or M ajorana’s, have a lens at their entrance point, they become useless for refuting this theory. It fell to the G rand M aster o f experimental optics, Albert A. Michelson, to devise an experiment that tested and refuted all versions o f the ballistic theory in one brilliant swoop (Sec. 1.3.6) — and I do not mean the well-known Michelson-Morley experiment, which does no contradict any of these versions.

Sec. 1.3.5

39

1.3.5. The Michelson-Morley Experiment
There is no need to go through the well know n Michelson-Morley experiment of 1881; it is available in any textbook, which probably also claims that it conclu­ sively disproved the existence o f an ether.
It did no such thing, o f course; it was perfectly consistent with an ether entrained by the earth, and M ichelson interpreted it that way, unabashedly holding to that view in his writings decades later, for example in his 1924 experi­ ment to be discussed below. T hat year, Tom aschek [1924] repeated the Michelson-Morley experiment with starlight, lending even more support to both the Einstein and the entrained-ether theory.
N or did it in any way refute the ballistic theory: it is obviously consistent with all o f its variants. A nd it is perfectly consistent with the gravitational hypothesis. To regard this experiment as a proof o f Einstein’s Second Postulate is one o f the ironies attached to it.
The im portance o f Michelson-Morley is above all historical, for its result stood in shocking contradiction to Airy’s experiment o f a mere 10 years earlier, which had put Fresnel’s partially entrained ether theory (the scientific community thought) on solid rock. The puzzle was to what extent the ether, whose existence nobody doubted, was entrained; and when Einstein and Ritz more than 20 years later each appeared with a theory that denied the very existence of an ether, many must have thought “a plague on both your houses.”
A part from its historical importance from this point of view, the experiment was also a milestone in that the M ichelson interferom eter used in it was the first that could have detected a fringe shift o f order /32 rather than only o f order /3. As used in 1881, the shift was close to the limits of detectability, but the technique was later perfected.
It is thus ironic that this experiment, which is perfectly consistent with four out o f the five theories discussed here, and which refuted nothing but the unentrained (or only partially entrained) versions o f the ether theory, should be held up in text­ books as proof of the Einstein theory or disproof o f classical physics.
But the saddest irony is that the nam e o f A lbert A. Michelson, in the minds of most Americans, should be linked only to this experiment, which was a minor gem in his stunning treasury. For M ichelson was a superstar in the field o f experimental optics; his wizardry has not been matched again by any single optician. His feats such as measuring the diameters o f distant stars by interferom etry have no place in this book, but even the two all but forgotten experiments discussed in the next two sections show the unm istakable hand o f the maestro. It is depressing that Americans should know no more about this man who at age sixteen buttonholed President G rant near the W hite H ouse to get into the Naval Academy (“I will make you proud o f me if I get the appointm ent!”) and who made good on his p ro ­ mise by bringing the United States its first Nobel Prize in 1907.

40

EINSTEIN M INUS ZERO

Sec. 1.3.6

1.3.6 Moving Mirrors
T o decide whether the wave or ballistic theory was correct, Michelson [1913] mounted two mirrors on the shaft of a m otor as shown in the figure and made

M ichelson’s experim ent with rotating m irrors [1913] C and D are mirrors m ounted on a m otor shaft rotating about O; the mirrors B
and E are stationary. S source, A beam splitter, T telescope.

them reflect the light in an interference loop. The length d from the stationary concave mirror at E to the m otor shaft O was 6.08 m, the distance between the mirrors was 26.5 cm, and the m otor speed could be varied continuously from 0 to 1800 rpm. Let V = c - r v be the velocity o f the light reflected from the m irror moving with velocity u, where r is an integer to be determined: if light pro­ pagates with constant velocity with respect to the laboratory, then r= 0 ; if the m irror acts as a new source, r = 1; if it is reflected like a tennis ball from a racket, r= 2. An elementary calculation then shows that the fringe shift is

_ v(Ti - T2) = kdc

A

vX

where X is the wavelength o f the light, the 7} are the time delays for the two directions round the loop, and k is 0, 4 or 8 for ballistic tennis-ball reflection, ballistic reradiation, and constant velocity with respect to the laboratory frame, respectively. The experiment gave k= 8, showing the velocity of light, within the error o f the experiment (about 2®7o) unaffected by the velocity o f the mirrors.
“Assuming that the effect is actually nil,” Michelson adds drily, “this interference method may be used to measure the velocity of light with an order of accuracy . . . of one part in 100,000.”
Gentlemen, meet Albert A. Michelson: the length o f his interference loop is more than 12 m, which is not trivial even with a laser; he kills three birds

Sec. 1.3.6

M OVING M IRRORS

41

with one stone (or rather kills two and gives lasting life to the third); and he walks away with a new, state-of-the-art measurem ent m ethod — all in one paper cover­ ing no more than 3 Vi small pages with large print. 1
Michelson’s little known experiment was the only one to refute the ballistic theory in its re-radiation version, but it was done in air. Therefore in 1964, Peter

Test of the “reradiation ballistic” theory in vacuum [Beckm ann and M andics 1965]. S slit, C L collim ator, M rotating m irror “accelerating” the light into Lloyd interferom eter A and through the window W. Length L was 4.08 m.
Mandics and I perform ed an experiment in a cham ber evacuated to 1O' 6 mm Hg [Beckmann and Mandics, 1964, 1965], The basic idea was similar to T olm an’s 1910 experiment (Sec. 1.3.4), but it was done with laser light, and the light was (by the ballistic hypothesis) “accelerated” by a m irror rotating in front o f the slit, so that there were no lenses that could have “slowed” the light by reradiation. The ballistic theory predicted a shift o f up to 0.7 o f one fringe, but in fact we observed no shift as the speed of the mirror was increased.
W ith hindsight, this was fortunate. By the relativity principle applied to the ballistic theory, it makes no difference whether the mirror moves in air or whether a wind blows against the m irror. In the latter case we can use the coefficient of drag (the refractive index o f air differs from 1 by an am ount o f order 10-3), and if evacuating the air had changed the null effect, it would either have ruined the rela­ tivity principle or established that the refractive index o f air is close to that of water.
W ith all this evidence against it (see also Sec. 1.3.4), 1 now consider the ballistic theory untenable. Other critics of the Einstein theory (see, for example, [W aldron 1977]) are still clinging to it, and I wish them luck, though 1 doubt they can under­ mine the Holy Grail from that direction.
1 There is, in my opinion, only one m an in the history o f experim ental optics to rival M ichelson, an d th at is, once again, Sir Isaac N ew ton. M ichelson w orked w ithout lasers, b ut N ew ton w orked with his bare hands. T he O plicks records his draw ing o f diffraction lobes clearly corresponding to those that Fresnel calculated three generations later with the aid of the integral nam ed after him. Today, these lobes are easily dem onstrated with a laser and a finely honed slit. W hat Newton used instead o f a slit was the edge of a kitchen knife; and w hat he used for a collim ated beam was the sun com ing through a small hole in the barn which he had otherwise blacked out — in rainy England of all places.

42

EINSTEIN M INUS ZERO

Sec. 1.3.7

There remains the puzzle of why, with the evidence available at the time, so many scientists either still adhered to a discredited ether theory, or accepted the Einstein theory rather than Ritz’s non-ether ballistic theory, which explained not only the propagation of light from moving sources, but also the decrease of electromagnetic force on a moving charged particle (increase in inertial mass), inertia, and gravitation (including the advance o f M ercury’s perihelion). Equally puzzling is the question why they accepted the flawed double-star refutation so easily.
Herbert Dingle [1960], a staunch adherent of the ballistic (Ritz) theory, thought that the double-star argum ent was accepted because scientists “were prepared to sacrifice almost anything rather than the electromagnetic equations, and a reason for shaking off a nuisance rather than a genuine test between two equally valid possibilities was what they sought.” T hat is close to what I will call the blunt side o f O ckham ’s razor: when two theories become incompatible (such as New ton’s and Maxwell’s), we sacrifice the simpler one, for we have a heavier investment of learning (and a cherished chunk o f snobbism) in the complicated one.
None o f which is to say that Ritz was right; I believe the experimental evidence against the ballistic theory o f light is now overwhelming (I do, o f course, use the same idea as his propagation des forces). It is, nevertheless, shameful that this genius is forgotten by all but a handful o f anti-Einsteinians: The W iley/Inter­ science Biographical Dictionary o f Scientists does not consider him worthy of inclusion among more than 1,000 scientists (such as Henry Ford), and a Soviet two-volume, 1000-page, fine-print encyclopedia of scientists (such as Trofim Lysenko) allots him 11 lines, mentioning only his work in spectroscopy and the “Ritz method” of solving variational problems.

1.3.7 The Michelson-Gale Experiment (1925)
The examples discussed above cover the main types of experiments on the velo­ city of light from moving sources or in moving media: there are more within the same type, but they do not essentially differ from the classes discussed above. All of them can be explained either by the Einstein theory, which assumes that the velocity o f light is constant with respect to everybody everywhere moving at any velocity, and by the hypothesis that it is constant with respect to the gravitational field through which it propagates. The agreement between the two is not surpris­ ing, because the “everybody everywhere” (my expression, not Einstein’s) has not been tested by anybody but observers at rest with respect to the earth’s gravi­ tational field.
There remains, however, a possible ambiguity: the earth’s gravitational field moves with the earth as it travels along its orbit; but does it also turn with the earth about its axis, or does the earth revolve within its own field? In other words, with respect to what is the velocity o f light constant on the rotating earth?

Sec. 1.3.7

THE M ICHELSON-GALE EXPERIM ENT

43

According to our hypothesis in Sec. 1.2, which regards the gravitational force as propagating from its source with a finite velocity, intuition suggests that the earth rotates within (with respect to) its own gravitational field, for once the force has been emitted, the earth rotates away under w hat it has emitted or what is propagated outw ard from it; the emitted or propagated agent is no longer under its control. Indeed, if the force remained under the control o f its source after being emitted, we would need an additional hypothesis, including the requirement that the controlling message travel faster than the originally transmitted force; and while I will readily accept velocities higher than c for observers moving toward a source through the local field, 1 know of no evidence of a velocity higher than c with respect to the local gravitational field.
Fortunately, we do not have to rely on such intuitive reasoning, for it is confirmed by another M ichelson sym phony. As we have seen in the case o f Hoek’s experiment, the use of a moving material medium thwarts the intent of any experi­ ment to measure a change o f the velocity o f light because the effect o f m otion is exactly canceled by the effect o f refraction — that is the essence o f the drag coeffi­ cient, and that is how H oek derived it from his experiment. (This applies, o f course, just as well to Airy’s experiment, which uses aberration, i.e. orbital m otion, rather than rotation about the earth’s axis.) The effect o f the earth’s motion can therefore be detected only by comparing light in vacuo (or air) to light

T he M ichelson-G ale experim ent [1925] is based on the ea rth ’s slower ro ta ­ tional velocity at higher latitudes. A n interference loop form ed by a spherical rectangle will therefore show a fringe shift (com pared with a rectangle with short east-to-west sides) because — in the classical conception — the velocity of light differs along the northern and southern sides o f the rectangle.
This can be done by exploiting the variation of the circumferential velocity of the earth’s surface, which rotates m ore slowly with increasing latitude: in a rect­ angular interference loop with east-west and north-south sides, the southern side o f the rectangle will (in the northern hemisphere) rotate slightly faster than the

44

EINSTEIN M INUS ZERO

Sec. 1.3.7

northern side. Michelson proposed the use of such an interference loop as early as 1904: If u, and u2 are the circumferential velocities o f the earth at the northern and southern sides of the (spherical) rectangle with sides /,, h, then the difference in time required for the two beams to complete the loop in opposite senses is

2I2V2

2liVi ^ 2(l?V2 - h v i )

m

ZV/ — 0

9

9

O ~

9

' '

c ~ v2 C ~ V\

C

so that after some elementary spherical trigonom etry the fringe shift is

4//iw siny2

f:A

(2)

where co is the angular velocity o f the earth’s rotation, I and h are the sides o f the rectangle, its geographic latitude, and X the wavelength o f the light.
The slow rotation o f the earth (00 = 73 m icroradians/sec) necessitates a large area Ih to give an appreciable fringe shift; and that in turn means taking the loop outdoors into the turbulent atmosphere. The experiment was not tried until 1923 near the Mount Wilson observatory, but the atmosphere and the resulting unsteadiness o f the fringes were too much even for an artist like Michelson, who had hoped to save the expense of evacuated pipes. Using such pipes, the experi­ ment was carried out successfully two years later, in the winter of 1924-25, at Clearing, Illinois [Michelson and Gale 1925],
The rectangle measured 2,010 feet from east to west and 1,113 feet from north to south, and was formed by straight and level 12-inch pipes connecting the four concrete boxes containing the mirrors and beamsplitters; the pipes were evacuated to half an inch o f mercury. The wavelength o f the light was 5,700 angstroms (5.7 x 10‘7 m). A total o f 269 m easurements was taken, usually in sets o f 20 for given conditions (weather, exchange of mirrors and beamsplitters, etc.). On the “hypothesis o f a fixed ether” (in Michelson’s words), i.e. the earth rotating in a stationary ether without entraining it, (2) yields a fringe shift o f 0.236; and the observed shift (yes, there was one) agreed with that value within the limits of the observational error.
The experiment at Clearing, Illinois, 60 years ago is surely the m ost grandiose interference experiment ever perform ed: its optical path length am ounted to something like 1014 wavelengths, traversed with w hat was then considered “monochromatic” light: the light from a carbon arc passed through a filter.
Michelson’s feat, which to my knowledge has never been repeated, is both a technical masterpiece, and one that provides fundamental insight into the optics and electromagnetics o f moving sources. Yet it rarely makes it into the textbooks — certainly not into the introductory ones.
However, the result does not contradict the Einstein theory, at least not when the rotation rather than uniform translation o f the coordinate system is invoked. The Einstein theory classes Michelson-Gale as a Sagnac-type experiment, so

Sec. 1.3.7

THE M ICHELSON-GALE EXPERIM ENT

45

named after G. Sagnac, who in 1913 dem onstrated a fringe shift by rotating an interferom eter (with a polygonal interference loop traversed in opposite senses) at high speed; in such cases Einstein’s general theory predicts a shift proportional to the angular velocity and to the area enclosed by the light path — not because the velocity o f the two beams is different, but because they each have their own time. In the present case, the general theory also predicts the shift (2).
There is, nevertheless, one significant difference between the two explanations, and that is that (2) follows from the Galileian principle o f in a few lines o f highschool algebra, whereas Einstein’s general theory does it with multidimensional complex tensors in space-time and non-Euclidean geodesics.

46

Sec. 1.4

1.4. Magnetic Force and Gravitational Field

The force exerted by one charge by the field o f another is given by the Lorentz force

F = g(E + v x B )

(1)

where E is the electric field strength, and v,, v2 are the velocities o f the two charges, the former the velocity of the charge q, the latter (to appear in a moment below) the velocity o f the charge that produces the field through which the first is moving — and for the moment we leave it open with respect to what rest standard these velocities are defined. B is the magnetic flux density given by

where Ec is the Coulom b field, i.e., the irrotational part o f the electric fieldstrength E . Relation (2) is a consequence o f the Maxwell equations; it will be derived in P art Two, but for the time being we will just regard (2) as a definition of B . The case of many charges moving in the field of many others then follows by superposition.
We define q as a scalar that modifies action at a distance. Let us look at this in a little more detail. Two spheres o f m atter that do not have any charge will attract each other gravitationally by the inverse square law. W hen they are given an electric charge, the form of the inverse square law remains unchanged, but the scalar value (including sign) of the force between the two bodies will change. If the ratio o f charge to mass is sufficiently large, the gravita­ tional field may be neglected, at least locally. There is no action at a distance known to us other than either gravitational or electromagnetic, and both obey the same basic force law — certainly at rest, and presumably also in m otion (the cor­ responding “gravimagnetic” field would usually be so weak as to escape direct detection). Defining charge or charge density as that which modifies the scalar value of a gravitational field, therefore, may be unusual, trivial or even inept; but it is perfectly consistent with what is usually understood by charge. Now let us return to the force equation. Formally, equations (1) and (2) are the same as used in the Einstein theory, but there is a fundam ental difference. The Einstein theory measures the velocities in (1) and (2) with respect to the observer, and the ether theory with respect to the ether. Before discussing with respect to what the two velocities are defined in the present theory, let me briefly go over some ground that is undisputed, yet not widely realized. In particular, few text­ books make the following point: The magneticforce between moving charges is so small compared with the elec­ tric force between them that (today) it is not measurable unless the latter is neutralized.

Sec. 1.4

M AGNETIC FORCE

47

Imagine an Einsteinian observer (at rest in the ether, to include that theory, too) observing two rigid rows o f electrons, or negatively charged tennis balls, moving

0V
©©©© ©©© —►
V
©©©©©©©© —►

©©©©© ©©
©©©©©©©© ©© ©© ©© ©©

©

(a)

(b)

Magnetic force and electrically neutralized charges

with the same velocity v. H e will be observing two equal electric currents, though not wire-bound ones, flowing in the same direction. W hat he will see is given by ( 1) and (2) with v, = v2= v; on resolving the double vector product, we have

F = q(l - f l 2)Ec

(4)

That is, he will see a strong electric force repelling the two rows o f equal charges from each other, diminished very slightly by a 0 2 times smaller magnetic force. I know o f no experiment, nor can I imagine one in the foreseeable future, that could realize these conditions and dem onstrate the slight decrease in repulsive force when two or more originally static charges are moved past an observer, without any charges of the opposite sign nearby.
To make this tiny force observable, we must first remove the electric force that overshadows it. The simple way to do this is to neutralize at least one o f the rows by a row o f stationary positive charges — stationary so as to keep a negative current flowing. T hat is, o f course, essentially the case o f a conductor, with the positive charges provided by the positive ion grid, and the electron flow forming the negative current. In other words, we are not able to demonstrate a magnetic force unless at least one o f the currents flows in a norm ally neutral conductor such as a wire.
A fter pointing to this undisputed, but rarely m entioned circumstance, let me now discuss with respect to what the proposed theory defines the velocities v, and v2 in (1) and (2). As always, it defines them with respect to the locally dominant force field. In the case o f the electric force between two charges, this is o f course the electric field o f the other (“source”) charge, through which the considered (“object”) charge q is moving. But in the case o f a magnetic force, this electric field must first be eliminated to m ake the magnetic force observable, so w hat is there left as the locally dom inant field?

48

EINSTEIN MINUS ZERO

Sec. 1.4

In the macroscopic world we live in, the positive charges, that is, the ion grid of the electrically neutral conductor, are at rest (or at best slowly moving) with respect to vast quantities o f m atter, which is likewise electrically neutral — the earth, in our neighborhood. The dom inant force field, therefore, is the gravita­ tional field through which the charge q in ( 1) and (2) is moving once the electric field has been neutralized.
This is the first time that we meet the gravitational field as the field o f m atter consisting of positive and negative electric charges that neutralize each other, or if you like, that almost neutralize each other, leaving the gravitational field as a “remainder” field. It is a concept that will remain with us for the rest o f the book, and will be discussed in more detail in P art Three.
M ore im portant at the m om ent, however, is my hope that the case o f the macroscopic magnetic force will make the idea o f a rest standard given by the local gravitational field appear less exotic than it may have seemed at first.
To return to the Lorentz force (1) with the magnetic field defined by (2), the pre­ sent theory defines the velocities occurring in them not with respect to an observer, as in the Einstein theory, but with respect to the dom inant local field, which in the case o f magnetic force due to wirebound currents is not the electric, but the gravitational field through which the charges are moving, as explained above.
From this it might be concluded that if the observer is at rest with respect to the gravitational field, which is usually the case, the two theories must be equivalent. Not so: we have already crossed the border from respectable orthodoxy into heresy. To see this, we first go back to some basic principles.
The Lorentz force (1) describes a force in terms of an interaction between a charge (q) and the field (E, for example) in its immediate environment. The charge that is the source o f that field has not the slightest effect on this interaction; in fact, that charge may no longer exist (it could have been discharged before the collapse o f its field arrived in the neighborhood o f <7). T hat is the first rule to remember if things get confusing.
Second, strictly speaking the fields such as E in (1) refer to all of the electric field in the immediate neighborhood o f the charge q, including that charge’s own field. However, it is clear from symmetry considerations that at rest or in uniform motion the net force on a charge by its own field is zero. It is therefore permissible, as is invariably done, to pretend that E is the field o f the other charge only, and that q behaves as if it had no field. T hat is the secdnd basic rule; but it applies only to uniform motion, including rest.
If the charge accelerates, partially catching up with its own field, the resulting asymmetry does produce a force on a charge by its own field. An easy way to see this is to recall that accelerating a charge means increasing its magnetic field, which will induce an electric field to oppose such a change, so that the charge is acted on by its own field. This “inertia o f the electromagnetic field” will be discussed in the next section, and in greater detail in Part Two, but will not be needed here.

Sec. 1.4

M AGNETIC FORCE

49

Both rules are still respectable physics, but once we define velocities with respect to the field, not with respect to an observer, we begin to obtain heretical results, as in the following example.
Let a charge q2 be at rest with respect to the observer (gravitational field, ether), and let an equal and opposite object charge have velocity v in that rest frame. Then in the Einstein (and ether) theory v2= 0, and there is no magnetic force. The moving charge does have a magnetic field, but in orthodox physics magnetic fields cannot affect stationary charges. Thus in orthodox electro­ magnetics we have

F = <7(E + v x B ) = q E

(5)

Now let the velocities be defined with respect to the local field through which the charge is traveling, or by the relativity principle, with respect to the field that is moving past the charge. Then v2 is the velocity o f the charge with respect to the field in its vicinity, which is clearly the electric field o f q, sweeping past it. Hence the velocity o f q2 with respect to the field surrounding it is v, and ( 1) and (2), after resolving the double vector product as before, yield

F = q E ( l - ( i 2)

(6)

which we may interpret either as a diminished electric attraction, or as the appearance o f a magnetic force. In either case, it is a result that differs from the prediction (5) of orthodox electromagnetics, and one that can, in principle, be subjected to an experimental test.
But only in principle. To test the difference, such an experiment would have to measure it with an accuracy o f 0 2 at the one m om ent when the velocity is perpen­ dicular to the radius vector joining the two charges. On the other hand,if the elec­ tric field that swamps the effect is removed by neutralizing the stationarycharge as discussed above, that is, by using a conductor with free electrons, (2) will indeed produce a zero magnetic field in the present theory, just as predicted by the Ein­ stein theory, for the neutralized charge has no field with which to act, and the moving charge has nothing to act on. (Polarization effects etc. are not peculiar to moving charges.)
Now if both charges move, with the electric field neutralized by the presence of a stationary charge of opposite sign near one of them, or more generally, if we perform the experiment with a w ire-bound current, then the force ( 1) between the moving charges is exactly what is predicted by the Biot-Savart Law — in all theories. T hat includes the present theory, for once the electric field has been neutralized, we fall back onto the remaining force field, which is gravitational, and which we also made the rest frame for the observer.
Thus, the original effect is too small to be measured, and trying to increase it will eliminate it. This is frustrating; but then, if the effect were easily measurable, the inaccuracy of orthodox electromagnetics would have been noted long ago.

50

EINSTEIN M INUS ZERO

Sec. 1.4

The reader well versed in conventional electrodynamics may be offended by the idea that a stationary charge (stationary with respect to the observer) can be affected by a magnetic field. Yet what offends the relativity principle from my point o f view is that a charge that is deflected when it travels through a magnetic field should sit still when a magnetic field sweeps past it. W hat possible difference can there be between the two cases?
None, of course, and the Einstein theory does ultimately obtain the same force in both cases. However, it does so by letting the observer see only an electric force in one case, and both an electric and magnetic force in the other, the two adding up to the same result. But to achieve this compliance with the relativity principle, space and time have to be distorted, and electric and magnetic fields have to be tied to the observer, instead of to the charges producing them.
By contrast, in the present theory electric and magnetic fields are quite indepen­ dent o f any observer, since the only velocity that is effective in producing a force is that o f one charge in the field o f the other. This relative velocity remains unchanged no matter what coordinate system an observer wishes to choose. The relativity principle is thus satisfied autom atically and naturally, w ithout time dila­ tions or space contractions.
As for the magnetic field, it is uniquely determined by the electric field through (2), and we may express it via that electric field (as we have done above) to eliminate an unnecessary complication rather than convert parts of it to rescue the relativity principle for an observer who has nothing to do with the problem.
It will be seen that the entire behavior o f a charge mdving through a magnetic field is based on only two foundations 1) the charge experiences a magnetic force q \ x B, where v is referred to the field, 2) it is subject to the the relativity prin­ ciple: its m otion through a field cannot be distinguished from a field sweeping past the charge.
Although the basic principle is simple, things can become confusing when we have to find B from (2), and 1 will therefore suggest two “magnetic rules” that should simplify possible complications.
Magnetic rule no. 1 has, in effect already been derived: W henever at least one set o f moving charges has been electrically neutralized — most often by the positive ion grid of the conductor in which the charges are moving — then the remainder field with respect to which all velocities are defined is the gravitational field, which also coincides with the laboratory fram e in all experiments on the point. Hence in this case, which includes all cases o f wirebound currents, Einsteinian electromagnetics remains valid. Note that, as shown in our thought experiment above, it is sufficient to neutralize only one o f the currents. For example, if we have an electron beam (with its electric field not neutralized) shooting through a magnetic field produced by wirebound currents in electrically neutral conductors, magnetic rule no. 1 will apply, so that mass, force, charge and everything else will obediently follow Einstein’s predictions.

Sec. 1.4

M AGNETIC FORCE

51

The difference between the present theory and orthodox electromagnetics arises only in interactions o f charges whose electric field has not been neutralized. These are also the cases that have not been tested experimentally: the force between two parallel electron beams, or the force o f one charge when moving in the field of another with no neutralizing charges present. The latter includes the case of planetary motion of the electron about the nucleus.
We need in principle consider only the case of two solitary charges, for the Max­ well equations are linear and therefore the case o f an arbitrary charge distribution follows by superposition. As far as this book is concerned, we shall mainly need the case of planetary motion.
Since the velocity o f a charge in a force field is defined as its velocity with respect to the lattice o f intersections o f lines o f force and equipotentials, it may be conve­ nient to think o f the radial E field o f a charge as draw n on the dial o f a clock. The source charge is in the center o f the clock, and 12 selected rays are labeled “one o’clock” to “noon.” W hen one charge moves through the dial (field) o f the other, we can then assign the sign o f the velocity depending on whether the m otion is clockwise or anticlockwise. (A moving charge has no magnetic field in the longitudinal direction, so radial m otion is irrelevant.)
The use o f this m ethod is trivial when the m otion o f the charge in the field o f the other is rectilinear; however, for planetary, circular m otion o f one charge about the other, we m ust remember that rotation o f a charge about its own axis (such as electron spin or the earth’s rotation) does not affect the field: we know from the Michelson-Gale [1925] experiment (Sec. 1.3.7) that the terrestrial gravitational field is no m ore affected by the earth’s rotation than is the plane o f Foucault’s pen­ dulum. Analogously, we assume any electron spin to be irrelevant to its E field. Therefore an electron moving clockwise through the nucleus dial from 2 to 4 o’clock will simultaneously have its field sweep past the nucleus with its 8 to 10

12

\ 12

• 6

6

C harge through field and field past charge

o’clock rays, and a quarter o f a period later with its 11 to 1 o’clock rays. Hence the velocity o f the electron in the field o f the nucleus is the same in sign and absolute value as the velocity with which the field of the electron sweeps past the nucleus.

52

EINSTEIN M INUS ZERO

Sec. 1.4

Thus, setting v, = v2= v, we find magnetic rule no. 2: The force between two solitary charges, none o f which is electrically neutralized by other charges, is

F = q (E + v x B ) = q (E - /32E f )

(7)

where E{. is the Coulom b field, i.e., the irrotational part o f E. The rotational part o f E, the Faraday field, is absent if there is no acceleration, and has no effect on the force between the two charges when the acceleration is perpendicular to the velocity (as it is in circular planetary m otion). In both o f these cases (7) reduces to

F = <7E(1 - /32)

(8)

The rule expressed by (7) or (8) am ounts to this: the magnetic field o f a moving electric charge, being uniquely determined by (2), can be eliminated as an unnecessary encum brance. The resulting modification o f the electric field is always such as to diminish the electric force (this is equally valid for repulsion).
The magnetic force between two moving charges is, o f course, the Biot-Savart force. It appears here as a simple modification o f the C oulom b force and, as always in the present theory, as a force independent o f the observer. This differs from the Einstein theory in a result that is not yet susceptible to m easurement. An Einsteinian observer located at the center o f mass in circular planetary motion will calculate a Biot-Savart force that is o f order f33 and acts in the opposite direction.

Sec. 1.5

53

1.5. Electromagnetic momentum
W hen hammer hits chisel, the elastic steel of the chisel transmits the force from the hamm er to the cutting edge of the chisel. In the same way, Faraday and Max­ well thought, the ether transm itted the force from one point charge to another, and Maxwell calculated the force per unit area (that is, the stress, such as pressure or tensile stress) exerted by a charge on a surface in the ether. He did so in much the same way that one would calculate the stress on the cross section of a rod.
The concept o f such an elastic ether has been abandoned, but since we have no reason to doubt the Maxwell equations (at low velocities, anyway), these calcula­ tions remain formally correct, for they are based on nothing else. The word “stress” is generally associated with forces between the particles o f a material medium, but though such stresses do occur in ponderable materials permeated by electromagnetic fields, there is obviously nothing under stress in a vacuum. This inappropriate word has remained in use, but in the present theory, which regards force as propagating with velocity c through the local force field, it might better be called the “force density” (in N /m 2), meaning the force transm itted through unit area perpendicular to its flow.
To calculate it, we start from the Maxwell equations for a region which contains charge and current distributions, but consists otherwise of vacuum:

V x E H— — = 0

( 1)

at

V x B - 4c/ ^at= ^ J

(2)

V ■E = —

(3)

r<>

V •B = 0

(4)

Cross-multiplying (1) by e0E, and (2) by B, then adding and rearranging using l/eo /io = c2, we obtain

f0 (V x E ) x E + —Ho (V x B) x B = J x B + f 0 <—)t ( E x B) (5)
By a simple, but somewhat longwinded calculation using (3) and (4) and given in many textbooks (e.g., [Stratton 1941]), the first two terms can be shown to equal

f0( V x E ) x E + — ( V x B ) x B = div 2S - r ,,E V ■E - — B V • B

^0

/•*()

= div 2S - pE

(6)

where 2S is the electromagnetic stress tensor. Its com ponents are o f no interest here; however, it is evident that 2S is a quantity that, when integrated over a closed sur­ face E, must yield the total force transm itted across it. Substituting (6)

54

EINSTEIN M INUS ZERO

Sec. 1.5

in (5), using J = pv, \g d V = q , and integrating (5) over a volume V , we therefore have by the Divergence Theorem

(7)

The first two terms on the right are obviously the electric and magnetic com­ ponents of the Lorentz force by which the corresponding fields act on a charge, and this is, in fact, how the Lorentz force is derived from the Maxwell equations.
But (7) also says that even if there are no charges or currents in the considered volume, so that the first two terms are zero, there is still a net force emanating from this “empty” volume, whenever it is perm eated by a time-varying elec­ tromagnetic field. Since force is the rate o f change o f m om entum , it follows that a m om entum

(8)

must be associated with an electromagnetic field. This phenom enon is sometimes called “inertia of the electromagnetic field.”
The physical meaning is the following. The m om entum o f an uncharged body m\ , when changed by external forces, seeks to stay constant and resists such a change. But so does an electromagnetic field, and quite independently of the momentum and inertia of the mechanical, Newtonian masses that carry its source charges and currents. A steady magnetic field, for example, is due to a steady current; if that field is changed (by changing the current) it will, by Faraday’s Law, induce an electric field that will seek to restore the current and its magnetic field to its previous value — its direction is given by Lenz’s Law, and the entire effect is known as self-inductance (mutual inductance if the field was changed by another current). The magnetic field, in effect, resists being changed.
Q uite similarly, a steady electric field is due to a steady charge distribution. If the field is changed (by moving the charges), the resulting displacement current dD /dl gives rise to a magnetic field B by (2), and the change in magnetic field induces an E directed against the displacement of the charges. The electric field, in effect, resists being changed.
But if the m om entum o f a field parallels the m om entum o f uncharged m atter, we would expect an inertial mass o f the field to parallel the inertial Newtonian or mechanical mass o f an uncharged body. This is indeed the case: we shall, in a moment, find the electromagnetic mass of a charged body as the factor multiply­ ing v in the expression for the m om entum (my) o f an electromagnetic field. In both cases, mechanical and electromagnetic, inertial mass is a measure o f a body’s resistance to having its mom entum changed.

Sec. 1.5

ELECTROM AGNETIC M OM ENTUM

55

This electromagnetic mass is no form al mathem atical trick. It is a physical reality that a charged body resists acceleration beyond the resistance offered to it by its Newtonian mass.
To see that the inertial mass o f an uncharged body is increased by an additional electromagnetic mass o f its field when that body is given a charge, consider an example that will be used several times in coming sections, the throwing o f a tennis ball. W hen it is uncharged, its N ew tonian (mechanical) inertial mass resists acceleration, and the work done by the throw er’s muscles in overcoming that resistance appears as kinetic energy o f the moving ball. But when the ball is elec­ trically charged, the ball offers additional resistance (in principle, that is, for the numerical am ount is actually very small): a moving charge has a magnetic field proportional to its velocity, and the change in magnetic field (from zero), by Fara­ day’s Law, induces an electric field opposing the acceleration o f the charge. The additional muscle work performed in overcoming this resistance appears as the energy o f the magnetic field in addition to the ball’s kinetic energy.
Now let us calculate this electromagnetic mass o f a body as the factor multiply­ ing its velocity to yield its electromagnetic m om entum . We consider a moving point charge, which by the Divergence Theorem is also equivalent to any spherical charge distribution with radial symmetry; by superposition, we may consider all (reasonable) charge distributions made up of such elementary spherical charges.
Substituting for H = B //i from (2), Sec. 1.4, in (8), resolving the double crossproduct, directing the x-axis along v, and omitting the terms that will integrate to zero because o f symmetry considerations, we find the momentum in the form

(9) Hence the electromagnetic or field mass, the factor multiplying v , is

( 10)

Let us now tem porarily assume that the Coulom b field remains spherically symmetrical when its source charge moves with respect to the rest frame (by letting the observer rest in the local force field or the ether, we need not yet differentiate between the three theories). In that case the three rectangular components of E will remain equal, as they were at rest, and we have

( 11)
where the constant K is defined in (2), Sec. 1.1. Substituting this in (10) and integrating over all space outside the charge (which we assume distributed over the surface of a sphere with radius R), we find the electromagnetic mass

56

EINSTEIN M INUS ZERO

Sec. 1.5

‘ €n

021

This is a result that we will also obtain in P art Two by several other methods. However, it is valid only to the extent that the assum ption o f field symmetry, on which (11) is based, is valid. This is evidently the case for slow (uniform ) velocities, which must merge continuously with the static case.
But for high velocities, the story is different. There is no direct experimental evidence available, and we must trust the Maxwell equations to provide the answer. To evaluate the expression for the electromagnetic mass (10) exactly, we must first examine what happens to the field o f a fast moving point charge.

Sec. 1.6

57

1.6. The Field of a Moving Charge
W e will now consider a seemingly simple problem . We take a point charge at rest with its concentric equipotential spheres and radial Coulom b field. W hat hap­ pens to this potential 0 and electric field E when the charge moves with uniform velocity v (directed along the x-axis) with respect to the rest frame? As before, we let the observer sit still in the local force field or in the ether for com parison; but we measure velocities with respect to the local force field in which the charge is moving.
We have but two tools to solve this simple problem: the Maxwell equations and the relativity principle.
The field vectors Eand B satisfying the Maxwell equations arederivable from a scalar potential 0 and a vector potential A. As shown intextbooks o f electro­ magnetism (and also in Part Two of this book), the relations are

E = —-77— —V 0

(1)

ilt

1 i)(b

/->,

B = V x A: V •A ' =—

(2)

cl at

where the potentials are solutions of the wave equations

72, 1 d 20

p

V 0 ca <H2

f„

(3>

2

1 (i2A

...

V A - ^ 2 - ^ 2~ = ^ P V

(4)

Next, we turn to the principle of relativity, which requires that the laws of physics, when properly form ulated, rem ain equally valid in all frames moving with uniform velocity with respect to each other. T hat means whatever the field distribution about a point charge, however it is affected by the particle’s velocity, and whatever that velocity is referred to, the field distribution must travel unaltered with the particle (“frozen to it”) whenever it moves with uniform velo­ city: otherwise we could — in principle, anyway — look at the distortion o f the field surrounding the particle, and w ithout reference to any rest standard, we could proclaim with w hat absolute velocity the particle is moving. The principle of relativity therefore requires the “freeze” condition: as the charge moves through the rest fram e with velocity v , each com ponent o f its field must satisfy the relation

f ( x , y. z. t) = f ( x -I- v x d t . y + v y d t . z + v z d t . t + dt.)

(5)

From this we have

at =

=

v/

161

58

EINSTEIN MINUS ZERO

Sec. 1.6

and using this relation twice over to eliminate the time derivative in (3), we find

where

< '-'> 0 +g +£ - £

ft= -

(8)

c

This is certainly different from the Poisson equation when the charge is at rest, due to the (1 - / 3 2) factor. Yet (7) is as valid as the relativity principle and the Maxwell equations.
There have been several attem pts to interpret this result in a way that is consistent with the experimental evidence without sacrificing either the relativity principle or the Maxwell equations, both o f which underlie (7). N ot all o f these attempts have been successful.
Here 1 would like to insert parenthetically that while I would not like to sacrifice the relativity principle, I lack the obligatory reverence toward the Maxwell equations: they are ether-begotten and tested without circularity only at low velocities. There are, however, two reasons why 1 am not ready to abandon them. The first is obvious: 1 have nothing better to offer. The other will be discussed in Sec. 1.10.1, which explains the fundamental reason why the Maxwell equations can very well survive without the elastic ether o f which they were born.
The two important methods of interpreting (7) are due, respectively, to Hendrik Lorentz and Albert Einstein.
Lorentz, who took v to mean the velocity with respect to the ether, noted that (7) is equivalent to the electrostatic case when the charge is at rest, provided x is replaced by Y, where

i =

(9)

vT ^

In that case (7) turns into

V 2<5=~ — <o

( 10)

where

" =

<M)

with go the rest charge density.

From this Lorentz concluded that electrons contract in the direction in which

they move through the ether. It was for this purpose that he introduced the

transform ation named after him, a very different purpose from that for which

Einstein used it. (This, incidentally, also explains the seeming paradox why the

man who provided the backbone of the Einstein theory remained irreconcilably

opposed to it to his death in 1928.)

Sec. 1.6

FIELD OF A M OVING CH ARG E

59

But the experimental evidence decided against the Lorentz theory. As the earth moved through the (unentraint J) ether, a m easurable torque would act on two charges attracting each other in the “ether wind” (as will be explained in Sec. 1.10.4), but the Trouton-N ob experiment [1903] failed to detect such a torque.
It is, however, im portant Lo note that Lorentz’s theory (or Ritz’s, for that m atter), did not contradict the relativity principle; it contradicted the experimental evidence. The contraction o f an electron (or in Fitzgerald’s view, o f any body) as it moves through an ether is no m ore violation o f relativity than rain drops being deform ed as they fall through the atm osphere; the fact that in reality there is no such contraction has as little to do with the relativity principle as the invalid theory that sailboats always move against the wind.
To Einstein, on the other hand, v was the velocity with respect to the observer; if he wanted all observers to see the same force between two charges, he had to build the contraction (9) into space itself, which in turn m ade the dilation o f time inevitable. This was consistent with the Maxwell equations using observer-referred velocities and squared with the experimental evidence, for there was no longer an ether to be refuted. W hether it satisfied the relativity principle, however, is a point less self-evident than the textbooks would have us believe. It certainly did not satisfy it for a world with undistorted space and time, nor for one that believed nature’s laws, including the relativity principle itself, to be rooted in nature, not in the observer observing it. Perhaps it is fair to say that the Einstein theory satisfies the relativity principle at a price that not everybody is willing to pay.
In the present theory, v is the velocity with respect to the local force field. There is no privileged standard o f rest, and since the effect-producing velocity in Maxwell’s equations and in the Lorentz force is held to be the vector difference between the velocities o f a charge and the local force field as both move through an arbitrary observer’s coordinate system, this effective velocity is unchanged and independent of any observer. It therefore satisfies the relativity principle automatically. (M ore on this in Sec. 1.10.4.)
N or is there any reason to abandon the Maxwell equations, which likewise become observer-independent (m ore on this, again, in Sec. 1.10.4). Judged by the present theory, the Trouton-N oble experiment (Sec. 1.10.4) produced a null effect because nothing moved. The two charges were at rest with respect to each other, i.e., at rest in each other’s fields.
To return to the basic problem , is there a good physical reason why a uniform charge on a sphere at rest should rearrange itself as implied by ( 11) when the charge moves?
Einstein did not need one; in effect, he regarded the Maxwell equations, with all velocities referred to the observer, as an axiom and distorted space and time to fit them. Lorentz, who regarded the ether “as endowed with a certain degree of substantiality, however different it may be from all ordinary m atter” [Lorentz 1915], followed Maxwell’s thinking o f actual stresses in the ether and thought of

60

EINSTEIN M INUS ZERO

Sec. 1.6

the electron contraction as the result of stresses by the ether on a deformable electron.
The present theory offers an entirely different physical explanation — but not at this point, for we need certain fundam ental relations that will not be derived until Part Two. For the time being, therefore, the reader is asked to accept (11) for no better reason than that it emerges from the Maxwell equations — a procedure that should present Einsteinians with no difficulty at all.
Now let us turn to the field produced by such a non-uniform charge distribution on a moving sphere. We can solve (7) by pretending that we are dealing with an electrostatic problem in a space in which the x coordinates have contracted in accordance with (9) — another just-as-if equivalence, and one that the Einstein theory regards as a physical reality. The charge distribution (11) will then lead to a potential
(12)

In the special case of a point charge, or its equivalent, a sphere with radially symmetrical charge distribution, this becomes

!*■'<) \ J j 2 + (1 - d 2 ) { y 2 + z 2 ) \ J { - d 2 s in 2 0
where 6 is the angle between the radius vector and the velocity.

The "bunching” o f the electric field in the direction perpendicular to the velocity o f a charge. In the Einstein theory, this is due to the contraction o f length in the direction o f the velocity referred to an observer, in the present theory, it is due to a redistribution o f the charge density determ ined by the velocity with respect to the traversed field.

Sec. 1.6

FIELD OF A M OVING CH ARG E

61

Thus the familiar concentric spherical equipotentials about a static point charge flatten into ellipsoids when the charge moves with high velocity. The Coulomb field is perpendicular to the equipotentials, so that if we take the velocity as the north-south axis o f a sphere, the electric lines o f force are less dense near the poles and denser near the equator — we shall refer to this effect as “bunching” of the lines o f force. T o calculate it, we take the gradient o f (13), obtaining

£ , , ( ! - /?2 )

(1 - /?2 sin2 <7)3/ 2 r °

(14)

This result is often needed in terms o f the field parallel and perpendicular to the velocity:

E „ ( l - ( 1 2) for r || v

Eo

f,or r l.v

(15)

Note that (11) implies only a rearrangement, not a change, of the total charge, which remains conserved:

/ / / " n //'/'

1 r- .

=

Po dV = q

(16)

The reader is reminded that equations (7) through (16), though not the text inbetween, are also formally valid in Einsteinian electromagnetics. There are, however, two substantial differences. First, v refers to the observer in the Einstein theory, but to the local force field in ours; second, the contraction (9) applies to space itself in the Einstein theory, but only to the charge density (and consequently to the field it produces) in ours.

62

Sec. 1.7

1.7. Mass and Energy

W ith the electric field strength o f a moving charge established, we are now ready to substitute in the expression for electromagnetic mass (10), Sec. 1.5.
We recall that the velocity was parallel to the x-axis; to substitute for Ey and Ez, we use (15), Sec. 1.6; then (10), Sec. 1.5, becomes

rrij =

=/ L

E?) y [ - ^ = = , y , z

-fl2 J

+E?)2 I

,y.z dxdydz

(1)

\y/T = ]P

On substituting £ = x /V (l —/32), this integrates to

l/lV,l 2

m f = ~ ~ r — -y

(2)

Grr/fv/l - f P

or using (12), Sec. 1.4,

mI ~ 7

I3 '

where rtijo is the electromagnetic mass at rest. This formula for the electromagnetic mass, associated with the resistance o f the
electromagnetic field to the acceleration o f its source charge, had been know n to the classics o f the late 19th century before the advent o f the Einstein theory, which derives the same type of formula for any mass, charged or neutral.
The resistance offered to change of momentum by uncharged matter evidently obeys the same law. First and forem ost, this is supported by the experimental evidence.
But there arc also theoretical reasons. It was shown by Page [1912] that C oulom b’s Law plus the Lorentz transform ation is enough to derive the Maxwell equations, and hence all the experimental evidence confirming the Einstein theory. But Coulom b’s Law is formally identical with Newton’s inverse square law, though its numerical values and dimensions may differ, and Page’s m ethod must therefore lead to formally identical results. Hence (3), which rests on nothing but the Maxwell equations and the relativity principle, must equally well hold for Newtonian, mechanical, electrically neutral mass.
By this Page-Coulom b-Lorentz argum ent (and, o f course, the experimental evidence), we may therefore write

m„ = -Ulfi()

(I 4a)\

v/ i^ 2

Sec. 1.7

MASS AND ENERGY

63

where m no is the neutral mass at rest. Since the total inertial reaction to an impressed force acting on a charged body
moving with velocity v is

d

d

— (m „v + m /v ) = — (m„ + rn /jv

(5)

we can combine the two masses into a single mass

m = m „ + rrif

(6)

without knowing what fraction is m ade up o f one kind or the other. Combining (3) and (4) in this way yields

mo

m = ——

(7)

The classics were aware o f (3), but unsure o f (4); they hoped that Newtonian mass would turn out to be constant, so that the two components couldeventually be separated by advancing technology, i.e., by acceleratingparticlesto sufficiently high velocities. Only Newton himself, who did not even know of the existence of electromagnetics, had guarded against the possibility o f mass being velocitydependent by never taking it out o f the m om entum entity when dealing with force.
Contem porary physics claims that the resolution o f mass into its two com ­ ponents (6) is unim portant, or even impossible. This is incorrect: in P art Two the resolution will be shown im portant and possible.
Equation (7), perhaps due to the abstract and overly mathematical character of the Einstein theory, has been im bued with a mystic rom anticism which it does not deserve. The quantity of matter does not, of course, increase with velocity. What increases is the inertial reaction or the resistance to a force changing a body’s m om entum . But that is nothing extraordinary: inertial reaction is a force, and there are many forces that are velocity-dependent — hydraulic or aerodynamic friction, for example, and the thrust by a ship or plane to overcome them.
Other formulas widely used in the experimental verification o f the Einstein theory follow from (7), and like (7) itself, they can again be derived w ithout use of the Lorentz transformation.
W riting (7) in theform

m 2( l —(I2 ) = const

(8)

and differentiating, we have after elementary manipulations

v 2 dm -|- vm dv = c2 dm

(9)

On the otherhand, the work done,

or energy (£ ) expended, in accelerating a

given quantity o f m atter from velocity 0 to velocity v is

,

64

EINSTEIN M INUS ZERO

Sec. 1.7

,, _ f v d(rnv)

fv

‘ ~ Jo ~ d t

=J V

fv = J V’2 d m + v m dv]

Substituting (9) for the integrand, this yields the kinetic energy

rm(v)

I'lkm = c2 /

dm = rnoC2[rn(v) - m(())]

J m(0)

= m„r:2 ( - r = L = = - 1 ]

( 10) ( 11)

which, as is easily seen by series expansion, reduces to the familiar V im v1 for

small f3. However, (11) shows that this kinetic energy is merely the difference o f

energies at velocityv and velocity 0, the latter indicating an energy associated with

a body at rest and

given by the zero-velocity terms in ( 11):

E 0 = rrioc2

(12)

The total energy o f m atter moving with velocity u is therefore

E = E kln + E () = m e2

(13)

Finally, to establish a relation between the energy of a moving momentum, we have from (13)

E 1 2 2 m?.

2 rrif.v2

7 * - - m?>f =

particleand its <14>

The last expression is clearly the square o f the m om entum p = mv; therefore the required relation is

! L - = moc2 + p 2

(15)

The relations that have been used most often in alleged proofs of the Einstein theory are (7), (13) and (15), as we shall see in more detail in Sec.1.9. None of them, as shown here, need the Lorentz transformation or the reformation of space and time.
Relation (13) has fascinated laymen, for it is often the only thing they know about the Einstein theory. But even some physics professors have romanticized “the equivalence of mass and energy.” A glance at (13) shows that this “equivalence” is a dimensional absurdity.
The interpretation o f (13) is rather simple. We have seen that the electro­ magnetic field resists acceleration o f the charge that is its source, so that it is responsible for part o f the inertial mass of the body carrying the charge. If we

Sec. 1.7

MASS AND ENERGY

65

cause the charge to disappear by discharging the body, the held disappears only from its immediate surroundings: it is radiated away. But the energy o f the held is radiated away with it, so that the conservation of energy demands that the inertial mass o f the body be decreased by a corresponding am ount. I see no reason to doubt that the same is true o f the N ew tonian part o f the inertial mass, though we cannot dem onstrate it by “dismassing” a body as we can discharge it, at least not in the macroscopic world.
This is the interpretation Einstein gave the relation (13); in fact, 1 know o f no simpler way to express it than Einstein himself did in a special paper devoted to the point [1905 b]. Emphasized by his own italics, Einstein’s statem ent is I f a body gives o ff the energy L in the fo rm o f radiation, its mass diminishes by L /c 1.
The classics came very close to deriving (13); in fact, it has been claimed that they were well aware o f it. In 1900, for example, H enri Poincare calculated the recoil experienced by a body radiating an energy E and found it by equating it to the mom entum of the radiated electromagnetic field, given by (8), Sec. 1.5. This led him to a form ula implying that the mass M associated with the radiated field equals E /c 2; however, this is not equivalent to Einstein’s form ula, as it deals only with the mass equivalent in radiation pressure, not with the mass lost as radiated energy. He may, however, have come closer in [Poincare 1904]. This and some other sources are discussed by Ives [1952], who claims that Einstein’s derivation is neither original nor correct. However, Ives’ vehement animosity toward Einstein may have driven him too far here, and the reader interested in the history o f the relation is cautioned to look up the original sources quoted by Ives before accept­ ing his interpretation of what they imply.

66

Sec. 1.8

1.8. The Modified Newton-Coulomb Law

Substituting the velocity dependent mass (7), Sec. 1.7, in Newton’s Second Law in the only form Newton ever stated and used it, we have

^ d

thq dv

mo dv

=

I t 1" 1 ,) = r r - > ) « v " + 7 T

where v0 is a unit vector in the direction o f the velocity, and u0 is a unit vector at right angles to it. The two terms in this form ula, which formally agrees with the formula for force in the Einstein theory, correspond to the components of acceleration directed along the velocity and perpendicular to it, respectively.
Thus, when the acceleration is norm al, or close to norm al, to the velocity (the magnetic force is always normal to it), then on com paring the second term to the second line in (15), Sec. 1.6, we have

dv

F i = <7(Eo_l + v x B ) = m 0 — u ()

(2)

dt

T hat is, not only does the Lorentz force remain valid, but the inertial reaction to it is given by the “mass times acceleration” form ula falsely attributed to Newton. In this “transversal” case it is correct.
The form ula is formally correct in the Einstein theory, too. Though this is usually known only to the more erudite Einstein scholars, the non-Newton for­ mula “force = mass times acceleration” is perfectly valid in the Einstein theory for “transversal m ass,” that is, for the effective inertial mass when the acceleration is norm al to the velocity — as can be seen from ( 1), which is obviously valid in the Einstein theory, too. But the agreement between these expressions andthose in the Einstein theory is only form al, for Einstein defines “rest” with respect to the observer, not the field. Therefore the field E has a different value for different observers. It is, in fact, well worth to take a little side trip into the Einstein theory to see what enorm ous complications hide behind its seemingly simple form ula

F = (/(E + v x B) = y l m v )

(3)

dt

even in the elementary case of two equal and opposite charges attracting each other when one is in motion. Let the charge 2 be at the origin o f the observer’s system at time t = 0 , and let the moving charge 1 move at right angles to the radius vector at that m om ent. M oreover, let the charge at the origin be so massive that the force acting on it makes it move only very slowly so that its mass M can be considered equal to its rest mass at all times, and the magnetic force is, in the Einstein theory, quite insignificant. Then on using (15), Sec. 1.6, the force o f the moving charge 1 with its “bunched up” field on the stationary charge 2 is

Sec. 1.8

THE MODIFIED NEW TO N-CO VLO M B LA W

67

o

©

fa)

(b)

Inequality o f action and reaction in the Einstein Theory: (a) force o f a moving

charge on a stationary one, (b) force of a stationary charge on a moving one.

F i2

qEi

J t z ip

(4 )

However, the force exerted by the rest field of charge 2 on the moving charge I is

F ‘21 = (7E 2 = t/Eo

(5)

which differs from (4) by a factor o f up to infinity. Action and reaction are thus no longer equal. (Note that we have never left the
observer’s system, so that this cannot be blamed onto a faulty transform ation.) Once again, it takes an erudite Einsteinian to resolve this paradox appearing in
this simplest o f all problems in the dom ain of moving charges. The explanation: There is no paradox. It is nowhere w ritten that the two forces must be equal.
Yes, it is, one might think. If they are not equal, then the custom ary derivation o f the conservation o f m om entum breaks down. Well, yes and no, say the Einsteinians. If the two bodies exerting a force on each other are in actual contact, as they are in the collision o f billiard balls, then we know where here and now is, and conservation of mom entum reduces to the classical meaning. But if we have action at a distance, then simultaneity is a concept that slides along slippery “world lines,” and the where, when and how o f a m om entum , at least some o f which is associated

68

EINSTEIN M INUS ZERO

Sec. 1.8

with a field stretching from here-now to four-dimensional eternity, becomes a somewhat nebulous concept. The conservation of momentum can be treated only in a generalized form involving H am ilton’s principle, and though its conservation, in a certain sense, is ultimately extruded from the goo o f four-vectors, the less than highly erudite Einsteinian will do better to use m om entum only in the context of (15), Sec. 1.7, which is valid in both the Einstein and the present theory.
W hatever happened to O ckham ’s razor? As one authority said, “It is known that Maxwell’s electrodynamics — as understood at the present time — when applied to moving bodies, leads to asymmetries which do not appear to be inherent in the phenom ena.”
The authority is Einstein, and the statem ent the opening sentence o f his classic paper [1905a], “As understood at the present time,” then meant the ether, and now means Einstein; but after 80 years the asymmetries have not been eliminated, they have only been replaced by such as the one we have just considered.
In the present theory, the asymmetry between action and reaction has been eliminated. The two forces are not only equal (and opposite), but utterly indis­ tinguishable: there is no way o f telling which o f the two charges is “genuinely” moving. In the Einstein theory, this is not possible, either; but this impossibility is tied to that of observing the simultaneity of two events that are not also coincident in space.
And that brings us to the crux of the matter: the Einstein theory (and all of con­ tem porary physics) deals in observables. But as the case o f the railroad track in the Introduction shows, it is better to deal in inferrables, and that is exactly what the present theory does.
For example, it is true that none o f us know what happened to the location of the sun in the last 8 minutes (which is roughly the time it takes for its light to reach us); but we have a pretty good idea where it is right now nevertheless, for we can use the laws o f nature to infer its present position, even though we cannot possibly measure it directly.
The planets orbiting the sun, and the electrons orbiting the nucleus, are, in fact, good examples of applying the present theory, for this will enable us to generalize the static Newton-Coulomb Law

F = % r ()

(6)

r1

to the case when the two masses or charges are in motion — one in the field of the other.
Consider the following problem: Is the bright disk in the sky a souvenir left by the sun where it was 8 minutes ago, or is that the direction to the real sun where it is now?
Let the sun S be at the origin of the coordinate system and let the earth be at £ , at a distance r, and let it move along a circular orbit with orbital (circum­

Sec. 1.8

THE MODIFIED NEW TON-COULOM B LA W

69

ferential) velocity v. Then during the time t= r /c that it takes for the light to propagate from sun to earth, the latter will have moved through a distance v t = 0r, where 0 = v /c . That is, it will have moved through an angle /3. Note that this angle is independent o f the distance r, which might as well be infinite, so that in determining the direction of the true sun we need consider only angles, not distances; in particular, parallel lines in our distorted figure can be considered identical. Now when the earth moves through the arriving light with its own

O£ *

1 * S'
A berration and delay. The delay and aberration angles b for the light (and gravitational force) reaching the earth from the sun equal about 100 m icro­ radians, or som e 12,000 times less than show n in the figure, so that the points
S and S ' both lie in the sun and are practically identical.
velocity, then by Sec. 1.3.1, the light will arrive from an aberrant angle, altering the geometrical angle (perpendicular to the velocity) by sin/3, or since /3 is small (3 X 10"4), by /3. The aberration is in the direction o f the velocity, so that the light will arrive in the direction from S', which is parallel to, and therefore identical with, the original direction to E u in which the light left the sun 8 minutes ago. In other words, to first order the effect o f the delay is canceled by the effect o f aberra­ tion, so that the bright spot in the sky is the location o f the real sun.
It is a sobering thought that when the professors are through arguing, they find the position o f the sun exactly where the janitors never doubted it to be; and if we apply this janitors’ principle to the fictitious celestial body L which travels at the same angular velocity as the earth but lies beyond it, so that its light — like the sun’s gravitational force — has the same direction as sunlight, but the opposite sense, then clearly everything that has been said about the propagation of light must equally well apply to the propagation o f force. In particular, the attractive gravitational force has the same aberrant direction as sunlight, that is, it arrives not exactly perpendicular to the earth’s velocity, but at an angle 90° —/3 to it. The

70

EINSTEIN M INUS ZERO

Sec. 1.8

same argum ent goes for electrons orbiting round the nucleus (at zero level in the hydrogen atom , where the electron is fastest, /3= 0.007).
This aberration of force results in a sharp difference from the conventional Newton-Coulomb Law (6), where for the case of an electron orbiting the atomic nucleus, the constant A"= 2.3 x 10~28. This form ula applies in all theories when the charges are at rest — whatever the pertinent theory understands by that. When they are moving with respect to that theory’s rest standard, then in the Einstein theory, as we have just seen, (6) applies to the proton attracting the electron, but not vice versa.
Now let us look at the dynamic N ew ton-Coulom b Law for the force between two moving charges from the point o f view o f the present theory. Even for the case o f a circular orbit, there will be three differences from the static case (6). First, the bunching o f the electric field strength in the transversal direction (seen by the Einsteinian observer if he sits on the nucleus, but observer-independent here); second, the magnetic force between moving charges as discussed in Sec. 1.4; and third, the effect of aberration just discussed.
Let us modify (6) for these points. The bunching o f the E lines o f force for E in the direction o f the acceleration when the latter is norm al to the velocity is given by (4), so that (6) becomes

Next, we correct for the magnetic force between the two charges; according to magnetic rule no. 2, i.e., (6), Sec. 1.4, we m ust multiply (7) by 1 —/3\ resulting in

and finally, we must correct for the aberrational angle under which the nucleus acts on the electron; that angle, from the discussion above, is d, so that

(9)

where 8 0 is a unit vector in the transversal direction in polar coordinates r, 8, and

for the case of a circular orbit also the unit vector in the direction of the velocity.

In both applications that will be o f interest, planetary systems and electron

orbits, d is small: about 7 x 10"J for the ground level in the hydrogen atom , and

about 1 .4 x 10"4 for M ercury, the fastest planet. For accuracy to second order,

we therefore use

cosd — \ / l - d 2 + 0 (0 * )

( 10)

and with the same accuracy, (9) simplifies to

F = ~ \ ( l —d 2)r0 + de„]

( 11)

Sec. 1.8

THE MODIFIED NEW TO N-COU LO M B L A W

71

which is an unusually radical departure from the conventional Coulom b-Newton Law (6), for a first-order term in 0 has appeared in it. For an attractive force, such as that o f the sun or o f the nucleus, the constant K is negative, so the 6 com­ ponent o f this force in the second term o f ( 11) is directed against the velocity. (This is not affected by a choice o f coordinates since for positive K the aberration always deviates from the “true” angle in the direction of the velocity.) The first thing that must therefore be explained is why the solar system, and all o f its atoms, do not collapse.
This would indeed be the case if the fi2 term were not present in the radial com­ ponent. In that case the orbiting body would do work in advancing against the force ( 11) in the direction o f its velocity, and since the system is closed, this work would have to be performed at the expense of the potential energy, i.e., by reduc­ ing the distance of the orbiting body from the attracting center. Quantitatively, an element o f work perform ed by the force ( 11) over a distance

ds = d r r 0 + r d O Q u

(12)

is

Fds =0

(13)

which equals zero since there cannot be any net energy change. Substituting (11) and (12) in (13) yields an elementary differential equation with
solution

r = roe - 09 = r Qe~ffut

(14)

where we have assumed j3 constant and equal to its instantaneous value over a few turns of this inward spiral.
However, the 02 term in (11), which has been ignored in thiscalculation, com­ pensates for this effect as follows. The path (14) corresponds to an effective force pushing the orbiting body toward the center given by

F i„ = - m f r 0 = - / ? 2w2r r 0

(15)

O n the other hand, the0 2 term in (11) is positive (for K isnegative for attrac­ tion), representing a force in the direction o f r0, or outw ard. O nusing the formula for the angular velocity o f an orbiting body (derived in textbooks of mechanics, and also in Part Two)

this term is

F 0uf = —ryl — r 0 = /?2mu>2r r 0

( 17)

which exactly cancels (15).

72

EINSTEIN MINUS ZERO

Sec. 1.8

However, apart from saving the solar system and its atom s from collapse, this term is o f no significance for the small values o f /3 in planetary and atom ic orbits, and we shall henceforth neglect it, leaving the modified Newton-Coulomb Law for circular orbits (and more generally for the force perpendicular to the velocity) in the form

F = ^ [ r 0 + /?e„]

(18)

The radial com ponent o f the force introduces no aberration, but gives rise to delay. Let two equal charges move away or toward each other uniformly along the straight line joining them and let their instantaneous (inferrable) distance from each other be r. Then during the time the force has propagated over the distance r, namely the time t = r/c, the distance between the charges will have increased by rt= rr/c. Therefore the force propagating from the source charge will act as from the point when it was emitted, not from the point where the object charge is at the time o f arrival. T hat is equivalent to modifying the distance between the charges by a factor (r/c). Thus the full version of the modified Newton-Coulomb Law (com parable to the Lienard-W iechert formula) to first order in 0 is

F = r n i - ; / c ) * M 1 " g 2 ) + '? e °l

(19)

However, we shall need this case only once, namely in the advance o f M ercury’s perihelion in Part Three. Otherwise the modified N ew ton-Coulom b Law we shall use throughout the book will be in the form (18).
The original Newton-Coulom b Law (6) will be seen identical with (11), (18) or (19) for c= oo, corresponding to Instant Action At a Distance (IAAD).
To summarize, the present theory assumes that forces propagate with velocity c from their sources, that Newton’s Laws and the Maxwell equations are valid when all velocities are referred to the local force field rather than to an observer, and that the relativity principle is valid in Euclidean space and unreform ed time. This leads formally to the same expressions for mass, momentum and energy, and to the same relations among these three as in the Einstein theory, but the corre­ sponding effects are rooted in the phenom ena themselves, independently o f any observer’s location or perceptions.
It will now be shown that the theory explained so far will explain all observed effects invoked as proofs of the Einstein Theory. The two additional effects that have hitherto remained unexplained will be discussed in Parts Two and Three.

Sec. 1.9

73

1.9 The Electromagnetic Evidence
To show that the proposed theory does not contradict the experimental evidence in the field o f electromagnetism, it must be shown that
1) the experiments confirming the Einsteinian form ulas for mass, energy and momentum are consistent with the proposed theory (in which these formulas are, as explained in Sec. 1.6, only form ally identical);
2) that the evidence purporting to dem onstrate time dilation has been mis­ interpreted;
3) that the electromagnetic equations o f moving m aterial media remain valid in the present theory; and
4) that the Maxwell equations and the Lorentz force are invariant to the Galileian transform ation once all velocities in them have been referred to the field rather than to the observer.
As for length contraction, there is no direct experimental evidence for it, and therefore no need to refute or reinterpret it. The indirect evidence comes from experiments such as that by Michelson and Morley; but as we have seen, this experiment is consistent with at least four different theories, o f which the proposed theory is one.
1.9.1. Mass, Momentum and Energy
The crucial relations of Einsteinian dynamics that have been confirmed by experiment are those involving mass, energy and their relation to momentum, given by (7), (11) and (15), Sec. 1.7, respectively. However, in the Einsteinian interpretation, v is understood to m ean the velocity with respect to the observer rather than with respect to the local force field.
All that needs to be shown, therefore, is that in all o f these experiments the observer was at rest with respect to the local force field, so that the experiments cannot decide whether the effective (effect-producing) velocity is that with respect to the observer or that with respect to the field.
For example, one o f the ways o f measuring mass at high velocities is to let a charged particle traverse a magnetic field at right angles. The Lorentz force q \ x B will curve the path o f the particle, which will balance this force with its inertial reaction (centrifugal force) m v 2/r. From the equality, the radius o f curva­ ture is
m o« ( 1)
g B s/l^p i

74

EINSTEIN M INUS ZERO

Sec. 1.9.1

and all quantities in this relation can be measured. The relation has been con­ firmed with protons for & as high as 0.81 [Zrelov et al. 1958].
In all o f these experiments using a magnetic field, the latter is, o f course, produced by wire-bound currents (perm anent magnets would also be electrically neutral). As explained in Sec. 1.4, for such magnetic fields the local force field is that o f the electrically neutral conductor, that is, the gravitational field in which the laboratory is also at rest. These experiments are therefore consistent with both theories and unable to test the difference between them.
The dependence o f kinetic energy on velocity

(2)

derived in (11), Sec. 1.7, has been confirmed, for example, by measuring the heat dissipated in the water tank in which the high-velocity electron beams o f linear accelerators are dum ped. A recent experiment by W altz et al. [1984] is impressive, not so much for its accuracy (an error o f 30% ) as for its high value o f j3 (0.9995) and o f the dissipated beam power (up to 3.5 kW).
In a linear accelerator, particles are accelerated by a series o f “gaps” across which the accelerating voltage is supplied by a radio-frequency traveling wave arriving at successive gaps simultaneously with (or just slightly ahead of) a bunch of particles. It might therefore be thought that the “velocity with respect to the local force field” should be the velocity o f the particles with respect to the traveling wave, which would be close to zero.
Not so: here and in all other cases, velocity with respect to the local field means the velocity of the particle with respect to the lattice of equipotential/line-of-force intersections in its immediate neighborhood, w ithout regard to how this field got there. The particles are accelerated in steps as they traverse the gaps, with no significant electromagnetic field or acceleration in their flight from gap to gap. During the short time that they traverse a gap, the electric field accelerating them is produced by the charges on the opposite ends of the stationary gap, not by the fields in other places, which are as irrelevant as their m utual relationship that con­ stitutes the traveling wave. The field within the stationary gap is nailed to it as securely as if it were produced by a battery switched on and off at the proper times. Thus the “local force field” is stationary in the laboratory frame, which is the rest fram e for the observer. Once again, this type o f experiment is consistent with both theories and unable to test the difference between them.

Sec. 1.9.2

C H AM PIO N 'S E X P E R IM E N T

75

1.9.2. Champion’s Experiment
It should be noted that a test o f a relation like (1) of the preceding section, though consistent with both theories, is not the most convincing thing in the world. The velocity is not directly measured, but inferred from the electro­ magnetics that the test is to confirm. In addition, the test assumes the conservation o f charge, for it is a m atter o f interpretation, not a m atter o f measurement, whether the square root divides the rest mass or multiplies the “rest charge,” which in some other theory (not Einstein’s or mine) might not be constant. This is an objection which applies to all experiments involving the mass-to-charge ratio, and this includes a large number, perhaps even a majority, of experiments claiming to prove the velocity dependence o f mass. In reality they prove nothing but the velocity dependence of the mass-to-charge ratio.
A clean (or at least, cleaner) experiment would demonstrate the mass-energymomentum relations independently of the value of charge or velocity used. There is such an experiment, an effect apparently first noted by Cham pion [1932], for in its simplest form it measures nothing but the change o f an angle — the angle o f the paths of two electrons after collision. In IAAD (instant-action-at-a-distance) mechanics, the tracks o f these electrons, which can be recorded in a cloud chamber, should be perpendicular, but at high velocities they were observed to conclude an acute angle. By 1935, velocities corresponding to (3= 0.968 had been achieved [Tonnelat 1959], and the good agreement with theory was widely inter­ preted as a confirmation of the Einstein theory.
The experiment confirms the relations derived in Sec. 1.7 when interpreted by the present theory. However, it is my belief that the Einstein theory comes through this test with less than flying colors, as discussed in the following.

(a) symbols and geometry for (1)

(b) repulsion at a distance

C h a m p io n ’s experim ent

Let two electrons (or billiard balls, for that matter) collide; let one of them be originally at rest at the origin, hit by the other with m om entum p. W e orient the x-axis along the path of one of the balls after collision and denote the momenta

76

EINSTEIN M INUS ZERO

Sec. 1.9.2

after collision with primes. Then from the conservation of momenta along the x and y axes we have

P i cos p = p'j + ;/2 cos 0 P i sin p = p\ sin0

Squaring and adding yields Pi —P'i + p' 2 + 2 p jp '2 cos 0

Using (15), Sec. 1.7, in the form

P2

2 2

^2 = 171 ~ m o

(3)

and from the conservation o f energy (dividing by c2)

rrii + rri2 — rn\ + rn'2

(4)

we find after some algebra

p'iPz cos 0 = c2(m'2 - m ())(m '| - m 0)

(5)

In IAAD mechanics, the masses on the right are all equal, so that the right side vanishes, whence cos0 = O and 9 = k/2 . But when mass is a function o f velocity as in (5), Sec. 1.6, both parentheses on the right are positive, as are the m om enta on the left; hence c o s9 is positive and 9 is acute (less than x /2 ).
Some more algebra will actually express tan (^ - 9) as a function o f /3 , and this agrees well with the measured data [Tonnelat 1958]. However, the derivation given here has gone far enough to confirm the difference between HAD mechanics and the expressions derived in Sec. 1.6. Thus the Cham pion effect, whether pur­ sued beyond this point or not, supports the present theory.
But its support o f the Einstein theory is questionable, notwithstanding the text­ books using the derivation given here as proof o f the Einsteinian mass dependence. The reason is that the validity o f the starting point (1) for colliding electrons is, from the Einsteinian point o f view, debatable. T o see this, let us go back to basics and recall where the conservation o f m om entum comes from.
Let a system of bodies (such as charged billiard balls) be subject to external forces (e.g., currents flowing nearby and friction on the billiard table). Let the external forces on the Arth body be F*, and let the internal forces by which any two bodies act on each other be Fgj and Fjg. Then since no body acts on itself, all * ii= o- and integrating over time from /, to t2, we have

f
k

J k

dt =

/

k

- m kx k (t2)} (6)

Sec. 1.9.2

TIME D ILA TIO N

77

provided the double sum on the left vanishes due to action and reaction being equal and opposite, i.e., provided that Fjk= Fkj- If there are no external forces, then the left side vanishes completely, and since the times r2 are arbitrary, the remainder states the conservation o f m om entum at all times.
But the words printed in italics, which are always valid in the present theory, do not necessarily hold in the Einstein theory. They hold for the collision of uncharged billiard balls, when m om entum is transferred by actual contact at a point where simultaneity holds for all observers. They do not hold for an electron, because it does not wait in space, nailed to its coordinates while Coulom b’s Law is suspended, until it is bodily hit by another electron. W hat happens is that an elec­ tron is repelled by the approaching electron at a distance; the two come to within a minimum distance, but not into contact, and they continue to repel each other as they recede from each other. During the entire process, which is studied in plasma physics and particle scattering, the two interact at a distance. But as we have seen in (4) and (5) of Sec. 1.8, the Einstein theory does not recognize the equality of action and reaction at a distance: the force exerted by a particle at rest on a moving particle is not at all the same as the force exerted by the moving particle on the one at rest.
I have no doubt that the Einstein theory can explain things as it always does. Perhaps the asym ptotes o f the curved trajectories are as good as straight lines from an equivalent collision; perhaps the whole thing can be conjured away in the opaque acrobatics o f four-vectors and world lines.
But the fact remains that the Einstein theory has some explaining to do; for a theory that does not recognize the equality of action and reaction cannot, without apology, invoke the conservation of momentum.

1.9.3. Time Dilation: Ives-Stilwell, Mesons and Clocks Around the Globe
According to Einstein, a clock ticks m ore slowly for an observer who passes it with velocity v than for one who is at rest with respect to it. (Both observers com­ pare its reading to their identically constructed electronic wristwatches, say).
We are offered three types of experimental proof for this phenomenon: the Einsteinian Doppler effect, the rate o f decay of fast moving mesons, and the transport of an accurate clock round the globe.
W hat all three techniques have in com m on is the failure to ask, let alone answer, the crucial question: is the m easured effect something that is dependent on the observer, or is it something that changes the clock?
To see the difference, imagine tw o identical pendulum clocks whose time readings are compared after one of them has been transported round the globe.
If flown eastward, the transported clock’s reading will be fast; flown westward, it will be slow.

78

EINSTEIN M INUS ZERO

Sec. 1.9.3

Time dilation? No: the period of a pendulum varies inversely as the square root o f the downward force on it, and that force is the vector sum o f gravitational attraction and the centrifugal force due to the earth’s rotation. As pointed out by Barnes [1983], the centrifugal force must necessarily, if only very slightly, increase when the clock is moved eastward, because its angular velocity about the earth’s center increases; and it must decrease when transported westward, against the earth’s rotation. This is an inherent change, one that an observer traveling with the clock (i.e. at rest with respect to it) could measure by com parison with an equally accu­ rate wrist watch — if it is unaffected by centrifugal force. I do not, o f course, propose this as an explanation of the alleged time dilations; I mention it as an illustration o f an inherent change in a clock which might easily be mistaken for a change in the flow o f time. In this and all other cases we must first check by a control experiment whether the rate of the clock has changed inherently, as measured by a co-traveling observer at rest with respect to the clock, before we check for any Einsteinian observer-dependent effects. In none o f the three techniques has this been done. Such a control experiment performed by observers (measuring instruments) traveling with hydrogen ions or mesons as they traverse a gravitational field at a significant fraction o f the velocity of light are beyond contemporary feasibility; so it is tacitly assumed, without the slightest proof, that there are no such inherent changes, and all observed changes are ascribed to the observer’s velocity. Consider the Ives-Stilwell [1938, 1941] experiment on the Doppler effect o f a fast moving source (light-emitting hydrogen ions in canal rays), which is concep-

lo spectrometer

diffraction grating
T he Ives-Stilwell experim ent [1938, 1941). T he ions S are generated to the left o f the figure, accelerated by the electrodes, and pass through a hole in them to the space on the right. Their light reaches the diffraction grating from an approaching source through the observation window directly, and from a receding source via the concave m irro r M , whose axis is only 7° off the velocity direction. T he grating is the disper­ sive element o f the spectroscope, whose telescope and photoplates are not shown.

Sec. 1.9.3

TIME D ILATIO N

79

tually the simplest o f the three types; it is also very impressive because its result depends only on a comparison of spectroscope readings, not on inferred velocities.
As shown in the figure, the grating o f the spectroscope is reached by the light emitted by fast moving hydrogen ions directly in the forward direction, and via the mirror in a direction making an angle of only 7° with the velocity. Thus the spectroscope measures the Doppler-shifted wavelengths of the radiation emitted by an approaching and a receding, yet identical, source.
The classical Doppler effect for a source moving with velocity v in a medium in which the observer is stationary (also applicable to the propagation o f light through a gravitational field) is found by elementary trigonom etry. The time difference between two successively received wave crests emitted with period T0 is

T = T0 + -c ~ -c

(1)

where R and ra re the distances of the source from the observer at the moments when the crests were emitted. F or X« r, the emitted and received wavelengths X0 and X are therefore related by

A = Ao(l - (3c o s 9)

(2)

where /? = v/c, and 0 is the angle m ade by v and the direction o f propagation to the receiver.
The Einsteinian Doppler effect, on the other hand, leads to

A = A °(l — ffcoag) ^ a ( i _ /geos0 + f/32)

(3)

Let us now write (2) and (3) for small as

\ = Ac, (1 - /3cos0 + k(i2)

(4)

where k = 0 in the classical, and k= Vi in the Einstein theory. The Ives-Stillwell experiment is based on the asymmetry o f (4): when the sign o f /3 is changed (i.e. the backw ard ray is considered instead o f the forw ard ray), and the two resulting wavelengths averaged, the terms in (3 will cancel, but the ones in /32 will remain. The Doppler-shifted wavelength is

AA -

= —/? cos 0 + k(32

(5)

Ao

and this measured by the spectroscope. The two Doppler-shifted lines, one from the approaching and one from the receding ray, correspond to +(3 and - j3 and are displaced to either side o f the “rest” spectral line. W hen the two shifts are averaged, we then have from (5)

A 2A — i[A A i + AA2] — k(32

(6)

80

EINSTEIN M INUS ZERO

Sec. 1.9.3.

We can now combine (5) and (6) into the relation

A 2A = fc(AA)2

(7)

in which the the bone of contention k can be checked by measuring only

wavelengths, unaffected by (reasonable) errors in voltage, velocity, or other error-

prone quantities: the spectroscope measures the two shifted lines as in (5), the

average as in (7) is then examined on the photographs under a microscope, and the

results are plotted for comparison with (7).

W orking on the borders o f the then feasible technology (mechanical

micrometer, all-day exposures), Ives and Stilwell established that k= '/i. This was

confirmed by M andelberg and W itten [1962] with the technological advance o f

two decades, and as this book was readied for press, news came that M acArthur et

al. [1986] had obtained further confirm ation at |(3= 0.84.

To Einsteinians this is proof o f time dilation; to me it is p ro o f that particles

traversing a gravitational field radiate, in their own rest frame, an inherent

frequency lowered by Zifi1 — reminiscent of the seeming time dilation o f the

pendulum clock. This is the frequency an observer sitting on a moving particle

would measure, and this is then Doppler-shifted by classical rules to yield the

result measured by Ives and Stilwell.

W ithout a check o f what frequency is radiated inherently in the source’s own

frame, the Ives-Stillwell result remains ambiguous, and would not have to be

accepted even if there were no alternative explanation.

But there is one.

When a hydrogen atom moves through a

gravitational field, then by our basic assump­

tion, force propagates with velocity c with

respect to the gravitational field; thus the

Coulomb force between nucleus and electron will be subject to delays and aberrations. The delay, if any, has no effect, since the radius of the electron orbit remains unchanged. In

r( e F *jc- ^' .
Fr

calculating the aberration, we ignore the aberration due to the electron’s orbital velo­ city — not because it is negligible, but because we are looking for the additional

N .. W <P\ Q. 1 -------------------------- --------------------- • V
Aberration and frequency shift

aberration that sets in when the atom moves

as a whole. From elementary geometry and

Sec. 1.3.1 we find the aberration angle

sill <= 3 cos ip

( 8)

Sec. 1.9.3

TIME D ILATIO N

81

where $ defines the electron’s instantaneous position on its orbit. The force in the radial direction (tow ard the nucleus) is therefore reduced by

c o s t = \ J 1 - /32c o sV % 1- /f2/4

(9)

where c o sV has been averaged over the orbit. This hasthe same effect as if the square of the charge were reduced by that am ount; we can therefore calculate force, energy or other quantities by using the fictitious charge

q = q0( l - f l 2/S )

(10)

where q0 is the equivalent “rest charge.” The radiated frequency (or energy quan­ tum hv) is proportional to the fourth power o f this charge [as derived in (18), Sec. 2.5, or found in any physics handbook], so that the inherently radiated frequency is

a = u0(l

(ID

and this is then shifted by the classical Doppler effect to yield exactly what was measured by Ives and Stilwell, or when inverted, to give the time difference on a transported cesium clock.
For radioactive decay [Frisch and Sm ith 1963], the timing mechanism is unknow n, but here again energy is proportional to the average frequency of disintegrations; the assum ption that ( 11) remains valid for this case is no more arbitrary than assuming (as is done in orthodox physics) that radiated frequencies or radioactive decay remain inherently unchanged when the corresponding atoms traverse a force field with a velocity approaching that of light.
Thus we again obtain the same results as in the Einstein theory. To summarize: the experimental evidence on alleged time dilation overlooks the crucial issue: is it time or the clock that is affected? It is a special case o f a more fundamental question: should physics seek to understand objective reality or should it describe an observer’s perceptions?

82

Sec. 1.10.1

1.10. Galileian Electrodynamics

1.10.1. The Maxwell Equations and the Lorentz Force
There is something puzzling about the Maxwell equations: they grew out o f Faraday’s concept o f lines o f force repelling each other as they weave their way through the ether; this concept has been totally abandoned, yet the Maxwell equa­ tions have remained valid through all types o f ether — elastic, rigid, stationary, entrained, partially entrained — and they remain valid in the Einstein theory. It is not the usual fate o f a flower to bloom on when the soil is removed or shown never to have existed.
The answer to the puzzle lies in w hat the com puter sages call “transportability:” a good com puter language, for example, will run under various operating systems when it has comparatively few routines that m ust be adapted to the system and is otherwise self-contained.
The Maxwell equations give the relations am ong four vectors, E, D, B, H. These vectors cannot manifest themselves until they act on charged matter; if the Maxwell equations were not tied to charge and matter, they would be a meaning­ less abstraction. But the bridge to charge and m atter is a narrow one (to be discussed in a mom ent), and the bridge from charge to force is given by a single equation, namely the Lorentz force

F = f/(E + v x B)

(1)

in which the magnetic induction B is not an independent vector, but is (for a moving point charge) derived from the electric field by

v x E

B= —

<2 >

Here, very typically, v is the velocity o f the moving charge q with respect to an unspecified rest standard: the stationary ether, the entrained ether, the observer, the field o f the other charge(s) — the Maxwell equations care no m ore than a transportable com puter program cares w hat operating system it is running under.
Thus, the Maxwell equations proper are a self-contained floating island that can be linked to the mainland o f charge and matter by various bridges involving various velocities, and this solves the puzzle why the island survived when the various mainlands went under.
Let us now look for the bridges that tie the floating abstractions to the firm mainland of charge and force. We write the Maxwell equations (for piecewise homogenous, isotropic, non-pathological media) as

Sec. 1.10.1

M AXW ELL EQUATIONS

83

(3)

(4)

(5) <<>

V B=0

( 6)

These four equations contain only two quantities that inconspicuously provide a bridge to charge and matter, namely the charge density g and the current density J . If we set these two to zero, we are in the well known textbook case of “a space without charges or currents,” the obvious implication being that we are in a space in the neighborhood o f charges and currents.
Note two points about these two quantities, without whose presence (immediate or distant) Maxwell’s equations become a meaningless torso:
First, both are velocity-dependent (the velocity-dependence o f g is attributed to that o f volume in the Einstein theory, and is caused by a charge redistribution in mine). They are are tied together, not by a natural law, but by a definition, the definition of current density:

J = pv

(7)

making the bridge even narrower. (A more general definition, expressing the invariance o f charge, is possible, but will not be needed here.)
Second, in the Einstein theory, these two quantities are not invariant to the Lorentz transform ation (even though the equations involving them are): charge density is charge per volume, where charge is an invariant, but volume has one dimension that shrinks with velocity. C urrent density is modified even more drastically in the Einstein theory, for velocity is transform ed in a more com­ plicated way than length. Hence the bridge (7) is no longer a simple relation in the Einstein theory, but a mask hiding an ugly complication.
There are three more quasi-bridges to charge and matter, namely the constituent equations

D = <E

(8)

(9) 0
to which one may add O hm ’s Law, if applicable,

J = aE

0°)

and all o f this, o f course, remains an abstraction until it has manifested itself by the force ( 1).

84

EINSTEIN M INUS ZERO

Sec. 1.10.1

The reason why I have called the constituent equations quasi-bridges from fields to charged or uncharged m atter is that for moving media they are not really bridges; they are no more than formulations o f the problem, as evident from the following.
Maxwell’s equations (3) through (6) are straightforw ard only for fields in unlimited free space; the behavior o f fields in m atter, and especially in moving m atter, has been pushed off into the constituent equations (8) through ( 10), which give the relation between the fields inside m atter (D, H) and the fields producing them, quite often in the free space outside it, (E, B). As long as this m atter is at rest, then we have nothing to talk about, for the “rest” permittivity e and the “rest” permeability n are quite uncontroversial, and we need neither Lorentz nor Galileo for electrodynamics where the flow o f charges as electric currents is the only thing that moves. (Let us not not complicate matters by the conductivity a, which can, in the most im portant case o f harmonic time variation, simply be absorbed into the imaginary part of a complex permittivity). But when we have a moving medium, what happens to the permittivity and permeability — what hap­ pens to the fields inside m atter — when we are at rest and the medium is moving?
To this one and only question of importance for moving media, the constituent equations answer with two shoulder-shrugging Greek symbols, e and /t, implying a relation between the fields in the moving medium and the fields outside produc­ ing them, but telling us nothing about what permittivity and permeability “on the move” stand for, much less how to find them.
The relation between the field D inside a medium and the field E outside it is not at all simple when the medium is moving, nor is it simple between H and B; in fact, it is best to abandon the concepts o f e and /t in (8) and (9), which become tensors, for D and E (just as B and H) inside a moving medium no longer have the same direction even if the medium is isotropic.
The Einstein theory supposedly first fully solved the problem of electromagnetic fields in moving media; and we shall examine this claim next.

Sec. 1.10.2

M OVING MEDIA

85

1.10.2. Electromagnetics of Moving Media
In his classic paper [1905a], Einstein showed that the Maxwell equations are Lorentz-invariant when applied to individual moving charges (with the effectproducing velocities assumed to be those with respect to an observer). But it fell to his disciple H erm ann Minkowski (1864-1909) to apply Einstein’s theory to moving m atter, that is, to find the relation between the “driving” fields E, B and the “driven” fields inside m atter, D, H. For moving m atter, such a relation is implied, but not explicitly given, by the constituent equations (8) through ( 10) o f the pre­ ceding section. Minkowski, the mathematician who introduced space-time (“Henceforth space by itself, and time by itself, are doomed to fade away into mere sh a d o w s.. . ”), found the solution in 1908 via six-vectors and their space­ time components; his derivation is given in [Sommerfeld 1964] or [Penfield and Haus, 1967], O ne o f the simpler ways o f writing the required relations is
( 1)
(2)

1 ---- — (I2 D± = f( l - ft2 )E ± + (<n - ro/zn) v x H

(3)

foMo

1 — ft2 D± ~ /i(l - ft2)H± - (<n - <t)fio) v x E

(4)

foMo

(J - pv)|| = o \ J 1 - ft2(E + v X B )|| (5)

where II and 1 denote the com ponents parallel and perpendicular to the velocity. However, to my knowledge these equations have never been verified to second
order, nor is it easy to do so: they cannot, like the mass dependence, be tested by particle accelerators, for they involve the fields in macroscopic, ponderable matter. The electrodynamics o f moving media based on (1) through (5) (or one of several other form ulations) is an esoteric, highly theoretical field, which is not without problems (such as the equivalence o f various form ulations, see [Penfield and H aus, 1967]) and which gives little physical insight.
O n the other hand, for slowly moving media (first-order ft), the corresponding equations can be derived without the Einstein theory, are verified, and do provide physical insight, for they rest on simple principles. For example, the polarization P in a moving m edium induced by an external stationary field E will result in charges that are bound within the medium, yet moving with respect to the rest frame, having the effect o f a current. Such a physically founded derivation is no longer easy to find, but it does exist: for non-m agnetic media, see [Becker 1964], while for the general case I know only of a Russian source [Tamm 1954],

86

EINSTEIN MINUS ZERO

Sec. 1.10.2

These derivations without use of the Lorentz transform ation lead to the same result as (1) through (5) with /32= 0, as they must, since the Galileian and Lorentz transformations merge under that condition. Thus for slowly moving media we have from (3) to (5)

(6)

n2 —1

B = /zH

^— v x E

(7)

J —p \ = rr(E + v x H)

(8)

Since these are the only experimentally verified equations for moving media, there is no need to attem pt the derivation o f (1) through (5).
However, we will pursue another point that will turn out to be instructive and relevant to our story: let us see whether we can simplify (6) through (8) further by looking for equivalent electromagnetic param eters e, n , a, that would express these equations in the more familiar form o f an equivalent stationary medium

D = ,'E

(9)

B = n'H

( 10)

J= o'E

( 11)

where, in (8) and ( 11), we have assumed the absence o f free charges (g = 0). The problem was posed and investigated some years ago in a brief paper
[Beckmann 1970]. The conditions under which such equivalent parameters of a stationary medium exist were then found to be

[hi < 1. E H = 0

( 12)

in which case the only v com ponent that need be considered is

V = V COS 0 S()

(13)

where s is a unit vector in the direction o f the Poynting vector E x H and 9 is the angle m ade by that vector and the velocity v.
The second condition in (12), E perpendicular to H, is fulfilled in the most important application of this type of problem, namely the propagation of electro­ magnetic waves, and some other problem s as well. If the fictitious, stationary medium is to be equivalent to the real, moving medium, the two media must obviously have the same impedance

Sec. 1.10.2

M OVING MEDIA

87

From (12), (13), (14), we have

v x E = cficosO ^^-H

( 15)

v x H = -r/)c o s< ? y ^ E

(16)

and when these two are substituted in (6) through (8) using n = c^J(ev), we obtain by comparison with (9) through (11) the required equivalent parameters

t' = , ( l

- ’jLzlncsll'j(17)

/ =,7, _iLzV™,,)

(18)

a' = o ( \ — (Incus 0) which solve the problem.
Now let us find the velocity o f propagation o f electromagnetic waves in this new, equivalent medium.
Using the first o f the conditions (12), we can write (17) and (18) as

where Hence

1 + k fin cos 0

H1 =

t

1 + K lin cos 0

* = 1 - ~n2l

(20) (21)
(22)

” = 1 + Kfin cos 0

(23)

and the velocity o f propagation is

u' = an' = — + K it c o s 0

(24)

n

This is the velocity o f propagation o f the electromagnetic wave through a medium which moves with velocity u with respect to the reference frame, such as that o f the laboratoryobserving the light propagating through the flowing water o f Fizeau’s 1851experiment; for (22) is nothing but Fresnel’s coefficient o f drag.
The Einstein theory derives this crucial coefficient as the first term in the series expansion of the velocity-addition theorem based on the Lorentz transformation, but as explained in Sec. 1.3.2, Hoek derived it in 1868 from the null effect in his experiment on purely optical grounds without the need for an ether.

88

EINSTEIN M INUS ZERO

Sec. 1.10.2

We now get the “etherless” Fresnel coefficient a second time from strictly electromagnetic considerations. There is no experimental evidence to decide whether the velocity v means the velocity of m atter with respect to the local field, as we assume, or whether it should be referred to the observer, as in the Einstein theory, for clearly in all Fizeau-like experiments the observer was at rest with respect to the field (for the case o f light, the gravitational field).
Once again the two theories lead to the same result, but once again, the electro­ magnetic derivation is based on tangible fields and impedances, whereas Einstein’s derivation (which, incidentally, runs into some difficulties for non-zero angles be­ tween the velocity o f the m edium and that o f the propagating wave) is based on an abstract addition o f velocities in redefined space and time.

1.10.3. Invariance of Relative Velocities
We must be careful about what is m eant by the statem ent that the Maxwell equations are invariant with respect to the Lorentz transform ation. It means that if we transform the Maxwell equations from one system o f space-time coordinates to another (each serving as the rest frame for an observer), we obtain equations of the same form in the new coordinate system when we use the Lorentz trans­ formation.
T hat statem ent is perfectly true; but if the effect-producing velocities are not those referred to an observer, then it is also trivial and irrelevant.
It is trivial because the Lorentz transform ation is founded in equations such as (7) and (9), Sec. 1.6 : if the Lorentz transform ation was made to fit the Maxwell equations, it is not surprising that the Maxwell equations fit the Lorentz trans­ form ation.
M ore im portant, it is irrelevant. The only velocities inherent in the Maxwell equations and the Lorentz force are those associated with current density (gv), with the magnetic field (v x Vc^/c2), and with the magnetic force (qv x B). If these velocities are responsible for the pertinent phenomena when they are referred to to the local field rather than to an observer, then in any observer’s coordinates they will remain invariant as the simple vector difference between charge velocity and field velocity in those coordinates, and the transform ation that preserves a simple difference in velocities is the Galileian transform ation. The same ultimate forces must therefore act in all moving observer’s frames; what pretty patterns o f equa­ tions for field vectors (other than forces) are preserved by what other pretty transformation does not matter.
We must, o f course, distinguish between velocities that affect phenom ena in which the observer is not involved — such as potential energy or hydraulic friction — and those that modify the transmission of information, force or energy to

Sec. 1.10.3 IN V A R IA N C E OF R E L A T IV E VELO CITIES

89

a moving observer. The latter category includes aberration and the Doppler effect in all its forms.
For example, if I walk tow ard a charged capacitor, I see its charges as currents flowing in my egocentric coordinates, and Einstein interprets this as part of the electric field changing into a magnetic field about these currents. But in the present theory I see not only a velocity of the charges (i.e., currents), but also a velocity of the electric field between the capacitor’s plates, and since in the present theory the effect-producing, determining, applicable velocity is that o f the charge on one plate in the field o f the other, the Galileian transform ation will m ake the effectproducing velocity equal to the difference between the two, which remains zero as before. This strengthens my conviction that the force between the plates of a capa­ citor, measured in unreform ed space and time, cares very little about observers observing that force, even if they travel past the capacitor in a spaceship at half the velocity of light.
As for electromagnetic Doppler effects and aberrations, which do involve the observer’s m otion, none o f the available evidence — such as the Ives-Stilwell experiment — contradicts the assum ption that the effect-producing velocity is again that with respect to the local force field, which (with present technology) means the gravitational field.
We have here treated the Doppler effect and aberration as phenom ena in which the observer’s m otion is directly involved as part o f the phenom enon. On the other hand, there is no a priori reason why the force m utually attracting two electric charges should have the slightest dependence on an observer observing it. Most people who have not studied the Einstein theory in detail would probably “instinc­ tively” agree with the last two statem ents, o f which the second contradicts the Einstein theory. It is, in fact, due to such contradictions that many people “instinctively” distrust the Einstein theory in spite o f eighty years o f its continued successes.
But instinct is not what science is m ade of, and it is not easy to find a nontautological and unam biguous criterion o f what distinguishes the Doppler effect from the force between two charges as far as dependence o f the observer’s velocity is concerned. But at low velocities (with no m ore than first order /3 significant) the experimental evidence, if nothing else, shows a very striking difference between the two: the observer’s velocity is what the Doppler effect is made of, but it is irrele­ vant to the force between two charges. And while I admit that this difference need not necessarily hold for high velocities also, I do m aintain that if it does not, then the burden o f pro o f — or at least the reponsibility to defend itself against rival theories — is on the theory which imposes such a difference on us.
Now let us turn to those phenom ena for which there is no reason to believe that they are in any way physically connected to the velocity o f an observer, although of course, that velocity will occur in the coordinates which he chooses to describe, but not to change, the phenom enon he is observing. As a criterion o f classifica­

90

EINSTEIN M INUS ZERO

Sec. 1.10.3

tion, I will use the laws governing the phenom enon at low velocities: am ong those not affected by the observer’s velocity are the force between two charges, hydraulic friction, the length o f all time intervals, and m any others. A m ong those that are velocity-dependent by this criterion are the Doppler effect, aberration, m om en­ tum, kinetic energy, and not many others.
By this criterion, if by nothing else, the forces and fields described by the M ax­ well equations belong to the first group — especially when it is realized that there is no experimental evidence to differentiate between velocities with respect to an observer and those with respect to the local field, such as the gravitational field for magnetic force between two currents.
Seen through the eyes of the proposed theory, therefore, the invariance of the Maxwell equations to the Lorentz transform ation with all velocities referred to the observer is undisputed, but immaterial, because that invariance involves an irrele­ vant velocity. The velocity that is relevant in the present theory is the velocity o f a charge with respect to the local force field it is traversing. A nd that velocity, being the (vector) difference between the velocity of the charge and that o f the traversed field in the observer’s coordinates is Ga///e/-invariant.
In other words, when the velocities in the Maxwell equations and Lorentz force are properly attributed, the equations remain valid in all inertial frames moving in unreformed space and time, and related to each other by the Galileian trans­ formation.
This will be shown in m ore detail below. However, we must first discuss a point that applies to any velocity, not necessarily high, but is often overlooked: the only measurable and observable quantity in all o f electromagnetics is force, usually expressed as the Lorentz force

F = r/(E + v x B)

( 1)

and that includes forces such as those activating the electrons in an antenna and those in the cones o f our retinas. W ithout a charge to multiply them and thus to convert them into a force, the vectors E, B, D, H, P, M are abstract aids to our im agination whose existence is inherently unprovable. Force is that which changes the momentum of matter, charged or not, such as the pointer of a measuring instrument. There is no way o f measuring or dem onstrating any quantity without first converting it to force. Therefore as long as force transform s correctly — in accordance with experience — it does not m atter that the abstractions expressing it, such as the field vectors by themselves, have certain properties, for example, invariance to a certain transformation. In particular, if the hypothesis that the effect-producing velocity — which means the force-producing velocity — is that of a charge with respect to the local field, then the Maxwell equations will be no less invariant to the Lorentz tranform ation with observer-defined velocities than before, but the only thing that m atters, the resulting force, will be Galilei-

Sec. 1.10.3 IN V A R IA N C E OF R E L A T IV E VELO CITIES

91

invariant, because it depends on the difference between the two velocities, and this relative velocity is conserved.
Referring velocities to a different type o f standard will, o f course, produce a different Lorentz force in cert":n circumstances, and the difference is in principle measurable. The fact that the difference has not been detected is not due to alter­ native interpretations of the same evidence, but due to the lack o f technology to detect the tiny difference under the circumstances — at least, as yet.
As we have seen, the Lorentz transform ation and the Einstein theory, which is based on it, cater to the modification of charge distribution and of the resulting electromagnetic field brought about by the m otion o f charges. This modification is predicted by the Maxwell equations, and there is no quarrel about its existence, only about its physical origin — that is, about the rest standard to which the velo­ city producing this effect should be referred. It is therefore clear that differences must be looked for in the interaction o f electric charges (or gravitational masses), in particular, in the simplest possible interaction at a distance — that of two charged particles described by the Newton-Coulomb Law.
We shall examine this question in a m om ent. However, it is also instructive to compare the two theories in cases that do not involve any action at a distance, that is, no electromagnetic or gravitational fields. These are also the cases where the Einstein theory is never applied, in part because the low velocities usually involved do not justify its application, but doubtlessly also because these cases reveal the staggering complexity of simple problems in space-time coordinates distorted to cater to something totally unconnected with the problem.
Consider, for example, the case of a windmill used to pump water. The power delivered by the windmill is know n to be proportional to the cube o f the wind velo­ city relative to the windmill, and to be quite unconnected with electromagnetism. We wish to describe the process in the coordinates of an observer with a general velocity v0 with respect to the windmill.
Let the wind velocity in the mill rest fram e be v; then in the present theory, for an observer going past the windmill at 99% of the speed of light, the wind blows with a velocity v + 0.99c, and the velocity o f the windmill in the observer’s frame is 0.99c. But the velocity producing the power is simply the difference between the wind and the mill velocities, which is v, for in the Galileian transform ation the velocity of the observer always cancels in the difference, and the power/velocity function remains unchanged.
In the Einstein theory, if the wind blows at an angle that will give both the wind and the blades the benefit o f the Lorentz transform ation, the observer sees the wind slowing and the blades shrinking; but the Lorentz-transformed force of the slower wind (moving denser and heavier air due to length contraction and mass increase) on the narrower blades acts in dilated time, lengthening it. On the pum p side, the water has become heavier, for its weight mg is affected by the vertical com ponent o f the observer’s velocity (and as a m atter of fact, by the other com­

92

EINSTEIN MINUS ZERO

Sec. 1.10.4

ponents as well); it has also become denser because the water column has con­ tracted (mercifully, this time due to the vertical com ponent o f the observer’s velo­ city only); and the upward velocity o f the water will be slowed to yield an altered power when multiplied by a modified force. For a given height above the water table, we can ordinarily express the power in gallons o f water per second; but in the Einstein theory, this becomes shrunk gallons of a weird liquid per dilated second.
Now all o f these stunningly complicated acrobatics are necessary to save a relativity principle (o f sorts) catering to the field o f a moving electric charge, which has nothing whatever to do with a windmill pumping water.
No such problems arise in a theory that does not refer (all) velocities to an observer: the force o f the wind on the blades is the same for all observers, moving or not, as is the power converted.
Since the effect-producing velocities o f electric charges are relative to the fields they traverse, not relative to any observers, electromagnetic forces must remain Galilei-invariant for the same reason as the force o f the wind on a mill. This will be shown next.

1.10.4. Invariance of the Maxwell Equations
Before we deal with the invariance o f the Maxwell equations to the Galilei transform ation once their velocities have been properly assigned, we must once more go back to the analogy of the wind moving a windmill.
W hen an observer measuring the windspeed is at rest with respect to the mill (as he usually is), it does not m atter whether he refers the wind velocity to himself or to the mill; in either case the windmill equation W= C u 3 linking the power W to the wind velocity v (with C the constant o f proportionality) will be confirmed by all measurements.
Now suppose that the observer falsely concludes th at the effect-producing velo­ city under all conditions is the velocity referred to himself, but that he correctly applies the relativity principle to predict what will happen when he moves with respect to the windmill. Then he will say: it matters not whether the air moves against me or 1 move against the air; therefore if I start running on a windless day, the windmill must begin to turn. He tries the experiment (which, alas, is far easier perform ed than its analogy o f an observer moving fast through an electromagnetic field) and finds the prediction wrong. If he does not want to sacrifice the relativity principle, he has two choices: he can abandon the false premise, or he can keep it alive by deform ing space and time in a transform ation o f coordinates that will properly cater to the false premise. For example, the transform ation might con­ firm the prediction that his running causes a force by the air on the windmill blades; but it would also cause his running to produce an eqally large torque

Sec. 1.10.4 IN V A R IA N C E OF M A X W E L L E Q U A T IO N S

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on the shaft in the opposite direction. T hat is why runners don’t move windmills. (Is that taking the spoof too far? I will remind you shortly, if you think so.)
But this is not a facetious spoof; it is a fairly good analogy o f relativity applied to electromagnetics, with the windmill representing a charge and the wind repre­ senting the field o f another charge. The reason why the Maxwell equations “are not invariant to the Galilei transform ation” is that the velocities implicitly occurring in them have been mistakenly referred to the observer as the rest stan­ dard. As soon as these velocities are recognized as velocities of charges with respect to the fields they traverse, the Galileian transform ation will work on the Maxwell equations as surely as it does on the wind and the windmill.
Before showing th at in detail, let me go one step further than saying the field vectors are abstractions which cannot manifest themselves until multiplied by charge to yield an observable force. I will now say that the field o f a single charge, unrelated to any other charge, is a meaningless pattern o f arrows and rings drawn on papers and blackboards.
This may sound radical, but we need only think back to the definition o f a line o f force, with which the electric field intensity is often plotted. It has the direction in which a small test charge, that is, a second charge or a charge other than that producing the field, would move if it were present at that point; and the density of the lines is proportional to the m agnitude o f the force that would act, again, on this test charge. Clearly a field without this second charge is the same type o f entity as a waterfall without water.
This is not hairsplitting or nitpicking, for the field o f a point charge changes very decidedly when it is in m otion, rather than at rest, with respect to this test charge, even though the Einsteinian observer thinks he is causing the change with his own motion. (He fails to ask about the velocity of the test charge, just as the analogous observer failed to ask about the velocity o f the windmill in his own coordinates.)
Now consider the Maxwell equations. Since they are linear, it is permissible to consider the case of only two moving charges; the case of many charges, up to and including a continuous charge distribution, then follows by superposition — at least in principle, though the actual superposition for charges moving in different directions can be quite dificult.
The two directly recognizable velocities in electrodynamics are the velocity of a charge in a magnetic field, which occurs in the Lorentz force, and the velocity of charges form ing a current, which occurs in the current density J = gv involved in the second Maxwell equation. To interpret them as “effect-producing” velocities, we must understand them to mean the difference between the velocity of the charge and that o f the field which it is traversing — just as a runner should take the difference between the wind and mill velocities that he sees in his own coordinates.

94

EINSTEIN M INUS ZERO

Sec. 1.10.4

Let an observer move with respect to at least one o f the two charges with velo­ city v0, and let him describe all phenom ena in a coordinate system in which he is the origin. All time derivatives in the Maxwell equations will remain unchanged, for there is no time dilation, and all space derivatives will rem ain unchanged because there is no length contraction, so that all curls and divergences remain un­ changed, too. His velocity is added to all charges and all fields in the same way, so that the difference remains unaltered and his own velocity, which cancels, becomes irrelevant. W hen it is understood which velocities are effect-producing, the M ax­ well equations are as invariant to the Galileian transform ation as the wind driving a mill.
Note that a charge density moving with respect to the observer’s coordinates, but not with respect to the local field, does not become a current density — at least not one that produces any electromagnetic effect such as a magnetic field. This is analogous to the runner on a windless day: he certainly feels a wind, but not one that will turn a windmill.
That takes care of all observers, who are thus condemned to observing without interfering. It also takes care o f the relativity principle, which is satisfied auto­ matically by relative, effect-producing velocities just as in the case o f the windmill.
There will, however, still be the physical effects arising when a charge moves through a field. (Analogy: the power-velocity law of the windmill.) These are phy­ sical effects predicted by the Maxwell equations; in the present theory they are the same for all observers in all inertial frames, and they have no particular bearing on the relativity principle, which they automatically satisfy.
In part, this point has already been dealt with in Sec. 1.6, where we obtained Poisson’s equation for the potential <t> in the form

j2 \

P

" - " f e + ;v + 5 ? = -r;

<»

implying that the concentric equipotential spheres about a moving charge flatten into ellipsoids. The Einstein theory attributes this to length contraction seen only by some observers; but the present theory must interpret this as a genuine effect visible to all observers whenever a charge traverses a force field. The reason, for the time being, is that the Maxwell equations say so; in P art Two we will find a direct physical explanation giving more insight.
This flattening o f the equipotentials was derived in Sec. 1.6 by eliminating the time derivative in the wave equation by means o f the relation

§r = -v-v /

<2 >

where / is any field com ponent o f the moving charge, “moving” meaning with respect to the local field, to the observer or to the ether depending on the cor­

Sec. 1.10.4 IN V A R IA N C E O F M A X W E L L E Q U A T IO N S

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responding theory. [Relation (2), incidentally, can be derived more cleanly than
has been done in Sec. 1.6 by direct use o f the Galileian transform ation.] However,
there is another way o f deriving this flattening o f the equipotentials, found some 17 years before the appearance o f the Einstein theory by the unique genius Oliver
Heaviside (1850-1925).1 Consider two charges at a rigidly fixed distance from each other, both traveling
with velocity v with respect to an observer (whom we will also put at rest with respect to the ether). In our theory, this velocity is irrelevant, for one charge is at
rest in the field o f the other; therefore the force between the two charges will be given by the static Coulom b Law and cannot be changed by the m otion of an observer or travel through the ether.
As in any other theory, it is of course assumed that the gravitational field is negligibly small compared with the electric field of the charges. This is not inconsistent with the explanation of a magnetic field produced by a wire-bound current (Sec. 1.4) or the explana­ tion of the Ives-Stillwell experiment (Sec. 1.9.3), for in both of these cases the electric field or its consequences are absent, leaving nothing but the gravitational field as the “remainder” field. In the case of a wirebound current, the electric field is neutralized by the positive iron grid; in the case of radiating hydrogen atoms, the electric force effectively disappears by averaging, since the crucial point is the effect of the gravitational field in addition to the electric field, which is present in both moving and stationary atoms and disappears by sub­ traction in the comparison.
However, Heaviside [1888, 1889], like all scientists o f the time, referred the velocities of charges to the ether, so that the force between the two co-traveling
charges was

F = q(E + v x B)

(3)

with v the velocity o f traversing the ether. Since all quantities here are constant in time, E has no curl, and is therefore a pure Coulom b field given as the gradient o f a potential 0. On the other hand, we have, by definition or derivation,

B _ _v_xV 0

(4 )

Hence

v x (v x V0) F = -q V0 +

-r/(l - / f 2)V 0

(5)

' T he m an w ho predicted the ionosphere, invented the O perational C alculus (“T he p ro o f is per­ form ed in th e lab o rato ry ” — 25 years ah ead o f the m athem aticians w ho found the p ro o f in the L aplace tran sfo rm atio n ), derived P oynting’s T heorem independently o f P oynting, invented and masterfully used the delta function, pioneered radio engineering, and has many astounding discoveries to his credit, had n o college education. T his is rem iniscent o f M ichael F araday, w ho had virtually no education at all.

96

EINSTEIN M INUS ZERO

Sec. 1.10.4

But this can be written as

(6)

where

<& = (1 - f12 )4>

(7)

is Heaviside’s “convection potential,” again showing the concentric, spherical equipotentials of the potential 0 of a charge at rest flattening into Heaviside ellip­ soids when the charge is moving.
If we direct a rectangular system of coordinates with the x axis along the velo­ city, the equipotentials of (7) are

+ (1 - (i2)(y2 + z 2) = const

(8)

The force exerted by the moving charges on each

other, by definition o f a gradient, is perpendicular to

the surface o f the equipotential ellipsoid at any

point. Hence the force between two charges sepa­

rated by a rigid distance and moving parallel to the

x axis through the ether will in general not be

directed along the line joining the two charges: it will

deviate from it as shown by the figure. T o see this,

imagine one of two like charges at opposite ends o f a

bar passing through the origin, and the other on the

Heaviside ellipsoid: the force will be perpendicular to the ellipse, so that except for the four points at the

M utual repulsion of two moving charges.

axial intercepts it will not be directed tow ard the

other charge at the origin.

But it is also evident from the figure that the bar would be subject to a torque.

For point charges this torque can easily be calculated by treating them as currents

(qvds) and using the Biot-Savart Law. For the charges on a parallel-plate capaci­

tor, the result differs only by a factor o f Zi \ the torque seeking to align the plates

perpendicular to the “ether wind” due to the translational (orbital) velocity o f the

earth should be

T = — 0 a ros2 0

(9)

2 r

where r is the separation o f the capacitor plates and 0 is the angle between the plates and the translational (orbital) velocity of the earth round the sun. The latter makes |8=10~4, so that (9) is sufficiently large to be measured if a charged capacitor is suspended from a torsion balance. This was tried in the famous experiment by Trouton and Noble [1903, 1904], but no torque was detected. Their experiment was the electromagnetic equivalent of the Michelson-Morley

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experiment in that it was also a second-order experiment in /3, and in that it also refuted the ether theory, at least in its unentrained version.
Einstein’s explanation was simple and similar to the one in the present theory: there is no ether; the velocity o f the tw o charges with respect to the observer (with respect to each other, in the present theory) is zero; nothing is moving, so there is no torque.
But now imagine that, contrary to the Trouton-Noble experiment, the two co­ traveling charges move with velocity v with respect to an observer. Then nothing changes in the explanation by the present theory: the velocity of one charge with respect to the other is still zero, and the velocity v is irrelevant. There is no more torque than in the previous case.
But what does the Einstein theory say? As long as we consider only electromagnetic forces, as we do here, there can be no difference between the ether theory and the Einstein theory when the ether is replaced by the observer as a standard o f rest. This must be true in general, and is easily checked in the present case, for the Heaviside ellipsoids and the TroutonNoble torque uses nothing but Maxwell’s equations and the Lorentz force. To make everything applicable to the Einstein theory, we need only refer v to an observer rather than to the ether. The Einstein theory must therefore predict exactly the same, non-zero torque as the ether theory. A nd it does: the entire calculation can be found in [Becker 1964, pp.397-401]. However, since the moving observer also causes mechanical forces to appear, the Einstein theory also predicts a mechanical torque of equal magnitude but opposite direction to which the bar is subject: contraction o f the bar in the direction o f the velocity is equivalent to a rotation o f the bar, so that the electric force on the charges shifts back into the direction o f the bar after all. (And this is also the point where I will remind readers, as I promised on p. 93, o f their possible dismay that I carried the spoof with the windmill too far.) So once again the Einstein theory scrapes through by the last twist com ­ pensating for all the previous ones, emerging with a result that was obvious from the outset. In general, the present theory may predict slightly different effects from those predicted by the Einstein theory when the velocity of a charge with respect to the traversed field differs from that with respect to the observer, though the example just discussed shows that this need not always be the case.

Sec. 1.11

98

1.11. Mercury, Mesons, Mossbauer and Miscellaneous
There are a few odds and ends left before the claim o f experimentally verified equivalence becomes fully valid.
For example, the advance o f M ercury’s perihelion should, for tidiness, appear in this “Einstein M inus Zero” part o f the book; however, since it involves only gravitation, it will be delayed to P art Three, Sec. 3.2, where readers may be sur­ prised to find that the “Einstein” form ula for the advance o f M ercury’s perihelion was derived by Paul Gerber in 1898, when Albert Einstein was nine years old.
The time dilation allegedly observed on mesons in the atmosphere has been discussed in Sec. 1.6.3. W ithout a control experiment o f the frequency measured in the moving frame, the argum ent is invalid; but it used to be circular as well, since it used to be based on quantities inferred from the Einstein theory, which they were supposed to prove.
This was particularly drole in the p ro o f that the ratio o f the mean free path L to the energy W o f the mesons is constant. The reason [Tonnelat 1959] is that in the moving system o f the mesons the fam iliar square root V(1 — /32) in the time dilation will cancel against the same square root in the length contraction. But in classical physics the ratio would be just as constant: not because the square roots cancel, but because they were never there in the first place.
The explanation o f the C om pton effect relies on quantum mechanics, not on the Einstein theory, which is often brought in quite unnecessarily.'
The bending o f light in a gravitational field follows immediately from our basic assum ption that the velocity o f light is constant with respect to the local gravita­ tional field in which it propagates. If the field is inhom ogenous, then by Ferm at’s principle it must bend: the fact that it bends towards the denser field implies that light propagates more slowly in denser gravitational fields, just as it does in denser material media.
Quite similarly, electromagnetic waves should propagate slightly more slowly at higher altitudes above the earth, where the gravitational field is less intense. A sufficiently precise standard o f a radiated frequency should therefore have a slightly different wavelength — slightly longer at higher altitudes, and a slightly different Doppler shift (which is a function o f the velocity o f propagation) if the source is moving. Regular sources, including lasers, have too broad a spectrum to detect such minute differences, but the Mossbauer effect, observed on gamma rays emitted in the radioactive decay o f certain isotopes at precise energy levels, can be used to detect such differences in Doppler shift for height differences as small as
1 G . Jo o s, Theoretical Physics (Blackie, L ondon 1947) is apolegetic about deriving it w ithout using the Einstein theory.