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 12 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 14 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.) 20 IN T R O D U C T IO N 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) 47rource 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 = 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 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

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

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=~ — 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 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 = 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 95 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 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 97 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.