SOME METHODOLOGICAL PROBLEMS OF THEELECTRODYNAMICS OF MOVING BODIES

Dragan Redzic

University of Belgrade

1

Table of contents

1. Recurrent topics in special relativity

1.1. Temptations

1.2. Miracles

1.3. Path toward understanding?

1.4. Relativity without Maxwell’s electrodynamics?

Notes

2. Electrodynamics of moving bodies and the Wilson-Wilson experiment

2.1. Einstein, Minkowski

2.2. Einstein and Laub, the Wilson-Wilson experiment

2.3. Review of recent reexaminations of the classical interpretation of

the Wilson-Wilson experiment

2.4. Electrodynamics of bodies in slow motion: with or without special

relativity?

Notes

3. A problem in electrodynamics of slowly moving bodies: Maxwell’s theory

versus relativistic electrodynamics

3.1. Setup of the problem

3.2. Solution in the framework of Maxwell’s theory

3.3. Solution in the framework of relativistic electrodynamics

3.4. Experiments

Notes

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1 Recurrent Topics in Special Relativity

1.1 Temptations

That Einstein’s special relativity - from its advent until today - continues

to be a live source of stupefaction and wonders for both laymen and profes-

sional physicists is well known.1 One of the reasons for a rather emotional,

almost passionate attitude toward that physical theory certainly lies in the

fact that its basic concepts (time, length, mass) are fundamentally different

from the corresponding concepts that have been used with enormous suc-

cess and without a trace of doubt by numerous generations of pre-relativistic

physicists (and laymen). Unfortunately, these different concepts have been

labeled with the same terms and so, thanks to the power of habit, created

an environment conducive to implanting the connotation of the old concepts

within that of the new ones. As a rule, that happens: terminological confu-

sion leads to confusion in sense. It is clear that new concepts need new terms,

but in addition to the fact that physicists too are doomed to a life-long use

of meta-language of everyday speech, the problem with physical concepts is

that they constantly evolve. We remind the reader of a relatively benign but

long-lived terminological problem concerning relativistic mass depending on

speed (Okun 1989, 1998, Strnad 1991, Sandin 1991, Redzic 1990a, 2002),

which, according to the present author, can be simply eliminated by using

Occam’s razor.2 Another less-known (and a lot less benign) terminological

and conceptual problem concerns relativistic tri-force and quadri-force with

differentiating “pure” and “impure” forces (cf Rindler 1991, Møller 1972,

Leiboviz 1969, Carini 1965, Kalman 1961, Redzic 1996). It is indicative,

one can say, that Rindler, in his rightly acclaimed book on special relativ-

ity, as the general form of the transformation law of relativistic tri-forces

presents equations in which, figuratively speaking, “monkeys and donkeys”

are mixed. To be a bit more precise, in the transformation law of quantities

3

that represent a ratio of spatial components of a quadri-vector in Minkowski

space and the corresponding relativistic factor gamma (i.e. in purely geo-

metric and kinematic relations), in Rindler appears also a time dependence

of the relativistic mass, obtained from the quadri-vector equation of motion

(a purely dynamic quantity).3 The result is, of course, a conceptual mess,

for both “pure” and “impure” forces.

When a traveler through relativity somehow escapes from the quicksand

of terminology, more dangerous temptations lurk, just like in fairy tales.4

Namely, it turns out that it is not sufficient to know of the FitzGerald-

Lorentz contraction and time dilatation, to brood over them for several

years and even to use them in everyday work; as Bridgman (1963) put it in

A Sophisticate’s Primer on Relativity, we all are groping our path toward

understanding basic concepts. As an illustration for this state of affairs might

serve the following simple problem, a little riddle with pictures suitable to

a primer on relativity.

Three small spaceships A, B and C drift freely in a region of space remote

from other matter, without rotation and without relative motion, with B and

C equidistant from A (Figure 1).

Figure 1

In one moment two identical signals from A are emitted toward B and

C. On reception of these signals the motors of B and C are ignited and they

accelerate gently along the straight line connecting them (Figure 2).

Let the ships B and C be identical, and have identical acceleration pro-

grammes. Then (as reckoned by an observer in A) the ships will have at

4

every moment the same velocity, and always be at the same distance from

one another. Let us suppose that a fragile thread connects two identical

projections placed exactly at the midpoints of the ships B and C before the

motors were started (Figure 3). If the thread with no stress is just long

enough to span the initial distance in question, then as the ships accelerate

the thread travels with them. Will the thread break when the ships B and

C reach a sufficiently high speed?

C

B

Figure 2

C

B

Figure 3

According to the testimony of a distinguished physicist John Bell (1976),

a polemic over this old problem that was started once between him and a

distinguished experimental physicist in the CERN canteen was eventually

passed on to a significantly broader forum for arbitration: the CERN Theory

Division. A clear consensus, testifies Bell, was eventually reached: the thread

would not break.

The answer is none the less wrong. Elementary explication, in Bell’s

formulation, goes as follows: “If the thread is just long enough to span the

required distance initially, then as the rockets speed up, it will become too

short, because of its need to FitzGerald contract, and must finally break.

It must break when, at a sufficiently high velocity, the artificial prevention

of the natural contraction imposes intolerable stress”. (Cf also Dewan and

Beran 1959, Evett and Wangsness 1960, Dewan 1963, Evett 1972.) It is

observed that the setup of the problem has been altered for several years.)

Here, we shall briefly paraphrase Bell’s remarkable comment on the described

situation which refers to the method of teaching special relativity.

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It is customary to emphasize the discontinuity, the radical breakup with

the primitive concepts of space and time. The result is often the complete de-

struction of the student’s trust in perfectly safe and useful concepts acquired

earlier. We neglect the fact that the chain of thought of the old pioneers-

wise men, FitzGerald, Larmor, Lorentz and Poincare, factually carried out

and freed from the “weak link” of Newton’s concepts of time and space,

predicts both time dilatation and length contraction and leads eventually to

the same conclusions as the Einstein’s theory. However, unexpected qual-

ities of rigid (in relativistic sense, cf Rindler 1991) sticks and clocks that

move do not appear as a dry consequence of certain abstract mathematical

transformations, achieved from logically entangled postulates, as is the case

in Einstein’s approach, but as a natural offspring of earlier physical ideas.5

It appeared to Bell that students who follow this longer, classical road, have

a stronger and more reliable intuition.

1.2 Miracles

It is time to mention a few of the host of small and big wonders of special

relativity. The small wonders are the methodological ones, the scientific

problems that have been solved earlier, before relativity, but in a tedious

and complicated way, and that can be solved by using special relativity sim-

ply and elegantly, merely by “pushing the button”, by “switching off” one

inertial frame of reference and “switching on” another. One of famous such

problems belongs to optics of perfect mirror in motion: what is the radiation

pressure of a monochromatic plane linearly polarized electromagnetic wave

on a planar perfect mirror, which is uniformly moving with velocity perpen-

dicular to the mirror’s plane. Max Abraham (1904) needed forty pages of

text for the solution of this problem, whereas Einstein (1905a) used only

three pages for the same thing in his epoch-making paper (honestly, rather

concise three pages, as pointed out by Arthur Miller (1981) in his rich and

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detailed monograph wherein Einstein’s Relativity Paper was analyzed sen-

tence by sentence). (The present author admits that he has not read the

Abraham’s article, but has read the Einstein’s, very meticulously. Heavy

reading.)

Another case of “methodological wonders” appears in electrodynamics of

moving bodies. It is well known that an isolated charged conducting sphere

of radius R at rest in laboratory (an inertial frame of reference), produces

in space outside the sphere the same electrostatic field as the corresponding

point charge at rest at the centre of the sphere. Following Maxwell (1891),

this point charge may be called the image of the conducting sphere. What

is the image of a conducting body moving uniformly at speed v and at the

same time having the shape and size of the sphere of radius R, as measured

in laboratory?7

Famous J. J. Thomson and Oliver Heaviside, undoubted authorities

in the field of Maxwell’s electrodynamics, and men able to recognize the

essence, dealt with this problem as well. However, Searle (1897) was the

first to find the correct solution: the image of a charged conducting sphere

in motion is a uniformly charged line; the ratio of the length of the line

and the diameter of the sphere is v/c. (The quest for the image of a mov-

ing sphere, a little cliff-hanger that takes place in London, Cambridge and

Dublin in late 19th century, has been sketched in an excellent monograph by

Max Jammer (1961). The main characters are Maxwellians, a small group of

eccentrics that will give much pain to historians of science (cf Brown 2001,

2003, Lorrain et al 2000).) In his article Searle uses the contemporary scien-

tific language (the sphere moves with respect to the ether). He doesn’t yet

know (and how could he?) that the bodies in uniform motion with respect

to the ether do not have the same shape as when at rest.8 In the historical

perspective, Searle’s cumbersome and complicated solution to the problem

arouses admiration. A simple and elegant solution based on the recipe of

special relativity has been recently published (Redzic 1992a, b).

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According to a nice metaphor by W. Rindler, a pure thought has the

power to leap ahead of the empirical frontier - a feature of all good phys-

ical theories, but rarely, Rindler emphasizes, on such a heroic scale as in

the case of special relativity. These new, unexpected and amazing physical

conclusions (“leaps ahead of the empirical frontier”) - these true and great

wonders of special relativity - all have the same powerful source: the concept

of time. Time as a measurable physical quantity in inertial frames of refer-

ence has exactly those peculiar traits as predicted by the Einstein’s theory.

Also, Einstein’s (1905a) definition of time and the principle of constancy of

the velocity of light,9 on its own completely benign, in combination with the

principle of relativity always give rise to the same dramatic effect: the feeling

of losing ground under one’s feet, disbelief and insecurity, and a perennial

question if it is possible that everything could be really so. Even when this

new concept of time is somehow ”swallowed” and the student of relativity

yielded to his destiny expects new relativistic wonders, the disbelief and

insecurity stay.10

And the miracles are numerous, and sometimes rather inconspicuous.

For example, a certain quality which is in an IFR purely spatial and time-

independent, can include dependence on time in another IFR. Such is the

case with the distance between the spaceships B and C in the problem dis-

cussed above (Dewan 1963).11 On the other hand, the following distances

are not of the same kind: a) the distance between two unconnected material

points that are moving at the same time with the same velocity (which can

be time-dependent) along the same line with respect to some IFR; b) the

distance between the ends of a rigid (in a relativistic sense!) stick moving

along its own direction.12 Also, the fundamental prediction of special rela-

tivity, notorious but not any less miracle over miracles: the period of a clock

that is uniformly moving with respect to an IFR is longer than the period

of identical clocks that are at rest with respect to the IFR, if the clocks

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at rest are Einstein-synchronized, when measured by the clocks at rest.13

Finally, a clock that travels around the globe in a commercial airplane and

comes back to the initial point is “younger” than an identical clock that

has not moved from that same point. The fact that this conclusion is not

just a casual intellectual game with Lorentz transformations (almost always

with an implicit assumption that the clock’s own time doesn’t depend on its

acceleration) is proven by a famous experiment in 1971, with macroscopic

caesium clocks (Hafele and Keating 1972, Cornille 1988).

1.3 Path toward understanding?

It seems that the feeling of discomfort that accompanies physicists (and lay-

men) about the slowing down of the clock in motion is a consequence of the

opacity of the usual relativistic method of inferring. Namely, features of a

certain physical system (e.g. a specific moving clock) are derived not from

the structure of that system described in the inertial frame with respect

to which the clock is in motion (“the laboratory”), but from the Lorentz

transformations that connect the two IFRs, the laboratory frame and the

clock’s rest frame. A natural question arises of what is the role of the

clock’s rest frame, with all of its Einstein-synchronized clocks (which, while

mutually identical, may of course be different from the observed “clock in

motion”). Is one reference frame (the laboratory) not quite sufficient? The

Lorentz transformations appear as “the Fates” whose power over destiny of

all physical systems (our moving clock included) is indubitable (as proven

by experiments), but quite puzzling. Even the creator himself of the theory

of relativity that will soon become the special one pointed out this funda-

mental limitation of “the principle of relativity, together with the principle

of constancy of the velocity of light” (Einstein 1907), that is, their purely

instrumental character.

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Maybe the previously described feelings of unease and powerlessness that

follow the understanding of basic results of special relativity could be atten-

uated, or even completely removed, if Einstein’s method were supplemented,

mutatis mutandis, with reasonings of FitzGerald, Lorentz, Poincare, in a way

suggested by Bell (1976, 1987). Here is a short sketch of Bell’s approach.

Let us suppose that natural laws known as Maxwell’s equations hold in

some inertial frame of reference (“the laboratory”). Since the coordinates of

position and time, such as x, y, z and t, always appear in the formulation

of all natural laws, it is above all necessary to define the meaning of these

fundamental quantities. Say that x, y, z and t have their usual meaning in

the laboratory. For example, with t we denote the reading of the synchro-

nized clocks that are at rest with respect to the laboratory. (Since Maxwell’s

equations imply the principle of constancy of the velocity of light, Einstein-

synchronization of an arbitrary number of clocks at rest with respect to the

laboratory is a trivially possible procedure.14,15 Let us now suppose that

a proton is at rest in laboratory, and that an electron rotates uniformly

around it on a circular trajectory of radius a under the action of the pro-

ton’s electrostatic field. Somehow the electron manages to maintain its own

energy constant. (Electron makes up for the energy lost as electromagnetic

radiation by absorbing the needed amount from some infinite reservoir of

energy, maybe vacuum?) In short, this hydrogen atom partly follows the

Bohr model. If we now expose this hydrogen atom to a constant and weak

electrostatic field, parallel to the plane of trajectory of the electron, the en-

tire system will accelerate in the direction of the field. Taking into account

that experiments show that the equation of motion of a charged particle in

the electromagnetic field has precisely the form suggested by Lorentz, and

applying this equation of motion on the electron and the proton that form

our hydrogen atom, we arrive at some unexpected conclusions. After turn-

ing off the external field, and after dying out of transient effects, the proton

10

moves with constant velocity vvv; the electron moves with respect to the pro-

ton (expressed, of course, through the laboratory coordinates x, y, z and t

) on an elliptical trajectory that is oblate in the direction of motion of the

system, with semi-axes a�

1− v2/c2 and a.16 The period of motion of the

electron on the ellipse around the proton in uniform translational motion is

1/�

1− v2/c2 times larger than the period of motion of the electron on the

circle of radius a centered at the proton at rest.17

The preceding analysis of the “hydrogen atom” in motion, carried out

completely in the laboratory frame, reveals, thus, that length contraction

in the direction of motion and time dilatation occur in this simple physical

model due to acceleration! Now, it seems, it is easier to accept that these

are universal phenomena, which will take place in every “stick” and “clock”

in uniform motion with respect to the laboratory. Also, now it is more ac-

ceptable that for the “observer” moving with same velocity vvv as the proton,

the trajectory of the electron around the stationary (for that “observer”)

proton is a circle of radius a (because his meters sticks are contracted by

the same factor�

1− v2/c2 as well in the direction of motion), and the pe-

riod of the electron’s rotation is the same as in the case when the proton

was at rest in the laboratory (because the seconds of the clock belonging to

the “observer in motion” are 1/�

1− v2/c2 times larger than the laboratory

seconds). Elliptical trajectory and a longer period are real for the “observer

in laboratory”; circular trajectory and a normal period are real for the “ob-

server in motion”. Since in physics real is what is reached by measuring

instruments, both “observers” are perfectly right.18

1.4 Relativity without Maxwell’s electrodynamics?

At this place, before entering electrodynamics of moving bodies, it is perhaps

worthwhile to make a small digression about the relationship between special

relativity, light and Maxwell’s electrodynamics. Although both light and

11

electrodynamics have played a central part in the historical development of

special relativity, the real basis of that theory, the Lorentz transformations,

in itself has nothing to do with Maxwell’s equations (Einstein 1935). In

addition to that, according to some authors, the principle of constancy of the

velocity of light has to be dethroned as one of the pillars of special relativity.

Rindler’s (1991) opinion is indicative in this connection: special relativity

would exist even if light and electromagnetism were somehow eliminated

from the nature.

Starting from the principle of relativity and the invariability of causal-

ity, Rindler proves that all inertial frames are related by either Galileo’s or

Lorentz’s transformations. If the transformations are Lorentz’s, then the

constant c which appears in them represents the smallest upper boundary

(the supremum) for the speed of particles in any inertial frame. At the same

time, the speed c can but does not have to be reachable by any physical

object. The possibility that the particles considered massless according to

contemporary opinion (photons, neutrinos, gravitons) may have a nonzero

mass was opened in this way (cf Vigier 1990). Thus, the constant c in the

Lorentz transformations would play the same role as the absolute zero of tem-

perature, the role of an inaccessible boundary.19 It seems, however, that the

alternative methods of clock synchronization, without light, which are indis-

pensable for Rindler’s argumentation, cannot be in accord with the principle

of relativity, nor “freed” from circular reasoning. The same objection goes

with a similar Mermin’s (1984) attempt to get the second postulate from

the principle of relativity. Mermin’s method of synchronization of distant

clocks by their “symmetric transport” (cf footnote 5 of his article) contains,

it seems, a hidden circular argument.

In this context, a recent demonstration of the power of relativistic kine-

matics should be mentioned. Feigenbaum and Mermin (1988) analyzed a me-

chanical version of the famous 1905 Gedankenexperiment, based on electro-

dynamic concepts (Einstein’s Lichtkomplex), which served Einstein (1905b)

12

to get to the equivalence between inertial mass and rest energy. These au-

thors reached the same fundamental conclusion, as well as the relativistic

expressions for energy and momentum of a free particle in the most general

form, by using solely relativistic kinematics and the laws of conservation of

energy and momentum in their most general form, without Maxwell’s elec-

trodynamics. Here, as in Einstein, the mass (the rest mass) appears in the

non-relativistic limit of kinetic energy, but Feigenbaum and Mermin get the

exact limit by calculating it, unlike Einstein who postulated it. Moreover,

they revitalized the problem of the integration constant in the expression for

the rest energy, which Einstein (1905b) “solved” by introducing the principle

of equivalence between inertial mass and rest energy. (It is well known that

Einstein was satisfied neither with that solution nor with the fact that the

mass-energy equivalence was obtained by using Maxwell’s theory (Einstein

1935).) Furthermore, Feigenbaum and Mermin showed that in the relativis-

tic expressions for momentum and kinetic energy of a free particle the same

mass-Lorentz scalar m appears. This is an important result for which Ein-

stein could find only a partial justification (Einstein 1935). Of course, it

would be hasty to conclude from the above discussion that the relativistic

kinematics is free of Maxwell’s electrodynamics (cf Jammer 2000, chapter

3), i. e. almighty.20

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Notes

[1] A fresh example are, so to say, circus attractions of special relativity such

as length dilatation and time contraction (Field 2000). But here we refer pri-

marily to prosaic situations such as the one in which Zapolsky (1988) found

himself: “not less than five” referees negated his conclusions paraphrased in

the present note [12]. This introductory chapter contains the inventory of

some recurrent topics in special relativity.

[2] It is perhaps worthwhile to mention that the usual formulation of the

relation between the rest mass and the Newtonian mass (“in all relativistic

equations the mass (the rest mass) is the usual Newtonian mass” (Okun

1998, Zigman 1997)) is not generally accepted. For example, Eriksen and

Vøyenly (1976) state that the classical and the relativistic concepts of mass

are “incommensurable” (cf Jammer 2000, pp 57-61).

[3] For two inertial frames of reference S and S� in the standard configura-

tion (S � is uniformly moving with respect to S along the common positive

x − x� axis with velocity veeex), and for a particle with rest mass m0 and

instantaneous velocity in the S frame uuu = (ux, uy, uz), the transformation

law of the x-component of the relativistic tri-force, fff = d(m0uuuγu)/dt, reads,

according to Rindler,

f�x =

fx − vdm/dt

1− uxv/c2=

fx − v[fff · uuu/c2 + (1/γu)dm0/dt]

1− uxv/c2

where m ≡ m0γu and γu ≡ (1 − u2/c

2)−1/2. However, taking into account

that the relativistic tri-force is not identically equal the time derivative of

the relativistic momentum of the particle, m0uuuγu, it is clear that the trans-

formation law of the x-component of the relativistic tri-force must have the

form

f�x =

fx − v[fff · uuu/c2 + (1/γ2

u/c2)Fα

Uα]

1− uxv/c2

14

where Fα = (F 0

, γufff) is the corresponding quadri-force, and Uα = γu(c,uuu)

is the quadri-velocity of the particle on which the force is acting. (Here,

of course, contra-variant components of these quadri-vectors appear; we use

the standard metrics (1, -1, -1, -1).) We remind the reader that, in the

general case, a quadri-force does not comply with the condition FαUα ≡ 0,

i. e. that the dependence of the particle’s rest mass on time is in the general

case given by c2γudm0/dt = F

αUα; the last equation is obtained from the

quadri-vector equation of motion, making use of the orthogonality of the

particle’s quadri-velocity and quadri-acceleration (Rosser 1964, Møller 1972,

Rindler 1991).

[4] Perhaps the mentioning of fairy tales in this context is not completely

devoid of sense. Some of the conclusions of special relativity touch the

archetypal dreams of humanity. Langevin’s (1911) Traveler (La Voyageuse

de Langevin, en francais) comes home young (biologically young, not just

young looking, note how language is a problem), after many years of inter-

stellar journey.

[5] Analyzing a simple model of the hydrogen atom, using classical elec-

trodynamics (which is a relativistic theory par excellence without knowing

that), Bell has shown that in that simple system, when it is moving, both

the FitzGerald-Lorentz contraction and time dilatation occur. According to

Bell (1976), the essence of his argument is not at all that Einstein was wrong

in his 1905 “kinematic” analysis, but instead that a more cumbersome, less

economic reasoning, “based on special assumptions on the structure of mat-

ter”, can lead to a fuller insight, in a similar way as statistical mechanics

can offer a much broader view than phenomenological thermodynamics. It

should be mentioned that Bell’s seminal essay gives only a sketch of the

approach to special relativity through ideas of FitzGerald, Larmor, Lorentz

and Poincare. As his only predecessor, Bell mentions a monograph by L.

Yanossy (1975).

15

[6] Recently, a solution to the same problem in the case of a uniformly

accelerated perfect planar mirror is published (Van Meter et al 2001). It is

a constant proper acceleration in question, of course.

[7] The conducting body has the shape of a prolate spheroid with semi-

axes R/

�1− v2/c2, R, R, as measured in the body’s proper frame. Due

to the FitzGerald-Lorentz contraction, the body is a sphere of radius R, as

measured in the laboratory. We remind our reader of the traps of language

in special relativity. Analyzing in 1905 how the shape of a body depends on

reference frame in which it is measured, Einstein occasionally used the verb

to observe (“betrachten”) instead of the verb to measure. Many years after,

if we do not take into account completely ignored Lampa (1924), physicists

(Terrell 1959, Weinstein 1959, cf also Rosser 1964) realized that a visible

shape (the one that can be seen by the eye, or photographed by a camera)

of a body whose speed is comparable to that of light does not coincide with

its measured shape. “If one saw an undistorted but rotated picture of a

moving cube, as predicted by the theory of special relativity, then, if one

calculated the dimensions of the cube allowing for the finite time of flight of

the light quanta from the various parts of the cube, one would deduce that

the length contraction had taken place.” (Here, “cube” is a body that has

the shape of a cube when at rest.) The moral of the story seems to have

been known to Democritus: things are not found therein where their picture

is.

[8] Today, post festum, it seems that Searle almost touched that discov-

ery. Namely, he recognized that the electromagnetic field outside a charged

conducting body in uniform motion at the speed v which has the shape

of a Heaviside ellipsoid (an oblate spheroid whose semi-axes bear the ra-

tio�

1− v2/c2 : 1 : 1, the shorter semi-axis being parallel to the direction

of motion) is identical to the field of a point charge in uniform motion at

the same speed as the ellipsoid, located at its centre (Searle 1897). (The

16

electric and magnetic fields of a point charge in uniform motion through

the ether were derived by Heaviside as early as 1888, without the Lienard

- Wiechert potentials, and without special relativity.) On the other hand,

the field outside a conducting sphere at rest is identical to the field of a

point charge at rest, located at the centre of the sphere. From the preceding

considerations, applying the principle of relativity to Maxwell’s electrody-

namics we infer (cf Redzic 2004a): a conducting body that has the shape

of a Heaviside ellipsoid when in motion is obtained by the motion of the

same conducting body which is a sphere when at rest (Figure 4)! Inference

too strange, unexpected, and even terrifying for pre-relativistic physicists

(excluding the brave FitzGerald whose 1889 speculations about deformation

of bodies in motion through the ether were immediately recognized by his

English contemporaries as “the brilliant baseless guess of an Irish genius”

(cf Brown 2001)).

[9] Einstein’s original formulation of the principle of constancy of the velocity

of light reads: “Any light ray moves in the ‘resting’ coordinate system with

the definite velocity c, which is independent of whether the light ray was

emitted by a resting or by a moving body. Herein is

velocity =light path

time interval,

where time interval is to be understood in the sense of the definition in §1.”

A lot of paper was consumed in clarifying this formulation of Einstein’s.

Namely, “time interval” is defined in Einstein’s §1 just by means of the ve-

locity of light, and thus one of the basic rules of valid definition is violated:

Definitio ne fiat in orbem (A definition must not be circular). In the view

of the present author, the circularity problem can be simply solved by refor-

mulating the principle of constancy of the velocity of light (cf the note 15

below).

17

y

n

O R/2 x

R

E

v B!

Q Q

E

E

z

*

Figure 4

A conducting sphere of radius R and with total charge Q at rest in the laboratoryframe creates the same field as a point charge Q at rest, located at the centre ofthe sphere. The electromagnetic field of a point charge Q in uniform motion withvelocity vvv = veeex is identical to the field of a conducting body having the shape of aHeaviside ellipsoid which is moving with the same velocity. Applying the principleof relativity to Maxwell’s electrodynamics we infer (Redzic 2004a): a chargedconducting body in motion having the shape of a Heaviside ellipsoid is obtainedby the motion of the same conducting body which is a sphere when at rest. Thefigure corresponds to the value v =

√3c/2, when γ = (1 − v2/c2)−1/2 = 2. The

field EEE∗ = EEE + vvv ×BBB is perpendicular to the surface of the Heaviside ellipsoid ata point arbitrarily close to the surface (Redzic 1992a).

[10] Perhaps the best illustration of this psychological situation is the exis-

tence of the journal Galilean Electrodynamics.

[11] It is not difficult to verify immediately that the statement is true by using

the corresponding Minkowski diagram. It is, however, somewhat more diffi-

cult to imagine that there is such a feature at all, due to our pre-relativistic

instincts.

[12] Differentiating of these distances is essential in the explanation of dis-

18

appearance of the electric field of steady currents in the framework of an el-

ementary but non-trivial model (Zapolsky 1988). In what follows we briefly

paraphrase Zapolsky’s argument.

The distance between two unconnected material points that are at rest

with respect to an IFR (the laboratory) is always Lorentz-contracted when

it is measured by an accelerated observer (in his co-moving IFR). On the

other hand, if the two material points are being uniformly accelerated with

respect to the laboratory, say along the line connecting them, starting at the

same moment of time from the state of rest with the same acceleration, the

distance between the two points, as measured in the lab, is always one and

the same. Insisting here on symmetry would be equally irrational as in the

much better known “twin paradox”. The result of measurement depends

essentially on who is accelerated with respect to the lab: the material points

or the observer. One might wonder does this prove that an accelerated meter

stick would also not be contracted. The answer is an emphatic “no!”. The

two material points are not connected, and are completely ignorant of one

another. A meter stick, however, is a system of bound atoms. If we try

to accelerate its two ends with the same acceleration, in the beginning they

would tend to behave in the same way as two unconnected material points.

This means that the stick would tend to extend itself as measured by two

observers “standing” at its ends. Each observer “sees” (at any instant of his

time) that the other observer is going away from him. The conclusion is that

restitutive forces in the stick will oppose the forces causing that the ends

of the stick move with same acceleration. If the internal forces can do that

no more, the stick breaks. (A version of this problem, reminds Zapolsky,

was nicely presented by Bell (1976).) It should be noted that the motion of

the stick we discuss here is not “a rigid body acceleration”, which is usually

defined in special relativity as the acceleration that causes no internal stress

(cf Rindler 1991). It is not difficult to show that this kind of acceleration

19

(in which the proper acceleration continuously changes from end to end of

the stick) leads to the Lorentz contraction; however, it is not the kind of

acceleration appearing in case of two independent material points (cf also

Nikolic 1999).

[13] The problem of reciprocity of the feature of the clock in motion was

the issue of the famous “duel” between Herbert Dingle (1962) and Max

Born (1963).According to Dingle, special relativity permits the following

argumentation. (In the present note, the same notation as Dingle and Born’s

will be used: K(x, t) is the “resting” reference frame, k(ξ, τ) is the one “in

motion”, where ξ = γv(x − vt), τ = γv(t − vx/c2), etc.). One k-second of

a clock at rest in the k frame equals 1/�

1− v2/c2 K-seconds of a clock

at rest in the K frame. One K-second of the clock at rest in the K frame

equals 1/�

1− v2/c2 k-seconds of the clock at rest in the k frame. It follows

that one k- second of a clock at rest in the k frame equals 1/(1 − v2/c

2)

k-seconds of the same clock. Overall conclusion: special relativity is not

a valid scientific theory since it contains a contradiction. Dingle addressed

to Professor Born to defend “the integrity of scientist” by replying to the

challenge.

Born’s counter-argument runs as follows. Dingle falsifies special relativ-

ity. According to Born, the correct relativistic argumentation reads: One

k-second of a clock at rest in the k frame equals 1/�

1− v2/c2 K-seconds

as measured by the system of Einstein-synchronized clocks at rest in the K

frame. One K-second of a clock at rest in the K frame equals 1/�

1− v2/c2

k-seconds, as measured by the system of Einstein-synchronized clocks at

rest in the k frame. Dingle’s inference does not follows from special rela-

tivity. (A compound event that takes place at various spatial points of the

K frame and has a duration of 1/�

1− v2/c2 K-seconds, and a compound

event that takes place at one spatial point of the K frame and has a duration

of 1/�

1− v2/c2 K-seconds must not be identified; those are two distinct

20

C

SH

TH

SH

C TH

x

B

O

!A

a

b

ct c"

Figure 5

C, C, section of light cone; SH, space calibration hyperbola; TH, time calibrationhyperbola; x, ct, conjugate diameters = axes in K; (ξ, cτ), conjugate diameters =axis in k; OA, represents the same time interval in K as OB in k:

OA ∼ OB

The clock at rest in K

�OA ∼ ctOa ∼ cτ

�Oa > OB ∼ OA

τ > t

The clock at rest ink

�OB ∼ cτOb ∼ ct

�Ob > OA ∼ OB

t > τ

On this Born’s figure, the axes of the “middle frame” for K and k are not rep-resented, for obvious reason; by convention, those axes are mutually orthogonal.The “middle frame” is moving at the speed v/(1 +

�1− v2/c2) to the right with

respect to K (and at the same speed to the left with respect to k).

straight lines in Minkowski space (Figure 5). Dingle has made the same kind

of error the student usually makes: two different quantities are denoted by

one and the same symbol.

The issue provoked a prolonged polemic in the Nature that lasted several

years. Numerous participants “accused” each other for elementary misun-

derstanding of basic concepts of special relativity. A consensus was never

21

reached. The present author pointed out the episode in the life of special

relativity just for illustrating the thesis that time dilatation also belongs to

relativistic miracles.

[14] The statement that the principle of constancy of the velocity of light

is already contained in Maxwell’s equations appears occasionally in the lit-

erature (Einstein 1905b, Bartocci and Mamone Capria 1991a). It is per-

haps worthwhile to mention that Maxwell’s equations are a sufficient but

not a necessary condition for the validity of the principle. A definition of

the space and time coordinates must precede the quest for the laws of na-

ture. Therefore any definition of the time coordinate based on a previously

discovered law of nature is nothing but a circulus vitiosus. In this sense

the principle of constancy of the velocity of light (also known as Einstein’s

second postulate) is essentially the first, fundamental, primordial principle

that conceptually precedes the principle of relativity. The assumption that

Maxwell’s equations apply in the lab takes for granted the validity of the

principle of constancy of the velocity of light or some other equivalent method

of clock-synchronization. The fact that Maxwell’s equations are consistent

with both principles is an excellent recommendation for the equations, but

nothing else.

[15] Einstein’s second postulate (1905a), cleaned up from the circular argu-

ment, states that in an IFR one way-two clock velocity of light, an immea-

surable quantity, always equals one clock-two way velocity of light which is

a measurable quantity and, as measurements reveal, a universal constant.

In this way, it is postulated that Einstein synchronization is a realizable

procedure, i. e. the meaning of time as a measurable physical quantity is

postulated. The fact that there exist physical laws (Maxwell’s equations)

consistent with the second postulate, which were discovered before physicists

began to deal with the problem of clock synchronization, is of course a good

recommendation for that postulate. Fortunately, the physical laws apply

22

in the pseudo-inertial reference frame tied for the Earth regardless of the

season.

[16] If vvv = vexexex, by a suitable choice of the initial moment t = 0 the equation

of the ellipse reads (xe − vt)2/(a

�1− v2/c2)2 + y

2e/a

2 = 1, where xe and ye

denote the electron’s coordinates.

[17] A few examples for time dilatation of a moving clock in the same

spirit, on the basis of classical electrodynamics, were presented by Jefimenko

(1996b).

[18] This implies that physical reality independent of the frame of reference

(“observer”) has no physical sense. (In the same way as the question whether

the FitzGerald-Lorentz contraction is real has no physical sense. Einstein

would begin his answer to the last question by the query: real with respect

to what?) Physical realities of various inertial “observers” may be almost

comically different, for the same “events”. Of course, the final outcomes

must be one and the same for all the “observers”. A vivid illustration of

the various physical realities provides Rindler’s length contraction paradox

(Rindler 1991, cf also Dewan 1963).

On the basis of the preceding considerations, one could infer that, since

“physical realities” of the same events corresponding to different “observers”

are not identical, this then means that the principle of relativity does not

apply. The inference would be false. The laws according to which the states

of physical systems undergo changes do really have the same form in all

inertial frames of reference (the principle of relativity does apply!), and dif-

ferent physical realities of different “observers” are a necessary consequence

of different initial conditions. This is clearly seen by passing from a passive

to an active interpretation of the Lorentz transformations (Bohm and Hiley

1985).

By the way, the thesis that both the FitzGerald-Lorentz contraction and

time dilatation are nothing but subjective phenomena was a continuing sub-

ject of lively discussion among physicists and philosophers. The thesis is

23

presumably a consequence of the mess about the concept of time. Namely,

it seems that Newton’s absolute time is perfectly consistent with the illusive

subjective feeling that thought “at one instant” can encompass everything,

the whole cosmos. That subjective feeling, unfortunately, is not consistent

with the time as a measurable physical quantity, because the time is in ac-

cord with Einstein’s theory, in inertial frames, of course. As a curiosity,

we mention that in the first, heroic years of the special theory of relativity,

when the discussions about the subjective nature of time were most lively, a

Yugoslavian physicist, Vladimir Varicak, also took part in them (cf Einstein

1911, Miller 1981).

[19] Bachman (1982) derived a relativistic Doppler formula for waves whose

phase velocity relative to the medium is u

f� = f0

�u + v0

u− vs

��1− v

2s/c

2

1− v20/c

2

�1/2

.

The equation expresses the frequency f� of the wave detected by the

“observer” through the proper frequency of the source f0, the phase velocity

u, the relativistic limiting speed c, the velocity of the source toward the

“observer” relative to the medium vs, and the velocity of the “observer”

toward the source relative to the medium v0. The formula is derived under

the assumption that the velocities of the source and detector are along the

line connecting them. If the phase velocity of the wave exactly equals c,

then, and only then, the detected frequency f� depends only on the velocity

of the source as measured in the proper frame of the detector, regardless

of the velocity of uniform motion of the medium relative to the source or

detector. Overall conclusion: if the phase velocity of the light is less than

c, the ether must exist; however, if the phase velocity of the light exactly

equals c, then the ether may but need not exist, and Occam’s razor solves

the problem (Mirabelli 1985).

24

[20] For example, in the case of the Doppler effect its power is limited.

Namely, kinematic derivations of the Doppler effect (French 1968, Peres

1987) are approximations. (These derivations lead to Einstein’s Doppler

formula which deals with the plane wave approximation (for a different look

at that formula see Schrodinger 1922, Redzic 1990b).) The present author is

aware of only one attempt of an exact kinematical treatment of the Doppler

effect (Rothenstein 2002).

25

2 Electrodynamics of moving bodies and theWilson-Wilson experiment

2.1 Einstein, Minkowski

As it was hinted above, the principle of relativity is essentially a meta-

principle (the term is Rindler’s (1991)). Like the well known biological

principle that ontogeny is a short and quick repetition of phylogeny, the

principle of relativity as well determines nothing but the general condition

that must be satisfied by “the laws according to which the states of physi-

cal systems change”. The law states what could be a physical law but the

principle is mute about which is a physical law (contrary to, for example,

Fermat’s principle). Physical laws are reached slowly and painfully.

Although the title of Einstein’s (1905a) epoch-making paper is “On the

Electrodynamics of Moving Bodies”, in that work, as is well known, electro-

dynamics of moving bodies is quite in second place.1 In the electrodynamic

part of the paper Einstein proved that the Maxwell-Hertz equations in vacuo

are Lorentz-covariant, and then applied the derived transformation laws for

the EEE- and BBB- fields to the case of a monochromatic plane linearly polarized

electromagnetic wave in vacuo. In this way he obtained the formulae express-

ing the Doppler principle and the light aberration “for arbitrary velocities.”2

Then he calculated the radiation pressure of a monochromatic plane linearly

polarized wave on a perfect planar mirror in uniform motion and also the

transformation law of the energy of a strange entity that he called the light

complex (Lichtkomplex).3 Only the last, tenth paragraph of the paper, enti-

tled “Dynamics of a (slowly accelerated) electron”, refers, in a certain sense,

to the electrodynamics of moving bodies. Applying the principle of relativ-

ity, Einstein derived the correct equation of motion of a point charge in the

electromagnetic field in the special case when the instantaneous velocity of

the charge is parallel to one of the coordinate axes; however, he interpreted

26

the obtained equation in a cumbersome way.4,5 Fundamental equations of

the phenomenological electrodynamics of moving bodies, on the ground-

work laid by Lorentz (1895) Poincare (1906) and Einstein, were constructed

by Hermann Minkowski (1908), whose ideas represent the starting point

of all subsequent researches in the field. Here we give how the essence of

Minkowski’s method was formulated by the famous physics teacher Arnold

Sommerfeld (1952).

“Minkowski’s logic was simple: The Maxwell equations for a state of rest

apply within the laboratory. Consider a point of space-time P of a body

moving with respect to the laboratory at the laboratory time t; let it have

the velocity vvv. Let P be transformed to rest by the introduction of the co-

ordinates x�, y

�, z

�, t� for the description of the processes in the neighborhood

of P, t. In this system Maxwell’s equations for a state of rest apply to the

quantities E�E�

E�,B

�B�

B�,D

�D�

D�,H

�H�

H�,J

�J�

J�, �

�:

curlEEE � = −∂BBB�

∂t�, curlHHH � = JJJ

� +∂DDD

�

∂t�,

divDDD� = ��, divBBB� = 0,

(AS1)

with material constants differing from those for vacuum

DDD� = εEEE

�, BBB

� = µHHH�, JJJ

� = σEEE� (AS2)

These constants have the same values as if the body were at rest with respect

to the laboratory, since it knows nothing of its motion. The operations

curl and div in (AS1) refer of course, just like the time t�, to the primed

system. Now the inverse Lorentz transformation is to be carried out, which

transforms the primed system back into the original one of the laboratory.

In the latter Eqs. (AS1) apply once more if all primes are omitted, in view

of the basic property of covariance of the Maxwell equations with respect

to the Lorentz transformations. However, Eqs. (AS 2), transformed to the

unprimed system, take on a new form.”

27

Minkowski’s physical ideas were simple indeed, which was not at all the

case with the mathematical “apparatus” he was using. Namely, fundamental

equations for the electromagnetic phenomena in moving bodies Minkowski

expressed through tensors in a (pseudo-) Riemannian four-dimensional Minkowski

space. Mathematical apparatus known today to every physics student was

used then for the first time.7 Physicists were far away from acquiring the

“tensorial mentality”.

In this work we shall not give an exposition of Minkowski’s theory which

was presented in detail in the classical monographs by Pauli (1958), Cullwick

(1959), Rosser (1964), Møller (1972). Instead, by using that theory we

shall attempt in the following chapter to analyze a simple problem from a

somewhat unusual perspective. In the present chapter we shall deal with

interpretations, some old some new, of the Wilson-Wilson experiment, one

of the crucial experiments of the electrodynamics of bodies in slow motion.

2.2 Einstein and Laub, the Wilson-Wilson experiment

Several months after the publication of the Minkowski paper, Einstein and

Laub (1908a,b) derived the same fundamental electromagnetic equations for

bodies in motion, now following Einstein’s “elementary path”. (Taking into

account, put the authors, that the work of Minkowski in the mathemati-

cal sense imposes too severe conditions before its reader, we find it useful

to derive the fundamental equations in an elementary way which, however,

essentially corresponds to Minkowski’s method.) Einstein and Laub imme-

diately applied the new tool of theoretical physics to an exotic system; their

inferences, as is usually the case with special relativity, were unusual.

Consider an infinite slab made of a linear, homogeneous and isotropic

dielectric of relative permittivity εr and relative permeability µr, where

εrµr > 1, which is uniformly moving at a speed v through the plate condenser

of infinite extent at rest. When a potential difference is applied between the

28

plates, the surface charge density on the plate which is at a higher potential

is positive when v < v∗ and negative when v > v∗, tending to infinity when

v tends to v∗, where v∗ ≡ c/√

εrµr denotes the velocity of of the electro-

magnetic waves in the magnetic dielectric when it is at rest. (The present

author still remembers that, after reading the above Einstein and Laub’s

conclusion for the first time, he had experienced a certain frisson mystique,

despite the fact that the system considered is impracticable. We remind our

reader of the fact that the human race has learnt, from 1905 until today,

that one should respect theoretical physics, even when it deals with such a

kind of problems.) If a constant magnetic field, parallel to the plates and

perpendicular to the velocity of the dielectric slab, is applied to the sys-

tem considered, and if the plates are connected by a thin conducting wire,

then a simple relationship between the electric displacement (DDD) and the

magnetic field strength (HHH) in the dielectric is obtained in the framework

of Minkowski’s theory of the first order in v/c. Lorentz’s non-relativistic

electron theory gives, however, a different relationship between DDD and HHH

for the same system.8 If there existed, Einstein and Laub wrote, dielectric

bodies with a considerable magnetic permeability, then it would be possible

to choose experimentally between the theories of Lorentz and Minkowski.

Such bodies, however, did not exist.

Whatever does not exist in the nature, appears occasionally in the labo-

ratories. In order to check up the theory of Einstein and Laub, a magnetic

dielectric “with considerable magnetic permeability” was created by Wilson

and Wilson (1913). Their recipe was as follows. They used small 1/8 in.

steel balls “and each one was coated thinly with sealing-wax. The coated

spheres were packed tightly and melted paraffin was poured into the empty

spaces between them so as to form a solid mass.” This magnetic dielectric,

with εr = 6 and µr = 3, filled the space between the plates of a cylindri-

cal condenser, made of brass; the length of the cylinder was 9 · 5 cm., and

29

the inner and outer diameters of the solid dielectric tube were respectively

2 cm. and 3 · 73 cm. The condenser was uniformly rotated at a speed of

about 6000 r.p.m., in the axial magnetizing field of a coaxial solenoid. An

electrometer was connected by means of stationary leads to brushes which

made contact with the inner and outer cylindrical plates of the rotating

condenser. In the experiment, the potential difference between the plates

of the condenser was measured. (Cullwick (1959) gave a detailed analysis

of the Wilson-Wilson experiment.) As it is well known, the results of the

experiment eliminated Lorentz’s theory.9 That was a triumph of both spe-

cial relativity and Minkowski’s phenomenological electrodynamics of moving

media.10

The reader has certainly noted that Wilson and Wilson, as it is usually

euphemistically said, somewhat modified the original “experimental set-up”

of Einstein and Laub. The uniform translation of an infinite slab, inacces-

sible to experimental verification, was replaced by the uniform rotation of

a long cylindrical tube made of magnetic insulator. From the viewpoint

of Minkowski’s theory, the substitution is perfectly legal: arbitrarily small

neighborhood of any rotating point of the dielectric is at rest in the corre-

sponding local IFR. The fact that in the local frame the material point of

the dielectric instantaneously at rest (its immediate neighborhood also being

instantaneously at rest) has a non-zero acceleration should not represent a

problem. Namely, one of the fundamental assumptions of Einstein’s theory

of relativity, both the special and the general one, is that both length con-

traction and time dilatation are determined only by the relativistic factor γ,

i. e. they do not depend on instantaneous acceleration (the clock hypothesis

and the stick hypothesis, cf Møller 1972).

30

2.3 Review of recent reexaminations of the classicalinterpretation of the Wilson-Wilson experiment

The conventional interpretation of the Wilson-Wilson experiment was re-

cently questioned by Pellegrini and Swift (1995). The authors pointed out

that the fundamental Minkowski’s hypothesis was that any material point

of the rotating cylinder may be treated as if it were in the local inertial

frame of reference (LI) in which the point is instantaneously at rest. Pelle-

grini and Swift (PS) argued that a correct analysis must take into account

the fact that a rotating frame is not an inertial frame. Their “corrected

analysis” borrowed from the general theory of relativity necessary tools for

dealing with electrodynamics in an accelerated frame of reference. The final

outcome of their analysis based on the assumed nature of a medium in mo-

tion differs from the result obtained by following the “elementary path” of

Einstein and Laub. Since the experiment was consistent with predictions of

Minkowski’s theory (which is incorrect!) one has, claim Pellegrini and Swift,

a fundamental conflict between theory and experiment.

Several authors questioned the validity of the PS argument (Burrows

1997, Weber 1997, Ridgely 1998). None of the critics found an error in

the PS calculation; instead, the critics contested their starting fundamental

physical assumptions [the use of an unacceptable coordinate system (Bur-

rows), errors in defining basic physical quantities (e. g. the current density)

in the rotating frame due to the problem of clock synchronization (Weber)].

Ridgely (1999) analyzed in detail the constitutive equations for the polar-

ization and magnetization in a uniformly rotating frame, starting from the

corresponding constitutive equations for DDD and BBB in the Lorentz-covariant

formulation (cf e. g. Pauli 1958, p 103, Griffiths 1999, p 545); transforming

back to the laboratory frame he obtained that, in the lab, the constitutive

equations inside the rotating cylinder have exactly the form predicted by

the “simple” Minkowski’s theory.

31

Krotkov et al (1999) gave a quite unexpected direction to the recent reex-

amination of the classical interpretation of the Wilson-Wilson experiment.

A justification of the specificity of their analysis needs some introductory

remarks.

As is well known, the essential difference between Minkowski’s and Lorentz’s

electrodynamics of moving media lies in the fact that only the former pre-

dicts that a magnetized medium in motion (with a non-zero magnetization

M�M�

M� in the proper inertial frame of the magnetic S

�) possesses, as measured

in the lab frame S, a non-zero polarization given by, in the framework of

first order theory, PPP = (1/c2)vvv ×M�M�

M�, where vvv denotes the velocity of the

considered point of the magnetic, as measured in the lab. This relationship

is usually derived by using relativistic transformations for the fields, which

are obtained as a consequence of the Lorentz-covariance of Maxwell’s equa-

tions (cf Rosser 1964). For ordinary media, consisting of atoms or molecules,

there is another, microscopic approach. Namely, according to the classical,

Amperian model, a magnetic dipole can be represented by a closed con-

ducting loop with a stationary (conduction) current, in its proper frame of

reference S�. In that frame, any, arbitrarily small segment of the current

loop is electrically neutral.11 In the lab frame S, however, in the current

loop that is now uniformly moving with velocity vvv there is a charge dis-

tribution over the loop and it possesses the corresponding electric dipole

moment ppp = (1/c2)vvv ×m�m�

m�, where m

�m�

m� denotes the magnetic dipole moment

of the loop in its proper frame S�.12 The appearance of charges inside the

current loop in uniform translation is a consequence of the relativistic trans-

formation law for the charge density; as Rosser (1964, 1993) pointed out,

the charge distribution stems, in the long run, from relativity of simultane-

ity. (The appearance of electric dipole moment of a current loop in motion

is, thus, a purely relativistic phenomenon, unknown in non-relativistic the-

ories.) Due to the Lorentz contraction, n0 of those magnetic dipoles per

32

m3 in S� takes the volume (1/γ)m3 as measured in S. Consequently, the

concentration of the corresponding electric dipoles in the S frame equals

n0γ, and thus the contribution to the polarization in the S frame due to

the motion of the magnetic is given by the expression PPP = γ(1/c2)vvv ×M�M�

M�,

since ppp = (1/c2)vvv ×m�m�

m� and PPP = n0γppp. It is clear that the “microscopic

approach”, based on the classical concepts, is somewhat problematic; in the

best case, ppp and m�m�

m� could only be the average values of the corresponding

quantum-mechanical operators.

The preceding considerations reveal that not only the theories of Minkowski

and Lorentz but also the modern analyses by Pellegrini and Swift and their

critics, all lie within the standard framework of the classical field theory,

i. e. they all use the usual method of the theory of continuous media

(the transition from micro- to macro-quantities by averaging over physically

infinitesimally small regions of space and time intervals). In the Wilson-

Wilson experiment, however, “magnetic dielectric” was constructed of small

steel balls of diameter about 3 mm embedded in the paraffin wax. Krotkov

et al (1999)point out that neither the LI nor the PS approaches are applica-

ble to this macroscopically inhomogeneous medium. The authors analyzed a

steel ball (a highly conductive and a highly permeable medium!) in uniform

rotation about an axis outside the ball, in a constant external magnetic field

B0B0B0 parallel to the rotation axis, and found, in the framework of the first

order theory in v/c, that the resulting electric dipole moment of the ball is

the sum of two terms: the first is the well known electric dipole moment

of a conducting ball in the effective electric field vvv ×B0B0B0, and the second is

the “relativistic” (1/c2)vvv×mmm term, where mmm is the magnetic dipole moment

of the ball and vvv is the velocity of its centre as measured in the lab. This

conclusion is reached by using only Maxwell’s equations in the lab frame,

without the use of special relativity, or any assumption on physics in the

ball’s proper frame.13 Krotkov et al claim that the result can be generalized

33

to the magnetic dielectric from the Wilson-Wilson experiment. Their argu-

ment is based on the fact that inside the material consisting of the host of

steel balls embedded in the wax the magnetization, and thus also the electric

dipole moment due to the motion of magnetic dipole, exist only in the steel

balls, where electric conductivity is high. (Needless to say, Krotkov et al did

not venture on finding the polarization and magnetization of the Wilson-

Wilson magnetic dielectric as a function of the electric and magnetic dipole

moments of the steel balls.) The final conclusion of those authors is that the

Wilson-Wilson experiment cannot detect a difference between the LI and

PS predictions since the composite steel-wax cylinder is highly conductive

in the regions with appreciable magnetization. In this way, claim Krotkov

et al, all models that take for granted Maxwell’s equations lead inevitably

to the LI results of Minkowski’s theory, in the case of the Wilson-Wilson

experiment.

The analysis made by Krotkov et al, regardless of the validity of their final

conclusions, pointed out the essential fact that in experiments of the Wilson-

Wilson type, whose objective is to make a choice between several classical

field theories, the rotating magnetic insulator must be (i. e. should be) a

microscopically homogeneous medium (we remind our reader that Rosser

(1964) suggested this long time ago).

An experiment with such a material has been recently performed by

Hertzberg et al (2001). Their “homogeneous” cylinder was made of yttrium-

iron-garnet “which is a magnetic insulator even on the molecular scale”.

Experimental results, very convincingly consistent with the LI predictions

of Minkowski’s theory, were for 6% different from the predictions of the

PS theory (the relative error of their measurement was 1%). The original

Wilson-Wilson experiment with the inhomogeneous dielectric constructed

from steel balls embedded in the wax was also repeated. In this case too the

results took sides of Minkowski’s theory.

34

2.4 Electrodynamics of bodies in slow motion: withor without special relativity?

It is a commonplace that relativistic effects disclose themselves only at speeds

close to that of light, faraway from the phenomena of everyday experience.

As it is picturesquely said, the true arena of special relativity is the exotic

kingdom of great speeds; at small speeds relativistic effects may be ignored.14

However, in experiments of the Wilson-Wilson type the maximum speeds of

the points of the rotating cylinder are of order of several meters per second.

We saw that a correct interpretation of the results of those experiments with-

out special relativity, that is without Minkowski’s theory, was not possible.

The answer to the query: electrodynamics of bodies in slow motion, at room

velocities, without or with special relativity, i. e. Minkowski’s theory, seems

to be obvious. Some difficulties, however, should be pointed out.

Before all, the query necessitates a certain explanation. At first sight

it is a pseudo-problem. As is well known, Maxwell’s equations for mate-

rial media (the so-called material equations) are Lorentz-covariant, and so

it seems that the problem is already solved, and that in favour of special

relativity. However, this is not so. Namely, the phenomenological elec-

trodynamics of moving bodies in an inertial frame of reference consists of

four Maxwell’s equations for material media + the Lorenz gauge condition +

Lorentz’s expression for the force acting on a point charge in the electromag-

netic field,q(EEE + vvv ×BBB)+ the constitutive equations.15 The question arises

whether a non-relativistic analysis, based on the use of Galilei transforma-

tion, is sufficient for a correct electrodynamic description of bodies in very

slow motion, in the framework of a first-order theory. Another problem, of

course, is whether Minkowski’s phenomenological relativistic electrodynam-

ics is correct at all. As Cullwick (1959, p 107) noted, Einstein and Laub were

not using electrodynamics of moving bodies but instead the electromagnetic

theory of bodies at rest, together with a mathematical application of special

35

relativity. Although there seems to be a consensus that Minkowski’s recipe

is valid in case of a uniform translational motion of a body, the consensus,

as far as the present author is aware, has no sound experimental basis.

On the basis of the considerations given in the preceding Section, one

could infer that there is no unambiguous answer to the above query. Ac-

cording to Krotkov et al (1999), electrodynamics of bodies in slow motion

does not necessitate special relativity. Howevere, their analysis deals with

macroscopically inhomogeneous bodies. In the view of the present author,

the conclusions reached by Krotkov et al (1999) are problematic. In case of

microscopically homogeneous (or inhomogeneous) bodies, the answer to the

query depends on the nature of bodies. Non-magnetic insulators in slow mo-

tion can be successfully described by using Lorentz’s non-relativistic theory

(Pauli 1958). When magnetic dielectrics are discussed, however, it seems

that in case of bodies in slow rotational motion, the query is “shifted” in the

sense of necessity of either special or general relativity. On the basis of the

experiment by Hertzberg et al (2001) one could infer that the question is

settled and this in favour of special relativity i. e. Minkowski’s recipe, and

also as a by-product that Lorentz’s theory is definitively eliminated (which

essentially could not be inferred on the basis of the original Wilson-Wilson

experiment, contrary to the generally accepted opinion). One should, how-

ever, remember the fact (already pointed out by Rosser (1964)) that there

are conceptual difficulties also in case of the electrodynamics of bodies at

rest, because macroscopic behaviour of a large number of micro-systems is

deduced from the classical (macroscopic) ideas about the micro-systems.

Taking into account the relatively complicated theory of the experiment by

Hertzberg et al, in the view of the present author (or, more precisely, follow-

ing his intuition) the question of whether special relativity is sufficient for

giving successful predictions in case of slowly rotating magnetic insulators

should be considered open.

36

To this topic also belong the standard didactic problems dealing with

electromagnetic phenomena in non-magnetic conducting bodies moving through

a constant externally applied magnetic field, the motion being a pure trans-

lation, a pure rotation, or a combination of the two motions. The present

author recently pointed out that even at room velocities special relativity,

that is Minkowski’s electrodynamics, seems to be indispensable for a cor-

rect derivation of basic inferences (cf Redzic 2004b, Bringuier 2004, Redzic

2004c), in case of a uniform translational motion of a conductor of arbitrary

shape, and also in the classical problem of a thin conducting ring uniformly

rotating about its diameter in a constant externally applied magnetic field

perpendicular to the rotation axis.

37

Notes

[1] In the whole Einstein’s paper, to the electrodynamics of moving bodies in

the usual sense only refers its introductory paragraph containing a very short

discussion on “the electrodynamic interaction between a magnet and a con-

ductor. The observed phenomenon in this case depends solely on the relative

motion of the magnet and the conductor...,” wrote Einstein. This example

has served to the author as an illustration for the thesis that “Maxwell’s elec-

trodynamics - as it is usually understood today - leads to asymmetries that

do not appear to be inherent to the phenomena ...” (Einstein 1905a). Many

years after Bartocci and Mamone Capria (1991a,b) argued that in Einstein’s

example of the interaction between a magnet and a conductor it is classically

interpreted Maxwell’s electrodynamics, and not relativistic electrodynamics,

that predicts a perfect (and not only to second order in v/c) symmetry, con-

trary to Einstein’s statement. While Rosser (1993) questioned the validity of

the interpretation of Maxwell’s electrodynamics proposed by Bartocci and

Mamone Capria, in the view of the present author their conclusions con-

cerning the interaction between a point charge and a current loop in relative

motion are correct (cf Redzic 1993). Einstein original example, however,

necessitates a more detailed analysis than that given by Miller (1981, pp

146–9), regardless of the error pointed out by Bartocci and Mamone Capria

(1991a, b). It is perhaps worthwhile to mention that the present author

recently pointed out a clear asymmetry in Maxwell’s electrodynamics which

is inherent to the phenomena and which, if properly understood, “opens the

door to special relativity” (Redzic 2004a).

[2] The formulae apply to arbitrary monochromatic plane wave, i. e. to

elliptic polarization. The special case of a circularly polarized wave was used

in Dodd’s (1983) attempt to interpret the Compton effect in the framework

of classical electrodynamics. The attempt contained a fatal flaw (Redzic

2000).

38

[3] Lichtkomplex is a mysterious quantity in the framework of Maxwell’s

electrodynamics and its appearance in Einstein’s paper is very strange. The

“mysterious stranger” will appear on the stage just one more time (Ein-

stein 1905b), now under a new name (Lichtmenge, the quantity of light),

and only then will its appearance in the first act become understandable.

The example of the light complex clearly shows that intuition is sometimes

more important than knowledge, and also that the role of logic in physical

sciences is sometimes very tricky (Stachel and Torretti 1982). It seems that

physics has not until very recently said its last word about the light complex

(Redzic and Strnad 2004). Namely, Einstein reached the correct final result

by making a methodological error.

[4] Planck (1906) was the first to derive and recognize the well known gen-

eral form of the relativistic equation of motion of a charged particle in the

electromagnetic field (“the Lorentz force equation”). While it seems that

Kaufmann’s latest measurements disprove the principle of relativity intro-

duced recently by Lorentz and, in a more general formulation, by Einstein

- argued Planck - one cannot exclude the possibility that a more detailed

elaboration of the experimental results will show that the principle (“such a

simple and general physical idea”) is consistent with observations ...

[5] It should be mentioned that Einstein obtained his electrodynamic results

without knowing of tensors in Minkowski’s space. In the view of the present

author, the grandeur of that scientific exploit, Einstein’s tour de force can be

adequately appreciated only by a researcher who trailed the same dangerous

mountain path (cf Schwartz 1977, Rosser 1964). Recently, Jefimenko (1996a)

derived in Einstein’s way the transformation law of the most renowned pure

relativistic tri-force, qEEE+qvvv×BBB. Jefimenko’s article is a natural complement

of Planck’s paper mentioned above (cf also French 1968, Rosser 1960). Of

course, for reaching a full insight it is indispensable to compare the pioneer

attempts with the modern derivation of the “Lorentz force equation” through

tensor calculus (cf e. g. Møller 1972).

39

[6] In this place Sommerfeld made the following remark: “The motion may

be variable in space and time and must merely be capable of quasi-stationary

treatment in the sense of Eq. (33.11). Thus vvv need not be a pure translation

and the body need not be rigid. only the fixed value of vvv in the space-time

point P, t enters in the following Lorentz’s transformations.” It should be

pointed out, however, that in the general case Sommerfeld’s remark does not

apply. For example, in case of an axially symmetric charged conducting body

that is uniformly rotating about its symmetry axis, the application of the

constitutive equation for the current density (the third equation (AS2))leads

to a contradiction (cf Redzic 2002, 2004b).

[7] Minkowski’s nomenclature is different from the present-day one. For

example, instead of tensors of the first (quadri-vectors) and second rank, he

speaks about space-time vectors of the first and second kind, respectively.

For the quadri-gradient he uses a nowadays forgotten symbol lor. And, of

course, he works in a complex space (ict!) whose metric is Euclidean.

[8] Einstein and Laub’s result reads Dz = (εrµr − 1)vHy/c2, and that of

Lorentz’s theory Dz = (εr− 1)µrvHy/c2, in the SI system of units. Here we

sketch how one can reach these results which refer to the system shown in

Figure 6. In Minkowski’s theory we start from the definition of the electric

displacement

DDD = ε0EEE + PPP , (1)

and the constitutive equation for the polarization of the “magnetic di-

electric” which in the lab frame, up to the second order terms in v/c, reads

(Rosser 1964)

PPP = ε0(εr − 1)(EEE + vvv ×BBB) + (1/c2)vvv ×MMM�, (2)

where M�M�

M� is the magnetization in the proper frame of the magnetic. A simple

analysis reveals that in the magnetic’s rest frame, in the first-order theory,

40

+ + + + + + + + + + +

_ _ _ _ _ _ _ _ _ _ _

Hx

z

v

v

v

Figure 6

The slab made of magnetic dielectric and the condenser’s plates all move withconstant velocity vvv = vexexex in a constant externally applied magnetic field whosemagnetic flux density is B0B0B0 = µ0Hyeyeyey. The magnetic dielectric in motion isalso electrically polarized. If the condenser’s plates are mutually connected bymeans of a stationary lead, a charge appears on the plates. The electric field inthe dielectric vanishes when the dielectric completely fills the space between theplates.

one has

MMM� =

(µr − 1)BBB0µr

µ0µr, (3)

where B0B0B0 = µ0Hyeyeyey is the magnetic flux density of the externally applied

magnetic field in the lab frame. Eq. (3) is obtained by using the continuity

of HHH and the relativistic transformation laws for the EEE- and BBB- fields. (By

the way, it is not difficult to verify that the magnetic flux density inside

the magnetic dielectric equals BBB = BBB� = µrB0B0B0, in both reference frames; the

result applies, of course, in the first order theory, and taking into account

that all relevant quantities are of the type vvv ×BBB.) From equations (1), (2)

and (3) we get

DDD = ε0εrEEE + ε0(εrµr − 1)vvv ×B0B0B0 (4)

41

Since the condenser’s plates are mutually connected by means of a stationary

lead through sliding contacts, and since a stationary state is established,

the potential difference between the plates is zero. If the gap between the

dielectric slab and the condenser’s plates vanishes, it follows that the electric

field inside the dielectric also vanishes, EEE = 0, and thus in this case we have

Dz = (εrµr − 1)vHy/c2. (5)

Of course, Dz = σf , where σf denotes the surface charge density over the

lower plate of the condenser. In Lorentz’s theory, however, the constitutive

equation for the polarization of the magnetic dielectric contains only the

first term on the right hand side of equation (2), PPP = ε0(εr− 1)(EEE +vvv×BBB),

so that in the same “experimental situation” we have

DzL = (εr − 1)µrvHy/c2. (6)

It should be mentioned that the original system discussed by Einstein and

Laub (1908a) is different from that ascribed to Einstein and Laub in the

literature in the following detail: in Einstein and Laub only the dielectric slab

is moving; the condenser’s plates connected by a stationary lead are also at

rest with respect to the lab. We discussed here the version usually presented

in the literature where the dielectric and the plates all move at the same

velocity and a stationary conducting wire is in contact with the plates by

means of brushes (Cullwick 1959, Rosser 1964). (This version is closer to the

Wilson- Wilson experiment where the cylindrical condenser rotates together

with dielectric.) Both versions give the same results, in the framework of

the first order theory. Interestingly, Cullwick claims that Einstein and Laub

identified without justification the magnetic field strength in the dielectric

(HHH) and the magnetic field strength of the externally applied magnetic field

H0H0H0. Fortunately, Cullwick states, this is an irrelevant second order effect. In

this place, however, otherwise very accurate Cullwick is wrong: namely, in

42

Einstein and Laub the condenser’s plates does not move, and consequently

there is no contribution to the vector HHH due to the convection current of

free surface charges. Einstein-Laub’s result (5) can be reached in another

(the third one) way, by using Cullwick’s (1959) “component field” method.

The present author, however, has not succeeded in reaching the result (6) of

Lorentz’s theory by that alternative method. Cullwick points out that there

is no consensus in the literature about what is the solution of the problem

according to Lorentz’s theory.

[9] [9] The measured potential difference according to Einstein - Laub’s the-

ory is proportional to the factor (1 − 1/εrµr) and according to Lorentz’s

theory to the factor (1 − 1/εr), which for the Wilson-Wilson magnetic di-

electric with εr = 6 and µr = 3 amounted to 0, 944 and 0, 83, respectively.

The average value of experimental results for that proportionality factor was

0, 96. In the analysis of the theory of the experiment (Cullwick pp 168-9)

the central part is played by equation (4) from the preceding note.

[10] As far as the present author is aware, among rare authors which warned

to caution in relation with the generally accepted interpretation of the

Wilson-Wilson experiment was ever sceptical O’Rahilly (1965), pp 606-613.

[11] The proper frame of a conducting loop is the reference frame in which

crystal lattice of the loop is at rest. The assumption that in the proper

frame any segment of a current loop with a stationary current is electrically

neutral is known in the literature as the Clausius postulate (O’Rahilly 1965,

vol. 2, p 589, Bartocci and Mamone Capria 1991a,b). This assumption

is found in many textbooks and therefore necessitates a comment. Some

time ago Matzek and Russell (1968) pointed out the fact that in case of

an infinite straight cylindrical conducting wire with a stationary current

the proper magnetic field of the current gives rise to a redistribution of the

current carriers i. e. to their concentrating towards the conductor axis (the

“self-induced pinch-effect”), as “observed” in the proper frame of the lattice.

43

Subsequent elaborations of the problem of finding the charge distribution in

a conductor with a stationary current for more realistic models, did not

provide a clear answer to the question: in what inertial frame is a current-

carrying conductor electrically neutral (Peters 1985, Gabuzda 1993, Redzic

1998). The problem will be also discussed in the next Chapter.

[12] This fundamental relation was exactly derived starting from the defini-

tion of the electric dipole moment, up to all orders in v/c, in case of a planar

closed current filament in a uniform translational motion in the proper plane,

for a rectangular loop whose direction of motion is parallel to one of its arms

(Panofsky and Phillips 1955, Rosser 1964, Blackford 1994), and for a circu-

lar loop (Rosser 1993). The crucial assumption was the Clausius postulate.

Taking into account that the derivations are based on the classical model of

magnetic dipole, it is necessary to mention how the relation ppp = (1/c2)vvv×m�m�

m�

is derived in the general case. One starts from the transformation law, from

the S� to the S frame, of the polarization vector of a magnetic dielectric in

motion: PPP = γ(1/c2)vvv×M�M�

M�, where only the contribution to the polarization

in S due to the magnetization of the magnetic in motion is taken into account

(cf Rosser 1964). Let the magnetic medium consists of n0 atomic magnetic

dipoles per cubic meter, each of the same dipole moment m�m�

m� = (m�

x,m�y,m

�z)

in the proper inertial frame S� in which the medium is at rest. The mag-

netization M�M�

M� in S

� is given by M�M�

M� = n0m

�m�

m�. Each of the atomic magnetic

dipoles possesses, by assumption, an electric dipole moment, identical for

all of them, ppp = (px, py, pz) as measured in the lab frame S with respect to

which the magnetic is uniformly moving with velocity vvv. Due to the Lorentz

contraction, n0 magnetic dipoles in m3 in S� occupy the volume (1/γ) m3,

where γ = (1 − v2/c

2)−1/2, as measured in S. Since the polarization in

S equals PPP = n0γppp, from the preceding equations one immediately finds

ppp = (1/c2)vvv ×m�m�

m�, without introducing any special assumption about the

structure of magnetic dipole. Rosser’s (1993) remark concerning the valid-

ity of the preceding elementary reasoning is worth mentioning: “According

44

to special relativity, this result should be true if the atomic magnetic dipole

moments arise from orbital electron motions or from electron spin or from a

combination of the two.”

Very soon, a proof has arisen that Rosser was right. Namely, this predic-

tion of special relativity was experimentally validated (Sangster et al 1995),

within an error of about 2%, as Krotkov et al pointed out (1999): “The mov-

ing magnetic dipole was a magnetically polarized thallium fluoride molecule

in a molecular beam that passed through a region of constant electric field

EEE. The experiment was planned as a measurement of the Aharonov-Casher

phase shift, which is (1/�)� b

a (mmm×EEE) · (vvv/c2)dt for a molecule traveling from

point a to point b. The integrand may be written as ((vvv/c2)×mmm)·EEE, which is

just the interaction energy between the ‘relativistic dipole’ ppp and the electric

field EEE. The measurement of this interaction energy for a molecule moving

at (essentially) constant velocity may be considered to be confirmation of

the Einstein-Laub analysis.”

[13] In the view of the present author, the central conclusion reached by

Krotkov et al (1999) can by no means be considered conclusive. Namely

if the motion of the sphere is uniform translation, then the second term in

their key equation (13) does vanish and then the electric dipole moment

of the sphere does have the value obtained by the authors. If, however,

the motion of the sphere is uniform rotation, then the second term in their

equation (13) neither vanishes nor has a simple interpretation, which implies

that a true expression for the electric dipole moment of the rotating sphere

contains some additional terms.

[14] One of rare exceptions is an excellent textbook by A. P. French (1968),

whose author passionately protests (on p 259) against this oversimplifica-

tion in the style unusual for textbook literature: “Who says relativity is

important only for velocities comparable to that of light?”

[15] In Lorentz’s non-relativistic electron theory, the constitutive equations

45

for a linear, isotropic medium in motion at low speeds read

MMM =µr − 1

µ0µrBBB + PPP × vvv, (∗)

PPP = ε0(εr − 1)(EEE + vvv ×BBB), (∗∗)

JJJ = σ(EEE + vvv ×BBB). (∗ ∗ ∗)

The constitutive equations, as well as Maxwell’s equations for material me-

dia in slow translational motion, can be obtained, as is well known, through

a non-relativistic reasoning from the corresponding equations that apply to

media at rest (Panofsky and Phillips 1955, Chapter 9). True, Lorentz’s

original theory was formulated with respect to the ether frame. However,

as Miller (1981) pointed out, researchers from the beginning of 20th cen-

tury had mainly “cavalierly” assumed that Lorentz’s theory applies in the

reference frame tied with the Earth, that is in the lab. This assumption

was also introduced in analysis of the Wilson-Wilson experiment (cf Cull-

wick 1959, pp 166-171). Lorentz’s theory understood in this way represents

“electrodynamics of bodies in slow motion without special relativity,” and

can be obtained from the formulae of relativistic electrodynamics in the limit

c −→ ∞ (of course, in case of the last limit only the constitutive equations

are in question).

It should be mentioned that various authors give different answers to

the question of what is the prediction of Lorentz’s theory in the case of the

Wilson-Wilson experiment (Cullwick 1959, p 170).

In relativistic electrodynamics, the constitutive equations for bodies in

slow motion differ from these given above by a relativistic term (1/c2)vvv×M�M�

M�

in equation (∗∗) for the polarization.

In relation with the preceding considerations, the idea is tempting of

an electrodynamics that would be Galilei-covariant. Such an electrodynam-

ics does exist, as Le Bellac and Levy-Leblond (1973) pointed out. Un-

fortunately, in that theory, there is no reference frame in which complete

Maxwell’s equations apply. Condensers don’t work. There is no light, etc.

46

3 A problem in electrodynamics of slowlymoving bodies: Maxwell’s theory versusrelativistic electrodynamics

3.1 Setup of the problem

In this Chapter we shall deal with the electromagnetic interaction between a

circular filamentary conducting loop with a stationary current in a uniform

slow translational motion and a point charge which is at rest or is uniformly

moving at the same velocity as the loop (Rindler 1989, Bartocci and Ma-

mone Capria 1991a, b, Redzic 1993). This problem, which seems to be one

of the simplest in electrodynamics of moving bodies, will be solved in the

laboratory reference frame in two ways: in the framework of classically inter-

preted Maxwell’s electrodynamics (henceforth, Maxwell’s theory), and in the

framework of relativistic electrodynamics. Basic assumptions of Maxwell’s

theory will be explicitly given. (The two methods of solving this and similar

problems represent electrodynamics of slowly moving bodies without and

with special relativity, respectively.) In addition, a variant of the problem

will be analyzed under the assumption that Maxwell’s theory applies in the

reference frame of the ether, its natural habitat. [In this interpretation, the

considered problem is a simple analogue of the famous Trouton-Noble ex-

periment, the theory of which, contrary to that of the present problem, is

very tricky (cf Teukolsky 1996).] Our presentation closely follows that of

Bartocci and Mamone Capria (1991a,b).

Setup of the problem in the framework of Maxwell’s theory.

Consider a filamentary circular current loop C with stationary current I

which moves with respect to the IFR S with constant velocity vvv = (v, 0, 0).

The loop C at the moment of time t = 0 is given by parametric equations

x = R cos θ, y = R sin θ, z = 0 (1)

47

What is the force acting at that instant on a charge q which is at rest in the

S frame at the point (0, 0, L), as is shown in Figure 7?

Figure 7

3.2 Solution in the framework of Maxwell’s theory

In what follows under Maxwell’s theory we shall strictly mean:

a) the system of four Maxwell’s equations

curlEEE = −∂BBB

∂t(2)

curlBBB = µ0

�ε0

∂EEE

∂t+ jjj

�(3)

divEEE = �/ε0 (4)

divBBB = 0 (5)

These equations reduce, in the standard notation, to the inhomogeneous

d’Alembert type equations for potentials Φ,AAA:

�Φ = −ρ/ε0 (6)

�AAA = −µ0jjj (7)

where � and jjj must satisfy charge conservation

divjjj = −∂�

∂t(8)

48

and the potentials AAA and Φ must satisfy the Lorenz1 gauge condition2

divAAA = − 1

c2

∂Φ

∂t, (c2 = 1/ε0µ0) (9)

(The electric and magnetic fields are expressed through the potentials by

the relations

EEE = −gradΦ− ∂AAA

∂t, BBB = curlAAA) (10)

b) The Lorentz force law

FFF = q

�−gradΦ− ∂AAA

∂t+ vvv × curlAAA

�(11)

c) An additional assumption that can be viewed as a restriction on the way

fields “originate” from sources: we assume that for given � and jjj, (6) and

(7) have a unique solution which is physically relevant, namely the one given

by the so-called retarded potentials3

Φ(rrr, t) =1

4πε0

�� (x�, y�, z�, t− | rrr − rrr

� | /c)

| rrr − rrr� | dx�dy

�dz

� (12)

AAA(rrr, t) =µ0

4π

�jjj (x�, y�, z�, t− | rrr − rrr

� | /c)

| rrr − rrr� | dx�dy

�dz

� (13)

[“Remark: In equation (3) it would be natural to add a term σ0EEE, where σ0

is the vacuum conductivity; it can be argued that the conventional present-

day choice of putting σ0 = 0 is not experimentally so well established as it

could be. Although this does not affect directly our argument, since in a

noncosmological context (such as the one we shall be dealing with) σ0 seems

to be really negligible, we wish to point out that the opposite view (i. e.,

σ0 > 0) has been recently gaining adherents.”4]

It is not difficult to verify that charge conservation (8) is a sufficient

condition for the retarded potentials Φ and AAA to satisfy the Lorenz gauge

condition (9).

Maxwell’s theory presented above applies, by assumption, in a given

inertial frame of reference S. However, we could add another hypothesis

49

which would ensure the validity of the theory in all reference frames S� linked

to S by a Lorentz transformation; this is the way relativistic electrodynamics

(RED) is obtained. The additional hypothesis reads

(�c, jjj) and (Φ/c,AAA) are contra-variant components of quadri-vectors of

Minkowski’s space-time. (14)

As is well known, this assumption enables us to write Maxwell’s equations

in a Lorentz-covariant form. It should be stressed that (14) is a fundamen-

tal physical assumption which is logically independent from the previous

(2)-(13). It is perfectly legitimate to consider the possibility of translating

Maxwell’s equations into space-time geometric terms as nothing more than

an interesting mathematical property, devoid of any physical content. In this

sense MT is formally covariant with respect to the Lorentz transformations.

As Bartocci and Mamone Capria pointed out, it is clear that without

more specific assumptions on the way simple physical systems have to be

modeled the theory so far described cannot get very far as a physical theory.

For our discussion we need to know for instance something about the electric

field produced by a current; we add a new hypothesis which we shall call the

Clausius postulate (CP)5:

Any segment of a conductor at rest with a stationary current is electri-

cally neutral.

Now we have almost all requisites necessary for solving our problem. An

essential detail, however, is missing. Namely, a circular loop in motion with

a stationary current I is not a clearly defined system. First, it should be

specified that “a stationary current I” refers to the proper frame of the

loop. Then, one should answer the question of what may be assumed in MT

about the behaviour of a loop in motion. Introducing the hypothesis that

both charge and lengths are preserved under motion, it is natural to take

for granted that charge and current densities in case of a loop in motion

are related with the corresponding densities for the loop at rest, �0 and jjj0,

50

according to the Galilean law of composition of velocities (Redzic 1993)

�(x, y, z, t) = �0(x− vt, y, z, t) (15)

jjj(x, y, z, t) = jjj0(x− vt, y, z, t) + �0(x− vt, y, z, t)vvv, (16)

where the notation is adapted to the present problem (the loop is moving

along the positive x-axis). Taking into account that a stationary current is

considered, i. e. that jjj0 = jjj0(x, y, z), using the continuity equation and the

CP we have

�(x, y, z, t) = �0(x− vt, y, z) = 0 (17)

jjj(x, y, z, t) = jjj0(x− vt, y, z). (18)

One can easily verify that � and jjj satisfy the continuity equation if the same

applies to the corresponding rest densities �0 and jjj0; in accord with our

definition of MT, we now only have to evaluate the retarded potentials (12)

and (13). The electric potential Φ obviously vanishes, whereas for the vector

potential AAA we have

AAA =µ0IR

4π

� 2π

0

(− sin θ, cos θ, 0)

Ddθ (19)

where

D2 = (x− vt−R cos θ)2 +

�1− β

2� �

(y −R sin θ)2 + z2�, (20)

and β ≡ v/c.7 From (19) we get the following expression for the magnetic

field BBB of our current loop C in motion

BBB = curlAAA =µ0IR

4π

��1− β

2�z

� 2π

0

cos θ

D3dθ, (1− β

2)z

� 2π

0

sin θ

D3dθ,

−� 2π

0

(x− vt−R cos θ) cos θ + (1− β2)(y −R sin θ) sin θ

D3dθ

�,

(21)

and for its electric field EEE the expression

EEE = −∂AAA

∂t= −µ0IvR

4π

� 2π

0

(x− vt−R cos θ)(− sin θ, cos θ, 0)

D3dθ. (22)

51

The required force acting by the electromagnetic field of the moving loop at

the instant t = 0 on the charge q at rest at the point (0, 0, L) is given by the

expression

FFF = −q

�∂AAA

∂t

�

x=0,y=0,z=L,t=0

=µ0

4πqIvR

2

� 2π

0

cos θ (− sin θ, cos θ, 0)�R2 + L2 − β2

�R2 sin2

θ + L2��3/2

dθ

(23)

Neglecting terms of second and higher orders in β in a series expansion of

the integrand in (23) we have

FFF ≈ (µ0qIvR2/4

�R

2 + L2�3/2

)eeey. (24)

One can easily verify that, in the same approximation,

BBB0,0,L,0 = (µ0IR2/2

�R

2 + L2�3/2

)exexex

which coincides with the exact expression for BBB of the current loop at rest,

at the same point.

In case the point charge q moves with the same velocity as the loop C,

so that its trajectory is given by

x = vt, y = 0, z = L (25)

for the force acting on q by the electromagnetic field of the loop at the

instant t = 0 (and, of course, at any instant) we get

FFF∗ = −q

∂AAA

∂t+ qvvv ×BBB ≈ −

�µ0qIvR

2/4(R2 + L

2)3/2�eyeyey, (26)

again up to the second order terms in β.8

3.3 Solution in the framework of relativistic electro-dynamics

We now obviously have to reformulate the problem: there is a stationary

current I, and the loop is circular, in the rest frame of the loop S�rf . Following

52

the standard procedure (first one evaluates the potential AAA�R in the S

�rf frame,

then one applies the transformation law), passing details, for the vector

potential of the electromagnetic field due to the moving loop in the S frame

we obtain

AAAR =µ0IR

4π

� 2π

0

(− sin θ,

�1− β2 cos θ, 0)

DRdθ, (27)

where

D2R = (x− vt−R

�1− β2 cos θ)2 +

�1− β

2� �

(y −R sin θ)2 + z2�, (28)

and the subscript R indicates that the solution belongs to relativistic elec-

trodynamics. Comparing equations (27) and (19) one could infer that, up to

the second order terms in β, there is no difference between the predictions

of RED and MT. However, this is not so. Namely, while according to the

CP each segment of the loop is electrically neutral in the S�rf frame, the

relativistic transformation law of charge and current densities implies that

there is a charge distribution in the current loop in motion, as measured in

the S frame. [As is well known, the presence of a charge distribution in the

current loop in motion is a purely relativistic effect and is a consequence, in

the long run, of the relativity of simultaneity (Rosser 1964).] In the case of

our filamentary circular loop of radius R, with stationary current I, mea-

sured of course in the S�rf frame, a rather simple calculation reveals that the

electric dipole moment of the loop in the S frame equals

ppp = −eeeyvIπR2/c

2, (29)

to all the orders in β (Rosser 1993). [Since the magnetic dipole moment of

the loop in the S�rf frame is m

�m�

m� = IπR

2eeez, and since vvv = veeex, obviously,

in this case the crucial relation ppp = vvv ×mmm�/c

2 is valid exactly.] The scalar

potential ΦR is readily obtained on the basis of the hypothesis (14), taking

into account that in the S�rf frame Φ�

R vanishes (CP): ΦR = vARx. Finally,

53

the force acting by the electromagnetic field of the moving loop on the charge

q which is at rest at (0, 0, L) is, according to RED, given by the expression

FFFR = −q(∂AAAR

∂t+ grad ΦR)x=0,y=0,z=L,t=0 ≈

≈ (µ0qIvR2/2

�R

2 + L2�3/2

)eeey,

(30)

up to the second order terms in β.9,10 In case q moves with the same velocity

as the loop, so that its trajectory is given by equation (25), the force on q

exactly vanishes,

F∗

F∗

F∗R = 0, (31)

which immediately follows from the fact that in the S�rf frame the corre-

sponding force vanishes, and from the relativistic transformation law of the

Lorentz force.

Comparing the corresponding equations (30) and (24), i. e. equations

(31) and (26), we come to a conclusion that even at extremely low speeds

the predictions of RED and MT are essentially different; note that equations

(24), (26) and (30) are correct up to the second order quantities in β, whereas

equation (30) is exact. As can be seen, this divergence in predictions arises

from the following two reasons. The first, in RED, in the transformation

law of the charge density

� = �� + vj

�x/c

2�≈ �

� + vj�x/c

2, (32)

there is a term vj�x/c

2, unknown in MT. (As is pointed out above, the ap-

pearance of that term is a consequence, in the long run, of the relativity of

simultaneity (Rosser 1964).) The second, not less important reason is the

Clausius postulate, which is used in both theories. Namely, if it were �� �= 0

in the S�rf frame, then the relation � = �

� would also apply in RED up to the

second order terms in β, for all reasonable values of drift velocity of current

carriers in S�rf , and the predictions of the two theories would coincide at low

speeds. Assuming the validity of the CP, our example reveals that in the

54

general case relativistic effects must not be ignored even at “room velocities”

of macroscopic systems in translational motion.

It seems that the preceding considerations are only of academic inter-

est. Namely, equations (24) and (30) i. e. (26) and (31) are presumably

inaccessible to experimental verification. However, as Bartocci and Mamone

Capria (1991a) pointed out, the situation is different if we go back to the

original Maxwell’s hypothesis that Maxwell’s theory is valid in the reference

frame of the ether. In this case the problem we discussed above suggests a

new experimentum crucis discriminating between RED and MT. According

to RED, the field due to a circular current loop of radius R with a stationary

current I which is at rest in a pseudo-inertial reference frame tied with the

Earth (the laboratory) exerts no force on a charge at the centre of the loop.

Maxwell’s theory, however, predicts that there exists a force −(µ0qIv/4R)eeey

on the charge q, as equation (26) reveals; this result applies under the pro-

viso that the velocity vvv of the Earth with respect to the ether is parallel

to the plane of the loop.11 (The force has the same unit vector as qvvv ×BBB,

where BBB is the magnetic flux density at the centre of the loop due to the

current in the loop.) “The predicted force depends both on the intensity and

on the direction of the current which should make it possible to separate a

nonzero effect from other disturbances due to constant fields existing in the

terrestrial reference frame, and to other sources of systematic errors. More-

over, by increasing I and q we might be able to observe an effect even if the

velocity of the laboratory is very small, as presumably it is, compared to c.

The possibility that the plane of the circuit does not contain the ‘absolute’

velocity makes no harm, because one can repeat the observations for various

choices of that plane, obtaining a maximum effect when this velocity lies in

the plane ...” (A simple analysis reveals that in the general case the force on

the stationary charge q at the centre of the loop equals −(µ0qIv/4R)eeey cos ψ,

where ψ denotes the acute angle between vvv and the plane of the cicuit; (−eeey)

55

is the unit vector of vvv×BBB.) Bartocci and Mamone Capria proposed to call a

possible experiment whose idea was presented above the Kennard-Marinov

experiment, intending thus to remember the name and the work of the two

enthusiasts in the field of classical electrodynamics, passionate adherents of

Maxwell’s original theory.12,13

3.4 Experiments

The suggested crucial Kennard-Marinov experiment, as far as the present

author is aware, has never been performed.14 Some experimental results,

however, have been published (Edwards et al (1976), Sansbury (1985)) dis-

proving the key assumption in the preceding analyses, the Clausius postu-

late. Edwards et al (1976) found that there is a nonzero electric potential

due to a stationary current in a closed superconducting coil, depending on

the square of the current intensity. The researchers observed just the de-

pendence they were expecting, led by a “non-obvious suggestion” that mag-

nitude of charge of current carriers is proportional to v2/c

2, where v is the

carriers’ speed. The subsequent attempts to explain the Edwards poten-

tial, as well as various fundamental theoretical conceptions related with it,15

have presumably all been made fruitless by recent experiments of Shishkin

et al (2002) which established that there is no Edwards potential, the ob-

served phenomenon being a consequence of the piezoelectric effect in the

teflon isolation of the superconducting coil used in experiments. Taking into

account delicacy of the interplay between theory and experiment (“experi-

ment is theory of theory” (Popper 1982)), as well as the fact that the quest

for the second-order effects is in question, any exclusiveness when reaching

conclusions would be irrational. In addition, it seems that one should also

listen lonely voices of those researchers in the field of electrodynamics which

are considered outsiders by the present-day scientific community (cf Maddox

1990).

56

Notes

[1] This is not a typographical error. Perusal of the most recent literature re-

veals that the Lorentz gauge is mainly replaced by the Lorenz gauge. Thus, a

long-lasting injustice toward the true author of that gauge condition, Ludwig

Lorenz, was corrected (cf Rohrlich (2002), and also O’Rahilly (1965)). Sci-

entific terminology is unfair occasionally. For example, Maxwell’s equations

are essentially Heaviside’s (Lorrain et al 2000, pp 486-7).

[2] As is well known, the potentials AAA and Φ that satisfy the Lorenz gauge

condition are not unique. By making a gauge transformation

Φ −→ Φ0 = Φ− ∂H

∂t, AAA −→ AAA

0 = AAA + gradH

one could get another solution of equations (6), (7) and (9) furnishing the

same fields EEE and BBB when �H = 0. As far as the problem of the sources

is concerned, and the existence of nonzero and nonsingular solutions of the

homogeneous wave equation we quote from B. H. Chirgwin, C. Plumpton

and C. W. Kilmister, Elementary Electromagnetic Theory, Vol. 3 Maxwell’s

Equations and Their Consequences (Pergamon Press, 1973), pp 549-550:

“How is one to interpret such a solution of Maxwell’s equations? There are

no singularities - that is, no sources of the field anywhere or at any time.

[...]. The existence of this kind of solution of Maxwell’s equations suggests

that Maxwell’s theory may be incomplete. It seems to lack some additional

restriction in order that fields originate only from sources like charges and

magnets. But we do not know how to modify the theory so as to rectify this

defect.”

[3] By the way, also this standard choice depends on the acceptance of other

“neutral”, from the point of view of the present consideration, hypotheses

about the way fields “originate” from sources, the behaviour at infinity of the

fields, and the lack of the physical relevance of the “anticipated” potentials.

57

[4] The hypothesis σ0 > 0 has been recently revived by R. Monti, who has

also shown its important large-scale consequences. For details, see, for in-

stance, R. Monti, “The electric conductivity of background space,” in Prob-

lems in Quantum Physics, Gdansk 1987 (World Scientific, 1988) or Vigier

(1990).

[5] See, for instance, O’Rahilly (1965), Vol. II, p 589. Clausius stated

that “a closed constant current in a stationary conductor exerts no force on

stationary electricity.” Compare note [11] in the preceding chapter, and the

last section of this chapter.

[6] Of course we might alternatively introduce some form of the FitzGerald-

Lorentz contraction hypothesis, but our aim here is to show some conse-

quences of MT in its most “classical” interpretation. We do not claim that

this version of MT does not require amendments in order to be proposed

as a realistic physical theory (cf French 1968, Panofsky and Phillips 1955);

for us MT is mainly a tool, with an obvious historical relevance, to analyze

some of the implications of the relativistic assumptions.

[7] A proof of equation (19) is based on the formal covariance of MT with

respect to the Lorentz transformations. As Bartocci and Mamone Capria

(1991a) give only a sketch of the proof, we present here a more complete

variant.

Having in mind that (Φ/c,AAA) formally looking are contra-variant com-

ponents of a quadri-vector, and using the fact that on the basis of the CP

Φ = 0, one has

A�x = γAx, A

�y = Ay, A

�z = Az,

where γ ≡ (1−β2)−1/2, and the primed and unprimed coordinates are related

by the standard Lorentz transformation

x� = γ (x− vt) , y

� = y, z� = z, t

� = γ�t− vx/c

2�.

58

Also

j�x = γjx0(x− vt, y, z) = γjx0(x

�/γ, y

�, z

�), j�y = jy, j

�z = jz,

using (17) and (18). Since

AAA�(rrr�, t�) =

µ0

4π

�j�j�j�(rrr�1, t

�ret)

| rrr� − rrr�1 | d

3r�

r�

r�1

and our final goal is to find the vector potential AAA, we can put AAA = AAA�, and

also

AAA�(rrr�, t�) =

µ0

4π

�jjj0(x�1/γ, y

�1, z

�1)dx

�1dy

�1dz

�1

[(x� − x�1)

2 + (y� − y�1)

2 + (z� − z�1)

2]1/2

The last two equations contain deliberate errors (the factor of γ!), but lead to

the correct result; the errors are motivated only by the economy of writing.

Introducing new variables x�1/γ = ξ

�, y

�1 = η

�, z

�1 = ζ

� we have

AAA�(rrr�, t�) =

µ0

4π

� +∞

−∞

jjj0(ξ�, η�, ζ �)dξ�dη

�dζ

�

{(x�/γ − ξ�)2 + (1/γ2) [(y� − η�)2 + (z� − ζ �)2]}1/2

It is now natural to introduce the corresponding polar cylindrical coordinates

by the relations

ξ� = ρ cos θ, η

� = ρ sin θ, ζ� = z.

In these coordinates

dξ�dη

�dζ

� = ρdρdθdz, jjj0 = Iδ(ρ−R)δ(z)eeeθ

and (19) is reached by an elementary calculation. QED

[8] One would expect that EEE + vvv ×BBB = 0 at the considered point, at the

instant t = 0, since in the so-called Galilean limit of RED the electric field in

the S�rf frame EEE

� = EEE + vvv ×BBB and equals zero. However, it is obvious from

equation (26) that EEE + vvv ×BBB �= 0 in MT. We point out that it is not very

obvious how to find the electric and magnetic fields EEE� and BBB

� in the S�rf

frame in the framework of MT. A possible prescription is due to Maxwell:

59

Let VVV be the instantaneous velocity of a charge q with respect to the

laboratory frame S (that is, in Maxwell, in the ether frame), and let VVV� be

the instantaneous velocity of the same charge with respect to an inertial

frame S� which is moving with velocity vvv = (vx, vy, vz) relatively to S. On

the basis of the Galilei transformation x� = x − vxt, y

� = y − vyt, z� =

z − vzt we have VVV = VVV� + vvv. We take that AAA

�(x�, y�, z�, t) = AAA(x, y, z, t), and

Φ�(x�, y�, z�, t) = Φ(x, y, z, t) where rrr� = rrr − vvvt. It is not difficult to verify

that now one has

−∂AAA

∂t− gradΦ = −∂AAA

�

∂t− grad�Φ� + (vvv · grad�)AAA�

and consequently

−∂AAA

∂t− gradΦ + VVV × curlAAA =

−∂AAA�

∂t− grad�Φ� + VVV

� × curl�AAA� + grad�(vvv ·AAA�),

that is EEE +VVV ×BBB = EEE�+VVV

�×BBB�+grad�(vvv ·AAA�); the meaning of the symbols

we used is obvious. Maxwell’s interpretation of this result is very interesting:

“[...] in all phenomena relating to closed (emphasis added by D. R.) circuits

and the currents in them, it is indifferent whether the axes to which we re-

fer the system be at rest or in motion,” that is, for these electrodynamical

systems the principle of relativity is valid in MT (an ideal example would be

the Faraday-Neumann-Lenz law of electromagnetic induction) (cf. Maxwell

(1891), vol. 2, p 601). It should be stressed, however, that Maxwell’s pre-

scription makes it possible to find AAA� and Φ� only when AAA and Φ are already

known. As can be seen from the example of equation (19), finding of AAA and

Φ can be a cumbersome task. Formally, the Faraday-Neumann-Lenz law of

electromagnetic induction is Galilei-invariant in the above, Maxwell’s, inter-

pretation. Essentially, of course, Maxwell is wrong here since in the S� frame

his E�E�

E� and B

�B�

B� fields do not satisfy Maxwell’s equations.

60

[9] The problem of finding the vector potential AAAR, i. e. ∂AAAR/∂t on the

z-axis at the instant t = 0 in RED, up to the second order quantities in β,

can be solved in another, less formal way based on intuition and symme-

try arguments. Namely, on the basis of the transformation law, within the

considered approximation, one has AAAR(x, y, z, t) = A�A�

A�R(x−vt, y, z), and A

�A�

A�R

at any point of space has only azimuthal component (the symmetry!, with

respect to the axis of the circular loop, of course), and can be found in the

immediate vicinity of the axis by applying Stokes’ theorem, taking that the

magnetic field of the circular current loop just near the axis has the same

value as on the axis (the last approximation is legal since we are looking for

the partial derivatives). The partial time derivative can be expressed through

the partial derivative over the x coordinate, ∂AAAR/∂t = −v∂AAAR/∂x, since the

convective (Eulerian) derivative of AAAR, dAAAR/dt = ∂AAAR/∂t + (vvv · grad)AAAR,

obviously vanishes in the problem we consider. This alternative, more beau-

tiful method of determining the solenoidal component of the electric field

of the loop in motion was proposed by Rosser (1993). [A historical remark

is in order. The vanishing of the convective derivative of the quantities de-

scribing an electromagnetic system in uniform translation, stationary in its

proper reference frame, was used by Heaviside (1889, 1892), and also by

Lorentz (1895) in his Versuch, for reducing some electrodynamic problems

to the electrostatic ones (that is for reducing the inhomogeneous d’Alembert

equation in case of a charge distribution in uniform translation to the Pois-

son equation, cf Panofsky and Phillips 1955, Jammer 1961, Miller 1981, pp

32-33; the method was recently “rediscovered” by Dmitriyev (2002).]

The irrotational component of the electric field, −gradΦR, can be eval-

uated directly, because one has exactly ΦR = vARx. On the other hand, as

is well known (Panofsky and Phillips 1955), the electric field of an electro-

static (in its proper frame) charge distribution that is uniformly moving at

speed v equals, up to the second order terms in β, the Coulomb field of the

61

same charge distribution that would be at rest in the instantaneous position

of the considered charge distribution in motion. This is the true meaning

of Rosser’s (1993) statement that when evaluating the irrotational compo-

nent of the electric field of the moving current loop retardation effects may

be ignored. [It is not difficult to verify that the contribution of the vector

potential to the electric field due to a charge distribution in uniform trans-

lation (this potential arises from the corresponding convection current) is a

second order quantity in β, and thus negligible.] Having in mind that the

Lorentz contraction is a second order effect, we come to a conclusion that the

charge distribution which, according to RED, exists on the current loop in

motion contributes to the irrotational component of the electric field of the

loop, −gradΦR, in a simple way, through the corresponding Coulomb field.

(As is mentioned above, AAAR ≈ A�A�

A�R, which means that within the consid-

ered approximation only the conduction current in the loop gives a relevant

contribution to the solenoidal component of the electric field; the convection

current arising from the charge distribution in motion may be ignored.) The

result reached in this way coincides with that evaluated directly, −gradΦR,

where ΦR = vARx.

The alternative method described above of finding −gradΦR was also

proposed by Rosser (1993). In the view of the present author, this method,

while efficacious, is conceptually tricky since it necessitates some non-obvious

steps (not even mentioned by Rosser). For example, the real charge distri-

bution on the current loop in motion vanishes in the S�rf frame; the real

distribution is replaced by an equivalent (in the sense of finding −gradΦR)

fictive charge distribution which is “one-component” (there is no current in

the S�rf frame for that distribution), and which “exists” in the S

�rf frame too.

Incidentally, it seems that the general form of the time-dependent Coulomb

law (Jefimenko 1989, Griffiths and Heald 1991, Lorrain et al 1988) would be

of no use here.

62

[10] The force by which the field of the charge q is acting on the loop in

motion at the instant t = 0 equals (−µ0qIvR2/4(R2 + L

2)3/2)eeey, which is

obtained by applying the Coulomb law, since q is stationary at (0, 0, L), and

the charge distribution over the loop in motion is given by a relatively simple

expression within the considered approximation. The preceding conclusion

contradicts Rosser’s (1993) statement that the force vanishes. (Professor

Geraint Rosser in a letter to the present author of 20. March 2003. agreed

that in his original paper a mistake was made, and that the force is indeed

given by the above expression.) Since the force by which the field of the

loop in motion is acting on the stationary q at the instant t = 0 is given

by equation (30), it seems that the principle of action and reaction is not

satisfied. The explanation is conventional: the electromagnetic interaction

between the current loop and the point charge is not a direct one. Namely,

as it is well known, there is a third “body”, a medium in the interaction, the

electromagnetic field, which also possesses a momentum, so that the total

linear momentum of the system the current loop + the point charge + the

electromagnetic field is conserved (cf Tamm 1979). Unfortunately, it is not

possible to verify by a direct calculation whether in this example the equation

dpfpfpf/dt =�−µ0qIvR

2/4(R2 + L

2)3/2�eeey applies at the instant t = 0, where

pfpfpf = ε0

�(EEE ×BBB) dV is the linear momentum of the total electromagnetic

field. Simple calculations seem to be reserved for exotic systems (cf Butoli

1989). (The situation is much more pleasing with illustrations for the angular

momentum of the electromagnetic field. Cf Griffiths (1989) and references

therein.)

[11] It is of some interest here to answer the question of what is the equation

of motion of a charged point, with a charge q and with a mass m, in the

electromagnetic field, in MT (in its original, “ether” variant, of course). It

would be natural to take the conventional form d(m0uuu)/dt = q(EEE + uuu ×BBB), where uuu is the instantaneous velocity of the particle, and assume that

63

the mass of the particle m0 is time-independent. However, appealing to

Kaufmann’s experiments (cf Miller 1981), one could postulate that in the

ether frame the equation of motion has the form

d(m0uuuγu)/dt = qEEE + quuu×BBB, (A)

by making the same assumption on the mass. (This postulate might, in the

long run, lead to discovery of special relativity, as Bell (1987) pointed out.)

By using the identity

d

dt(m0uuuγu) = γ

3um0

du

dteueueu + γum0u

deueueu

dt,

where eueueu = uuu/u, one obviously has d(m0uuuγu)/dt = d(m0uuu)/dt, up to second

order terms in βu (the last equation is exact for u = 0). In this way, for

the problem discussed here it is irrelevant which one of the two equations of

motion we use. By the way, in RED equation (A)is Lorentz-covariant if and

only if the fundamental assumption is valid (starting from Einstein (1905a)

and Planck (1906) always tacit) that the mass of the particle, m0, is a time-

independent Lorentz-scalar. (This neglected fact was recently pointed out

by the present author (Redzic 2002).) As can be seen, the fundamental

assumption on the time-independence of the Lorentz-scalar m0, together

with the “spatial” equation of motion (A) imply the “zeroth” component

d(m0c2γu)/dt = qEEE · uuu,

of the corresponding quadri-vectorial equation of motion. Thus, the quadri-

vectorial “Lorentz force equation” is tantamount to equation (A) comple-

mented by the assumption that m0 is a time-independent Lorentz-scalar.

[12] It happens sometimes that ideas and discoveries of “old” physicists sink

into oblivion. The present author feels that it is indispensable to point out

the fact neglected by Bartocci and Mamone Capria (1991a,b): the here dis-

cussed problem of the electrodynamical interaction of the charge and the cur-

rent loop at relative rest, that are moving with respect to the ether, was also

64

the topic of discussions among the physicists in late 19th century (Budde,

FitzGerald, Lorentz, see Miller (1981)). Somewhat unexpectedly, their solu-

tion to the problem coincides, in a certain sense, with what we think today

to be the correct solution, and represents another illustration of Wigner’s

statement that sometimes intuition is more important than knowledge. The

basic idea of the “old” physicists was simple: “it is highly improbable that

anything depends on the absolute motion” (FitzGerald 1882), that is on the

motion with respect to the ether; physical effects depend only on the relative

motion between ponderable bodies and on their mutual relative positions.

One recognizes here the principle of relative motion which, together with the

assumption of the validity of Galilei transformation, is in classical mechan-

ics tantamount to the principle of relativity (“identical systems in any two

inertial frames behave in the same way under the same initial conditions”).

[The equivalence of the two principles is presumably the reason for ignoring

the principle of relative motion in textbooks devoted to classical mechanics;

an exception is V. I. Arnold, Mathematical Methods of Classical Mechan-

ics, transl. K. Vogtmann and A. Weinstein (Springer, New York, 1978) p

10. Relationships among the principle of relative motion, the corresponding

principle of slow relative motion, MT and RED were analyzed in detail by

Bartocci and Mamone Capria (1991a). They have shown that the principle

of relative motion does not apply in MT nor in RED; the principle of slow

relative motion, however, applies in RED but not in MT.]

The “old” physicists instinctively applied the powerful principle of rela-

tivity to MT. Since in the considered problem MT predicts a nonzero force

(our equation (26)) depending on unobservable speed v (the speed of the

system with respect to the ether) Budde (1880) and Lorentz (1895, 1904,

1912) (these references are given in Miller (1981), pp 176-7) postulated that

charges are induced on the current loop in exactly that amount which is

needed to cancel the electrodynamic force due to the absolute motion of

65

the loop and the point charge. Their result for density of charges induced

on the loop, up to second order terms in β, reads �i = vvv · JJJr/c2, where JJJr

denotes the conduction current density in the proper frame of the loop and

the charges!

The present author has become aware of this adherence of the “old”

physicists to the principle of relativity also in the domain of electromagnetic

phenomena only very recently, after the publication of his speculation that

Maxwellians were on the threshold of a discovery of special relativity (Redzic

2004a).

[13] Cf. E. H. Kennard, “On unipolar induction - another experiment and

its significance as evidence for the existence of the aether, “Philos. Mag.

33, 179-190 (1917), and also Stefan Marinov, The Thorny Way of Truth, I-

IX, International Publishers “East-West”, Graz, 1982-1991. Having in mind

the preceding note, it seems that Bartocci and Mamone Capria should not

have ignored Budde, FitzGerald, Lorentz, the true authors of the proposed

experimentum crucis.

[14] A variant of that experiment has been realized, now in the context of

a different “philosophy” (Bartocci et al 2001); as the authors cautiously

mention, preliminary experimental results indicate a violation of the local

Lorentz-invariance.

[15] Numerous references are given in a paper by Shishkin et al (2002). For

example, Ivezic (1990) attempted to explain the Edwards I2- potential, by

analyzing the classical illustration of the relationship between electromag-

netism and special relativity, an infinite straight cylindrical conductor with

a stationary current (Feynman et al 1964, Ugarov 1979, Purcell 1985, and

perhaps most completely French 1968), introducing an ad hoc assumption on

the Lorentz-contraction of the distance between electrons-current carriers in

the laboratory reference frame. It is difficult to discuss the validity of Ivezic’s

attempt because of the obviously didactical nature of the considered model

66

(an infinite one-dimensional system). (A somewhat more realistic model

of an infinite current-carrying wire implies a self-induced pinch-effect and

leads to new dilemmas (Matzek and Russell 1968, Gabuzda 1993, Redzic

1998).) On the one hand, the analysis presented by Zapolsky (1988) gives

a theoretical justification of the Clausius postulate in the framework of an

elementary (but nontrivial) model of a circular current loop with a station-

ary current, and thus annuls indirectly Ivezic’s assumption. On the other

hand, the present author agrees with Bartlett and Edwards (1990) that what

Ivezic considers a fatal defect of the standard relativistic electrodynamics

(Lorentz - non-invariance of the macroscopic charge of a segment of a closed

current-carrying loop) is essentially a natural and necessary consequence of

the relativity of simultaneity and the Clausius postulate. A discussion on

this topic, in which also some other authors took part, lasted some time in

the same journal, without reaching some new essential conclusion.

67

References

Abraham M 1904,“Zur Theorie der Strahlung und des Strahlungsdruckes,”

Ann. Phys., Lpz. 14 236-287

Bachman R A 1982,“Relativistic acoustic Doppler effect,” Am. J. Phys.

50 816-8. A useful reference is also Reynolds R E 1990,“Doppler effect for

sound via classical and relativistic space-time diagrams,” Am. J. Phys. 58

390-4

Bartlett D F i Edwards W F 1990,“Invariance of charge to Lorentz trans-

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