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Dragan Redˇi´ zc
University of Belgrade
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
Recurrent Topics in Special Relativity
That Einstein's special relativity - from its advent until today - continues to be a live source of stupefaction and wonders for both laymen and professional 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 success 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 confusion 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, Redˇi´ 1990a, 2002), zc 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, Redˇi´ 1996). It is indicative, zc one can say, that Rindler, in his rightly acclaimed book on special relativity, 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. of course. and have identical acceleration programmes. When a traveler through relativity somehow escapes from the quicksand of terminology. in Rindler appears also a time dependence of the relativistic mass. 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).4 Namely. to brood over them for several years and even to use them in everyday work. in purely geometric and kinematic relations). Figure 1 In one moment two identical signals from A are emitted toward B and C. obtained from the quadri-vector equation of motion (a purely dynamic quantity). for both "pure" and "impure" forces. as Bridgman (1963) put it in A Sophisticate's Primer on Relativity. we all are groping our path toward understanding basic concepts. more dangerous temptations lurk. it turns out that it is not sufficient to know of the FitzGeraldLorentz contraction and time dilatation. Three small spaceships A. Then (as reckoned by an observer in A) the ships will have at 4 . a little riddle with pictures suitable to a primer on relativity. As an illustration for this state of affairs might serve the following simple problem. a conceptual mess. with B and C equidistant from A (Figure 1). B and C drift freely in a region of space remote from other matter. without rotation and without relative motion.3 The result is. Let the ships B and C be identical. just like in fairy tales.e.
was eventually reached: the thread would not break. Elementary explication. and must finally break. testifies Bell. Will the thread break when the ships B and C reach a sufficiently high speed? B B C C Figure 2 Figure 3 According to the testimony of a distinguished physicist John Bell (1976). Evett 1972. the artificial prevention of the natural contraction imposes intolerable stress". (Cf also Dewan and Beran 1959.) It is observed that the setup of the problem has been altered for several years. in Bell's formulation. 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.every moment the same velocity. A clear consensus. 5 . because of its need to FitzGerald contract. then as the ships accelerate the thread travels with them. The answer is none the less wrong. goes as follows: "If the thread is just long enough to span the required distance initially. then as the rockets speed up. Evett and Wangsness 1960. at a sufficiently high velocity. It must break when. and always be at the same distance from one another.) Here. Dewan 1963. we shall briefly paraphrase Bell's remarkable comment on the described situation which refers to the method of teaching special relativity. it will become too short. 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.
the scientific problems that have been solved earlier.2 Miracles It is time to mention a few of the host of small and big wonders of special relativity. as is the case in Einstein's approach. rather concise three pages. We neglect the fact that the chain of thought of the old pioneerswise men. before relativity. Max Abraham (1904) needed forty pages of text for the solution of this problem. but in a tedious and complicated way. Lorentz and Poincar´. 1. have a stronger and more reliable intuition. by "switching off" one inertial frame of reference and "switching on" another. whereas Einstein (1905a) used only three pages for the same thing in his epoch-making paper (honestly. unexpected qualities of rigid (in relativistic sense. Larmor. but as a natural offspring of earlier physical ideas. classical road. factually carried out e and freed from the "weak link" of Newton's concepts of time and space. cf Rindler 1991) sticks and clocks that move do not appear as a dry consequence of certain abstract mathematical transformations. predicts both time dilatation and length contraction and leads eventually to the same conclusions as the Einstein's theory. achieved from logically entangled postulates.It is customary to emphasize the discontinuity. FitzGerald. the radical breakup with the primitive concepts of space and time. The small wonders are the methodological ones.5 It appeared to Bell that students who follow this longer. which is uniformly moving with velocity perpendicular to the mirror's plane. merely by "pushing the button". 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. as pointed out by Arthur Miller (1981) in his rich and 6 . and that can be solved by using special relativity simply and elegantly. The result is often the complete destruction of the student's trust in perfectly safe and useful concepts acquired earlier. However.
as measured in laboratory?7 Famous J. The main characters are Maxwellians. 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. produces in space outside the sphere the same electrostatic field as the corresponding point charge at rest at the centre of the sphere. (The quest for the image of a moving sphere. J. Lorrain et al 2000). a little cliff-hanger that takes place in London. but has read the Einstein's.) In his article Searle uses the contemporary scientific language (the sphere moves with respect to the ether). Heavy reading. 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. a small group of eccentrics that will give much pain to historians of science (cf Brown 2001. A simple and elegant solution based on the recipe of special relativity has been recently published (Redˇi´ 1992a.) Another case of "methodological wonders" appears in electrodynamics of moving bodies. has been sketched in an excellent monograph by Max Jammer (1961). (The present author admits that he has not read the Abraham's article. 2003. b). However. zc 7 . Following Maxwell (1891). very meticulously.8 In the historical perspective. dealt with this problem as well. Searle's cumbersome and complicated solution to the problem arouses admiration. and men able to recognize the essence. Thomson and Oliver Heaviside. the ratio of the length of the line and the diameter of the sphere is v/c. this point charge may be called the image of the conducting sphere. undoubted authorities in the field of Maxwell's electrodynamics. Cambridge and Dublin in late 19th century. It is well known that an isolated charged conducting sphere of radius R at rest in laboratory (an inertial frame of reference). Searle (1897) was the first to find the correct solution: the image of a charged conducting sphere in motion is a uniformly charged line.detailed monograph wherein Einstein's Relativity Paper was analyzed sentence by sentence).
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. a certain quality which is in an IFR purely spatial and timeindependent. For example. b) the distance between the ends of a rigid (in a relativistic sense!) stick moving along its own direction. Time as a measurable physical quantity in inertial frames of reference has exactly those peculiar traits as predicted by the Einstein's theory.a feature of all good physical theories. Rindler emphasizes. but rarely. unexpected and amazing physical conclusions ("leaps ahead of the empirical frontier") .9 on its own completely benign.12 Also. These new. can include dependence on time in another IFR.all have the same powerful source: the concept of time. Such is the case with the distance between the spaceships B and C in the problem discussed above (Dewan 1963). disbelief and insecurity.these true and great wonders of special relativity .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. Rindler. the fundamental prediction of special relativity.According to a nice metaphor by W. Einstein's (1905a) definition of time and the principle of constancy of the velocity of light.10 And the miracles are numerous. in combination with the principle of relativity always give rise to the same dramatic effect: the feeling of losing ground under one's feet. 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. a pure thought has the power to leap ahead of the empirical frontier . and a perennial question if it is possible that everything could be really so. on such a heroic scale as in the case of special relativity. Also. and sometimes rather inconspicuous. if the clocks 8 .
may of course be different from the observed "clock in motion"). when measured by the clocks at rest.3 Path toward understanding? It seems that the feeling of discomfort that accompanies physicists (and laymen) about the slowing down of the clock in motion is a consequence of the opacity of the usual relativistic method of inferring. the laboratory frame and the clock's rest frame.g. 1. features of a certain physical system (e. with all of its Einstein-synchronized clocks (which. with macroscopic caesium clocks (Hafele and Keating 1972. A natural question arises of what is the role of the clock's rest frame. but from the Lorentz transformations that connect the two IFRs. 9 .13 Finally. together with the principle of constancy of the velocity of light" (Einstein 1907). their purely instrumental character. that is. 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). 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. 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 quite puzzling.at rest are Einstein-synchronized. Even the creator himself of the theory of relativity that will soon become the special one pointed out this fundamental limitation of "the principle of relativity. Namely. Cornille 1988). while mutually identical. 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.
Poincar´. with reasonings of FitzGerald.Maybe the previously described feelings of unease and powerlessness that follow the understanding of basic results of special relativity could be attenuated. with t we denote the reading of the synchronized clocks that are at rest with respect to the laboratory.14. or even completely removed. this hydrogen atom partly follows the Bohr model. Lorentz. we arrive at some unexpected conclusions. Somehow the electron manages to maintain its own energy constant. z and t have their usual meaning in the laboratory. If we now expose this hydrogen atom to a constant and weak electrostatic field. and that an electron rotates uniformly around it on a circular trajectory of radius a under the action of the proton's electrostatic field. always appear in the formulation of all natural laws. Since the coordinates of position and time. it is above all necessary to define the meaning of these fundamental quantities. mutatis mutandis. (Electron makes up for the energy lost as electromagnetic radiation by absorbing the needed amount from some infinite reservoir of energy. Let us suppose that natural laws known as Maxwell's equations hold in some inertial frame of reference ("the laboratory"). and after dying out of transient effects. 1987). if Einstein's method were supplemented. the proton 10 . y. and applying this equation of motion on the electron and the proton that form our hydrogen atom. Here is a short sketch of Bell's approach.15 Let us now suppose that a proton is at rest in laboratory. in a way e suggested by Bell (1976. y. z and t. such as x. Say that x. maybe vacuum?) In short. (Since Maxwell's equations imply the principle of constancy of the velocity of light. the entire 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. Einsteinsynchronization of an arbitrary number of clocks at rest with respect to the laboratory is a trivially possible procedure. parallel to the plane of trajectory of the electron. After turning off the external field. For example.
Since in physics real is what is reached by measuring instruments.18 1. that length contraction in the direction of motion and time dilatation occur in this simple physical model due to acceleration! Now. Although both light and 11 . before entering electrodynamics of moving bodies. y. light and Maxwell's electrodynamics. circular trajectory and a normal period are real for the "observer in motion".16 The period of motion of the electron on the ellipse around the proton in uniform translational motion is 1/ 1 − v 2 /c2 times larger than the period of motion of the electron on the circle of radius a centered at the proton at rest. Elliptical trajectory and a longer period are real for the "observer in laboratory".17 The preceding analysis of the "hydrogen atom" in motion. thus. now it is more acceptable that for the "observer" moving with same velocity v as the proton. carried out completely in the laboratory frame. of course. which will take place in every "stick" and "clock" in uniform motion with respect to the laboratory. both "observers" are perfectly right. it is perhaps worthwhile to make a small digression about the relationship between special relativity.4 Relativity without Maxwell's electrodynamics? At this place. through the laboratory coordinates x. it is easier to accept that these are universal phenomena. 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 − v 2 /c2 as well in the direction of motion). with semi-axes a 1 − v 2 /c2 and a. the electron moves with respect to the proton (expressed. Also.moves with constant velocity v . z and t ) on an elliptical trajectory that is oblate in the direction of motion of the system. reveals. and the period 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 − v 2 /c2 times larger than the laboratory seconds). it seems.
Feigenbaum and Mermin (1988) analyzed a mechanical version of the famous 1905 Gedankenexperiment. the principle of constancy of the velocity of light has to be dethroned as one of the pillars of special relativity. the speed c can but does not have to be reachable by any physical object. then the constant c which appears in them represents the smallest upper boundary (the supremum) for the speed of particles in any inertial frame. according to some authors. a recent demonstration of the power of relativistic kinematics should be mentioned.electrodynamics have played a central part in the historical development of special relativity. In addition to that. it seems. Rindler's (1991) opinion is indicative in this connection: special relativity would exist even if light and electromagnetism were somehow eliminated from the nature. which served Einstein (1905b) 12 . gravitons) may have a nonzero mass was opened in this way (cf Vigier 1990). without light. cannot be in accord with the principle of relativity. The same objection goes with a similar Mermin's (1984) attempt to get the second postulate from the principle of relativity. the Lorentz transformations. Thus.19 It seems. neutrinos. the role of an inaccessible boundary. in itself has nothing to do with Maxwell's equations (Einstein 1935). Starting from the principle of relativity and the invariability of causality. Rindler proves that all inertial frames are related by either Galileo's or Lorentz's transformations. At the same time. In this context. a hidden circular argument. The possibility that the particles considered massless according to contemporary opinion (photons. based on electrodynamic concepts (Einstein's Lichtkomplex). Mermin's method of synchronization of distant clocks by their "symmetric transport" (cf footnote 5 of his article) contains. nor "freed" from circular reasoning. which are indispensable for Rindler's argumentation. the real basis of that theory. however. the constant c in the Lorentz transformations would play the same role as the absolute zero of temperature. that the alternative methods of clock synchronization. If the transformations are Lorentz's.
Here. Moreover. unlike Einstein who postulated it. which Einstein (1905b) "solved" by introducing the principle of equivalence between inertial mass and rest energy. as in Einstein. Feigenbaum and Mermin showed that in the relativistic expressions for momentum and kinetic energy of a free particle the same mass-Lorentz scalar m appears. i. it would be hasty to conclude from the above discussion that the relativistic kinematics is free of Maxwell's electrodynamics (cf Jammer 2000. without Maxwell's electrodynamics. by using solely relativistic kinematics and the laws of conservation of energy and momentum in their most general form. they revitalized the problem of the integration constant in the expression for the rest energy. the mass (the rest mass) appears in the non-relativistic limit of kinetic energy.) Furthermore. but Feigenbaum and Mermin get the exact limit by calculating it. Of course. These authors reached the same fundamental conclusion. This is an important result for which Einstein could find only a partial justification (Einstein 1935). almighty.to get to the equivalence between inertial mass and rest energy. as well as the relativistic expressions for energy and momentum of a free particle in the most general form. e. chapter 3). (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).20 13 .
so to say. uy . fx = law of the x-component of the relativistic tri-force. circus attractions of special relativity such as length dilatation and time contraction (Field 2000). f = d(m0u γu )/dt. Zigman 1997)) is not generally accepted. it is clear that the transformation law of the x-component of the relativistic tri-force must have the form fx = 2 f fx − v[f · u /c2 + (1/γu /c2 )F α Uα ] 1 − ux v/c2 f fx − vdm/dt fx − v[f · u /c2 + (1/γu )dm0 /dt] = 1 − ux v/c2 1 − ux v/c2 14 . reads. [3] For two inertial frames of reference S and S in the standard configuration (S is uniformly moving with respect to S along the common positive e x − x axis with velocity ve x ). However. For example. where m ≡ m0 γu and γu ≡ (1 − u2 /c2 )−1/2 . taking into account that the relativistic tri-force is not identically equal the time derivative of the relativistic momentum of the particle. [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 ˇ 1998. m0u γu . pp 57-61). But here we refer primarily 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]. uz ). and for a particle with rest mass m0 and instantaneous velocity in the S frame u = (ux . This introductory chapter contains the inventory of some recurrent topics in special relativity.Notes [1] A fresh example are. Eriksen and Vøyenly (1976) state that the classical and the relativistic concepts of mass are "incommensurable" (cf Jammer 2000. the transformation according to Rindler.
but instead that a more cumbersome. i. that the dependence of the particle's rest mass on time is in the general case given by c2 γu dm0 /dt = F α Uα . "based on special assumptions on the structure of matter". γuf ) is the corresponding quadri-force. Lorentz and Poincar´. making use of the orthogonality of the particle's quadri-velocity and quadri-acceleration (Rosser 1964. in a similar way as statistical mechanics can offer a much broader view than phenomenological thermodynamics. Larmor. a quadri-force does not comply with the condition F α Uα ≡ 0. and U α = γu (c. It should be mentioned that Bell's seminal essay gives only a sketch of the approach to special relativity through ideas of FitzGerald. Rindler 1991).where F α = (F 0 . both the FitzGerald-Lorentz contraction and time dilatation occur. we use the standard metrics (1. e Yanossy (1975). using classical electrodynamics (which is a relativistic theory par excellence without knowing that). Bell has shown that in that simple system. (Here. e. -1. -1. [4] Perhaps the mentioning of fairy tales in this context is not completely devoid of sense. in the general case. Bell mentions a monograph by L. Langevin's (1911) Traveler (La Voyageuse de Langevin. not just c young looking. Some of the conclusions of special relativity touch the archetypal dreams of humanity. of course. According to Bell (1976). can lead to a fuller insight. -1). 15 . en fran¸ais) comes home young (biologically young. note how language is a problem). when it is moving. less economic reasoning. after many years of interstellar journey. As his only predecessor.) We remind the reader that. [5] Analyzing a simple model of the hydrogen atom. contra-variant components of these quadri-vectors appear. Møller 1972. u ) is the quadri-velocity of the particle on which the force is acting. the last equation is obtained from the quadri-vector equation of motion. the essence of his argument is not at all that Einstein was wrong in his 1905 "kinematic" analysis.
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. (The 16 to the FitzGerald-Lorentz contraction. It is a constant proper acceleration in question. as predicted by the theory of special relativity. physicists (Terrell 1959. as . of course." (Here. Weinstein 1959.) The moral of the story seems to have been known to Democritus: things are not found therein where their picture is. We remind our reader of the traps of language in special relativity. post festum. "cube" is a body that has the shape of a cube when at rest.[6] Recently. Due measured in the laboratory. Namely. Many years after. located at its centre (Searle 1897). Einstein occasionally used the verb to observe ("betrachten") instead of the verb to measure. as measured in the body's proper frame. it seems that Searle almost touched that discovery. [8] Today. if we do not take into account completely ignored Lampa (1924). "If one saw an undistorted but rotated picture of a moving cube. one would deduce that the length contraction had taken place. R. [7] The conducting body has the shape of a prolate spheroid with semi axes R/ 1 − v 2 /c2 . or photographed by a camera) of a body whose speed is comparable to that of light does not coincide with its measured shape. R. 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. a solution to the same problem in the case of a uniformly accelerated perfect planar mirror is published (Van Meter et al 2001). 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 − v 2 /c2 : 1 : 1. cf also Rosser 1964) realized that a visible shape (the one that can be seen by the eye. then. the body is a sphere of radius R. Analyzing in 1905 how the shape of a body depends on reference frame in which it is measured.
" velocity = A lot of paper was consumed in clarifying this formulation of Einstein's. Namely.electric and magnetic fields of a point charge in uniform motion through the ether were derived by Heaviside as early as 1888. From the preceding considerations. "time interval" is defined in Einstein's §1 just by means of the velocity of light. the circularity problem can be simply solved by reformulating the principle of constancy of the velocity of light (cf the note 15 below). unexpected.) On the other hand. time interval where time interval is to be understood in the sense of the definition in §1. and thus one of the basic rules of valid definition is violated: of the present author. 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)). applying the principle of relativity to Maxwell's electrodynamics we infer (cf Redˇi´ 2004a): a conducting body that has the shape zc 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. located at the centre of the sphere.Wiechert potentials. Definitio ne fiat in orbem (A definition must not be circular). [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. In the view 17 . without the Li´nard e . the field outside a conducting sphere at rest is identical to the field of a point charge at rest. and without special relativity. which is independent of whether the light ray was emitted by a resting or by a moving body. Herein is light path .
located at the centre of the sphere. zc [10] Perhaps the best illustration of this psychological situation is the existence of the journal Galilean Electrodynamics. It is. [12] Differentiating of these distances is essential in the explanation of dis18 . due to our pre-relativistic instincts. Applying the principle of relativity to Maxwell's electrodynamics we infer (Redˇi´ 2004a): a charged zc conducting body in motion having the shape of a Heaviside ellipsoid is obtained by the motion of the same conducting body which is a sphere when at rest. however. when γ = (1 − v 2 /c2 )−1/2 = 2. The field E ∗ = E + v × B is perpendicular to the surface of the Heaviside ellipsoid at a point arbitrarily close to the surface (Redˇi´ 1992a). [11] It is not difficult to verify immediately that the statement is true by using the corresponding Minkowski diagram. The electromagnetic field of a point charge Q in uniform motion with e velocity v = ve x is identical to the field of a conducting body having the shape of a Heaviside ellipsoid which is moving with the same velocity.y R E E* n E v´B O R/2 Q z Q x Figure 4 A conducting sphere of radius R and with total charge Q at rest in the laboratory frame creates the same field as a point charge Q at rest. somewhat more difficult to imagine that there is such a feature at all. The √ figure corresponds to the value v = 3c/2.
(A version of this problem. It is not difficult to show that this kind of acceleration 19 . in the beginning they would tend to behave in the same way as two unconnected material points. reminds Zapolsky. If we try to accelerate its two ends with the same acceleration. was nicely presented by Bell (1976). starting at the same moment of time from the state of rest with the same acceleration. Each observer "sees" (at any instant of his time) that the other observer is going away from him. is always one and the same.appearance of the electric field of steady currents in the framework of an elementary but non-trivial model (Zapolsky 1988). The result of measurement depends essentially on who is accelerated with respect to the lab: the material points or the observer. If the internal forces can do that no more. say along the line connecting them. as measured in the lab. 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). One might wonder does this prove that an accelerated meter stick would also not be contracted. The conclusion is that restitutive forces in the stick will oppose the forces causing that the ends of the stick move with same acceleration. This means that the stick would tend to extend itself as measured by two observers "standing" at its ends. the stick breaks. A meter stick. In what follows we briefly paraphrase Zapolsky's argument.) It should be noted that the motion of the stick we discuss here is not "a rigid body acceleration". the distance between the two points. is a system of bound atoms. Insisting here on symmetry would be equally irrational as in the much better known "twin paradox". which is usually defined in special relativity as the acceleration that causes no internal stress (cf Rindler 1991). The answer is an emphatic "no!". however. The two material points are not connected. On the other hand. and are completely ignorant of one another. if the two material points are being uniformly accelerated with respect to the laboratory.
Dingle addressed that one k.). and a compound event that takes place at one spatial point of the K frame and has a duration of 1/ 1 − v 2 /c2 K-seconds must not be identified. One K-second of the clock at rest in the K frame equals 1/ 1 − v 2 /c2 k-seconds of the clock at rest in the k frame. k(ξ. Dingle falsifies special relativity. According to Born.(in which the proper acceleration continuously changes from end to end of the stick) leads to the Lorentz contraction. however. Overall conclusion: special relativity is not to Professor Born to defend "the integrity of scientist" by replying to the challenge. Dingle's inference does not follows from special relativity. the same notation as Dingle and Born's will be used: K(x. τ = γv (t − vx/c2 ). It follows k-seconds of the same clock. One k-second of a clock at rest in the k frame equals 1/ 1 − v 2 /c2 K-seconds of a clock at rest in the K frame. as measured by the system of Einstein-synchronized clocks at rest in the k frame.According to Dingle. t) is the "resting" reference frame. etc. τ ) is the one "in motion". it is not the kind of acceleration appearing in case of two independent material points (cf also Nikoli´ 1999). where ξ = γv (x − vt). (In the present note.second of a clock at rest in the k frame equals 1/(1 − v 2 /c2 ) . One K-second of a clock at rest in the K frame equals 1/ 1 − v 2 /c2 k-seconds. Born's counter-argument runs as follows. those are two distinct 20 a valid scientific theory since it contains a contradiction. the correct relativistic argumentation reads: One k-second of a clock at rest in the k frame equals 1/ 1 − v 2 /c2 K-seconds as measured by the system of Einstein-synchronized clocks at rest in the K frame. c [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). special relativity permits the following argumentation. (A compound event that takes place at various spatial points of the K frame and has a duration of 1/ 1 − v 2 /c2 K-seconds.
The issue provoked a prolonged polemic in the Nature that lasted several years. conjugate diameters = axes in K. the axes of the "middle frame" for K and k are not represented. The "middle frame" is moving at the speed v/(1 + 1 − v 2 /c2 ) to the right with respect to K (and at the same speed to the left with respect to k). A consensus was never 21 . (ξ. cτ ). for obvious reason. by convention.ct b C a SH A B ct TH C SH x O x TH Figure 5 C. those axes are mutually orthogonal. Numerous participants "accused" each other for elementary misunderstanding of basic concepts of special relativity. conjugate diameters = axis in k. time calibration hyperbola. OA. Dingle has made the same kind of error the student usually makes: two different quantities are denoted by one and the same symbol. x. C. straight lines in Minkowski space (Figure 5). ct. space calibration hyperbola. SH. section of light cone. represents the same time interval in K as OB in k: OA ∼ ct Oa > OB The clock at rest in K Oa ∼ cτ τ OB ∼ cτ Ob > OA The clock at rest ink Ob ∼ ct t OA ∼ OB ∼ OA > t ∼ OB > τ On this Born's figure. T H.
a universal constant. primordial principle that conceptually precedes the principle of relativity. cleaned up from the circular argument. i. it is postulated that Einstein synchronization is a realizable procedure. as measurements reveal. Therefore any definition of the time coordinate based on a previously discovered law of nature is nothing but a circulus vitiosus. [14] The statement that the principle of constancy of the velocity of light is already contained in Maxwell's equations appears occasionally in the literature (Einstein 1905b. The fact that Maxwell's equations are consistent with both principles is an excellent recommendation for the equations. which were discovered before physicists began to deal with the problem of clock synchronization. In this sense the principle of constancy of the velocity of light (also known as Einstein's second postulate) is essentially the first. always equals one clock-two way velocity of light which is a measurable quantity and.reached. the physical laws apply 22 . [15] Einstein's second postulate (1905a). e. In this way. 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. but nothing else. Fortunately. The fact that there exist physical laws (Maxwell's equations) consistent with the second postulate. fundamental. the meaning of time as a measurable physical quantity is postulated. Bartocci and Mamone Capria 1991a). It is perhaps worthwhile to mention that Maxwell's equations are a sufficient but not a necessary condition for the validity of the principle. states that in an IFR one way-two clock velocity of light. A definition of the space and time coordinates must precede the quest for the laws of nature. is of course a good recommendation for that postulate. 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. an immeasurable quantity.
this then means that the principle of relativity does not apply. since "physical realities" of the same events corresponding to different "observers" are not identical. where xe and ye denote the electron's coordinates. [18] This implies that physical reality independent of the frame of reference ("observer") has no physical sense. e [16] If v = vex . The inference would be false. by a suitable choice of the initial moment t = 0 the equation 2 of the ellipse reads (xe − vt)2 /(a 1 − v 2 /c2 )2 + ye /a2 = 1.in the pseudo-inertial reference frame tied for the Earth regardless of the season. 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. By the way. The thesis is 23 . On the basis of the preceding considerations. [17] A few examples for time dilatation of a moving clock in the same spirit. 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!). (In the same way as the question whether the FitzGerald-Lorentz contraction is real has no physical sense. one could infer that. were presented by Jefimenko (1996b). This is clearly seen by passing from a passive to an active interpretation of the Lorentz transformations (Bohm and Hiley 1985). 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. for the same "events". and different physical realities of different "observers" are a necessary consequence of different initial conditions. Of course. cf also Dewan 1963). on the basis of classical electrodynamics. the thesis that both the FitzGerald-Lorentz contraction and time dilatation are nothing but subjective phenomena was a continuing subject of lively discussion among physicists and philosophers.
if the phase velocity of the light exactly equals c. a Yugoslavian physicist. unfortunately. [19] Bachman (1982) derived a relativistic Doppler formula for waves whose phase velocity relative to the medium is u f = f0 The equation expresses the frequency f of the wave detected by the "observer" through the proper frequency of the source f0 . and only then. we mention that in the first. Namely. If the phase velocity of the wave exactly equals c. of course. Vladimir Vari´ak. the whole cosmos. then the ether may but need not exist. the ether must exist. That subjective feeling. The formula is derived under the assumption that the velocities of the source and detector are along the line connecting them. the phase velocity u. the relativistic limiting speed c. is not consistent with the time as a measurable physical quantity. Overall conclusion: if the phase velocity of the light is less than c. heroic years of the special theory of relativity.presumably a consequence of the mess about the concept of time. and the velocity of the "observer" toward the source relative to the medium v0 . regardless of the velocity of uniform motion of the medium relative to the source or detector. however. the velocity of the source toward the "observer" relative to the medium vs . in inertial frames. and Occam's razor solves the problem (Mirabelli 1985). As a curiosity. because the time is in accord with Einstein's theory. Miller 1981). also took part in them (cf Einstein c 1911. the detected frequency f depends only on the velocity of the source as measured in the proper frame of the detector. 24 . u + v0 u − vs 2 1 − vs /c2 2 1 − v0 /c2 1/2 . when the discussions about the subjective nature of time were most lively. then. it seems that Newton's absolute time is perfectly consistent with the illusive subjective feeling that thought "at one instant" can encompass everything.
in the case of the Doppler effect its power is limited.) The present author is o zc aware of only one attempt of an exact kinematical treatment of the Doppler effect (Rothenstein 2002). kinematic derivations of the Doppler effect (French 1968. 25 . Redˇi´ 1990b). 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 Schr¨dinger 1922. Namely.[20] For example.
and B . Physical laws are reached slowly and painfully. and then applied the derived transformation laws for the E . tenth paragraph of the paper. however. in that work. in a certain sense. Although the title of Einstein's (1905a) epoch-making paper is "On the Electrodynamics of Moving Bodies". electrodynamics of moving bodies is quite in second place. to the electrodynamics of moving bodies. entitled "Dynamics of a (slowly accelerated) electron". refers.3 Only the last. for example. the principle of relativity is essentially a metaprinciple (the term is Rindler's (1991)). Fermat's principle). Minkowski As it was hinted above.1 In the electrodynamic part of the paper Einstein proved that the Maxwell-Hertz equations in vacuo are Lorentz-covariant."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). 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 physical systems change". as is well known. Like the well known biological principle that ontogeny is a short and quick repetition of phylogeny.2 2. he interpreted 26 . In this way he obtained the formulae expressing the Doppler principle and the light aberration "for arbitrary velocities. Applying the principle of relativity.1 Electrodynamics of moving bodies and the Wilson-Wilson experiment Einstein. 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. The law states what could be a physical law but the principle is mute about which is a physical law (contrary to.fields to the case of a monochromatic plane linearly polarized electromagnetic wave in vacuo.
Consider a point of space-time P of a body moving with respect to the laboratory at the laboratory time t.the obtained equation in a cumbersome way. (AS1) apply once more if all primes are omitted. Let P be transformed to rest by the introduction of the coordinates x . B . The operations curl and div in (AS1) refer of course. ∂t (AS1) with material constants differing from those for vacuum E D = εE .5 Fundamental equations of the phenomenological electrodynamics of moving bodies. were constructed e by Hermann Minkowski (1908). D ∂D . "Minkowski's logic was simple: The Maxwell equations for a state of rest apply within the laboratory. let it have the velocity v . D . : E curlE = − B ∂B . ∂t D divD = . y . H . E J = σE (AS2) These constants have the same values as if the body were at rest with respect to the laboratory. In this system Maxwell's equations for a state of rest apply to the quantities E . z . take on a new form. (AS 2). Eqs. t for the description of the processes in the neighborhood of P. just like the time t . whose ideas represent the starting point of all subsequent researches in the field. H curlH = J + B divB = 0. on the groundwork laid by Lorentz (1895) Poincar´ (1906) and Einstein.4." 27 . Here we give how the essence of Minkowski's method was formulated by the famous physics teacher Arnold Sommerfeld (1952). since it knows nothing of its motion. in view of the basic property of covariance of the Maxwell equations with respect to the Lorentz transformations. which transforms the primed system back into the original one of the laboratory. J . Now the inverse Lorentz transformation is to be carried out. to the primed system. transformed to the unprimed system. However. In the latter Eqs. t. H B = µH .
2 Einstein and Laub. Namely. Instead. which is uniformly moving at a speed v through the plate condenser of infinite extent at rest. however. some old some new.Minkowski's physical ideas were simple indeed. Cullwick (1959). homogeneous and isotropic dielectric of relative permittivity εr and relative permeability µr . Rosser (1964). that the work of Minkowski in the mathematical sense imposes too severe conditions before its reader. Einstein and Laub (1908a. where εr µr > 1. now following Einstein's "elementary path". one of the crucial experiments of the electrodynamics of bodies in slow motion. 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. were unusual. the Wilson-Wilson experiment Several months after the publication of the Minkowski paper. Consider an infinite slab made of a linear.7 Physicists were far away from acquiring the "tensorial mentality". When a potential difference is applied between the 28 . we find it useful to derive the fundamental equations in an elementary way which. 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). In the present chapter we shall deal with interpretations. of the Wilson-Wilson experiment. by using that theory we shall attempt in the following chapter to analyze a simple problem from a somewhat unusual perspective. Møller (1972). as is usually the case with special relativity. (Taking into account. 2. essentially corresponds to Minkowski's method. which was not at all the case with the mathematical "apparatus" he was using. their inferences. put the authors.) Einstein and Laub immediately applied the new tool of theoretical physics to an exotic system.b) derived the same fundamental electromagnetic equations for bodies in motion.
plates. steel balls "and each one was coated thinly with sealing-wax.8 If there existed. In order to check up the theory of Einstein and Laub. did not exist. The coated spheres were packed tightly and melted paraffin was poured into the empty spaces between them so as to form a solid mass. and if the plates are connected by a thin conducting wire. Such bodies. Lorentz's non-relativistic electron theory gives. that one should respect theoretical physics. D then a simple relationship between the electric displacement (D ) and the H magnetic field strength (H ) in the dielectric is obtained in the framework of Minkowski's theory of the first order in v/c. however. dielectric bodies with a considerable magnetic permeability. after reading the above Einstein and Laub's conclusion for the first time. filled the space between the plates of a cylindrical condenser. despite the fact that the system considered is impracticable. tending to infinity when √ v tends to v∗." This magnetic dielectric. (The present author still remembers that. parallel to the plates and perpendicular to the velocity of the dielectric slab. a magnetic dielectric "with considerable magnetic permeability" was created by Wilson and Wilson (1913). with εr = 6 and µr = 3. is applied to the system considered. even when it deals with such a kind of problems. appears occasionally in the laboratories. and 29 . Whatever does not exist in the nature. then it would be possible to choose experimentally between the theories of Lorentz and Minkowski. made of brass. where v∗ ≡ c/ εr µr denotes the velocity of of the electromagnetic waves in the magnetic dielectric when it is at rest. We remind our reader of the fact that the human race has learnt. Their recipe was as follows. the surface charge density on the plate which is at a higher potential is positive when v < v∗ and negative when v > v∗. Einstein and Laub wrote. he had experienced a certain frisson mystique. They used small 1/8 in. from 1905 until today. the length of the cylinder was 9 · 5 cm.) If a constant magnetic field.. however. a different relationship between D and H for the same system.
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. and 3 · 73 cm. The uniform translation of an infinite slab. was replaced by the uniform rotation of a long cylindrical tube made of magnetic insulator. 30 . In the experiment. From the viewpoint of Minkowski's theory.. i. the substitution is perfectly legal: arbitrarily small neighborhood of any rotating point of the dielectric is at rest in the corresponding local IFR. (Cullwick (1959) gave a detailed analysis of the Wilson-Wilson experiment. as it is usually euphemistically said. the results of the experiment eliminated Lorentz's theory. they do not depend on instantaneous acceleration (the clock hypothesis and the stick hypothesis.m. both the special and the general one. Namely. the potential difference between the plates of the condenser was measured.9 That was a triumph of both special relativity and Minkowski's phenomenological electrodynamics of moving media.) As it is well known. one of the fundamental assumptions of Einstein's theory of relativity. 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. in the axial magnetizing field of a coaxial solenoid. The condenser was uniformly rotated at a speed of about 6000 r.the inner and outer diameters of the solid dielectric tube were respectively 2 cm. is that both length contraction and time dilatation are determined only by the relativistic factor γ. e. cf Møller 1972).p.10 The reader has certainly noted that Wilson and Wilson. inaccessible to experimental verification. somewhat modified the original "experimental set-up" of Einstein and Laub.
the critics contested their starting fundamental physical assumptions [the use of an unacceptable coordinate system (Burrows). The final outcome of their analysis based on the assumed nature of a medium in motion differs from the result obtained by following the "elementary path" of Einstein and Laub. Ridgely (1999) analyzed in detail the constitutive equations for the polarization and magnetization in a uniformly rotating frame. Ridgely 1998). in the lab. p 545). Their "corrected analysis" borrowed from the general theory of relativity necessary tools for dealing with electrodynamics in an accelerated frame of reference. g. the constitutive equations inside the rotating cylinder have exactly the form predicted by the "simple" Minkowski's theory.2. a fundamental conflict between theory and experiment. 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. Since the experiment was consistent with predictions of Minkowski's theory (which is incorrect!) one has. 31 . None of the critics found an error in the PS calculation. starting from the corresponding constitutive equations for D and B in the Lorentz-covariant formulation (cf e. Griffiths 1999. transforming back to the laboratory frame he obtained that.3 Review of recent reexaminations of the classical interpretation of the Wilson-Wilson experiment The conventional interpretation of the Wilson-Wilson experiment was recently questioned by Pellegrini and Swift (1995). Pauli 1958. errors in defining basic physical quantities (e. Pellegrini and Swift (PS) argued that a correct analysis must take into account the fact that a rotating frame is not an inertial frame. g. p 103. the current density) in the rotating frame due to the problem of clock synchronization (Weber)]. claim Pellegrini and Swift. instead. Weber 1997. Several authors questioned the validity of the PS argument (Burrows 1997.
from relativity of simultaneity. which . the charge distribution stems. in its proper frame of reference S . a non-zero polarization given by. As is well known.12 The appearance of charges inside the current loop in uniform translation is a consequence of the relativistic transformation law for the charge density. there is another. 1993) pointed out.Krotkov et al (1999) gave a quite unexpected direction to the recent reexamination of the classical interpretation of the Wilson-Wilson experiment. (The appearance of electric dipole moment of a current loop in motion is. in the long run. arbitrarily small segment of the current loop is electrically neutral. where m denotes the magnetic dipole moment of the loop in its proper frame S . a magnetic dipole can be represented by a closed cone ducting loop with a stationary (conduction) current. Namely. consisting of atoms or molecules. in the framework of v first order theory. unknown in non-relativistic theories. For ordinary media. This relationship are obtained as a consequence of the Lorentz-covariance of Maxwell's equations (cf Rosser 1964). as measured in the lab. where v denotes the velocity of the considered point of the magnetic. however.) Due to the Lorentz contraction. n0 of those magnetic dipoles per 32 is usually derived by using relativistic transformations for the fields. any. microscopic approach. according to the classical. In that frame. as measured in the lab frame S. in the current loop that is now uniformly moving with velocity v there is a charge distribution over the loop and it possesses the corresponding electric dipole v moment p = (1/c2 )v × m . thus. A justification of the specificity of their analysis needs some introductory remarks.11 In the lab frame S. a purely relativistic phenomenon. Amp`rian model. the essential difference between Minkowski's and Lorentz's electrodynamics of moving media lies in the fact that only the former predicts that a magnetized medium in motion (with a non-zero magnetization M in the proper inertial frame of the magnetic S ) possesses. P = (1/c2 )v × M . as Rosser (1964.
This conclusion is reached by using only Maxwell's equations in the lab frame. i. Krotkov et al (1999)point out that neither the LI nor the PS approaches are applicable to this macroscopically inhomogeneous medium. v p since p = (1/c2 )v × m and P = n0 γp .13 Krotkov et al claim that the result can be generalized 33 v m the "relativistic" (1/c2 )v ×m term. and found. It is clear that the "microscopic approach". The authors analyzed a steel ball (a highly conductive and a highly permeable medium!) in uniform rotation about an axis outside the ball. in the quantum-mechanical operators. in the framework of the first order theory in v/c. Consequently.m3 in S takes the volume (1/γ)m3 as measured in S. based on the classical concepts. without the use of special relativity. in a constant external magnetic field B0 parallel to the rotation axis. is somewhat problematic. p and m could only be the average values of the corresponding . 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. or any assumption on physics in the ball's proper frame.to macro-quantities by averaging over physically infinitesimally small regions of space and time intervals). however. In the WilsonWilson experiment. and thus the contribution to the polarization in the S frame due to v the motion of the magnetic is given by the expression P = γ(1/c2 )v × M . all lie within the standard framework of the classical field theory. and the second is of the ball and v is the velocity of its centre as measured in the lab. where m is the magnetic dipole moment best case. the concentration of the corresponding electric dipoles in the S frame equals n0 γ. they all use the usual method of the theory of continuous media (the transition from micro. e. 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 v × B0 . "magnetic dielectric" was constructed of small steel balls of diameter about 3 mm embedded in the paraffin wax.
exist only in the steel balls. e. all models that take for granted Maxwell's equations lead inevitably to the LI results of Minkowski's theory. Their argument 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.) 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. An experiment with such a material has been recently performed by Hertzberg et al (2001). whose objective is to make a choice between several classical field theories. 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. The analysis made by Krotkov et al. claim Krotkov et al. the rotating magnetic insulator must be (i. regardless of the validity of their final conclusions. were for 6% different from the predictions of the PS theory (the relative error of their measurement was 1%). 34 . where electric conductivity is high. pointed out the essential fact that in experiments of the WilsonWilson type. Their "homogeneous" cylinder was made of yttriumiron-garnet "which is a magnetic insulator even on the molecular scale". In this way. Krotkov et al did not venture on finding the polarization and magnetization of the WilsonWilson magnetic dielectric as a function of the electric and magnetic dipole moments of the steel balls. (Needless to say. very convincingly consistent with the LI predictions of Minkowski's theory. in the case of the Wilson-Wilson experiment. should be) a microscopically homogeneous medium (we remind our reader that Rosser (1964) suggested this long time ago). Experimental results.to the magnetic dielectric from the Wilson-Wilson experiment.
faraway from the phenomena of everyday experience. At first sight it is a pseudo-problem. should be pointed out. this is not so. Another problem. i. at room velocities. e. Maxwell's equations for material media (the so-called material equations) are Lorentz-covariant. and so it seems that the problem is already solved.14 However. that is without Minkowski's theory. together with a mathematical application of special 35 tion. As it is picturesquely said. without or with special relativity. however. Before all.4 Electrodynamics of bodies in slow motion: with or without special relativity? It is a commonplace that relativistic effects disclose themselves only at speeds close to that of light. p 107) noted. We saw that a correct interpretation of the results of those experiments without special relativity. at small speeds relativistic effects may be ignored. Namely. seems to be obvious. 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. However. was not possible. Minkowski's theory.2.15 The question arises whether a non-relativistic analysis. of course. As Cullwick (1959. The answer to the query: electrodynamics of bodies in slow motion. Some difficulties. As is well known.q(E + v × B )+ the constitutive equations. is whether Minkowski's phenomenological relativistic electrodynamics is correct at all. is sufficient for a correct electrodynamic description of bodies in very . the phenomenological electrodynamics 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 electromagE netic field. in the framework of a first-order theory. the true arena of special relativity is the exotic kingdom of great speeds. based on the use of Galilei transformaslow motion. the query necessitates a certain explanation. and that in favour of special relativity. Einstein and Laub were not using electrodynamics of moving bodies but instead the electromagnetic theory of bodies at rest.
In the view of the present author. Minkowski's recipe. remember the fact (already pointed out by Rosser (1964)) that there are conceptual difficulties also in case of the electrodynamics of bodies at rest. has no sound experimental basis. the answer to the query depends on the nature of bodies. 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. their analysis deals with macroscopically inhomogeneous bodies. one could infer that there is no unambiguous answer to the above query. the conclusions reached by Krotkov et al (1999) are problematic. the consensus. One should. 36 . as far as the present author is aware. electrodynamics of bodies in slow motion does not necessitate special relativity. Taking into account the relatively complicated theory of the experiment by Hertzberg et al. 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. e. Howevere.relativity. On the basis of the considerations given in the preceding Section. however. contrary to the generally accepted opinion). the query is "shifted" in the sense of necessity of either special or general relativity. When magnetic dielectrics are discussed. According to Krotkov et al (1999). however. more precisely. Non-magnetic insulators in slow motion can be successfully described by using Lorentz's non-relativistic theory (Pauli 1958). it seems that in case of bodies in slow rotational motion. in the view of the present author (or. because macroscopic behaviour of a large number of micro-systems is deduced from the classical (macroscopic) ideas about the micro-systems. following 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. Although there seems to be a consensus that Minkowski's recipe is valid in case of a uniform translational motion of a body. In case of microscopically homogeneous (or inhomogeneous) bodies.
the motion being a pure translation. or a combination of the two motions. Bringuier 2004. a pure rotation. 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 .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. in case of a uniform translational motion of a conductor of arbitrary shape. Redˇi´ zc zc 2004c). seems to be indispensable for a correct derivation of basic inferences (cf Redˇi´ 2004b. that is Minkowski's electrodynamics. The present author recently pointed out that even at room velocities special relativity.
and not relativistic electrodynamics. pp 146–9).Notes [1] In the whole Einstein's paper.leads to asymmetries that do not appear to be inherent to the phenomena . regardless of the error pointed out by Bartocci and Mamone Capria (1991a. "opens the door to special relativity" (Redˇi´ 2004a).b) argued that in Einstein's example of the interaction between a magnet and a conductor it is classically interpreted Maxwell's electrodynamics. zc [2] The formulae apply to arbitrary monochromatic plane wave... The attempt contained a fatal flaw (Redˇi´ zc 2000). While Rosser (1993) questioned the validity of the interpretation of Maxwell's electrodynamics proposed by Bartocci and Mamone Capria... contrary to Einstein's statement. Einstein original example. in the view of the present author their conclusions concerning the interaction between a point charge and a current loop in relative motion are correct (cf Redˇi´ 1993). if properly understood. however. 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. b). zc necessitates a more detailed analysis than that given by Miller (1981. Many years after Bartocci and Mamone Capria (1991a." wrote Einstein. that predicts a perfect (and not only to second order in v/c) symmetry. 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 conductor.. This example has served to the author as an illustration for the thesis that "Maxwell's electrodynamics . The observed phenomenon in this case depends solely on the relative motion of the magnet and the conductor.as it is usually understood today . e. i. 38 . 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. to elliptic polarization." (Einstein 1905a).
.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 . qE +qv ×B . Recently. 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. The example of the light complex clearly shows that intuition is sometimes more important than knowledge. and only then will its appearance in the first act become understandable.argued Planck .[3] Lichtkomplex is a mysterious quantity in the framework of Maxwell's electrodynamics and its appearance in Einstein's paper is very strange. the quantity of light). now under a new name (Lichtmenge. It seems that physics has not until very recently said its last word about the light complex (Redˇi´ and Strnad 2004). by Einstein . Jefimenko's article is a natural complement of Planck's paper mentioned above (cf also French 1968. The "mysterious stranger" will appear on the stage just one more time (Einstein 1905b). While it seems that Kaufmann's latest measurements disprove the principle of relativity introduced recently by Lorentz and. Namely. and also that the role of logic in physical sciences is sometimes very tricky (Stachel and Torretti 1982). Rosser 1964). [5] It should be mentioned that Einstein obtained his electrodynamic results without knowing of tensors in Minkowski's space. Einstein reached the correct final result zc by making a methodological error. [4] Planck (1906) was the first to derive and recognize the well known general form of the relativistic equation of motion of a charged particle in the electromagnetic field ("the Lorentz force equation"). Einstein's tour de force can be adequately appreciated only by a researcher who trailed the same dangerous mountain path (cf Schwartz 1977. 39 . g. in a more general formulation. Møller 1972). Rosser 1960). Jefimenko (1996a) derived in Einstein's way the transformation law of the most renowned pure E v B relativistic tri-force. In the view of the present author. Of course.. the grandeur of that scientific exploit.
Here we sketch how one can reach these results which refer to the system shown in displacement D = ε0E + P . the application of the constitutive equation for the current density (the third equation (AS2))leads to a contradiction (cf Redˇi´ 2002. that in the general case Sommerfeld's remark does not apply. For example. and that of Lorentz's theory Dz = (εr − 1)µr vHy /c2 .11). For example. (33. instead of tensors of the first (quadri-vectors) and second rank. And." It should be pointed out. only the fixed value of v in the space-time point P. in case of an axially symmetric charged conducting body that is uniformly rotating about its symmetry axis. in the SI system of units. t enters in the following Lorentz's transformations. he works in a complex space (ict!) whose metric is Euclidean. reads (Rosser 1964) E v P = ε0 (εr − 1)(E + v × B ) + (1/c2 )v × M . A simple analysis reveals that in the magnetic's rest frame.[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. (2) where M is the magnetization in the proper frame of the magnetic. of course. In Minkowski's theory we start from the definition of the electric and the constitutive equation for the polarization of the "magnetic dielectric" which in the lab frame. Thus v need not be a pure translation and the body need not be rigid. (1) Figure 6. 2004b). [8] Einstein and Laub's result reads Dz = (εr µr − 1)vHy /c2 . zc [7] Minkowski's nomenclature is different from the present-day one. however. in the first-order theory. he speaks about space-time vectors of the first and second kind. 40 . For the quadri-gradient he uses a nowadays forgotten symbol lor. respectively. up to the second order terms in v/c.
(3) µ0 µr where B0 = µ0 Hyey is the magnetic flux density of the externally applied M = magnetic field in the lab frame. in both reference frames.) From equations (1). (2) and (3) we get v D = ε0 εrE + ε0 (εr µr − 1)v × B0 41 (4) . The electric field in the dielectric vanishes when the dielectric completely fills the space between the plates. one has B (µr − 1)B 0 µr . of course. a charge appears on the plates. If the condenser's plates are mutually connected by means of a stationary lead. in the first order theory. Eq. The magnetic dielectric in motion is also electrically polarized.and B . and taking into account that all relevant quantities are of the type v × B .z x H _ _ _ _ _ _ _ _ _ _ _ v v + + + + + + + + + + + v Figure 6 The slab made of magnetic dielectric and the condenser's plates all move with e constant velocity v = vex in a constant externally applied magnetic field whose magnetic flux density is B0 = µ0 Hyey . it is not difficult to verify that the magnetic flux density inside the magnetic dielectric equals B = B = µr B0 . (By the way. the result applies.fields. (3) is obtained by using the continuity of H and the relativistic transformation laws for the E .
where σf denotes the surface charge density over the lower plate of the condenser. E = 0.Since the condenser's plates are mutually connected by means of a stationary lead through sliding contacts. and thus in this case we have Dz = (εr µr − 1)vHy /c2 . in the framework of the first order theory. Rosser 1964). the potential difference between the plates is zero. P = ε0 (εr − 1)(E + v × B ). so that in the same "experimental situation" we have DzL = (εr − 1)µr vHy /c2 . otherwise very accurate Cullwick is wrong: namely. Cullwick claims that Einstein and Laub identified without justification the magnetic field strength in the dielectric H (H ) and the magnetic field strength of the externally applied magnetic field H0 . Fortunately. in 42 .) Both versions give the same results. 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. In this place. If the gap between the dielectric slab and the condenser's plates vanishes. the condenser's plates connected by a stationary lead are also at rest with respect to the lab. (This version is closer to the Wilson. this is an irrelevant second order effect. (5) Of course. however. it follows that the electric field inside the dielectric also vanishes. and since a stationary state is established. (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. Dz = σf . however. Interestingly. In Lorentz's theory. the constitutive equation for the polarization of the magnetic dielectric contains only the E first term on the right hand side of equation (2). Cullwick states.Wilson experiment where the cylindrical condenser rotates together with dielectric.
which for the Wilson-Wilson magnetic dielectric with εr = 6 and µr = 3 amounted to 0. [9] [9] The measured potential difference according to Einstein . pp 606-613. and consequently there is no contribution to the vector H due to the convection current of free surface charges. 2. Cullwick points out that there is no consensus in the literature about what is the solution of the problem according to Lorentz's theory. [11] The proper frame of a conducting loop is the reference frame in which crystal lattice of the loop is at rest. respectively. [10] As far as the present author is aware. by using Cullwick's (1959) "component field" method. Einstein-Laub's result (5) can be reached in another (the third one) way. 944 and 0. 83. 43 . This assumption is found in many textbooks and therefore necessitates a comment.Laub's theory is proportional to the factor (1 − 1/εr µr ) and according to Lorentz's theory to the factor (1 − 1/εr ). 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. has not succeeded in reaching the result (6) of Lorentz's theory by that alternative method. The average value of experimental results for that proportionality factor was 0. as "observed" in the proper frame of the lattice. 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. p 589. Bartocci and Mamone Capria 1991a. 96. however. e. 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). vol.Einstein and Laub the condenser's plates does not move.b). 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. to their concentrating towards the conductor axis (the "self-induced pinch-effect"). The present author.
mz ) in the proper inertial frame S in which the medium is at rest.Subsequent elaborations of the problem of finding the charge distribution in a conductor with a stationary current for more realistic models. an electric dipole moment. The magnetization M in S is given by M = n0m . p = (px . in case of a planar closed current filament in a uniform translational motion in the proper plane. as measured in S. Gabuzda 1993. identical for all of them. The crucial assumption was the Clausius postulate. by assumption. Rosser 1964. from the preceding equations one immediately finds structure of magnetic dipole. from the S to the S frame. without introducing any special assumption about the ity of the preceding elementary reasoning is worth mentioning: "According 44 . One starts from the transformation law. did not provide a clear answer to the question: in what inertial frame is a currentcarrying conductor electrically neutral (Peters 1985. up to all orders in v/c. Due to the Lorentz contraction. pz ) as measured in the lab frame S with respect to which the magnetic is uniformly moving with velocity v . n0 magnetic dipoles in m3 in S occupy the volume (1/γ) m3 . and for a circular loop (Rosser 1993). py . [12] This fundamental relation was exactly derived starting from the definition of the electric dipole moment. it is necessary to mention how the relation p = (1/c2 )v ×m is derived in the general case. Since the polarization in p S equals P = n0 γp . 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). Taking into account that the derivations are based on the classical model of v m magnetic dipole. Redˇi´ zc 1998). my . The problem will be also discussed in the next Chapter. of the polarization vector of a magnetic dielectric in v M motion: P = γ(1/c2 )v ×M . for a rectangular loop whose direction of motion is parallel to one of its arms (Panofsky and Phillips 1955. Blackford 1994). Rosser's (1993) remark concerning the validv p = (1/c2 )v × m . Each of the atomic magnetic dipoles possesses. Let the magnetic medium consists of n0 atomic magnetic dipoles per cubic meter. each of the same dipole moment m = (mx . where γ = (1 − v 2 /c2 )−1/2 .
" Very soon. then the second term in their equation (13) neither vanishes nor has a simple interpretation. 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. The experiment was planned as a measurement of the Aharonov-Casher b m v phase shift. If. 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. Namely. 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. [14] One of rare exceptions is an excellent textbook by A. whose author passionately protests (on p 259) against this oversimplification 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. which is . as Krotkov et al pointed out (1999): "The moving magnetic dipole was a magnetically polarized thallium fluoride molecule in a molecular beam that passed through a region of constant electric field E . which is (1/) a (m × E ) · (v /c2 )dt for a molecule traveling from just the interaction energy between the 'relativistic dipole' p and the electric field E . a proof has arisen that Rosser was right. which implies that a true expression for the electric dipole moment of the rotating sphere contains some additional terms. The integrand may be written as ((v /c2 )×m )·E . within an error of about 2%. P. French (1968). 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." [13] In the view of the present author. the motion of the sphere is uniform rotation. this prediction of special relativity was experimentally validated (Sangster et al 1995).to special relativity. however. the constitutive equations 45 v m E point a to point b.
can be obtained. However. isotropic medium in motion at low speeds read µr − 1 M = B + P × v. researchers from the beginning of 20th century had mainly "cavalierly" assumed that Lorentz's theory applies in the reference frame tied with the Earth. In relativistic electrodynamics." and can be obtained from the formulae of relativistic electrodynamics in the limit c −→ ∞ (of course. as is well known. in that theory. p 170). etc. Une fortunately. the idea is tempting of an electrodynamics that would be Galilei-covariant.for a linear. E J = σ(E + v × B ). as Le Bellac and L´vy-Leblond (1973) pointed out. Lorentz's theory understood in this way represents "electrodynamics of bodies in slow motion without special relativity. This assumption was also introduced in analysis of the Wilson-Wilson experiment (cf Cullwick 1959. True. as well as Maxwell's equations for material media in slow translational motion. Condensers don't work. µ0 µr E P = ε0 (εr − 1)(E + v × B ). 46 . pp 166-171). (∗) (∗∗) (∗ ∗ ∗) The constitutive equations. in case of the last limit only the constitutive equations are in question). Chapter 9). 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. that is in the lab. Lorentz's original theory was formulated with respect to the ether frame. through a non-relativistic reasoning from the corresponding equations that apply to media at rest (Panofsky and Phillips 1955. There is no light. there is no reference frame in which complete Maxwell's equations apply. In relation with the preceding considerations. as Miller (1981) pointed out. Such an electrodynamics does exist. the constitutive equations for bodies in v M slow motion differ from these given above by a relativistic term (1/c2 )v ×M in equation (∗∗) for the polarization.
1 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. Basic assumptions of Maxwell's theory will be explicitly given. Redˇi´ 1993).) 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. [In this interpretation. 0). and in the framework of relativistic electrodynamics. 0. the considered problem is a simple analogue of the famous Trouton-Noble experiment. Setup of the problem in the framework of Maxwell's theory. b. the theory of which. its natural habitat. respectively.] Our presentation closely follows that of Bartocci and Mamone Capria (1991a.b). y = R sin θ. 47 z=0 (1) . which seems to be one zc of the simplest in electrodynamics of moving bodies. Consider a filamentary circular current loop C with stationary current I which moves with respect to the IFR S with constant velocity v = (v. This problem. The loop C at the moment of time t = 0 is given by parametric equations x = R cos θ. is very tricky (cf Teukolsky 1996). contrary to that of the present problem.3 A problem in electrodynamics of slowly moving bodies: Maxwell's theory versus relativistic electrodynamics Setup of the problem 3. Bartocci and Mamone Capria 1991a. (The two methods of solving this and similar problems represent electrodynamics of slowly moving bodies without and with special relativity. will be solved in the laboratory reference frame in two ways: in the framework of classically interpreted Maxwell's electrodynamics (henceforth. Maxwell's theory).
in the standard notation.What is the force acting at that instant on a charge q which is at rest in the S frame at the point (0. to the inhomogeneous d'Alembert type equations for potentials Φ. A : Φ = −ρ/ε0 A A = −µ0j where and j must satisfy charge conservation j divj = − 48 ∂ ∂t (8) (6) (7) .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 B ∂B E curlE = − ∂t E ∂E B curlB = µ0 ε0 +j ∂t E divE = /ε0 B divB = 0 (2) (3) (4) (5) These equations reduce. L). as is shown in Figure 7? Figure 7 3. 0.
t) = r dx dy dz 4π | r − r | (12) (13) ["Remark: In equation (3) it would be natural to add a term σ0E . Although this does not affect directly our argument. z . t− | r − r | /c) r Φ(r . However. t) = dx dy dz 4πε0 | r − r | µ0 j (x . in a given inertial frame of reference S. c2 ∂t (c2 = 1/ε0 µ0 ) (9) (The electric and magnetic fields are expressed through the potentials by A ∂A .and the potentials A and Φ must satisfy the Lorenz1 gauge condition2 A divA = − the relations E = −gradΦ − b) The Lorentz force law A ∂A F = q −gradΦ − A + v × curlA ∂t (11) 1 ∂Φ . y . where σ0 is the vacuum conductivity. 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. t− | r − r | /c) A(r . z . namely the one given by the so-called retarded potentials3 1 (x . σ0 > 0) has been recently gaining adherents. Maxwell's theory presented above applies.. y . e. we could add another hypothesis 49 . (6) and (7) have a unique solution which is physically relevant. ∂t A B = curlA ) (10) c) An additional assumption that can be viewed as a restriction on the way fields "originate" from sources: we assume that for given and j ."4 ] It is not difficult to verify that charge conservation (8) is a sufficient condition for the retarded potentials Φ and A to satisfy the Lorenz gauge condition (9). it can be argued that the conventional presentday choice of putting σ0 = 0 is not experimentally so well established as it could be. by assumption.
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 electrically neutral. Then. however. a circular loop in motion with a stationary current I is not a clearly defined system. An essential detail. 50 . j ) and (Φ/c. A ) are contra-variant components of quadri-vectors of Minkowski's space-time. As Bartocci and Mamone Capria pointed out. is missing. 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. The additional hypothesis reads (c. 0 and j 0 . this assumption enables us to write Maxwell's equations in a Lorentz-covariant form. For our discussion we need to know for instance something about the electric field produced by a current.which would ensure the validity of the theory in all reference frames S linked to S by a Lorentz transformation. it should be specified that "a stationary current I" refers to the proper frame of the loop. 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. Introducing the hypothesis that both charge and lengths are preserved under motion. 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. (14) As is well known. Now we have almost all requisites necessary for solving our problem. First. devoid of any physical content. one should answer the question of what may be assumed in MT about the behaviour of a loop in motion. It should be stressed that (14) is a fundamental physical assumption which is logically independent from the previous (2)-(13). Namely. this is the way relativistic electrodynamics (RED) is obtained. In this sense MT is formally covariant with respect to the Lorentz transformations.
t) = 0 (x − vt. y. ∂t 4π D3 0 51 (22) . e. z. using the continuity equation and the CP we have (x. z. z. t) = j 0 (x − vt. y. y. t) v j (x. Taking into account that a stationary current is considered. z). D3 0 and for its electric field E the expression A ∂A µ0 IvR 2π (x − vt − R cos θ)(− sin θ. y. The electric potential Φ obviously vanishes. 0) dθ D (19) and β ≡ v/c. 0) E =− =− dθ. cos θ. z). we now only have to evaluate the retarded potentials (12) and (13). z. (15) (16) where the notation is adapted to the present problem (the loop is moving along the positive x-axis). y. 3 4π D D3 0 0 (21) 2π (x − vt − R cos θ) cos θ + (1 − β 2 )(y − R sin θ) sin θ − dθ . (1 − β )z dθ. i. whereas for the vector potential A we have µ0 IR A= 4π where 2π 0 (− sin θ. z. y. y. z. y. t) = 0 (x − vt. in accord with our definition of MT. t) = j 0 (x − vt. (20) field B of our current loop C in motion 2π 2π µ0 IR cos θ sin θ 2 2 A B = curlA = 1−β z dθ. z. z) = 0 j (x. y. (17) (18) One can easily verify that and j satisfy the continuity equation if the same applies to the corresponding rest densities 0 and j 0 .7 From (19) we get the following expression for the magnetic D2 = (x − vt − R cos θ)2 + 1 − β 2 (y − R sin θ)2 + z 2 . y. t) + 0 (x − vt. that j 0 = j 0 (x. cos θ.according to the Galilean law of composition of velocities (Redˇi´ 1993) zc (x. t)v .
y=0. at any instant) we get F ∗ = −q A ∂A v + qv × B ≈ − µ0 qIvR2 /4(R2 + L2 )3/2 ey . In case the point charge q moves with the same velocity as the loop C. in the same approximation. ∂t (26) again up to the second order terms in β. L) is given by the expression F = −q = A ∂A ∂t µ0 qIvR2 4π x=0. y = 0.0. in the rest frame of the loop Srf . 3/2 e B 0. so that its trajectory is given by x = vt.8 3. cos θ. of course.L.0 = (µ0 IR2 /2 R2 + L2 )ex at the same point.3 Solution in the framework of relativistic electrodynamics We now obviously have to reformulate the problem: there is a stationary current I. which coincides with the exact expression for B of the current loop at rest. Following 52 . z=L (25) for the force acting on q by the electromagnetic field of the loop at the instant t = 0 (and.z=L. (24) cos θ (− sin θ. 0) 3/2 dθ R2 + L2 − β 2 R2 sin2 θ + L2 (23) One can easily verify that. 0.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. and the loop is circular.t=0 2π 0 Neglecting terms of second and higher orders in β in a series expansion of the integrand in (23) we have 3/2 e F ≈ (µ0 qIvR2 /4 R2 + L2 )e y .
while according to the CP each segment of the loop is electrically neutral in the Srf frame.] The scalar potential ΦR is readily obtained on the basis of the hypothesis (14). (29) to all the orders in β (Rosser 1993).] In the case of our filamentary circular loop of radius R. up to the second order terms in β. as measured in the S frame. for the vector potential of the electromagnetic field due to the moving loop in the S frame we obtain µ0 IR AR = 4π where 2 DR = (x − vt − R 2π (− sin θ. obviously. with stationary current I. Finally. [As is well known. then one applies the transformation law). there is no difference between the predictions of RED and MT. [Since the magnetic dipole moment of e the loop in the Srf frame is m = IπR2e z . mea sured of course in the Srf frame. a rather simple calculation reveals that the electric dipole moment of the loop in the S frame equals e p = −e y vIπR2 /c2 . 53 . taking into account that in the Srf frame ΦR vanishes (CP): ΦR = vARx . (28) relativistic transformation law of charge and current densities implies that there is a charge distribution in the current loop in motion. this is not so. 0 1 − β 2 cos θ. However. the 1 − β 2 cos θ)2 + 1 − β 2 (y − R sin θ)2 + z 2 . DR (27) and the subscript R indicates that the solution belongs to relativistic electrodynamics. Namely. Comparing equations (27) and (19) one could infer that. in this case the crucial relation p = v × m /c2 is valid exactly. and since v = ve x . the standard procedure (first one evaluates the potential A R in the Srf frame. of the relativity of simultaneity (Rosser 1964). 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. passing details. 0) dθ.
Assuming the validity of the CP. L) is. Namely. equations (31) and (26).t=0 ≈ ∂t 3/2 e ≈ (µ0 qIvR2 /2 R2 + L2 )e y .y=0. the force on q exactly vanishes.9. in RED. i. this divergence in predictions arises from the following two reasons. unknown in MT. and the predictions of the two theories would coincide at low speeds. e. (32) there is a term vjx /c2 .z=L. so that its trajectory is given by equation (25). As can be seen. of the relativity of simultaneity (Rosser 1964).) The second. which is used in both theories.10 In case q moves with the same velocity as the loop. if it were = 0 in the Srf frame. for all reasonable values of drift velocity of current carriers in Srf . not less important reason is the Clausius postulate. Comparing the corresponding equations (30) and (24). (As is pointed out above. given by the expression F R = −q( A ∂A R + grad ΦR )x=0. (26) and (30) are correct up to the second order quantities in β. 0. The first. in the transformation law of the charge density = γ + vjx /c2 ≈ + vjx /c2 . in the long run. (31) which immediately follows from the fact that in the Srf frame the corre- sponding force vanishes. we come to a conclusion that even at extremely low speeds the predictions of RED and MT are essentially different.the force acting by the electromagnetic field of the moving loop on the charge q which is at rest at (0. F ∗ R = 0. our example reveals that in the 54 . according to RED. (30) up to the second order terms in β. the ap- pearance of that term is a consequence. then the relation = would also apply in RED up to the second order terms in β. and from the relativistic transformation law of the Lorentz force. whereas equation (30) is exact. note that equations (24).
because one can repeat the observations for various choices of that plane. this result applies under the proviso that the velocity v of the Earth with respect to the ether is parallel v to the plane of the loop. compared to c. obtaining a maximum effect when this velocity lies in the plane .) "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. e where ψ denotes the acute angle between v and the plane of the cicuit. where B is the magnetic flux density at the centre of the loop due to the current in the loop. by increasing I and q we might be able to observe an effect even if the velocity of the laboratory is very small. However.11 (The force has the same unit vector as qv × B .. however. 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. as presumably it is. 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. predicts that there exists a force −(µ0 qIv/4R)e y on the charge q. According to RED. It seems that the preceding considerations are only of academic interest.general case relativistic effects must not be ignored even at "room velocities" of macroscopic systems in translational motion. as equation (26) reveals. e Maxwell's theory. as Bartocci and Mamone Capria (1991a) pointed out. In this case the problem we discussed above suggests a new experimentum crucis discriminating between RED and MT.. Moreover. The possibility that the plane of the circuit does not contain the 'absolute' velocity makes no harm. Namely. (26) and (31) are presumably inaccessible to experimental verification. equations (24) and (30) i. and to other sources of systematic errors. (−e y ) 55 . e." (A simple analysis reveals that in the general case the force on e the stationary charge q at the centre of the loop equals −(µ0 qIv/4R)e y cos ψ.
4 Experiments The suggested crucial Kennard-Marinov experiment. have been published (Edwards et al (1976). Edwards et al (1976) found that there is a nonzero electric potential due to a stationary current in a closed superconducting coil.B is the unit vector of v ×B . 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). led by a "non-obvious suggestion" that magnitude of charge of current carriers is proportional to v 2 /c2 . the Clausius postulate.) Bartocci and Mamone Capria proposed to call a possible experiment whose idea was presented above the Kennard-Marinov enthusiasts in the field of classical electrodynamics. has never been performed. as well as the fact that the quest for the second-order effects is in question.14 Some experimental results. passionate adherents of Maxwell's original theory.13 experiment. the observed phenomenon being a consequence of the piezoelectric effect in the teflon isolation of the superconducting coil used in experiments. as well as various fundamental theoretical conceptions related with it. intending thus to remember the name and the work of the two 3. The researchers observed just the dependence they were expecting. as far as the present author is aware. depending on the square of the current intensity. Sansbury (1985)) disproving the key assumption in the preceding analyses.15 have presumably all been made fruitless by recent experiments of Shishkin et al (2002) which established that there is no Edwards potential. any exclusiveness when reaching conclusions would be irrational. The subsequent attempts to explain the Edwards potential. however. where v is the carriers' speed. 56 . In addition. Taking into account delicacy of the interplay between theory and experiment ("experiment is theory of theory" (Popper 1982)).12.
Kilmister. and also O'Rahilly (1965)). and the existence of nonzero and nonsingular solutions of the homogeneous wave equation we quote from B. Perusal of the most recent literature reveals that the Lorentz gauge is mainly replaced by the Lorenz gauge. a long-lasting injustice toward the true author of that gauge condition.]. the potentials A and Φ that satisfy the Lorenz gauge condition are not unique. pp 549-550: "How is one to interpret such a solution of Maxwell's equations? There are no singularities . [. hypotheses about the way fields "originate" from sources. Plumpton and C. H. As far as the problem of the sources is concerned. [2] As is well known. W." [3] By the way.. But we do not know how to modify the theory so as to rectify this defect. ∂t A −→ A 0 = A + gradH one could get another solution of equations (6). C. 3 Maxwell's Equations and Their Consequences (Pergamon Press. Thus. no sources of the field anywhere or at any time. Maxwell's equations are essentially Heaviside's (Lorrain et al 2000.Notes [1] This is not a typographical error. The existence of this kind of solution of Maxwell's equations suggests that Maxwell's theory may be incomplete. 57 . Vol. was corrected (cf Rohrlich (2002). By making a gauge transformation Φ −→ Φ0 = Φ − ∂H . Ludwig Lorenz. For example. Elementary Electromagnetic Theory. also this standard choice depends on the acceptance of other "neutral". pp 486-7). and the lack of the physical relevance of the "anticipated" potentials. from the point of view of the present consideration. the behaviour at infinity of the fields. Chirgwin. (7) and (9) furnishing the same fields E and B when H = 0. Scientific terminology is unfair occasionally. 1973)..that is. It seems to lack some additional restriction in order that fields originate only from sources like charges and magnets.
we present here a more complete variant. [7] A proof of equation (19) is based on the formal covariance of MT with respect to the Lorentz transformations. Monti. Az = Az .[4] The hypothesis σ0 > 0 has been recently revived by R." Compare note [11] in the preceding chapter. "The electric conductivity of background space. with an obvious historical relevance. Vol. and the primed and unprimed coordinates are related by the standard Lorentz transformation x = γ (x − vt) . to analyze some of the implications of the relativistic assumptions. Ay = Ay . II. for us MT is mainly a tool. As Bartocci and Mamone Capria (1991a) give only a sketch of the proof. for instance. Having in mind that (Φ/c. . but our aim here is to show some consequences of MT in its most "classical" interpretation. For details. 58 t = γ t − vx/c2 . and the last section of this chapter. Gdansk 1987 (World Scientific. [5] See. who has also shown its important large-scale consequences. p 589." in Problems in Quantum Physics. [6] Of course we might alternatively introduce some form of the FitzGeraldLorentz contraction hypothesis. for instance. A ) formally looking are contra-variant components of a quadri-vector. 1988) or Vigier (1990). O'Rahilly (1965). Panofsky and Phillips 1955). R. one has Ax = γAx . Clausius stated that "a closed constant current in a stationary conductor exerts no force on stationary electricity. z = z. where γ ≡ (1−β 2 )−1/2 . y = y. see. Monti. 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. and using the fact that on the basis of the CP Φ = 0.
QED [8] One would expect that E + v × B = 0 at the considered point. tret ) 3 dr1 | r − r 1 | and our final goal is to find the vector potential A . y. j z = jz . A possible prescription is due to Maxwell: 59 . ζ = z. the errors are motivated only by the economy of writing. t ) = 4π r j (r 1 . t ) = 4π −∞ {(x /γ − ξ )2 + (1/γ 2 ) [(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. However. it is obvious from obvious how to find the electric and magnetic fields E and B in the Srf equation (26) that E + v × B = 0 in MT. and also µ0 r A (r . z1 = ζ we have µ0 +∞ j 0 (ξ . but lead to the correct result.Also jx = γjx0 (x − vt. η . y . at the instant t = 0. z) = γjx0 (x /γ. since in the so-called Galilean limit of RED the electric field in the Srf frame E = E + v × B and equals zero. and (19) is reached by an elementary calculation. Since µ0 r A (r . We point out that it is not very frame in the framework of MT. using (17) and (18). e j 0 = Iδ(ρ − R)δ(z)e θ η = ρ sin θ. In these coordinates dξ dη dζ = ρdρdθdz. Introducing new variables x1 /γ = ξ . we can put A = A . j y = jy . y1 . z ). t ) = 4π j 0 (x1 /γ. z1 )dx1 dy1 dz1 [(x − x1 )2 + (y − y1 )2 + (z − z1 )2 ]1/2 The last two equations contain deliberate errors (the factor of γ!). ζ )dξ dη dζ r A (r . y1 = η .
in Maxwell. the Faraday-Neumann-Lenz law of electromagnetic induction is Galilei-invariant in the above. y. vy .] in all phenomena relating to closed (emphasis added by D. and that now one has Φ (x . ∂t V B V v A that is E +V ×B = E +V × B + grad (v ·A ). t) = Φ(x. Essentially. finding of A and Φ can be a cumbersome task. z. the meaning of the symbols we used is obvious. and let V be the instantaneous velocity of the same charge with respect to an inertial frame S which is moving with velocity v = (vx .. however. z = z − vz t we have V = V + v . interpretation. y = y − vy t. vz ) relatively to S. R. 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.) circuits . Maxwell (1891). Maxwell's interpretation of this result is very interesting: and the currents in them.. z . y . vol. y . that Maxwell's prescription makes it possible to find A and Φ only when A and Φ are already known. p 601). y. z . Formally. of course. It is not difficult to verify A A ∂A ∂A v A − − gradΦ = − − grad Φ + (v · grad )A ∂t ∂t and consequently − A ∂A A − gradΦ + V × curlA = ∂t A ∂A v − − grad Φ + V × curlA + grad (v · A ). in the ether frame). z. 2. On the basis of the Galilei transformation x = x − vx t. Maxwell is wrong here since in the S frame his E and B fields do not satisfy Maxwell's equations. t) = A (x. it is indifferent whether the axes to which we refer the system be at rest or in motion. It should be stressed. We take that A (x . t).Let V be the instantaneous velocity of a charge q with respect to the laboratory frame S (that is. As can be seen from the example of equation (19). 60 "[." that is. Maxwell's. t) where r = r − v t.
and can be found in the immediate vicinity of the axis by applying Stokes' theorem. The partial time derivative can be expressed through A A the partial derivative over the x coordinate. pp 32-33. more beauA A v A convective (Eulerian) derivative of A R . can be eval- . within the considered approximation. dA R /dt = ∂A R /∂t + (v · grad)A R . −gradΦR . can be solved in another. i. z. one has A R (x. ∂A R /∂t on the z-axis at the instant t = 0 in RED. t) = A R (x − vt. the electric field of an electrostatic (in its proper frame) charge distribution that is uniformly moving at speed v equals. tiful method of determining the solenoidal component of the electric field of the loop in motion was proposed by Rosser (1993). 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 Poisson equation. up to the second order terms in β. on the basis of the transformation law. y.] uated directly. with respect to the axis of the circular loop. and A R at any point of space has only azimuthal component (the symmetry!. cf Panofsky and Phillips 1955. Jammer 1961. The vanishing of the convective derivative of the quantities describing an electromagnetic system in uniform translation. was used by Heaviside (1889. stationary in its proper reference frame. e. as is well known (Panofsky and Phillips 1955). Namely. [A historical remark is in order. 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). ∂A R /∂t = −v∂A R /∂x. 1892). On the other hand. z). Miller 1981. y. the Coulomb field of the 61 The irrotational component of the electric field. This alternative. less formal way based on intuition and symmetry arguments. the method was recently "rediscovered" by Dmitriyev (2002). of course). since the obviously vanishes in the problem we consider. up to the second order quantities in β.A [9] The problem of finding the vector potential A R . and also by Lorentz (1895) in his Versuch. because one has exactly ΦR = vARx .
62 . 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. [It is not difficult to verify that the contribution of the vector potential to the electric field due to a charge distribution in uniform translation (this potential arises from the corresponding convection current) is a second order quantity in β. exists on the current loop in motion contributes to the irrotational component of the electric field of the loop. where ΦR = vARx . Lorrain et al 1988) would be of no use here. 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 Srf frame for that distribution). −gradΦR . This is the true meaning of Rosser's (1993) statement that when evaluating the irrotational component of the electric field of the moving current loop retardation effects may be ignored. and which "exists" in the Srf frame too. Griffiths and Heald 1991. according to RED. the real charge distri- while efficacious.same charge distribution that would be at rest in the instantaneous position of the considered charge distribution in motion. In the view of the present author. Incidentally. A R ≈ A R . (As is mentioned above. through the corresponding Coulomb field.) The result reached in this way coincides with that evaluated directly.] Having in mind that the Lorentz contraction is a second order effect. and thus negligible. The alternative method described above of finding −gradΦR was also proposed by Rosser (1993). is conceptually tricky since it necessitates some non-obvious bution on the current loop in motion vanishes in the Srf frame. −gradΦR . this method. in a simple way. steps (not even mentioned by Rosser). For example. it seems that the general form of the time-dependent Coulomb law (Jefimenko 1989. we come to a conclusion that the charge distribution which. the convection current arising from the charge distribution in motion may be ignored.
since q is stationary at (0. a medium in the interaction. The explanation is conventional: the electromagnetic interaction between the current loop and the point charge is not a direct one. L). (The situation is much more pleasing with illustrations for the angular momentum of the electromagnetic field. in MT (in its original. of course). the electromagnetic field. Cf Griffiths (1989) and references therein. it seems that the principle of action and reaction is not satisfied. The preceding conclusion contradicts Rosser's (1993) statement that the force vanishes. "ether" variant. with a charge q and with a mass m. agreed that in his original paper a mistake was made. so that the total linear momentum of the system the current loop + the point charge + the electromagnetic field is conserved (cf Tamm 1979). Namely. where u is the instantaneous velocity of the particle. 0. where E pf = ε0 (E × B ) dV is the linear momentum of the total electromagnetic field. in the electromagnetic field. Simple calculations seem to be reserved for exotic systems (cf Butoli 1989).) [11] It is of some interest here to answer the question of what is the equation of motion of a charged point. Unfortunately. and assume that 63 E would be natural to take the conventional form d(m0u )/dt = q(E + u × . and that the force is indeed given by the above expression. It B ).) 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). and the charge distribution over the loop in motion is given by a relatively simple expression within the considered approximation. March 2003. which also possesses a momentum. which is obtained by applying the Coulomb law.[10] The force by which the field of the charge q is acting on the loop in e motion at the instant t = 0 equals (−µ0 qIvR2 /4(R2 + L2 )3/2 )e y . it is not possible to verify by a direct calculation whether in this example the equation p dpf /dt = −µ0 qIvR2 /4(R2 + L2 )3/2 e y applies at the instant t = 0. there is a third "body". (Professor Geraint Rosser in a letter to the present author of 20. as it is well known.
lead to discovery of special relativity. However. was also 64 . (A) by making the same assumption on the mass. that are moving with respect to the ether. is a timeindependent Lorentz-scalar.) By using the identity e d du deu 3 (m0u γu ) = γu m0 eu + γu m0 u . up to second order terms in βu (the last equation is exact for u = 0). (This postulate might. for the problem discussed here it is irrelevant which one of the two equations of motion we use. The present author feels that it is indispensable to point out the fact neglected by Bartocci and Mamone Capria (1991a. In this way. (This neglected fact was recently pointed out by the present author (Redˇi´ 2002). together with the "spatial" equation of motion (A) imply the "zeroth" component E d(m0 c2 γu )/dt = qE · u . one could postulate that in the ether frame the equation of motion has the form E u d(m0u γu )/dt = qE + qu × B . Thus. in the long run. dt dt dt where eu = u /u. 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. of the corresponding quadri-vectorial equation of motion. as Bell (1987) pointed out.) As can be seen.b): the here discussed problem of the electrodynamical interaction of the charge and the current loop at relative rest. appealing to Kaufmann's experiments (cf Miller 1981). By the way. the quadrivectorial "Lorentz force equation" is tantamount to equation (A) complemented by the assumption that m0 is a time-independent Lorentz-scalar. one obviously has d(m0u γu )/dt = d(m0u )/dt. the fundamental zc assumption on the time-independence of the Lorentz-scalar m0 .the mass of the particle m0 is time-independent. m0 . [12] It happens sometimes that ideas and discoveries of "old" physicists sink into oblivion.
1912) (these references are given in Miller (1981). physical effects depend only on the relative motion between ponderable bodies and on their mutual relative positions. is in classical mechanics tantamount to the principle of relativity ("identical systems in any two inertial frames behave in the same way under the same initial conditions"). in a certain sense. FitzGerald. K.] The "old" physicists instinctively applied the powerful principle of relativity to MT. Arnold. One recognizes here the principle of relative motion which. Weinstein (Springer. Mathematical Methods of Classical Mechanics. Lorentz. Somewhat unexpectedly. 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. New York. transl. the principle of slow relative motion. MT and RED were analyzed in detail by Bartocci and Mamone Capria (1991a). The basic idea of the "old" physicists was simple: "it is highly improbable that anything depends on the absolute motion" (FitzGerald 1882). and represents another illustration of Wigner's statement that sometimes intuition is more important than knowledge. They have shown that the principle of relative motion does not apply in MT nor in RED. with what we think today to be the correct solution. an exception is V. see Miller (1981)). [The equivalence of the two principles is presumably the reason for ignoring the principle of relative motion in textbooks devoted to classical mechanics.the topic of discussions among the physicists in late 19th century (Budde. their solution to the problem coincides. Relationships among the principle of relative motion. 1978) p 10. 1904. together with the assumption of the validity of Galilei transformation. 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 . however. the corresponding principle of slow relative motion. I. Vogtmann and A. applies in RED but not in MT. that is on the motion with respect to the ether.
the loop and the point charge. 33. now in the context of a different "philosophy" (Bartocci et al 2001).another experiment and its significance as evidence for the existence of the aether. by c analyzing the classical illustration of the relationship between electromagnetism and special relativity. "On unipolar induction . it seems that Bartocci and Mamone Capria should not have ignored Budde. up to second order terms in β. where J r 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. The Thorny Way of Truth. "Philos. For example. Purcell 1985. introducing an ad hoc assumption on the Lorentz-contraction of the distance between electrons-current carriers in the laboratory reference frame. as the authors cautiously mention. and perhaps most completely French 1968). Having in mind the preceding note. reads i = v · J r /c2 . and also Stefan Marinov.potential. Kennard. Ugarov 1979. 1982-1991. Graz. [13] Cf. Ivezi´ (1990) attempted to explain the Edwards I 2 . FitzGerald. 179-190 (1917). Mag. International Publishers "East-West". [15] Numerous references are given in a paper by Shishkin et al (2002). an infinite straight cylindrical conductor with a stationary current (Feynman et al 1964. Their result for density of charges induced on the loop. after the publication of his speculation that Maxwellians were on the threshold of a discovery of special relativity (Redˇi´ zc 2004a). IIX. [14] A variant of that experiment has been realized. the true authors of the proposed experimentum crucis. E. Lorentz. preliminary experimental results indicate a violation of the local Lorentz-invariance. It is difficult to discuss the validity of Ivezi´'s c attempt because of the obviously didactical nature of the considered model 66 denotes the conduction current density in the proper frame of the loop and . H.
A discussion on this topic. the present author agrees with Bartlett and Edwards (1990) that what Ivezi´ considers a fatal defect of the standard relativistic electrodynamics c (Lorentz . On the other c hand.(an infinite one-dimensional system). 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 stationary current. in which also some other authors took part.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. and thus annuls indirectly Ivezi´'s assumption. Gabuzda 1993. 67 . Redˇi´ zc 1998). without reaching some new essential conclusion.) On the one hand. lasted some time in the same journal. (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.
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