SYMMETRIES AND ASYMMETRIES

IN CLASSICAL AND RELATIVISTIC

ELECTRODYNAMICS
 



 
 

(U. Bartocci - M. Mamone Capria)


 






1 - Introduction
 
 

As is well known, Einstein special relativity (SR) is built upon two postulates: the principle of relativity (PR), and the invariance of light velocity. However, by making some "natural" assumptions on the isotropy and the homogeneity of space and time, it is possible to obtain the second postulate from the first one, at least if one admits that the speed of light is the least upper bound of the physically possible speeds of a material body(1).

So the really important postulate is PR, although this seems to be not sufficiently appreciated by anti-relativists who readily accept it while being critical of the second one (by the way, if one accepts the validity of Maxwell's electromagnetism then the invariance of light velocity directly follows from Maxwell's equations plus PR).

The relativity principle only seems natural when we forget about the peculiarity of asserting "no matter what is your reference frame" for optical phenomena: in principle, it would instead be reasonable to claim that light does have a preferred reference frame (like sound), and of course this has been historically the original approach to the question.

What evidence led Einstein to the contrary viewpoint? Apart from the "unsuccessful attempts to discover any motion of the earth relatively to the 'light medium'", so sparingly referred to in the introductory lines of his 1905 relativity paper, Einstein laid the main emphasis on the following induction experiment:
 
 

"Consider, for example, the reciprocal electrodynamic interaction of a magnet and a conductor. The observable phenomenon here depends only on the relative motion of the conductor and the magnet, whereas the customary conception draws a sharp distinction between the two cases in which either the one or the other of these bodies is in motion"(2).
 
 

We want to analyse the relationship between this kind of experimental evidence and the principle of relativity; as we shall see, it is by no means an obvious one.
 
 
 
 

2 - Principle of Relative Motion versus Principle of Relativity
 
 

To start with, we might tentatively state a principle of relative motion (PRM) in the following terms:
 
 

"In an inertial frame R(Oxyzt), the forces acting on the bodies of an isolated system depend only on the relative velocities and positions".
 
 

What is meant by this is that to compute the forces acting on a given system we do not need to know the velocity field, but just all the differences between the velocity vectors. Unfortunately this statement is not "neutral" with respect to a choice between different physical theories; in fact in SR the difference between two velocities is not necessarily a possible (physical) velocity, so it is not reasonable to require that these differences play such an important role. Still worse, we have the unpleasant phenomenon that while in classical mechanics the sum of the velocity field of a physical system with a fixed velocity is equal to the velocity field of the same system in another reference frame, this need not be true in SR, even if the norms of the resulting vectors never exceed the light speed. Thus, in view of a comparison as to the validity of a PRM between Maxwell's electrodynamics in the classical, aether-based interpretation (what we shall call Maxwell theory: MT), and relativistic electrodynamics (RED) we must modify the given formulation of PRM. A more convenient one is:
 
 

"Let an isolated physical system S , which can be thought of as composed of two subsystems S1, S2 , be given in an 'admissible' reference frame R , where S1 is at rest and S2 is moving with constant velocity v; then the forces acting on S as computed in R are the same if we interchange the roles of S1 and S2 so that S2 is at rest and S1 is moving with velocity -v".
 
 

An "admissible" reference frame is one for which the theory to be tested can be supposed to hold true (in MT only the aether frames are admissible, in RED all the inertial frames).
 
 

As for the principle of relativity we can formulate it briefly as follows:
 
 

"In any two inertial frames R and R' identical systems behave in the same way".
 
 

Here an "inertial frame" is any reference frame moving with constant velocity with respect to an admissible frame.

More explicitly, suppose that the state of a certain physical system is defined by means of s functions fi(x1,...,xr,t), i=1,...,s, where the xj's , j=1,..r, are generalized (Lagrangian) coordinates and t is the time parameter; PR asserts that if the values of f1,...,fs and their derivatives are assigned for a fixed t , and if (force) laws exist by which the fi's can be determined in some time interval, then, as long as the data are the same, the same functions are obtained in different inertial frames.

What is the relationship between PRM and PR? PRM says something about the way physical laws must look to be valid in some reference frame; PR says that physical laws must be so expressed that they are valid in all inertial frames. It is clear that PRM and PR are logically independent, unless we are told how to transform the basic physical quantities from one inertial system to another.

In classical mechanics the Galilei transformation bridges the gap between PR and PRM in a simple way: PR implies PRM.

In fact in order to see the subsystem at rest, say S1, as moving with velocity -v, and S2 as being at rest, one has only to transfer to an inertial frame having velocity v with respect to R. In this case we can also recover our original formulation of PRM , which happens to be equivalent to PR; this is why textbooks on mechanics often fail to state explicitly PRM(3).
 
 

(Rather surprisingly even textbooks on relativity as a rule omit to discuss, or indeed to formulate, PRM(4)).
 
 

Thus the kind of experimental evidence indicated by Einstein in the quotation above is exactly what we should expect if we believe classical physics to be correct. Of course the well-known trouble is that "classical physics" in this sense does not include Maxwell's theory!
 
 

We shall show that:
 
 

- MT is inconsistent with PRM, even if we weaken PRM by allowing that discrepancies of second order or more in b = v/c , where v = çvç , can be neglected (this is what we call the low velocity principle of relative motion: LPRM);
 
 

- RED is also inconsistent with PRM, but it implies LPRM.
 
 

As a consequence, even for low velocities MT provides different predictions from RED, and this makes it possible to devise "crucial" experiments which are definitely not optical (we shall discuss such an experiment at the end of this paper). This fact is at variance with what seems to be a rather widespread belief, namely that SR makes a difference ONLY for "high" speeds (i.e. close to c)(5). Our claims about the contrast between classical electrodynamics and SR may look strange at first, because textbooks often insist on the first being "naturally" relativistic. It may be true that "within the Galilean framework, Maxwell's theory was a rather unnatural and complicated construct", while "within relativity [...] it is one of the two or three simplest possible theories of a field of force"(6), but it should be made clear that the difference is not one of aesthetic taste; on the contrary, the difference is between two scientific theories which make different empirical predictions.
 
 
 
 

3- Maxwell Theory and Relativistic Electrodynamics
 
 

We shall strictly mean by Maxwell theory (MT)(7):
 
 

- the system of the four Maxwell equations:
 
 

(1) curl(E) = -B/t
 
 

(2) curl(B) = m0(e0 E/t + j)
 
 

(3) div(E) = r/e0
 
 

(4) div(B) = 0 ,
 
 

which reduce, in standard notation, to the DAlembert wave equations for the potentials F , A:
 
 

(5) F = -r/e0 ,
 
 

(6) A = -m0j ,
 
 

where the charge density r(x,y,z,t) and the density current j(x,y,z,t) must satisfy the charge continuity equation:
 
 

(7) div(j) = -¶r/¶t ,
 
 

and the potentials F , A are linked by the Lorentz gauge condition:
 
 

(8) div(A) = -c-2¶F/t (where: c-2 = e0m0) ;
 
 

the electric and magnetic fields E, B are expressed in terms of the potentials by the relations:
 
 

(9) E = -ÑF - A/t ; B = curl(A) ;
 
 

- the Lorentz force law:
 
 

(10) F = q(-Ñ F - A/ t + v´ curl(A)) ;
 
 

- a further assumption which can be viewed as a restriction on the way fields originate from sources(8): we assume that for a given r and j, (5) and (6) have a unique solution which is physically relevant, namely the one given by the so called Liénard-Wiechert retarded potentials:
 
 

(11) 4pe0F(x,y,z,t) = INT [r-1r (x,y,z,t-r/c)dxdydz] ,
 
 

(12) 4pA(x,y,z,t) = m0*INT [r-1j(x,y,z,t-r/c)dxdydz]
 
 

(where: r(x,y,z,t;x,y,z,t) = [(x-x)2+(y-y)2+(z-z)2]1/2).
 
 
 
 

Remark - In equation (2) it would be natural to add a term s0E [s0 = sigma zero], where s0 is the vacuum conductivity; it can be argued that the conventional present-day choice of putting s0 = 0 is not experimentally so well established as it could be. Although this does not affect directly our argument, since in a non-cosmological context (such as the one we shall be dealing with) seems to be really negligible, we wish to point out that the opposite view (i.e. s0 > 0 ) has been recently gaining adherents(9).
 
 

The MT introduced above must be thought of as valid in a given reference frame R , the "aether frame" (unique up to spatial rotations and translations and up to a time homothety), where the ordinary concepts of classical mechanics are used. However, we can alternatively add another hypothesis which ensures the validity of the theory in all reference frames R' linked to R by a Lorentz transformation; this is the way RED is obtained. This further hypothesis is:
 
 

(13) (j,r c) and (cA,F) are 4-vectors of the Minkowski space-time.
 
 

As is well known, this assumption enables us to write the Maxwell's equations in a Lorentz invariant form. The point we wish to stress is that (13) is a genuine physical assumption which is logically independent of the previous (1) to (12) , and that it is perfectly legitimate to consider the possibility of translating the Maxwell's equations into space-time geometric terms as nothing more than an interesting mathematical property.
 
 
 
 

4 - PRM is not valid in MT
 
 

Let us go back to Einstein's example. The same experimental setting had already been examined by A. Foppl(10), who showed in one particular instance that the forces in the two cases were equal indeed. Foppl however was not rash in drawing any inference as to the general validity of PRM in electrodynamics. He discussed the case in which the conductor and the magnet were uniformly moving in the aether with zero relative velocity, and he mentioned that no effect had been experimentally detected. Furthermore he offered an example in which motion with respect to the aether did make a difference: the system of two charged particles q1 and q2, uniformly moving with the same velocity v = (v,0,0) (this corresponds in our definition of PRM to one of the two subsystems being empty). When the charges are at rest, say q1 is in the origin of the coordinate system, while the test charge q2 occupies the position (0,L,0), with L > 0, the force acting on q2 is simply the Coulomb force:
 
 

(14) F = (q1*q2/4pe0L2)*iy
 
 

(we denote by ix, iy, iz the positive orthonormal basis of the reference frame).

If both particles move, then each creates a convection current, and a magnetic field arises which must be taken into account; the resulting force is then
 
 

(15) F = (q1*q2/4pe0L2)*sqr(1-b2)*iy,
 
 

which explicitly depends on b = v/c, and so PRM is violated in MT. Foppl concluded that this was one of the difficulties that contemporary electrodynamics had to face.

Before dealing in detail with the relativistic interpretation, we must say that the last formula is exactly the 3-force expression we get by Lorentz transforming the Coulomb force as computed in the inertial frame in which the two charges are at rest. It follows that both RED and MT are inconsistent with PRM, and in this single case they are so in the same way(11).
 
 
 
 

5 - LPRM is valid in RED
 
 

As far as LPRM is concerned, it is easy to prove its validity in RED. The basic point is that, since PR holds, to compute the force acting on S in the reverse velocity setting is tantamount to Lorentz transforming the force computed in the original set up. This gives, for an elementary Lorentz transformation
 
 

(16) x' = (x-vt)/sqr(1-b2) , y'=y , z'=z , t'=(t-vx/c2)/sqr(1-b2) ,
 
 

the formula
 
 

(17) F' = (fx, fy*sqr(1-b2), fz*sqr(1-b2)) ,
 
 

which implies the identity of F and F' up to second order terms in b (we shall write F ÷ F' ).

It is interesting to note that, as it is clear from this proof, it is never true in RED that F' = F rigorously, except for trivial cases. In other words, what RED predicts for Einstein's and most other similar examples is an asymmetric, not symmetric, behaviour.
 
 
 
 

6 - LPRM is not valid in MT
 
 

What is even more remarkable at this point is that LPRM is not valid in general in classically interpreted electrodynamics (what we have called MT), so Einstein's problem of reconciling the different explanations that MT gives of symmetric results must be considered somewhat marginal, because (approximately) symmetric results are not the rule in MT (of course this is not an argument against SR).

As an example we want to compute in the aether frame R the force acting at the instant t=0 on a charge q as an effect of the presence of a circular stationary current I in the following four cases:
 
 

(a) - the circuit C , given by parametric equations
 
 

(18) x = Rcos(q) , y = Rsin(q) , z = 0 ,
 
 

is at rest, while the charge q is moving with constant nonzero velocity v = (v,0,0):
 
 

(19) x = vt , y= 0 , z = L ;
 
 

(b) - the circuit is moving with velocity v, the charge is at rest;
 
 

(c) - both the circuit and the charge are moving with the same velocity v;
 
 

(d) - both the circuit and the charge are at rest.
 
 

In the first case the force is
 
 

(20) F(a) = qv´ B(a) ,
 
 

where B(a) is the usual Biot-Savart field, which implies, if the current is oriented according to the parametrization (18),
 
 

(21) F(a) = -(m0qIvR2/2(R2+L2)3/2)*iy
 
 

which is the formula that can be found in most textbooks.
 
 

However, by comparing (20) with (10) we see that a new principle is involved here: in fact, if the vanishing of the term A/t can be taken as the mathematical expression of the stationarity assumption, how do we know that ÑF = 0 ?
 
 

The new principle is what we call the Clausius postulate(12):
 
 

CP - For a stationary circuit the charge density is zero.
 
 

We postpone a discussion of the validity of this commonly accepted assumption to the end of this paper. We notice that even in RED, CP is more or less tacitly assumed to be valid, and in the next paragraph we shall follow this practice.
 
 

Case (b) requires some more care. What can we assume in MT about the behaviour of moving circuits? In the hypothesis that no change in the shape can occur(13), it is natural to assume that the charge and current density of the moving circuit should be related to the corresponding rest densities r0 and j0 through translation, in agreement with the Galileian addition law. Taking also into account CP, we have:
 
 

(22) r(xyzt) = r0(x-vt,y,z) = 0 ,
 
 

(23) j(xyzt) = j0(x-vt,y,z) + r0*v = j0(x-vt,y,z) (since r0 = 0) .
 
 

It is easy to check that r and j verify the continuity equation (8) if the corresponding rest densities do, and so in accordance with our definition of MT we have just to compute the retarded potentials (11) and (12) . The electric potential F(b) is again zero, while for A(b) we have:
 
 

(24) A(b) = (m0IR/4p)*INT [0,2p ] [((-sin(q),cos(q),0)/D)dq],
 
 

where
 
 

(25) D2 = (x-vt-Rcos(q))2+(1-b2)*[(y-Rsin(q))2+z2] .
 
 

From (24) we obtain for the components of the magnetic field B(b) = curl(A(b)), the expressions:
 
 

(26)' B(b)x = (m0IR/4p)*(1-b2)*z*INT [0,2p ] [(cos(q)/D3)dq],
 
 

(26)'' B(b)y = (m0IR/4p)*(1-b2)*z*INT [0,2p ] [(sin(q)/D3)dq],
 
 

(26)''' B(b)z = -(m0IR/4p)*INT [0,2p ] [[(x-vt-Rcos(q))*cos(q)/D3+

+(1-b2)*(y-Rsin(q))*sin(q)/D3]dq] .
 
 

A proof of (24) runs as follows. The current density (23) can be expressed in terms of the Lorentzian coordinates (16) as
 
 

j'(x',y',z',t') = j0(x'*sqr(1-b2),y',z') ,
 
 

and the retarded potential A' in these coordinates is:
 
 

A'(x',y',z') = (m0/4p)*INT [r'-1j0(x1*sqr(1-b2),y1,z1)dx1dy1dz1] .
 
 

With the coordinate change
 
 

X' = x'1*sqr(1-b2) , Y' = y'1 , Z' = z'1 ,
 
 

we can also write
 
 

A' = (m0/4p)*INT [r'-1j0(X',Y,Z)dXdYdZ]
 
 

where now
 
 

r'2 = [(x'*sqr(1-b2)-X')2 + (1-b2)*(y'-Y')2 + (1-b2)*(z'-Z')2] ,
 
 

and this vector potential
 
 

A' = (m 0I/4p)*INT [0,2p ] [r'-1(-sin(q),cos(q),0)dq] ,
 
 

where
 
 

r'2 = [(x'*sqr(1-b2)-Rcos(q))2 + (1-b2)*(y'-Rsin(q))2 + (1-b2)*z'2] ,
 
 

is a solution of the equation
 
 

 'A' = -m0*j' .
 
 

Finally, since ' =  , our formula follows.
 
 

One can now do the required computation for the force F(b). If the charge q is at rest in the position (0,0,L) in the istant t=0, while the circuit is moving (case b), we have
 
 

(27) F(b) = -q*A(b)/t =

= -(m0qIvR2/4p)*INT [0,2p ] [r-3(-cos(q)sin(q),cos2(q),0)dq] ,
 
 

where
 
 

r2 = [R2cos2(q) + (1-b2)*R2sin2(q) + (1-b2)*L2] ,
 
 

and omitting in the series expansion of (27) all terms of second order or more in b :
 
 

(28) F(b) ÷ (m0qIvR2/4(R2+L2)3/2)*iy ,
 
 

which is in intensity half the previous (21).
 
 

In the third case, since a direct computation shows that, at the instant t=0 , B(b) ÷ B(a) , we get that F(c) is, up to second order in b , the sum of the force strengths computed for the two previous cases:
 
 

F(c) = -q*A(b)/t + qv´B(b) ÷ F(b) + qv´B(a) ,
 
 

that is to say
 
 

(29) F(c) ÷ -(m0qIvR2/4(R2+L2)3/2)*iy .
 
 

Finally the fourth case is easily dealt with by referring again to CP: we get
 
 

(30) F(d) = 0 .
 
 

Comparing (28) with (21), or, more strikingly, (30) with (29) gives the rather unexpected asymmetry result: LPRM does not hold in MT.
 
 
 
 

7 - Maxwell Theory versus Relativistic Electrodynamics
 
 

We turn now to the predictions of RED concerning the four cases discussed in the previous paragraph in the classical electromagnetism framework. Clearly since the shape of a material body in SR changes relative to the reference frame, we must restate the requirement that the current be circular and stationary by specifying that these conditions have to hold in the rest frame of C . Let us now denote by FR(i) the relativistic 3-force acting on the test charge in the case i = a, b, c, d . We have:
 
 

(31) FR(a) = F(a) ,
 
 

(32) FR(b) ÷ -FR(a) ÷ 2F(b) ,
 
 

(33) FR(c) = FR(d) = 0 .
 
 

The second and third line follow from the validity of LPRM in RED, and from (28) and (21).

There is a remarkable difference between FR(b) and F(b), which is at the origin of the quite definite divergence of the relativistic and classical predictions in the third case. RED foresees a zero force, while MT predicts the nonzero force (29).

For a comparison we give also the relativistic vector potential AR(b) in the case of a moving circuit:
 
 

(34) AR(b) = (m0IR/4p)*INT [0,2p ] [DR-1(-sin(q),sqr(1-b2)*cos(q),0)dq] ,
 
 

where
 
 

(35) DR2 = (x-vt-R*sqr(1-b2)*cos(q))2+(1-b2)*[(y-Rsin(q))2+z2] .
 
 

Comparing (34) and (24) would suggest that there can be no difference up to second order in b between RED and MT, but this is wrong because in the classical case one has div(A) = 0, because of the Lorentz gauge condition (8) and the vanishing of the electric potential F(b), while in the relativistic set up one has a "small" nonzero div(A) which multiplied by -c2 gives a nonzero time derivative for the relativistic electric potential FR(b), which then appears to be nonzero and even not "small"(14).
 
 
 
 

8 - A Proposed Experiment
 
 

The aforesaid divergence can be elaborated in view of the proposal of a new (as far as we know) "crucial" experiment discriminating between RED and MT, which we propose to call the Kennard-Marinov experiment(15).

By specializing (29) to the case that the charge q is situated right in the centre of the circuit (L = 0) we get
 
 

(36) F(c) ÷ -(m0qIv/4R)*iy ,
 
 

which depends both on the intensity and on the direction of the current.

This should make it possible to separate a nonzero effect from other disturbances due to constant fields existing in the terrestrial reference frame, and to other sources of systematic errors. Moreover, by increasing I and q we might be able to observe an effect even if the velocity of the laboratory is very small, as presumably it is, compared to c. One could either do direct measures of the force acting on the charge q, or of the voltage across a thin linear conductor T placed along a radius with an extremity in the centre of C and the other close to the current wire(16).

The possibility that the plane of the circuit does not contain the "absolute" velocity makes no harm, because one can repeat the observations for different choices of that plane, obtaining a maximum effect when this velocity lies in the plane (and the sensor T , which can be rotated around the centre, is orthogonal to the velocity direction).
 
 

Something more should be said about the Clausius postulate. Recently experimental reports(17) have been published questioning its validity. In one case a dependence on the square of the current intensity, and no dependence on its direction, has been found for the electric potential created by a stationary conducting circuit. Of course a charge situated in the centre of a circular circuit should not be affected by the force arising from this nonzero electric potential, apart from asymmetries in the circuit. The problem would be more complicated for the aforesaid voltage measure, since no general explicit formulae have been given for the effect conjectured to contradict CP (which we shall call the Edwards effect), and so quantitative predictions are difficult to make. However, a dependence of the voltage on the position of the plane of the circuit and on the direction of the current should make it possible to isolate the "drift" effect from the Edwards effect, and it is not completely unlikely that the possible existence of such an effect (though presumably much smaller than the Edwards effect) may have been simply overlooked in those experimental accounts which give no information at all about the dependence of their results on the orientation of the experimental apparatus.

We feel that "electrodynamical" (charges-and-currents, that is) experiments to test SR have been undeservedly neglected in favour of optical experiments. SR is such a fundamental theory in physics that any inquiry aiming at ascertaining the extent of its empirical accuracy should be welcome(18).
 
 
 
 

Acknowledgements - The authors thank most heartily Prof. L. Mantovani and Prof. S. Marinov for very useful discussions. Prof. J.P. Wesley was helpful in making a few improvements in the text.
 
 



Footnotes


 




(1) See R. Torretti, "Relativity and Geometry", Pergamon Press, 1983, p. 76-82, for details.
 
 

(2) We quote from the translation given by A.I. Miller, "Albert Einstein's Special Theory of Relativity", Addison-Wesley, Reading, 1981, p. 392. In an unpublished 1919 essay Einstein wrote: "The phenomenon of the electromagnetic induction forced me to postulate the (special) relativity principle" (A.I. Miller, p. 145).
 
 

(3) For an exception see V.I. Arnold, "Mathematical Methods of Classical Mechanics", transl. by K. Vogtmann and A. Weinstein, Springer-Verlag, NY, 1978, p. 10.
 
 

(4) For instance in a recent textbook on the foundations of relativity (Ref. 1) R. Torretti says that "In the preamble to [the 1905 relativity paper] Einstein argues persuasively for [PR] "by pointing out the "startling and altogether unwarranted asymmetries in the description of otherwise undistinguishable phenomena" (p. 50). Also M. Friedman in his "Foundations of Space-Time Theories" (Princeton University Press, 1983, p. 5) misses the distinction completely. As for Miller's book (Ref. 2), in which plenty of detailed historical information can be found also concerning PRM , something more needs to be said. At p. 164 Miller writes that "Einstein did not attempt to reduce mechanics to an atomistically-based theory of electromagnetism, but, rather, proposed the "conjecture" that, to order v/c, both disciplines obeyed a principle of relative motion applied to inertial reference systems", thus equating PR (Einstein's own "conjecture") with LPRM. We should point out that this equation is logically questionable because LPRM would be true in classical electromagnetism if we prescribed the relativistic rule of Lorentz-transforming from the moving frame to the aether frame without attaching to the computations made in the moving frame any independent physical meaning (of course we should renounce to use translations to get the charge and current densities of a moving conductor, which is the more natural procedure, cf. Sec. 6). Apart from this relatively minor point, in his treatment of Foppl's discussion of a case of magnet and conductor's interaction (cf. Ref. 10) Miller fails to recognize that the symmetry achieved in MT in that specific instance requires no approximation at all, such as that "electromagnetic radiation is neglected" as he mistakenly claims (his (3.3), p. 146 does not need the assumption div(A) = 0 ). So the total effect of Miller's analysis is to blur the distinction between PRM and PR by implying that only LPRM is really at stake in both MT and SR.
 
 

(5) For an exception see A.P. French, "Special Relativity", The MIT Introductory Physics Series, Norton, NY, 1968, p. 259. Of course, from the aether-theoretic point of view, it is well possible that high absolute velocities imply "real modifications" in the shape of bodies, masses, and so on, which would require adjustments of the classical theory similar to the "relativistic" ones. Nevertheless, we claim that it is in the low-velocity case that relativistic and classical predictions are certainly different.
 
 

(6) W. Rindler, "Essential Relativity", Springer-Verlag, NY, 1977, 2nd ed., p. 97. According to M. Bunge ("Foundations of Physics", Springer-Verlag, Berlin, 1967, p. 197) RED "is not a new theory but a reformulation of CEM [classical electromagnetism], which was relativistic without knowing it".
 
 

(7) Throughout this paper we shall use the MKSQ system of units.
 
 

(8) This expression gives the "right" behaviour of fields E and B at infinity, and excludes the so-called advanced potentials as having no possible physical meaning. The integration is made on the whole space, and one gets smooth solutions if one starts from smooth data for r and j . This uniqueness does not exclude of course the possibility of making a gauge transformation F * = F - H/t , A* = A + ÑH , which would give (for any choice of the function H(x,y,z,t)) another solution of (5), (6) and (8) furnishing the same fields E , B , provided that H = 0 . As far as the problem of the sources is concerned, and the existence of nonzero and nonsingular solutions of the homogeneous wave equation, we quote from B.H. Chirgwin, C. Plumpton, C.W. Kilmister, "Elementary Electromagnetic Theory", Vol. 3 ("Maxwell's Equations and their Consequences"), Pergamon Press, 1973, p. 549-550: "How is one to interpret such a solution of Maxwell's equations? There are no singularities - that is, no sources of the field anywhere or at any time. [...] The existence of this kind of solution of Maxwell's equations suggests that Maxwell's theory may be incomplete. It seems to lack some additional restriction that will serve that fields originate only from sources like charges and magnets. But we do not know how to modify the theory so as to rectify this defect".
 
 

(9) The hypothesis s0 > 0 has been recently revived by R. Monti , who has also shown its important large scale consequences. For details see for instance: R. Monti, "The electric conductivity of background space", Problems in Quantum Physics, Gdansk 1987, World Scientific, 1988; or: J.P. Vigier, "Evidence For Nonzero Mass Photons Associated With a Vacuum-Induced Dissipative Red-Shift Mechanism", IEEE Transactions on Plasma Science, 18, 1, 1990.
 
 

(10) In his Einfuhrung in die Maxwell'sche Theorie der Elektricitaet, 1894, which was probably one of Einstein's sources (cf. A.I. Miller, Ref. 2, p. 145 ff.). Maxwell ("A Treatise on Electricity and Magnetism", 3rd Ed., 1892,601) proves that "in all phenomena relating to closed circuits and the currents in them, it is indifferent whether the axes to which we refer the system be at rest or in motion", that is for these electrodynamical systems PR is valid in MT. In fact if we take
 
 

A' = A(x'+Vt,t) , F ' = F(x'+Vt,t)
 
 

to be the vector and scalar potential in an inertial frame R' moving with constant velocity V with respect to the aether frame R, then the expression for the "electromagnetic intensity" in R' is
 
 

-ÑF ' - A'/t + v'´curl(A) + Ñ<A',V> ,
 
 

so we have the same form as in the aether frame up to a gradient. This is why there is no hope to find in MTa numerical "asymmetry" if we deal only with closed circuits.
 
 

(11) We could add that if the test charge q2 is at the instant t=0 in the position (L,0,0) , then MT predicts the same result as (15), while RED takes also into account the length contraction in the direction of the x-axis.
 
 

(12) See for instance: A. O'Rahilly, "Electromagnetic Theory - A Critic Examination of Fundamentals", Dover, New York, 1965, vol. II, p. 589. Clausius stated that: "a closed constant current in a stationary conductor exerts no force on stationary electricity".
 
 

(13) Of course we might alternatively introduce some form of the Lorentz-Fitzgerald contraction hypothesis, but our aim here is to show some consequences of MT in its most "classical" interpretation. We do not claim that this version of MT does not require amendments in order to be proposed as a realistic physical theory (cf. also Ref. 5); for us it is mainly a tool, with an obvious historical relevance, to analyse some of the implications of the relativistic assumptions.
 
 

(14) On the same subject cf. also by the same authors "Some remarks on classical electromagnetism and the principle of relativity", to appear in Am. J. Phys, 1991. Needless to say, the presence of a nonzero electric potential is easily explained according to therelativistic postulates as a consequence of the length contraction (see for instance R.P. Feynman, "The Feynman Lecture Notes in Physics", Addison-Wesley, Reading, Mass., 1964, Vol. II, 13-6). Furthermore, we note that even if one obtains j(b) by simply "translating" j0 = j(a), then A(b) is not the simple translation of the vector potential A(a), that is to say A(b) is different from A(a)(x-vt,y,z), because of the retarded time effects which are inherent to the D'Alembert's equation; one has only indeed:
 
 

A(b) ÷ AR(b) ÷ A(a)(x-vt,y,z).
 
 

(15) With this choice we intend to remember the name and the work of E.H. Kennard (see for instance: "On Unipolar Induction - Another Experiment and its Significance as Evidence for the Existence of The Aether", Phil. Mag.,33, 1917, p. 179-190), and the multifarious activity of Stefan Marinov in favour of classical electrodynamics (cf. his many volumes work "The Thorny Way of Truth", International Publishers "East-West", Austria, 1982-1991, in press).
 
 

(16) In order to measure this voltage, and not to close the circuit going from the sensor T to the measuring apparatus, one could connect one end of T to the plate of a condenser C, and the other plate to earth. The accumulated charge of C would be a measure of the required voltage (one could also make use of a charge amplifier).
 
 

(17) W.F. Edwards, C.S. Kenyon, D.K. Lemon, "Continuing investigation into possible electric fields arising from steady conduction currents", Physical Review D, 14, 4, 1976, pp. 922-938; R. Sansbury, "Detection of a force between a charged metal foil and a current-carrying conductor", Rev. Sci. Instruments, 56, 3, 1985, pp. 415-417.
 
 

(18) For another, conceptually very simple "electrodynamical" experiment to test RED which has never been performed properly, see J. Maddox, "Stefan Marinov's seasonal puzzle", Nature, 346, 12 July 1990, p. 103.
 
 
 
 

Perugia, November 1990