**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**) = m_{0}_{(}e_{0}¶_{
}_{E/}¶t
+ **j**)

(3) div(**E**) = r/e_{0}

_{(4) div(B) = 0 ,}

_{which reduce, in standard notation, to the D’Alembert wave equations
for the potentials }F , **A**:

(5) F = -r/e_{0}_{
,}

_{(6) }**A****
= -**m_{0}_{j
,}

_{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} = e_{0}m_{0}_{)
;}

_{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) 4pe_{0}F(x,y,z,t)
= INT [r^{-1}r^{ }^{(x’,y’,z’,t’-r/c)dx’dy’dz’]
,}

^{(12) 4}p**A****(x,y,z,t)
= **m_{0}_{*INT
[r-1j(x’,y’,z’,t’-r/c)dx’dy’dz’]}

_{(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 (c**A**,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*q2/4pe_{0}_{L2)*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*q2/4}pe_{0}_{L2)*sqr(1-}b^{2}^{)*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-b^{2}^{)
, y'=y , z'=z , t'=(t-vx/c2)/sqr(1-}b^{2}^{)
,}

^{the formula}

^{(17) F' = (fx, fy*sqr(1-}b^{2}^{),
fz*sqr(1-}b^{2}^{))
,}

^{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)** = q**v**´
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)** = -(m_{0}_{qIvR2/2(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 r_{0}_{ and
j0 through translation, in agreement with the Galileian addition
law. Taking also into account CP, we have:}

_{(22) }r(xyzt) = r_{0}_{(x-vt,y,z)
= 0 ,}

_{(23) j(xyzt) = j0(x-vt,y,z) + }r_{0}_{*v
= j0(x-vt,y,z) (since }r_{0}_{
= 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)** = (m_{0}_{IR/4}p)*INT^{
}_{[0,2}p ] [((-sin(q),cos(q),0)/D)dq],

where

(25) D^{2} = (x-vt-Rcos(q))^{2}+(1-b^{2}^{)*[(y-Rsin(}q))^{2}+z^{2}]
.

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

(26)' **B(b)**x = (m_{0}_{IR/4}p)*(1-b^{2}^{)*z*INT
[0,2p ] [(cos(}q)/D^{3})dq],

(26)'' **B(b)**y = (m_{0}_{IR/4}p)*(1-b^{2}^{)*z*INT
[0,2p ] [(sin(}q)/D^{3})dq],

(26)''' **B(b)**z = -(m_{0}_{IR/4}p)*INT^{
}_{[0,2}p ] [[(x-vt-Rcos(q))*cos(q)/D^{3}+

+(1-b^{2}^{)*(y-Rsin(}q))*sin(q)/D^{3}]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') = **j**_{0}(x'*sqr(1-b^{2}^{),y',z')
,}

^{and the retarded potential A' in these coordinates is:}

^{A'(x',y',z') = (}m_{0}_{/4}p)*INT
[r'^{-1}**j**_{0}(x’_{1}*sqr(1-b^{2}^{),y’1,z’1)dx’1dy’1dz’1]
.}

^{With the coordinate change}

^{X' = x'1*sqr(1-}b^{2}^{)
, Y' = y'1 , Z' = z'1 ,}

^{we can also write}

^{A' = (}m_{0}_{/4}p)*INT
[r'^{-1}**j**_{0}(X',Y’,Z’)dX’dY’dZ’]

where now

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

^{and this vector potential}

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

where

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

^{is a solution of the equation}

'A' = -m_{0}_{*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
=

= -(m_{0}_{qIvR2/4}p)*INT^{
}_{[0,2}p ] [r^{-3}(-cos(q)sin(q),cos^{2}(q),0)dq]
,

where

r^{2} = [R^{2}cos^{2}(q)
+ (1-b^{2}^{)*R2sin2(}q)
+ (1-b^{2}^{)*L2] ,}

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

(28) **F(b)** ÷ (m_{0}_{qIvR2/4(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
+ q**v**´B(b) ÷ **F(b)**
+ q**v**´B(a) ,

that is to say

(29) **F(c)** ÷ -(m_{0}_{qIvR2/4(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) = (}m_{0}_{IR/4}p)*INT^{
}_{[0,2}p ] [D_{R}^{-1}(-sin(q),sqr(1-b^{2}^{)*cos(}q),0)dq]
,

where

(35) D_{R}^{2} = (x-vt-R*sqr(1-b^{2}^{)*cos(}q))^{2}+(1-b^{2}^{)*[(y-Rsin(}q))^{2}+z^{2}]
.

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 -c^{2}
gives a nonzero time derivative for the relativistic electric potential
F_{R}_{(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) ÷ -(}m_{0}_{qIv/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 s_{0} > 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'+**V**t,t) , F
' = F(x'+**V**t,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" **j**_{0} = **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)** ÷ **A _{R}(b)** ÷

(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