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§ 3. -- IRREVERSIBLITY AND RETARDED POTENTIALS
Translated (1980) from
Recherches critiques sur l'Électrodynamique Générale,
Latest update 16 Oct 2016
Annales de Chimie et de Physique, Vol. 13, p. 145, 1908.
Now I intend to examine more closely the hypothesis which relates
(IX) and (X) to retarded potentials formulas (XII) and (XIII) and to show that
the transformation of the latter ones from the first ones is immediate, but that
the inverse proposition is far from being true.
First we have to recognize the fundamental importance of the formulas. In
contrast to mechanical phenomena, (Oeuvres 333) electromagnetic
phenomena are, in general, irreversible because of radiation. We can, by this
motive, even hope to get by their means a more complete interpretation of
irreversible physical phenomena. But Lorentz's equations don't change when we
change the direction of time. They contain the affirmation of reversibility,
whereas for retarded potentials and elementary actions, the positive and negative
directions of time play entirely different rôles. We insert, as in Helmholtz's
irreversible cycles, a speed which is hypothetically impossible to change the
sense, the speed with which waves constantly move away from the
bodies that generated them. It is from this, that electromagnetic irreversibility
is derived. This additional hypothesis which precedes must therefore be
examined with care.
With f(x,y,z,t) a continuous function, proportional to the electric
density ρ(x,y,z,t) and ϕ another function having the
property of continuity of potential in all space and at infinity and satisfying
everywhere the equation
it is easy to verify that the integral
is a solution of (3). In fact, lets isolate a small volume
τo around point xyz.
We'll be able to differentiate the integral under the summation sign
relative to the rest of space for which xyz is an external point. In
applying the operation
(1/c²)(∂²/∂t²) - Δ, we find
zero. Inside τo the
∂²/∂t² operation is performed again,
under the summation sign, and gives a result closing to zero with
τo. To do the
Δ operation lets put r, being very small, into τo.
The -Δ operation applied to the first term gives 4πf(x,y,z,t)
according to Poisson's theorem. In the second term the denominator r
is missing. The result closes to zero with τo. We
therefore obtain equation (3). I don't insist on the condition of continuity
lightly, in that we have to insure that there are derivatives of f.
We demonstrate that
and, more generally,
are solutions of (3), provided that the arbitrary functions
satisfy the relation
The solution φ1 corresponds to waves which move away in all directions
from the electric charges that generated them; φ2 is for waves coming from
infinity converging on the same points. In contrast to φ1, which depends
only on previous states, φ2 depends on subsequent states. The solution
φ3 contains both kinds of waves. And finally, φ4 corresponds
to waves whose centers of disturbance may be located in pure ether, where f = 0.
Experience shows, and Lorentz admits, that only φ1 waves can exist, and
besides, we will see that contrary hypotheses would involve inadmissible consequences,
such as the possibility of perpetual motion. We conclude, to begin with, that
Lorentz's equations (and the result extends to those of Maxwell and Hertz)
accept an infinite number of solutions, satisfying all conditions, but incompatible
We readily find such solutions, and are led to set them aside a priori, each
time we calculate, for example, the electrical oscillations (Oeuves 335) of a system
(a conducting sphere, Hertz's excitor, an oscillating electron, etc.).
Let's discuss the hypothesis by which we believe we can eliminate these solutions. We
show that we have for all solutions of (3) inside a closed surface S
where we have set, as before,
Suppose that at any instant t=0 situated
in the past we had everywhere, or at least at great distances,
ϕ = 0, ∂ϕ/∂t = 0; we could, for all positive values of t and at all
points xyz, choose for S a sphere of center xyz and radius R > ct
big enough so that all terms of the surface integral will be zero. Formula (4) will
remain. This reasoning calls for the following remarks:
1° The terms for uniform translational or rotational motion which enter into
electromagnetic theory never satisfy the relative conditions at t = 0.
This theory therefore remains excluded. More generally, the authors who have
used this reasoning, once the formula was set up, didn't bother to verify in the
equations they dealt with if this condition is fulfilled. It isn't in most cases. Now
we have seen that the formula must be absolutely general.
2° If for t = 0 there is only a very weak field at very great distances, this
field, if it is for a convergent wave, could acquire a few moments later a great
intensity at a given point in space. It is not sufficient therefore to suppose a weak
field for the moment t = 0 for all space (or at least at great distances).
It has to be rigorously zero (which is an hypothesis of inadmissible character in
physics) or that had been previously excluded in the convergent waves, which
would be a petition of principle. In the case of sound (and this analogy can
(Oeuvres 336) mislead easily) friction destroys the wave entirely after a few
moments, and the reasoning is practically the same. It is not the same though for
ether and we should expect, a priori, to find a similar state of affairs to
the one we observe at seaside, where in addition to the divergent waves produced
by solid bodies of the shore, there are others
which constantly come in from the sea and are not produced by solid bodies. So
if formula (4) is not rigorous, we would have to expect at each moment the sudden
creation of an intense field, a kind of electromagnetic wave coming from infinity
or which diverges from a region of ether through which it just converged.
3° Only solar and stellar radiation, which has been creating an oscillating
electromagnetic field throughout the universe daily for an extremely long time,
could oblige moving the instant t = 0 back beyond all limits of cognizance.
A hypothesis so fundamental must not present this inadmissible character.
4° Lets examine what must have happened before the moment t = 0. We
will find, in changing c into -c, the analogous formula
The same reasoning will give
that is to say that before the instant t = 0 the waves were convergent.
The bodies became excited by the radiation and thus produced perpetual
It is useless to dwell [on this] any longer. The hypothesis that we start
from rest (or the unimportant modifications that were presented) is not admissible
as a basis for the general law for retarded potentials. It isn't even for the
particular cases. Let's consider a Hertzian oscillator. At instant t = 0 a
spark jumps. The magnetic field, at first nil [at rest] everywhere, is disturbed;
but, after a short time, the system is at a new state (Oeuvres 337) of rest. It is not
exactly at rest, any more than it was before the experiment. It is in that
state only sensibly. If we begin our reasoning during the first state of rest,
there would only be converging waves. Why do we choose the first state of rest but
have no inclination for doing a new experiment wherein the second state of rest plays
the same role as in the first? It is because the elements away from the immediate area,
which are inaccessible to the experiment, play a preponderant role in the hypothesis.
If they were to send us convergent waves, our approximate, based on the
close-in elements, would soon cease to give an approximation, even a coarse one.
Fortunately, we know, a priori, by means of a lengthy experiment, that
the distant waves diverge. This is what allows us to ignore them and
also renders the demonstration unnecessary. If ether had viscosity, similar to that of
air, very different considerations would ensue and there wouldn't be any reason to be
surprised by irreversibility, since we would have introduced it in the equations
More generally, any theory which makes more specialized hypotheses about the
initial state than the law of potentials requires, will not be admissible. It would
exclude some actual phenomena and would allow, for t < 0, impossible
solutions. Therefore, what are the initial conditions, necessary and sufficient, for
(4) to hold?
φ = φ1 + ψ
We will have
in all space; and ψ = 0 at infinity. For ψ to be constantly nil, it is necessary and
sufficient that for t = 0 we have
Ψ = 0, ∂ψ/∂t = 0 in all space. This says that for formula (4) to be applicable
over all time, it is necessary and sufficient that it hold for two instants,
t = 0, and t = dt infinitely
close, a statement about which there is evidently nothing to be drawn from Maxwell's
point of view.
Can we replace the initial state hypothesis by some other equivalent general condition?
Lorentz(1) doesn't make use of the hypothesis. He simply admits that the surface integral
in (5) is nil when S moves away indefinitely. After having written (4), he
continues, "This solution is not therefore the general integral of (3). There will also be,
for example, solutions corresponding to a movement of waves which would be directed
towards the element of volume instead of moving away from it. We will reject them from
the theory by admitting, once and for all, that charged elements are the only points of
origin for disturbances. We exclude also the states of ether which are completely
independent of the charged substance; which if non-existent, the ether would remain
continually at rest."
But let's apply these ideas, without change, to formula (6). As the surface integral
vanishes we will get convergent waves. But neither method of proceeding is admissible.
These surface integrals, considered as functions of x,y,z,t, are general
solutions of (6). They would not therefore approach zero for a given value
of x,y,z,t when the surface moves away indefinitely. They have an invariable
value, finite or nil, according to whether the chosen
(1)Elektronetheorie, p 158; see also WIECHERT, Arch. neerl.,
1900, p 549, and P. HERTZ, Untersuchungen uber unstetige Bervegunen eins
Elektrons, Inau. Diss., Gottinge, 1904, p 5 and 12, and note. In his first thesis
of 1892 Lorentz contented himself with verifying that satisfactory equations exist.
solution is finite or nil. There is nothing to draw from such identities. Finally, there is
no precise sense to attach to this proposition; the disturbances that are dependent on
ether are the only ones to be excluded. If we could use, a priori, formula
φ2 or φ3, which depend only on the material,
we could write
φ1 = φ2 + ϕ,
(1/c2)(∂2ϕ/∂t2 - Δϕ =0,
and we would still have the right to say that ϕ is independent of the
material since its differential equation and its conditions of continuity are also
independent. In this manner of examination it is
φ1 which would contain a term that is
independent of the material. Finally, will we say that the state at a given point is
determined by the "previous history" of the material only? Solutions other than (3)
are still possible.
The inadequacy of these statements, and others analogous, is that they hold that,
the decomposition of a field of waves is a mathematical operation which can be done
in an infinite number of ways. But the character of this operation is doubly
artificial from the viewpoint of Maxwell's ideas because consideration of the origin of
the waves demands consideration of the entire field during a finite interval of time,
whereas Maxwell saw an essential advantage in his theory, precisely from the fact that
it makes it unnecessary to consider elementary actions and the field's origin and only
concerns itself with the immediate surroundings of the point under consideration. We
see that these statements are useless, and that to eliminate the physically impossible
solutions it only requires adoption a priori the formulae for retarded
potentials which distinguish elementary actions like the classical theories did, and
to prove that they satisfy the equations is to say that they can replace them completely,
whereas the inverse is not the case.
In bringing these results closer to the ones in the preceding section, we
see that in the last analysis it is the formula of elementary actions,
and not the system of partial differential equations, which is the complete
and exact expression of Lorentz's theory.
We have to add the hypothesis of absolute coordinates. We have just seen
that ether, instead of playing an independent role, and even preponderant,
as we should have expected, since it is supposed to be the reservoir of all
electromagnetic energy, steals away once more. Its only role will consist
in providing, though contradicting experience, a system of absolute
We will evidently have the same difficulties in passing from any kind of
system of partial differential equations, reversible at least
wherein it concerns pure ether, to irreversible solutions that the
experiment demands. In Hertz's theory, even this seems impossible.
The partial differential equations and the notion of ether are
essentially inappropriate to express the comprehensive laws for
the propagation of electrodynamic actions.
Copyright © 1980, 2000 Robert S. Fritzius
Installed May 8, 2000