 7. – ANALOGY BETWEEN ETHER
AND ELASTIC BODIES
WALTER RITZ
Translated (1980) from Recherches critiques sur l'Ėlectrodynamique
Générale,
Annales de Chimie et de Physique, Vol. 13, p. 145, 1908.
Annales 186 (Oeuvres 351)
Maxwell’s and Lorentz’s equations take, in the case of pure ether, a form
remarkably analogous to the ones for the equations of elasticity. What is the
real significance of this analogy?
The electric vector E is
satisfied in ether by the equations
and likewise for H. This is
the immediate consequence of the fundamental equations (I) through (IV).
On the other hand, the
components of displacement, x, h, z, assumed to be small with
respect to a point in the elastic body, A and B being constants, and m the density; we have
(Oeuvres 352)
Electromagnetic theory
shows, as we know, that E is identical to Fresnel’s vector, H to Neuman’s
vector (parallel to the plane of polarization). This identity with systems
(14) and (15) leads to an elastic
Annales 187
theory of light. To do that,
we have to admit either the incompressibility of ether, that is to say the
condition
or the condition A + B = 0. In
both cases, the identification is immediate.
These two ways of explaining the non-existence of longitudinal waves
have been admitted. In each of these hypotheses we could again choose between
Fresnel’s theory which leads to identifying the speed
of ether with E, or Neumann’s theory which replaces E with H.
What are the general
conditions, necessary and sufficient, such that a physical phenomenon,
characterized by a vector, will follow the laws expressed in (15)? I say that they are the following:
1. The phenomenon is
reversible.
2. x, h, z satisfy a system of three
partial differential equations which are of second order at most, and at
least are linear to a first approximation.
3. The medium is isotropic
and homogeneous.
Indeed, in considering
reversibility, the equations don’t have first derivatives with respect to
time; we will be able to solve them by means of their second derivatives
which are vector components.
Considering homogeneity, the right hand terms will have constant
coefficients, and considering isotropy they will be the summations of vector
components that were obtained by differentiation of x, h, z with respect
Annales 188
to x, y, z. But Burkhardt(Oeuvres 353) has determined all of
these vectors. When we admit only the
first and second derivatives, only three exist which are linearly
independent, namely
and we will have, a, b, c being constants,
In changing the signs of x, h, z, and keeping in mind the complete
isotropy, we find a = 0.
Therefore only system (15) remains, Q.E.D.
Condition(1) is satisfied
by all mechanical phenomena (and we have seen that in reality it shouldn’t be
satisfied by electromagnetic equations which correspond to irreversible
phenomena); (2) and (3) are satisfied by the phenomena of diffusion, heat
propagation, and others which certainly don’t present any natural connection
between them. We will conclude that
the analogy between (14) and (15) is a lot less characteristic than we would
be inclined to believe at first. We
won’t conclude that there’s any real physical connection between the two
orders of phenomena unless the analogy carries over even beyond this general
analytical
Annales 189
form. But this is precisely
not the case. Indeed, the hypothesis
of the nil speed of propagation for longitudinal waves (A + B = 0) is not, as
Green and Cauchy have previously said, admissible for a finite elastic body;
such a body would not offer any resistance to compression, its equilibrium
would be unstable. It is only recently
that Lord Kelvin’s gyrostatic ether has permitted us to devise such
systems. On the other hand, the
hypothesis of incompressibility (Oeuvres 354) calls for the introduction to
the equations one of Lagrange’s factors, performing the role of
pressure. The identification is no
longer possible in the cases where the pressure is constant. Finally, the limiting conditions demanded
by Optics are not the same as in the theory of elasticity.
I don’t believe
therefore that we should consider a priori these analogies as
indicative of profound physical agreement between the two domains. If we adopt this conclusion, we won’t be
overly amazed at the difficulties and strangeness that accompany all attempts
made to extend these analogies of pure ether (where Maxwell’s equations
express only the fact of uniform propagation) to reciprocal actions of
electric charges and the ether, expressed in the general equations (I) thru
(VI). For this part of the question, I
can do no more that return to the chapter which Poincaré has dedicated in his
Leçons to the most remarkable, it seems, of
these attempts: that of Larmor.
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