In the previous episode, we saw how both Young and Fresnel had put forward the view that
the vibrations of light are performed at right angles to its direction of propagation. Fresnel
also pointed out that this might be explained by assuming the aether possesses the power
of resisting attempts to distort its shape and could be likened to an Elastic sold. After
Fresnel death, this concept was developed further and this is what we will examine in
this episode.
The idea that the aether is composed of an elastic solid has one rather glaring problem.
How is it possible that the planets in their orbital motions are able to journey through
it at immense speeds without encountering any perceptible resistance?
The first person to address this was Sir George Gabriel Stokes. He had suggested that the
aether may behave both as a solid and a liquid. For objects with very rapid vibrations, like
light, it would behave like an elastic solid but for those moving with comparatively slower
speeds, it would act like a liquid.
In 1828 Poisson succeeded in solving the differential equations to determine the wave motions possible
in an elastic solid. This revealed that there are two types of waves created in elastic
solid. Transverse and longitudinal. The longitudinal waves cause dilation and condensations, or
compressions and rarefaction just like sound waves. Whereas transverse waves will not cause
these just a distortion to the shape while maintaining a constant density.
Up and until this point the investigations had only considered isotropic bodies. In 1828
Augustine-Louis Cauchy would extend this to examine light in a crystalline substance.
The general equations he obtained were extensive, containing 21 constants. From this, he came
up with 2 theories. In the first, the assumption is that the vibrations which constitute light
were executed parallel to the plane of polarisation, and the second where the would-be executed
at right angles to the plane of polarisation.
Both equations would give a reasonable overall equivalence for light in a crystal but they
would also show that a 3rd type of longitudinal wave would also be created. This presented
a rather large problem with this concept. Cauchy felt that the existence of these vibrations
would ultimately be demonstrated experimentally. A further objection to this theory is that
the relations between the constants do not appear to admit any simply physical interpretation,
being evidently assumed for the sole purpose of forcing the formulae into some degree of
conformity with the results of experiment. His work at this point did also not account
for reflection or refraction.
In order to explain refraction, it is necessary to assign a cause for the existence of refractive
indices. In other words an explanation for the variations in the velocity of light from
one body to another. Previously we have discussed that Huygens had suggested that transparent
bodies consist of hard particles which interact with the aethereal matter, modifying its elasticity.
Cauchy in his initial papers followed this lead and assumed the density of the aether
is the same in all media, but that its rigidity varies from one medium to another.
Cauchy would go on to force a theory for the reflection and in so doing would have to compromise
his earlier theories. In order to make his equations fit Fresnel’s ratio of the intensity
of the reflected ray, he had to adopt his second theory that the vibrations were executed
at right angles to the plane of polarisation of the light.
The theory he advanced was therefore encumbered with many difficulties. His decisions to select
certain conditions was evidently to secure the fulfilment of Fresnel’s sine law and
tangent law, but the results are inconsistent with the true boundary conditions, which were
given later by Green. Later he would even abandon his earlier assumption that the density
of the aether is the same in all material bodies.
These features bring out the weakness of Cauchy’s methods of attacking the problem. His object
was to derive the properties of light from a theory of the vibrations of elastic solids.
At the outset, he had already in his possession the differential equations of motion of the
sold, which were to be his starting and the equations of Fresnel his goal. It only remained
to supply the boundary conditionsat an interface, which required in the discussion of relection,
and the relations between the elastic constants of the solid, which are required in the optics
of the crystals.
The problem with this approach is that it will allow for multiple solutions. Within
the space of 10 years, the great French mathematician produced two distinct theories of crystal-optics
and 3 theories of reflection, almost all yielding correct or nearly correct final formulae and
yet mostly irreconcilable with each other, and involving incorrect boundary conditions
and improbable relations between elastic constants.
At around the same time, James MacCullagh and Franz Neumann were also attempting to
tackle the same problem in an almost identical approach. They felt that the great objection
to Fresnel’s theory of reflection was its failure to provide for the continuity of the
normal component of displacement at the interface between two media. If the aether were an elastic
solid then there could be no discontinuity. The only way they could achieve this was by
assuming the vibration of the aether were parallel to the plane of polarisation. In
order to obtain the sine and tangent law, they had to assume that the inertia of the
luminiferous medium is the same everywhere. The differences in the behaviour of different
substances would be due to differences in its elasticity. Sadly these theories did not
advance much on the quasi mechanical theories of their predecessors.
This much-needed reformation would eventually come from Green in 1837. Instead of designing
boundary equations for the express purpose of yielding Fresnel’s sine and tangent laws,
he set to work to determine the conditions which are actually satisfied at the interfaces
of real elastic solids. What he was able to determine was that the boundary conditions
that must be satisfied at the interface between two elastic media in contact meant that the
three components of displacement and stress across the interface are equal in the two
media.
It was now straightforward to obtain the boundary conditions for reflection and refraction.
He found thatr the vibrations of the aetherial particles is executed at right angles to the
plane of incidence, the intensity of the reflected light obeys Fresnel’s law provided the rigidity
is assumed to be the same for all media, but the inertia to vary from one medium to another.
Green’s conclusions confirmed the hypotheses of Fresnel that the vibrations were executed
at right angles to the plane of polarisation, and that the optical differences between media
are due to the different densities of aether within them.
The only thing left was for Green to explain the cause where the incidence light is polarised
at right angles to the plane of incidence, so the motion of the aethereal particles is
parallel to the intersection of the plane of incidence with the front of the wave. In
this case, it is impossible to satisfy the boundary conditions without assuming that
longitudinal vibrations are generated by the act of reflection.
Green avoided this difficulty by adopting Fresnel’s suggestion that the resistance
of the aether to compression may be very large in comparison with the resistance to distortion,
as is the case when we examine real examples of elastic solids like jelly. The problem
now was that the ratio of the intensity of the transverse wave. it does not agree with
Fresnel’s tangent formula and his theory of reflection does not harmonise well with
the elastic solid theory of crystal optics. This means that the vibrations of a Greenian
solid do create an exact parallel to the vibrations which constitute light.
MacCullagh keen to seek a solution to these problems presented his own theory to the Royal
Irish Academy in 1839. From Greens investigations, he had concluded that it was not possible
to explain optical phenomena satisfactorily by comparing the aether to an elastic solid
of the ordinary type, which resists compression and distortion. He saw that the only hope
was to device a medium which should be as strictly conformable to dynamical laws as
Green’s elastic solid, and yet should have its properties specially designed to fulfil
the requirements of the theory of light.
In a normal elastic solid the potential energy of strain depends only on the change of size
and shape of the volume elements, their compression and distortion. For MacCullagh’s new medium,
on the other hand, the potential energy would depend only on the rotation of the volume
elements. This immediately meant that no longitudinal waves would be created.
MacCullagh’s work was regarded with doubt by his own and the succeeding generation of
mathematical physicists, and can scarcely be said to have been properly appreciated
until FitzGerald drew attention to it 40 years later. There can be no doubt that MacCullagh
really solved the problem of devising a medium whose vibrations, calculated in accordance
with the correct laws of dynamics, should have the same properties as the vibrations
of light.
The hesitation which was felt in accepting the rotationally elastic aether arose mainly
from the lack of any example of a body which had such properties.
This would be removed by Lord Kelvin in 1889 who designed mechanical models possessing
rotational elasticity.
If we imaging a structure formed of spheres, arranged so that it is at the centre of a
tetrahedron formed by its 4 closest neighbours. Each sphere is connected to these 4 by rigid
bars which have spherical caps at their ends so as to slide freely on the spheres. Such
a structure would, for small deformations, behave like an incompressible perfect fluid.
Now attach to each bar a pair of gyroscopically mounted flywheels, rotating with equal and
opposite angular velocities with their exes in the line of the bar. This will give the
whole structure a quasi-elasticity exactly as MacCullagh had envisioned.
Based on the idea of both Green and MacCullgh, Cauchy published a third theory in 1839. In
this theory, he took up the remark by Green that the longitudinal wave might be avoided
in either of two ways. Firstly supposing its velocity to be extremely fast compared to
light or extremely slow. Cauchy chose the latter for this theory. This would require
him to have an elastic medium of negative compressibility. This implied that no work
is required to be done in order to give the medium any small irrotational disturbance.
An example of such a medium is given by a homogenous foam, free from air, and held from
collapse by adhesion to a containing vessel. This type of aether was called a contractile
aether.
The great advantage of this type of aether is that it overcomes the difficulty of securing
continuity of the normal components of displacement at an interface between two media. Lord Kelvin
would, later on, devote much of his attention to this type of aether. He assumed that in
space devoid of normal matter the aether is practically incompressible but in the space
occupied by solids and liquids, it has negative compressibility. This assumption is based
on the idea that material atoms move through space without displacing the aether. A concept
that Lord Kelvin had remarked, contradicts the old scholastic axiom that two different
portions of matter cannot simultaneously occupy the same space. He thought the aether to be
attracted and repelled by atoms, and thereby to be condensed and rarefied.
It is impossible to avoid noticing throughout Kelvin’s work evidence of the deep impression
which was made upon him by the writings of Green. The same may be said of Kelvin’s
friend and contemporary Stokes; and indeed, it is no exaggeration to describe Green as
the real founder of that ‘Cambridge School’ of natural philosophers, of which Kelvin,
Stokes, Rayleigh, Maxwell, Lamb, J.J Thomson, Larmor and Love were the most illustrious
members in the latter half of the nineteenth century.
The attempt to represent the properties of the aether by those of an elastic sold lost
some of its interest after the rise of the electromagnetic theory of light. In 1867,
before the electromagnetic hypothesis had attracted much attention, an elastic solid
theory which was in many aspects preferable to its predecessors was presented to the French
Academy by Joseph Boussinesq. Up and until this time there had been two different views
of how the light was propagated in the aether, one who viewed it as being caused by variations
in the inertia of the medium, and the other that this was due to variations in the elastic
properties. Boussinesq took the view that the aether is exactly the same in all material
bodies as in interplanetary space. The same everywhere both in inertia and elasticity.
The optical properties of matter are due to the interaction between the aether and the
material particles. These material particles were disseminated in the aether much like
dust particles floating in the air.
He viewed that all aetherial processes are to be represented by two kinds of equations,
of which one kind expresses the invariable equations of motion of the aether, while the
other kind expresses the interactions between aether and matter.
Many years afterwards these ideas were revived in connection with the electromagnetic theory,
in this form they are indeed of fundamental importance.