Riemann's Equation

Riemann in [45] his 1861 lectures remarked that the scalar-potential φ and vector-potential a, corresponding to his own law of force between electrons, satisfy the equation:

an equation which is satisfied also by the potentials of L. Lorenz[46]

This appeared to Riemann to indicate that φ might represent the density of an aether, of which a represents the velocity.

On this hypothesis the electric and magnetic forces correspond to second derivates of the displacement—a circumstance which makes it somewhat difficult to assimilate the energy possessed by the electromagnetic field to the energy of the model.

We must now proceed to consider those models in which the aether is represented as composed of more than one kind of constituent: of these Maxwell’s model of 1861-2, formed of vortices and rolling particles, may be taken as the type. Another device of the same class was described in 1885 by FitzGerald[47].

This was constituted of a number of wheels, free to rotate on axes fixed perpendicularly in a plane board; the axes were fixed at the intersections of two systems of perpendicular lines.

Each wheel was geared to each of its four neighbours by an indiarubber band. Thus all the wheels could rotate without any straining of the system, provided they all had the same angular velocity; but if some of the wheels were revolving faster than others, the indiarubber bands would become strained,

The wheels in this model play the same part as the vortices in Maxwell’s model of 1861-2: their rotation is the analogue of magnetic force; and a region in which the masses of the wheels are largo corresponds to a region of high magnetic permeability.

The indiarubber bands of FitzGerald’s model correspond to the medium in which Maxwell’s vortices were embedded; and a strain on the bands represents dielectric polarization, the line joining the tight and slack sides of any band being the direction of displacement.

A body whose specific inductive capacity is large would be represented by a region in which the elasticity of the bands is feeble. Lastly, conduction may be represented by a slipping of the bands on the wheels.

Such a model is capable of transmitting vibrations analogous to those of light. For if any group of wheels be suddenly set in rotation, those in the neighbourhood will be prevented by their inertia from immediately sharing in the motion; but presently the rotation will be communicated to the adjacent wheels, which will transmit it to their neighbours; and so a wave of motion will be propagated through the medium. The motion constituting the wave is readily seen to be directed in the place of the wave, i.e. the vibration is transverse.

The axes of rotation of the wheels are at right angles to the direction of propagation of the wave, and the direction of polarization of the bands is at right angles to both these directions.

The elastic bands may be replaced by lines of governor balls:[48] if this be done, the energy of the system is entirely of the kinetic type.[49]

Models of types different from the foregoing have been suggested by the researches of Helmholtz and W. Thomson on vortex-motion. The earliest attempts in this direction, however, were intended to illustrate the properties of ponderable matter rather than of the luminiferous medium.

A vortex existing in a perfect fluid preserves its individuality throughout all changes, and cannot be destroyed; so that if, as Thomson[50] suggested in 1867, the atoms of matter are constituted of vortex-rings in a perfect fluid, the conservation of matter may be immediately explained.

The mutual interactions of atoms may be illustrated by the behaviour of smoke-rings, which after approaching each other closely are observed to rebound: and the spectroscopic properties of matter may be referred to the possession by vortex-rings of free periods of vibration.[51]

There are, however, objections to the hypothesis of vortex-atoms. It is not easy to understand how the large density of ponderable matter as compared with aether is to be explained; and further, the virtual inertia of a vortex-ring increases as its energy increases; whereas the inertia of a ponderable body is, so far as is known, unaffected by changes of temperature.

It is doubtful whether vortex-atoms would be stable.

W. Thomson[52] wrote in 1905: “If any motion be given within a finite portion of an infinite incompressible liquid, originally at rest, its fate is necessarily dissipation to infinite distances with infinitely small velocities everywhere; while the total kinetic energy remains constant.

After many years of failure to prove that the motion in the ordinary Helmholtz circular ring is stable, I came to the conclusion that it is essentially unstable, and that its fate must be to become dissipated as now described.”

The vortex-atom hypothesis is not the only way in which the theory of vortex-motion has been applied to the construction of models of the aether, It was shown in 1880 by W. Thomson[53] that in certain circumstances a mass of fluid can exist in a state in which portions in rotational and irrotational motion are finely mixed together, so that on a large scale the mass is homogeneous, having within any sensible volume an equal amount of vortex-motion in all directions. To a fluid having such a type of motion he gave the name vortex-sponge.

Five years later, FitzGerald[54] discussed the suitability of the vortex-sponge as a model of the aether.

Since vorticity in a perfect fluid cannot be created or destroyed, the modification of the system which is to be analogous to an electric field must be a polarized state of the vortex motion, and light must be represented by a communication of this polarized motion from one part of the medium to another.

Many distinct types of polarization may readily be imagined: for instance, if the turbulent motion were constituted of vortex-rings, these might be in motion parallel to definite lines or planes; or if it were. constituted of long vortex filaments, the filaments might be bent spirally about axes parallel to a given direction.

The energy of any polarized state of vortex-motion would be greater than that of the unpolarized state; so that if the motion of matter had the effect of reducing the polarization, there would be forces tending to produce that motion.

Since the forces due to a small vortex vary inversely as a high power of the distance from it, it seems probable that in the case of two infinite planes, separated by a region of polarized vortex-motion, the forces due to the polarization between the planes would depend on the polarization, but not on the mutual distance of the planes—a property which characteristic of plane distributions whose elements attract according to the Newtonian law.

It is possible to conceive polarized forms of vortex-motion which are steady so far as the interior of the medium is concerned, but which tend to yield up their energy in producing motion of its boundary—a property parallel to that of the aether, which, though itself in equilibrium, tends to move objects immersed in it.

In the same year Hicks[55] discussed the possibility of transmitting waves through a medium consisting of an incompressible fluid in which small vortex-rings are closely packed together. The wave-length of the disturbance was supposed large in comparison with the dimensions and mutual distances of the rings.

The translatory motion of the latter was supposed to be 30 slow that very many waves can pass over any one before it has much changed its position. Such a medium would probably act as a fluid for larger motions.

The vibration in the wavefront might be either swinging oscillations of a ring about a diameter, or transverse vibrations of the ring, or apertural vibrations; vibrations normal to the plane of the ring appear to be impossible. Hicks determined in each case the velocity of translation, in terms of the radius of the rings, the distance of their planes, and their cyclic constant.

The greatest advance in the vortex-sponge theory of the aether was made in 1887, when W. Thomson[56] showed that the equation of propagation of laminar disturbances in a vortex-sponge is the same as the equation of propagation of luminous vibrations in the aether. The demonstration, which in the circumstances can scarcely be expected to be either very simple or very rigorous, is as follows:—

Let (u, v, w) denote the components of velocity, and p the pressure, at the point (x, y, z) in an incompressible fluid. Let the initial motion be supposed to consist of a laminar motion {f(y), 0, 0}, superposed on a homogeneous, isotropic, and finegrained distribution (u′0, v0, w0): so that at the origin of time the velocity is {f(y) + u′0, v0, w0}: it is desired to find a function f(y, t) such that at any time t the velocity shall be {f(y, t) + u′, v, w}, where u′, v, w, are quantities of which every average taken over a sufficiently large space is zero.

Substituting these values of the components of velocity in the equation of motion

there results

Take now the xz-averages of both members. The quantities ∂u′/∂t, ∂u′/∂x, v, ∂p/∂x have zero averages; so the equation takes the form

if the symbol A is used to indicate that the xz-average is to be taken of the quantity following. Moreover, the incompressibility of the fluid is expressed by the equation

whence

When this is added to the preceding equation, the first and third pairs of terms of the second member vanish, since the x-average of any derivate ∂Q/∂x vanishes if Q is finite for infinitely great values of x; and the equation thus becomes

From this it is seen that if the turbulent motion were to remain continually isotropic as at the beginning, f (y, t) would constantly retain its critical value f(y). In order to examine the deviation from isotropy, we shall determine A∂(u′v)/∂t, which may be done in the following way:—Multiplying the u- and v-equations of motion by v, u′ respectively, and adding, we have

Taking the xz-average of this, we observe that the first term of the first member disappears, since A . v is zero, and the first term of the second member disappears, since A . ∂(u′v)/∂x is zero. Denoting by 1 3 R2 the average value of u2, v2, or ω2, so that R may be called the average velocity of the turbulent motion, the equation becomes

where

Let p be written (), where p′ denotes the value which p would have if f were zero. The equations of motion immediately give

and on subtracting the forms which this equation takes in the two cases, we have

which, when the turbulent motion is fine-grained, so that f(y, t) is sensibly constant over ranges within which u′, v, ‘ω pass through all their values, may be written

Moreover, we have

for positive and negative values of u′, v, ω are equally probable; and therefore the value of the second member of this equation is doubled by adding to itself what it becomes when for u′, v, ω we substitute -u′, -v, -ω; which (as may be seen by inspection of the above equation in Δ2p) does not change the value of p′. Comparing this equation with that which determines the value of Q, we have

or substituting for

The isotropy with respect to x and z gives the equation

But by integration by parts we obtain the equation

and by the condition of incompressibility the second member may be written

So we have

On account of the isotropy, we may write

and, therefore,

The deviation from isotropy shown by this equation is very small, because of the smallness of ∂f(y,t)/∂y. The equation is therefore not restricted to the initial values of the two members, for we may neglect an infinitesimal deviation from (2/9) R2 in the first factor of the second member, in consideration of the smallness of the second factor. Hence for all values of t we have the equation

which, in combination with (1), yields the result

the form of this equation shows that laminar disturbances are propagated through the vortex-sponge in the same manner as waves of distortion in a homogeneous elastic solid.