The FitzGerald contraction of matter

The FitzGerald contraction of matter as it moves through the aether might conceivably be supposed to affect in some way the optical properties of the moving natter.

For instance, transparent substances might become doubly refracting.

This was tested by:

• Lord Rayleigh in 1902[78]
• D. B. Brace in 1904[79]

They could not detect any double refraction comparable with the proportion (w/c)2 of the single refraction.

The FitzGerald contraction of a material body cannot therefore be of the same nature as the contraction which would be produced in the body by pressure, but must be accompanied by such concomitant changes in the relations of the molecules to the aether that an isotropic substance does not lose its simply refracting character.

The hypothesis of contraction originally had no direct connection with electric theory.

• By this time, it has a direct connection.

Lorentz [80] had obtained the equations of a moving electric system by applying a transformation to the fundamental equations of the aether.

In the original form of this transformation, quantities of higher order than the first in w/c were neglected.

But in 1900, Larmor[81] extended the analysis so as to include small quantities of the second order, and thereby discovered a remarkable connexion between the equations of transformation and the equations which represent FitzGerald’s contraction.

After this, Lorentz[82] went further. He obtained the transformation in a form which is exact to all orders of the small quantity w/c. In this form we shall now consider it.

The fundamental equations of the aether are

div d = 4 pi c2 p

It is desired to find a transformation from the variables x, y, z, t, ρ, d, h, v, to new variables x1, y1, z1, t1, ρ1, d1, h1, v1, such that the equations in terms of these new variables may take the same form as the original equations, namely:

Evidently one particular class of such transformations is that which corresponds to rotations of the axes of coordinates about the origin: these may be described as the linear homogeneous transformations of determinant unity which transform the expression (x2 + y2 + z2) into itself.

These particular transformations are, however, of little interest, since they do not change the variable t.

But in place of them consider the more general class formed of all those linear homogeneous transformations of determinant unity in the variables x, y, z, ct, which transform the expression (x2 + y2 + z2 - c2t2)[errata 2] into itself: we shall show that these transformations have the property of transforming the differential equations into themselves.

All transformations of this class may be obtained by the combination and repetition (with interchange of letters) of one of them, in which two of the variables—say, y and z—are unchanged. The equations of this typical transformation may easily be derived by considering that the equation of the rectangular hyperbola

(in the plane of the variables x, ct) is unaltered when any pair of conjugate diameters are taken as new axes, and a new unit of length is taken proportional to the length of either of these diameters. The equations of transformation are thus found to be

where a denotes a constant.

The simpler equations previously given by Lorentz[83] may be derived from these by writing w/c for tanh a, and neglecting powers of w/c above the first.

By an obvious extension of the equations given by Lorentz for the electric and magnetic forces, it is seen that the corresponding equations in the present transformation are

The connexion between ρ and ρ1 may be obtained in the following way.

If a charge e is attached to a particle which occupies the position (ξ, η, ζ)`` at the instant t, an equal charge will be attached to the corresponding point (ξ1, η1, ζ1)at the corresponding instantt1`, in the transformed system

In this way, a charge e attached to an adjacent particle (ξ + Δξ, η + Δη, ζ + Δζ) at the instant t will give rise in the derived system to a charge e′ at the place

at the instant

that is to say, at the place

at the instant {{Wikimath|(t1 - sinh a. Δξ/c). Thus at the instant t1, this charge will occupy the position

The charges corresponding to those in the original system which were at the instant t contained in a volume ΔξΔηΔζ will therefore in the derived system at the instant t1, occupy a volume

or,

Thus if ρ1 denote the volume-density of electric charge in the transformed system, we shall have

this equation expresses the connexion between ρ1 and ρ. We have moreover

and similarly

and

When the original variables are by direct substitution replaced by the new variables in the differential equations, the latter take the form

that is to say, the fundamental equations of the aether retain their form unaltered, when the variables are subjected to the transformation which has been specified.

We are now in a position to show the connexion of this transformation with FitzGerald’s hypothesis of contraction. Suppose that two material particles are moving along the axis of x with velocity w = c tanh α. From the relation

it follows that

{\displaystyle v_{x_{1}}}, is zero for each of the particles, which implies that they are at rest relative to the new axes. Let x1, and x′1 denote their coordinates with respect to this latter system; then the coordinates of one particle at the instant t1, referred to the original axes, will be given by the equations

and the coordinates of the other particle will be given by

so that at time t the latter particle will have the coordinate x′′, where

which gives

This equation shows that the distance between the particles in the system of measurement furnished by the original axes, with reference to which the particles were moving with velocity w, bears the ratio (1 - w2/c2) 1 2 :1 to their distance in the ​system of measurement furnished by the transformed axes, with reference to which the particles are at rest.

But according to FitzGerald’s hypothesis of contraction, when a material body is in motion relative to the aether, in a direction parallel to the axis of x, its dimensions parallel to this direction contract in precisely this ratio; so that the equation of the body, in terms of the coordinates x1, y1, z1, which move with it, is unaltered.

Thus the hypothesis of FitzGerald may be expressed by the statement that the equations of the figures of ponderable bodies are covariant with respect to those transformations for which the fundamental equations of the aether are covariant.

The covariance holds with respect to all linear homogeneous transformations in the variables (x, y, z, t), of determinant unity, which transform the expression (x2 + y2 + z2 - c2t2) into itself.

This group comprises an infinite number of transformations; so that there are an infinite number of sets of variables resembling (x1, y1, z1, t1), of which any one set (xr, yr, zr, tr) can be derived from any other set (xs, ys, zs, ts) by a transformation of the group; among the sets wo must of course include the original set of coordinates (x, y, z, t).

But hitherto we have proceeded on the assumption that the original set (x, y, z, t) is entitled to a primacy among all the other sets, since the axes (x, y, z, t) have been supposed to possess the special property of having no motion relative to the aether, and the time represented by the variable t has been understood to be a definite physical quantity.

The other sets of variables (xr, yr, zr, tr) have been regarded merely as symbols convenient for use in problems relating to moving bodies, but not as corresponding to physical entities in the same degree as (x, y, z, ct). We must now inquire whether this view is justified.