Space Transformations

The theory of the motionless aether was hampered by one difficulty: it could not explain the negative result of the Michelson-Morley experiment.[62]

In the issue of “Nature” for June 16th, 1892,[63] Lodge mentioned that Fitzgerald suggested that the dimensions of material bodies are slightly altered when they are in motion relative to the aether.

Five months later, this hypothesis was adopted by Lorentz and was generally accepted.

Let us first see how it explains Michelson’s result. On the supposition that the aether is motionless,

One of the two portions of the divided beam of light completes its journey in less time than the other by w2l/c2:

• w is the velocity of the earth
• c the velocity of light
• l the length of each arm.

This would be exactly compensated if the arm which is pointed in the direction of the terrestrial motion were shorter than the other by an amount w2l/2c2; as would be the case if the linear dimensions of moving bodies were always contracted in the direction of their motion in the ratio of (1 - w2/2c2) to unity. This is FitzGerald’s hypothesis of contraction.

Since for the earth the ratio w/c is only

30 km./sec. 300 , 000 km./sec. , {\displaystyle {\frac {30{\text{km./sec.}}}{300,000{\text{km./sec.}}}},}

the fraction w2/c2 is only one hundred-millionth.

Several further contributions to the theory of electrons in motionless aether were made in a short treatise[65] which was published by Lorentz in 1895.

One of these related to the explanation of an experimental result obtained some years previously by Th, des Coudres,[66] of Leipzig. Des Coudres had observed the mutual inductance of coils in different circumstances of inclination of their common axis to the direction of the earth’s motion, but had been unable to detect any effect depending on the orientation.

Lorentz now showed that this could be explained by considerations similar to those which Budde and FitzGerald[67] had advanced in a similar case; a conductor carrying a constant electric current and moving with the earth would exert a force on electric charges at relative rest in its vicinity, were it not that this force induces on the surface of the conductor itself a compensating electrostatic charge, whose action annuls the expected effect.

The most satisfactory method of discussing the influence of the terrestrial motion on electrical phenomena is to transform the fundamental equations of the aether and electrons to axes moving with the earth. Taking the axis of x parallel to the direction of the earth’s motion, and denoting the velocity of the earth by w, we write

so that (x1, y1, z1) denote coordinates referred to axes moving with the earth. Lorentz completed the change of coordinates by introducing in place of the variable t a “local time” t1 defined by the equation

It is also necessary to introduce, in place of d and h, the electric and magnetic forces relative to the moving axes: these are[68]

and in place of the velocity v of an electron referred to the original fixed axes, we must introduce its velocity v1, relative to the moving axes, which is given by the equation

The fundamental equations of the aether and electrons, referred to the original axes, are

where F denotes the ponderomotive force on a particle carrying a unit charge.

By direct transformation from the original to the new variables it is found that, when quantities of order w2/c2 and wv/c2 are neglected, these equations take the form

where div1 d1 stands for

Since these have the same form as the original equations, it follows that when terms depending on the square of the constant of aberration are neglected, all electrical phenomena may be expressed with reference to axes moving with the earth by the same equations as if the axes were at rest relative to the aether.

In the last chapter of the Versuch, Lorentz discussed those experimental results which were as yet unexplained by the theory of the motionless aether. That the terrestrial motion exerts no influence on the rotation of the plane of polarization in quartz[69] might be explained by supposing that two independent effects, which are both due to the earth’s motion, cancel each other; but Lorentz left the question undecided.

Five years later Larmor[70] criticized this investigation. He concluded that there should be no first-order effect.

Lorentz[71] maintained his position against Larmor’s criticism.

Although the physical conceptions of Lorentz had from the beginning included that of atomic electric charges, the analytical equations had hitherto involved ρ, the volume-density of electric charge; that is, they had been conformed to the hypothesis of a continuous distribution of electricity in space.

It might hastily be supposed that in order to obtain an analytical theory of electrons, nothing more would be required than to modify the formulae by writing e (the charge of an electron) in place of ρdxdydz. That this is not the case was shown[72] a few years after the publication of the Versuch.

Consider, for example, the formula for the scalar potential at any point in the aether,

where the bar indicates that the quantity underneath it is to have its retarded value.[73]

This integral, in which the integration is extended over all elements of space, must be transformed before the integration can be taken to extend over moving elements of charge. Let de′ denote the sum of the electric charges which are accounted for under the heading of the volume-element dx′dy′dz′ in the above integral. This quantity de′ is not identical with

For, to take the simplest case, suppose that it is required to compute the value of the potential-function for the origin at the time t, and that the charge is receding from the origin along the axis of x with velocity u.

The charge which is to be ascribed to any position x is the charge which occupies that position at the instant t - x/c; so that when the reckoning is made according to intervals of space, it is necessary to reckon within a segment (x2 – x1) not the electricity which at any one instant occupies that segment, but the electricity which at the instant (t - x3/c) occupies a segment (x2 – x′1), where x′1 denotes the point from which the electricity streams to x1, in the interval between the instants (t - x2/c) and (t - x1/c). We have evidently

For this case we should therefore have

In the general case, it is only necessary to replace u by the component of velocity of the electric charge in the direction of the radius vector from the point at which the potential is to be computed. This component may be written

where r is measured positively from the point in question to the charge, and v denotes the velocity of the charge. Thus

and therefore

where the integration is extended over all the charges in the field, and the bars over the letters imply that the position of the charge considered is that which it occupied at the instant

Meanwhile the unsettled problem of the relative motion of earth and aether was provoking a fresh series of experimental investigations.

The most interesting of these was due to FitzGerald,[74] who shortly before his death in February, 1901, commenced to examine the phenomena manifested by a charged electrical condenser, as it is carried through space in consequence of the terrestrial motion.

On the assumption that a moving charge develops a magnetic field, there will be associated with the condenser a magnetic force at right angles to the lines of electric force and to the direction of the motion: magnetic energy must therefore be stored in the medium, when the plane of the condenser includes the direction of the drift; but when the plane of the condenser is at right angles to the terrestrial motion, the effects of the opposite charges neutralize each other.

FitzGerald’s original idea was that, in order to supply the magnetic energy, there must be a mechanical drag on the condenser at the moment of charging, similar to that which would be produced if the mass of a body at the surface of the earth were suddenly to become greater.

Moreover, it was conjectured that the condenser, when freely suspended, would tend to move so as to assume the longitudinal orientation, which is that of maximum kinetic energy[75]: the transverse position would therefore be one of unstable equilibrium.

For both effects a search was made by FitzGerald’s pupil Trouton:[76] in the experiments designed to observe the turning couple, a condenser was suspended in a vertical plane by a fine wire, and charged.

If the plane of the condenser were that of the meridian, about noon there should be no couple tending to alter the orientation, because the drift of aether due to the earth’s motion would be at right angles to this plane; at any other hour, a couple should act.

The effect to be detected was extremely small; for the magnetic force due to the motion of the charges would be of order w/c, where w denotes the velocity of the earth; so the magnetic energy of the system, which depends on the square of the force, would be of order (w/c)2; and the couple, which depends on the derivate of this with respect to the azimuth, would therefore be likewise of the second order in (w/c).

No couple could be detected. As the energy of the magnetic field must be derived from some source, there seems to be no escape from the conclusion that the electrostatic energy of a charged condenser is diminished by the fraction (w/c)2 of its amount when the condenser is moving with velocity w at right angles to its lines of electrostatic force.

To explain this diminution, it is necessary to admit FitzGerald’s hypothesis of contraction. The negative result of the experiment may be taken to indicate[77] that the kinetic potential of the system, when the FitzGerald contraction is taken into account as a ​constraint, is independent of the orientation of the plates with respect to the direction of the terrestrial motion.

It may be remarked that the existence of the couple, had it been observed, would have demonstrated the possibility of drawing on the energy of the earth’s motion for purposes of terrestrial utility.