On General Relativity

by Harry Bateman

On General Relativity.

To the Editors of the Philosophical Magazine.

Gentlemen,—

THE appearance of Dr. Silberstein's recent article^[1] on "General Relativity without the Equivalence Hypothesis" encourages me to restate my own views on the subject. I am perhaps entitled to do this as my work on the subject of General Relativity was published before that of Einstein and Kottler, and appears to have been overlooked by recent writers. In 1909 I proposed a scheme of electromagnetic equations^[2] which are covariant for all transformations of co-ordinates which are biuniform in the domain we are interested in. These equations were similar to Maxwell's equations, except that the familiar relations B = μH, D = kE of Maxwell's theory were replaced by more general equations, which implied that two fundamental integral forms were reciprocals with regard to a quadratic differential form

$\sum \sum g_{m,n}dx_{m}dx_{n}$

which was assumed to be invariant for all transformations of co-ordinates. The coefficients of the quadratic form were regarded as characteristics of the medium supporting the electromagnetic field and of the motion of the medium and its parts. The vanishing of the quadratic form was regarded as the condition that two neighbouring particles should be in positions such that a disturbance starting from one at the associated time should arrive at the other at its associated time^[3].

The idea that the coefficients of the quadratic form might be considered as characteristics of the mind interpreting the phenomena was also entertained^[4], and it was suggested that a correspondence or transformation of co-ordinates might be considered as a crude mathematical symbol for a mind.

The phenomena here considered were those occurring in the brain and body; and although the correspondence by which the universe is reconstructed, so to speak, may be totally different^[5] from the type contemplated here, yet it was thought that some of the general conclusions might still be valid if a transformation of co-ordinates was adopted as a working model of the correspondence. It was thought, for instance, that there was an analogy between the relativity principle that the earth's motion in space cannot be detected from experiments with terrestrial objects: and the interesting fact that we are unaware of the flow of blood and other processes taking place in our own bodies so long as they take place in the normal way. It was thought that the correspondence by which the elementary processes in the brain are interpreted may be adjusted in such a way that some of the changes are obscured.

Again, if we assume that the nature of an electromagnetic field depends on the type of fundamental quadratic form which determines the constitutive relations, and thus depends indirectly on a transformation which filters the coefficients of this quadratic form, this dependence may be a symbol for the relation between physical and mental phenomena instead of giving the influence of gravitation on light as in Einstein's theory.

Einstein and others have attempted to formulate a set of equations of motion which will cover all physical phenomena; but the present writer does not feel inclined to accept them as final, because in his opinion the true equations of motion should be capable of accounting for the phenomena of life, which after all are the most important physical phenomena.

To make my position more definite, let us consider one of the methods by which the equations of motion of an electron are obtained in the usual electromagnetic theory. The principle is adopted that at each instant the integral over the electron of the total force on each element must be zero. Now before this principle can be used to write down the equations of motion we must know the design of the electron, and we must know the way in which the motions of the different elements are co-ordinated. This co-ordination or organization of the motions of the elements may be represented mathematically by a sequence of infinitesimal transformations, by which some of the features of the design are preserved. The design of the electron and the co-ordinated motion of its parts may, perhaps, be specified by a quadratic differential form in four variables, which determines a mapping of the interior of the electron on the interior of a stationary sphere; but I doubt if this is sufficiently general. A knowledge of this quadratic differential form is necessary then before we can write down the equations of motion of the electron as a whole. What we usually regard as the equations of motion of matter need then to be supplemented by geometrical conditions which specify the design and organization of each elementary portion of matter. Furthermore, when this design and organization is assumed to be known, the ordinary equations of motion may be regarded as a consequence of the electromagnetic laws and the above-mentioned principle.

It must be confessed, however, that this principle does not seem satisfactory for a fundamental principle, and is probably a consequence of some deep underlying principles which are the true equations of motion. These new principles should indicate the reason for a similarity of design of the different electrons. One of the fundamental facts of life is that a good design is copied, and that there is a certain characteristic of the design of an object and its surrounding medium, depending perhaps on the closeness of fit of an imperfect correspondence, which determines the extent to which the design of the object is copied and preserved in the surrounding medium. This may be called the value of the design in relation to the medium, and it is a quantity which I feel must be taken into account in the true equations of motion, and a number assigned to it at each instant. As an example of standardization, the Ford motor-car is not in it with the electron; and, according to the above view, we must regard the design of the electron as one of very great value in relation to the surrounding medium.

Returning to our generalized scheme of electromagnetic equations, and looking at matters from the point of view of physical optics, it may be remarked that the scheme of constitutive relations mentioned above is not sufficiently general to cover the case of a doubly-refracting crystalline medium^[6]. To remedy this defect we may use a biquadratic integral form instead of a quadratic differential form to specify the constitutive relations. The vanishing of the biquadratic integral form may perhaps be regarded as the condition for action of a moving curve on a particle, a type of condition that seems natural if we regard moving Faraday tubes as fundamental. With this generalized theory it is possible for the elementary wave surface in a medium to be a general Kummer surface, a surface of which Fresnel's wave surface is a particular case. It is doubtful whether this generalized theory is sufficiently general for all purposes, and the above example is given just to emphasize that the absolute calculus of Ricci and Levi Civita can be used to develop a theory of generalized relativity on many lines in addition to that adopted by Einstein.

Going back to the case in which a quadratic form is sufficient to determine the optical properties of a medium, we may remark that if Einstein's idea of the gravitational equations is accepted, it is still by no means certain that his quadratic form from which the gravitational equations are derived is the same as the quadratic form which determines the optical properties of the medium. Indeed, the example which I considered on p. 262 of my first paper would seem to indicate that this was not the case. It should be mentioned that in the first seven equations in this example there is a misprint, $\epsilon \mu$ should be replaced by $(\epsilon \mu )^{-1}$. On the above view Einstein's idea of an influence of gravitation on light is simply an hypothesis, but a very interesting and reasonable one. It may be remarked, however, that in the theory of surfaces there are two fundamental quadratic forms, and we may perhaps expect something similar in general relativity.

With regard to possible extensions of the idea of relativity it may be worth while to consider transformations analogous to the contact transformations of dynamics in which the co-ordinates x, y, z, t and the component velocities u, v, w correspond to a new set $\left(x_{1}y_{1}z_{1}t_{1}u_{1}v_{1}w_{1}\right)$ in such a way that the differential equations

${\frac {dx_{1}}{u_{1}}}={\frac {dy_{1}}{v_{1}}}={\frac {dz_{1}}{w_{1}}}=dt_{1}$

are a consequence of the equations

${\frac {dx}{u}}={\frac {dy}{v}}={\frac {dz}{w}}=dt$

This may be secured by making a single quadratic form, such as

${\begin{array}{cc}\left(dx^{2}+dy^{2}+dz^{2}-c^{2}dt^{2}\right)\left(c^{2}-u^{2}-v^{2}-w^{2}\right)\\+\left(c^{2}dt-udx-vdy-wdz\right)^{2},&\left(c^{2}>u^{2}+v^{2}+w^{2}\right)\end{array}}$

an invariant^[7]. Various other quadratic forms consisting of sums of squares may, of course, be adopted instead.

H. Bateman.

Throop College.

Pasandena, Cal.

Aug. 10th, 1918.

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1. Phil. Mag. July 1918
2. Proc. Lond. Math. Soc. ser. 2, vol. viii. p. 223 (1910).
3. Loc. cit. p. 225. See also Amer. Journ. of Math. vol. xxxiv. p. 340 (1912).
4. Memoirs of the Manchester Literary and Philosophical Society (1910). A full account of my ideas has not yet been published, owing to the difficulty of eliminating vagueness.
5. The term correspondence is used here in a very general sense, and is by no means restricted to the familiar one to one correspondence of entities of the same type, such as points. We should say, for instance, that there is a correspondence between the disturbance running along a telephone-wire and the sound-waves which produce it, because we can pass from one to the other by mathematical equations of a definite type, or rather by solving the equations and the boundary conditions. A correspondence is, moreover, regarded as an entity which may have real existence and be capable of growth and variation.
6. Proc. London Math. Soc. ser. 2, vol. viii. p. 375. See also p. 261 of my first paper.
7. This is a positive definite quadratic form in the variables dx - udt, dy - vdt, dz - wdt, and so can only vanish when all these quantities are zero.