The Gravitational Hypothesis
Einstein flatly denies Newton’s hypothesis that there is an absolute system (and, indeed, many others before him had found it difficult to admit that so insignificant a part of the universe as our fixed star system should have such a privileged position as that accorded to it in the Newtonian Mechanics). In any system, he says, we have no reason to distinguish between the so-called real gravitational force and the so-called fictitious centrifugal forces—if we wish so to express it gravitational force is fictitious force.[1] A particle moving in the neighborhood of material bodies moves according to a law of inertia—a physical law expressible, therefore, in a manner quite independent of the choice of coordinates. The law of inertia is that a particle left to itself moves along the geodesics or shortest lines in the space. If the particle is remote from other bodies the space has the Euclidean character and we have Newton’s law of inertia; otherwise the particle is in a space of a non-Euclidean character (the space being always the four-dimensional space) and the path of the particle is along a geodesic in that space. Einstein, in order to make the theory more concrete, makes a certain stipulation as to the nature of the gravitational space which stipulation is expressed, as are all physical laws, by means of a tensor equation—and this is sometimes called his law of gravitation.
Perhaps it will be well, in exemplification, to explain why light rays, which pass close to the sun, should be bent according to the new theory. It is assumed that light rays travel along certain geodesics known as minimal geodesics. The sun has an intense gravitational field near it—or, as we now say, the departure of the four-dimensional space from the Euclidean is very marked for points near the sun—but for points so remote as the earth this departure is so small as to be negligible. Hence the form of the geodesics near the sun is different from that near the earth. If the space surrounding the sun were Euclidean the actual paths of the light rays would appear different from geodesics or straight-lines. Hence Einstein speaks of the curvature of the light rays due to the gravitational field of the sun; but we must not be misled by a phrase. Light always travels along geodesics (or straight lines—the only definition we have of a straight line is that it is a geodesic); but, owing to the “distortion” of the space they traverse, due to the sun, these geodesics reach us with a direction different from that they would have if they did not pass through the markedly non-Euclidean space near the sun.
The consideration of the fundamental four-dimensional space as being non-Euclidean where matter is present gives a possibility of an answer to the world old question: Is space finite or infinite? Is time eternal or finite? The fascinating possibility arises that the space may be like the two-dimensional surface of a sphere which to a limited experience seems infinite in extent and flat or Euclidean in character. A new Columbus now asks us to consider other possibilities in which we should have a finite universe—finite not only as to space measurement but as to time (for the space may be such that all of the four coordinates of its points are bounded in magnitude). However, although Einstein speaks of the possibility of a finite universe, we do not, personally, think his argument convincing. Points on a sphere may be located by the Cartesian coordinates of their stereographic projections on the equatorial plane and these coordinates, which might well be those actually measured, are not bounded.
The Special Relativity Theory
In our account of the Einstein theory we have not followed its historical order of development for two reasons. Firstly, the earlier Special Relativity Theory properly belongs to a school of thought diametrically opposed to that furnishing the “General Theory of Relativity” and, secondly, the latter cannot be obtained from the former by the process of generalization as commonly understood. Einstein, when proposing the earlier theory, adopted the position of the empiricist so that to him the phrase, a point in space, had no meaning without a material framework of reference in which to measure space distances. When he came to investigate what is meant by time and when he asked the question “what is meant by the statement that two remote events are simultaneous?” it became evident that some mode of communication between the two places is necessary; the mode adopted was that by means of light-signals. The fundamental hypothesis was then made that the velocity of such signals is independent of the velocity of their source (some hypothesis is necessary if we wish to compare the time associated with events, when one material reference-system is used, and the corresponding time when another in motion relative to the first is adopted). It develops that time and space measurements are inextricably interwoven; there is no such thing as the length of a body or the duration of an event but rather these are relative to the reference-system.[2] Minkowski introduced the idea of the space of events—of four dimensions—but this space was supposed Euclidean like the three-dimensional space of his predecessors. To Einstein belongs the credit of taking from this representation a purely formal mathematical character and of insisting that the “real” space—whose distances have a physical significance—is the four-dimensional space. But we cannot insist too strongly on the fact that in the gravitational space of the general theory there is no postulate of the constancy of velocity of a light-signal and accordingly no method of assigning a time to events corresponding to that adopted in the special theory. In this latter theory attention was confined to material systems moving with uniform velocity with respect to each other and it developed that the velocity of light was the ultimate velocity faster than which no system could move—a result surprising and a priori rather repugnant. It is merely a consequence of our mode of comparing times of events; if some other method—thought transference, let us say—were possible the velocity of this would be the “limiting velocity.”
In conclusion we should remark that the postulated equivalence of “gravitational” and “centrifugal” forces demands that anything possessed of inertia will be acted upon by a gravitational field and this leads to a possible identification of matter and energy. Further our guiding idea (a) will prompt us to say, following the example of Faraday in his electrical researches, that the geodesics of a gravitational space have a physical existence as distinct from a mere mathematical one. The four-dimensional space we may call the ether, and so restore this bearer of physical forces to the position it lost when, as a three-dimensional idea in the Special Relativity Theory, it had to bear an identical relation to a multitude of relatively moving material systems. The reason for our seemingly paradoxical title for an essay on Relativity will be clear when it is remembered that in the new theory we consider those space-time properties which are absolute or devoid of reference to any particular material reference-frame. Nevertheless, although the general characteristics of the theory are thus described, without reference to experiment, when the theory is to be tested it is necessary to state what the four coordinates discussed actually are—how they are determined by measurement. It is our opinion that much remains to be done to place this portion of the subject on a satisfactory basis. For example, in the derivation of the nature of the gravitational space, surrounding a single attracting body, most of the accounts use Cartesian coordinates as if the space were Euclidean and step from these to polar coordinates by the formulæ familiar in Euclidean geometry. But these details are, perhaps, like matters of elegance, if we shall be allowed to give Einstein’s quotation from Boltzmann, to be left to the “tailor and the cobbler.”