a few inferences from the general principle of relativity

the considerations of section xx show that the general principle of relativity puts us in a position to derive properties of the gravitational field in a purely theoretical manner. let us suppose, for instance, that we know the space-time “course” for any natural process whatsoever, as regards the manner in which it takes place in the galileian domain relative to a galileian body of reference k. by means of purely theoretical operations (i.e. simply by calculation) we are then able to find how this known natural process appears, as seen from a reference-body k′ which is accelerated relatively to k. but since a gravitational field exists with respect to this new body of reference k, our consideration also teaches us how the gravitational field influences the process studied.

for example, we learn that a body which is in a state of uniform rectilinear motion with respect to k (in accordance with the law of galilei) is executing an accelerated and in general curvilinear motion with respect to the accelerated reference-body k′ (chest). this acceleration or curvature corresponds to the influence on the moving body of the gravitational field prevailing relatively to k. it is known that a gravitational field influences the movement of bodies in this way, so that our consideration supplies us with nothing essentially new.

however, we obtain a new result of fundamental importance when we carry out the analogous consideration for a ray of light. with respect to the galileian reference-body k, such a ray of light is transmitted rectilinearly with the velocity c. it can easily be shown that the path of the same ray of light is no longer a straight line when we consider it with reference to the accelerated chest (reference-body k′). from this we conclude, that, in general, rays of light are propagated curvilinearly in gravitational fields. in two respects this result is of great importance.

in the first place, it can be compared with the reality. although a detailed examination of the question shows that the curvature of light rays required by the general theory of relativity is only exceedingly small for the gravitational fields at our disposal in practice, its estimated magnitude for light rays passing the sun at grazing incidence is nevertheless 1.7 seconds of arc. this ought to manifest itself in the following way. as seen from the earth, certain fixed stars appear to be in the neighbourhood of the sun, and are thus capable of observation during a total eclipse of the sun. at such times, these stars ought to appear to be displaced outwards from the sun by an amount indicated above, as compared with their apparent position in the sky when the sun is situated at another part of the heavens. the examination of the correctness or otherwise of this deduction is a problem of the greatest importance, the early solution of which is to be expected of astronomers.1

in the second place our result shows that, according to the general theory of relativity, the law of the constancy of the velocity of light in vacuo, which constitutes one of the two fundamental assumptions in the special theory of relativity and to which we have already frequently referred, cannot claim any unlimited validity. a curvature of rays of light can only take place when the velocity of propagation of light varies with position. now we might think that as a consequence of this, the special theory of relativity and with it the whole theory of relativity would be laid in the dust. but in reality this is not the case. we can only conclude that the special theory of relativity cannot claim an unlimited domain of validity; its results hold only so long as we are able to disregard the influences of gravitational fields on the phenomena (e.g. of light).

since it has often been contended by opponents of the theory of relativity that the special theory of relativity is overthrown by the general theory of relativity, it is perhaps advisable to make the facts of the case clearer by means of an appropriate comparison. before the development of electrodynamics the laws of electrostatics were looked upon as the laws of electricity. at the present time we know that electric fields can be derived correctly from electrostatic considerations only for the case, which is never strictly realised, in which the electrical masses are quite at rest relatively to each other, and to the co-ordinate system. should we be justified in saying that for this reason electrostatics is overthrown by the field-equations of maxwell in electrodynamics? not in the least. electrostatics is contained in electrodynamics as a limiting case; the laws of the latter lead directly to those of the former for the case in which the fields are invariable with regard to time. no fairer destiny could be allotted to any physical theory, than that it should of itself point out the way to the introduction of a more comprehensive theory, in which it lives on as a limiting case.

in the example of the transmission of light just dealt with, we have seen that the general theory of relativity enables us to derive theoretically the influence of a gravitational field on the course of natural processes, the laws of which are already known when a gravitational field is absent. but the most attractive problem, to the solution of which the general theory of relativity supplies the key, concerns the investigation of the laws satisfied by the gravitational field itself. let us consider this for a moment.

we are acquainted with space-time domains which behave (approximately) in a “galileian” fashion under suitable choice of reference-body, i.e. domains in which gravitational fields are absent. if we now refer such a domain to a reference-body k′ possessing any kind of motion, then relative to k′ there exists a gravitational field which is variable with respect to space and time.2 the character of this field will of course depend on the motion chosen for k′. according to the general theory of relativity, the general law of the gravitational field must be satisfied for all gravitational fields obtainable in this way. even though by no means all gravitational fields can be produced in this way, yet we may entertain the hope that the general law of gravitation will be derivable from such gravitational fields of a special kind. this hope has been realised in the most beautiful manner. but between the clear vision of this goal and its actual realisation it was necessary to surmount a serious difficulty, and as this lies deep at the root of things, i dare not withhold it from the reader. we require to extend our ideas of the space-time continuum still farther.

1 by means of the star photographs of two expeditions equipped by a joint committee of the royal and royal astronomical societies, the existence of the deflection of light demanded by theory was first confirmed during the solar eclipse of 29th may, 1919. (cf. appendix iii.) ↑

2 this follows from a generalisation of the discussion in section xx.