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 possessing any kind of motion, then relative to there exists a gravitational field which is variable with respect to space and time. This follows from a generalisation of the discussion in [chapter:XX]Section XX. The character of this field will of course depend on the motion chosen for . According to the general theory of relativity, the general law oi 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