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47
Relativity: The Special and General Theory
Albert Einstein: Relativity
Part II: The General Theory of Relativity
A
Few Inferences
from the General Principle
of Relativity
The considerations of Section 20 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 K¹ which is accelerated
relatively to K. But since a gravitational field exists with respect to this new body of reference K¹,
our consideration also teaches us how the gravitational field influences the process studied.
For example, we Wayrn 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 K¹ (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 K¹). 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 genernal 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 unlinlited 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).