The Inclusion of Gravitation
With the idea of investigating the problem from the very bottom, Einstein now undertook a broader and more daring point of view. In the first place he said that there is no apparent reason in the great scheme of world events why any one special system of coordinates should be fundamental to the description of phenomena, just as in the special theory a ray of light would appear the same whether viewed from a fixed system or a system moving with constant velocity with respect to the ether. This makes the very broad assumption that no matter what system of coordinates we may use, the mathematical expressions for the laws of nature must be the same. In Einstein’s own words, then, the first principle of this more general theory of relativity must be the following:
“The general laws of nature are expressed through equations which hold for all systems of coordinates, that is, they are covariant with respect to arbitrary substitutions.”[3]
But this was not enough to include gravitation so Einstein next formulated what he was pleased to call his “equivalence hypothesis.” This is best illustrated by an example. Suppose that we are mounting in an elevator and wish to investigate the world of events from our moving platform. We mount more and more rapidly, that is with constant acceleration, and we appear to be in a strong gravitational field due to our own inertia. Suppose, on the other hand, that the elevator descends with an acceleration equal to that of gravity. We would now feel certain that we were in empty space because our own relative acceleration has entirely destroyed that of the earth’s gravitational field and all objects placed upon scales in an elevator would apparently be without weight.
Applying this idea, then, Einstein decided to do away with gravitation entirely by referring all events in a gravitational field to a new set of axes which should move with constant acceleration with respect to the first. In other words we are going to deal with a system moving with uniform acceleration with respect to the ether, just as we considered a system moving with uniform velocity in the special theory.
The next step in the construction of this complicated theory is to reduce these two hypotheses to the language of mathematics and this was accomplished by Einstein with the help of M. Grossmann by means of the theory of tensors.
On account of the very great intricacy of the details, we must content ourselves with the mere statement that this really involved the generalization of the famous expressions known as Laplace’s and Poisson’s equations, on the explicit assumption that these two equations would still describe the gravitational field when we are content to use a first approximation to the truth. The set of ten differential equations which Einstein got as a result of his generalization he called his field equations of gravitation.[4]
[1] Dr. Davis went rather fully into the algebra of the Michelson-Morley experiment. But Dr. Russell has covered the same ground in a form somewhat more advantageous from the typographical viewpoint, and the point is not one which it is profitable to discuss twice; so we eliminate this part of Dr. Davis’ text.—Editor. [↑]