Page:The Meaning of Relativity - Albert Einstein (1922).djvu/106

Introducing (88) and (70) into this expression, we see that the only terms that remain are those in which third derivatives of the $g^{\mu \nu }$ enter. Since the $g_{\mu \nu }$ are to be replaced by $-\delta _{\mu \nu }$, we obtain, finally, only a few terms which may easily be seen to cancel each other. Since the quantity that we have formed has a tensor character, its vanishing is proved for every other system of co-ordinates also, and naturally for every other four-dimensional point. The energy principle of matter (97) is thus a mathematical consequence of the field equations (96).

In order to learn whether the equations (96) are consistent with experience, we must, above all else, find out whether they lead to the Newtonian theory as a first approximation. For this purpose we must introduce various approximations into these equations. We already know that Euclidean geometry and the law of the constancy of the velocity of light are valid, to a certain approximation, in regions of a great extent, as in the planetary system. If, as in the special theory of relativity, we take the fourth co-ordinate imaginary, this means that we must put

$g_{\mu \nu }=-\delta _{\mu \nu }+\gamma _{\mu \nu }$

(98)

in which the $\gamma _{\mu \nu }$, are so small compared to 1 that we can neglect the higher powers of the $\gamma _{\mu \nu }$ and their derivatives. If we do this, we learn nothing about the structure of the gravitational field, or of metrical space of cosmical dimensions, but we do learn about the influence of neighbouring masses upon physical phenomena.

Before carrying through this approximation we shall