Page:The Meaning of Relativity - Albert Einstein (1922).djvu/108

of the $\gamma _{\mu \nu }$. We shall now assume that the system of co-ordinates is so chosen that the four relations—

$0={\frac {\delta \gamma _{\mu \nu }'}{\delta x_{\nu }}}={\frac {\delta \gamma _{\mu \nu }}{\delta x_{\nu }}}-{\frac {1}{2}}{\frac {\delta \gamma _{\sigma \sigma }}{\delta x_{\mu }}}$

(100)

are satisfied. Then (96a) takes the form

${\frac {\delta ^{2}\gamma _{\mu \nu }}{\delta x_{\alpha }^{2}}}=2\kappa T_{\mu \nu }^{*}.$

(96b)

These equations may be solved by the method, familiar in electrodynamics, of retarded potentials; we get, in an easily understood notation,

$\gamma _{\mu \nu }=-{\frac {\kappa }{2\pi }}\int {\frac {T_{\mu \nu }^{*}(x_{0},y_{0},z_{0},t-r)}{r}}dV_{0}.$

(101)

In order to see in what sense this theory contains the Newtonian theory, we must consider in greater detail the energy tensor of matter. Considered phenomenologically, this energy tensor is composed of that of the electromagnetic field and of matter in the narrower sense. If we consider the different parts of this energy tensor with respect to their order of magnitude, it follows from the results of the special theory of relativity that the contribution of the electromagnetic field practically vanishes in comparison to that of ponderable matter. In our system of units, the energy of one gram of matter is equal to 1, compared to which the energy of the electric fields may be ignored, and also the energy of deformation of matter, and even the chemical energy. We get an approximation that is fully sufficient for our purpose if