Page:The Meaning of Relativity - Albert Einstein (1922).djvu/128

Since the relation $R_{\mu \nu }={\frac {2}{a^{2}}}g_{\mu \nu }$ is universally co-variant, and since all points of the manifold are geometrically equivalent, this relation holds for every system of co-ordinates, and everywhere in the manifold. In order to avoid confusion with the four-dimensional continuum, we shall, in the following, designate quantities that refer to the three-dimensional continuum by Greek letters, and put

$P_{\mu \nu }=-{\frac {2}{a^{2}}}\gamma _{\mu \nu }.$

(120)

We now proceed to apply the field equations (96) to our special case. From (119) we get for the four-dimensional manifold,

$\left.{\begin{aligned}R_{\mu \nu }&=P_{\mu \nu }{\text{ for the indices }}1{\text{ to }}3\\R_{14}&=R_{24}=R_{34}=R_{44}=0\end{aligned}}\right\}$

(121)

For the right-hand side of (96) we have to consider the energy tensor for matter distributed like a cloud of dust. According to what has gone before we must therefore put

$T^{\mu \nu }=\sigma {\frac {dx_{\mu }}{ds}}{\frac {dx_{\nu }}{ds}}$

specialized for the case of rest. But in addition, we shall add a pressure term that may be physically established as follows. Matter consists of electrically charged particles. On the basis of Maxwell's theory these cannot be conceived of as electromagnetic fields free from singularities. In order to be consistent with the