Page:The Meaning of Relativity - Albert Einstein (1922).djvu/110

Then we get for $T^{\mu \nu }$ and $T_{\mu \nu }$ the values

$\left.{\begin{matrix}0&0&0&0\\0&0&0&0\\0&0&0&0\\0&0&0&-\sigma \end{matrix}}\right\}.$

(104)

For $T$ we get the value $\sigma$, and, finally, for $T_{\mu \nu }^{*}$ the values,

$\left.{\begin{matrix}{\frac {\sigma }{2}}&0&0&0\\0&{\frac {\sigma }{2}}&0&0\\0&0&{\frac {\sigma }{2}}&0\\0&0&0&-{\frac {\sigma }{2}}\end{matrix}}\right\}.$

(104a)

We thus get, from (101),

$\left.{\begin{aligned}\gamma _{11}=\gamma _{22}=\gamma _{33}&=-{\frac {\kappa }{4\pi }}\int {\frac {\sigma dV_{0}}{r}}\\\gamma _{44}&=+{\frac {\kappa }{4\pi }}\int {\frac {\sigma dV_{0}}{r}}\end{aligned}}\right\}$

(101a)

while all the other $\gamma _{\mu \nu }$ vanish. The least of these equations, in connexion with equation (90a), contains Newton's theory of gravitation. If we replace $l$ by $ct$ we get

${\frac {d^{2}x_{\mu }}{dt^{2}}}={\frac {\kappa c^{2}}{8\pi }}{\frac {\delta }{\delta x_{\mu }}}\left\{\int {\frac {\sigma dV_{0}}{r}}\right\}.$

(90b)

We see that the Newtonian gravitation constant $K$, is connected with the constant $\kappa$ that enters into our field equations by the relation

$K={\frac {\kappa c^{2}}{8\pi }}.$

(105)