Page:The Meaning of Relativity - Albert Einstein (1922).djvu/105

of the gravitational field we therefore get the equation

$R_{\mu \nu }-{\frac {1}{2}}g_{\mu \nu }R=-\kappa T_{\mu \nu }.$

(96)

Equation (95) is a consequence of this equation, $\kappa$ denotes a constant, which is connected with the Newtonian gravitation constant.

In the following I shall indicate the features of the theory which are interesting from the point of view of physics, using as little as possible of the rather involved mathematical method. It must first be shown that the divergence of the left-hand side actually vanishes. The energy principle for matter may be expressed, by (83),

$0={\frac {\delta {\mathfrak {T}}_{\sigma }^{\alpha }}{\delta x_{\alpha }}}-\Gamma _{\sigma \beta }^{\alpha }{\mathfrak {T}}_{\alpha }^{\beta }$

(97)

in which

${\mathfrak {T}}_{\sigma }^{\alpha }=T_{\sigma \tau }g^{\tau \alpha }{\sqrt {-g}}$.

The analogous operation, applied to the left-hand side of (96), will lead to an identity.

In the region surrounding each world-point there are systems of co-ordinates for which, choosing the $x_{\mu }$-co-ordinate imaginary, at the given point,

$g_{\mu \nu }=g^{\mu \nu }=-\delta _{\mu \nu }{\begin{cases}=1&{\text{if }}\mu =\nu \\=0&{\text{if }}\mu \neq \nu ,\end{cases}}$

and for which the first derivatives of the $g_{\mu \nu }$ and the $g^{\mu \nu }$ vanish. We shall verify the vanishing of the divergence of the left-hand side at this point. At this point the components $\Gamma _{\sigma \beta }^{\alpha }$ vanish, so that we have to prove the vanishing only of

${\frac {\delta }{\delta x_{\sigma }}}\left[{\sqrt {-g}}g^{\nu \sigma }\left(R_{\mu \nu }-{\frac {1}{2}}g_{\mu \nu }R\right)\right].$