Page:The Meaning of Relativity - Albert Einstein (1922).djvu/107

transform (96). We multiply (96) by $g^{\mu \nu }$ summed over the $\mu$ and $\nu$; observing the relation which follows from the definition of the $g^{\mu \nu }$,

$g_{\mu \nu }g^{\mu \nu }=4$

we obtain the equation

$R=\kappa g^{\mu \nu }T_{\mu \nu }=\kappa T.$

If we put this value of $R$ in (96) we obtain

$R_{\mu \nu }=-\kappa \left(T_{\mu \nu }-{\frac {1}{2}}g_{\mu \nu }T\right)=-\kappa T_{\mu \nu }^{*}.$

(96a)

When the approximation which has been mentioned is carried out, we obtain for the left-hand side,

$-{\frac {1}{2}}\left({\frac {\delta ^{2}\gamma _{\mu \nu }}{\delta x_{\alpha }^{2}}}+{\frac {\delta ^{2}\gamma _{\alpha \alpha }}{\delta x_{\mu }\delta x_{\nu }}}-{\frac {\delta ^{2}\gamma _{\mu \alpha }}{\delta x_{\nu }\delta x_{\alpha }}}-{\frac {\delta ^{2}\gamma _{\nu \alpha }}{\delta x_{\mu }\delta x_{\alpha }}}\right)$

or

$-{\frac {1}{2}}{\frac {\delta ^{2}\gamma _{\mu \nu }}{\delta x_{\alpha }^{2}}}+{\frac {1}{2}}{\frac {\delta }{\delta x_{\nu }}}\left({\frac {\delta \gamma _{\mu \alpha }'}{\delta x_{\alpha }}}\right)+{\frac {1}{2}}{\frac {\delta }{\delta x_{\mu }}}\left({\frac {\delta \gamma _{\nu \alpha }'}{\delta x_{\alpha }}}\right)$

in which has been put

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

(99)

We must now note that equation (96) is valid for any system of co-ordinates. We have already specialized the system of co-ordinates in that we have chosen it so that within the region considered the $g_{\mu \nu }$ differ infinitely little from the constant values $-\delta _{\mu \nu }$. But this condition remains satisfied in any infinitesimal change of co-ordinates, so that there are still four conditions to which the $\gamma _{\mu \nu }$ may be subjected, provided these conditions do not conflict with the conditions for the order of magnitude