Page:The Meaning of Relativity - Albert Einstein (1922).djvu/116

to the equation (96), but base it upon a principle of variation that is equivalent to this equation. I shall indicate the procedure only in so far as is necessary for understanding the method.

In the case of a statical field, $ds^{2}$ must have the form

$\left.{\begin{aligned}ds^{2}&=-d\sigma ^{2}+f^{2}dx_{4}^{2}\\d\sigma ^{2}&=\sum \limits _{1-3}\gamma _{\alpha \beta }dx_{\alpha }dx_{\beta }\end{aligned}}\right\}$

(109)

where the summation on the right-hand side of the last equation is to be extended over the space variables only, The central symmetry of the field requires the $\gamma _{\mu \nu }$ to be of the form.

$\gamma _{\alpha \beta }=\mu \delta _{\alpha \beta }+\lambda x_{\alpha }x_{\beta }$

(110)

$f^{2}$, $\mu$ and $\lambda$ are functions of $r={\sqrt {x_{1}^{2}+x_{2}^{2}+x_{3}^{2}}}$ only. One of these three functions can be chosen arbitrarily, because our system of co-ordinates is, a priori completely arbitrary; for by a substitution

${\begin{aligned}x_{4}'&=x_{4}\\x_{\alpha }'&=F(r)x_{\alpha }\end{aligned}}$

we can always insure that one of these three functions shall be an assigned function of $r'$. In place of (110) we can therefore put, without limiting the generality,

$\gamma _{\alpha \beta }=\delta _{\alpha \beta }+\lambda x_{\alpha }x_{\beta }.$

(110a)

In this way the $g_{\mu \nu }$ are expressed in terms of the two quantities $\lambda$ and $f$. These are to be determined as functions of $r$, by introducing them into equation (96), after