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

104 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 {{MathForm2|(109)| $$ \left. \begin{align} ds^2 & = -d\sigma^2 + f^2dx_4^2 \\ d\sigma^2 & = \sum\limits_{1-3}\gamma_{\alpha\beta}dx_\alpha dx_\beta \end{align} \right\} $$}} 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.

$$f^2$$, $$\mu$$ 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

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,

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