Page:The Meaning of Relativity - Albert Einstein (1922).djvu/112

the naturally measured lengths and times. Since $ds^{2}$, on the other hand, is known in terms of the co-ordinates $x_{\nu }$ employed in finite regions, in the form

$ds^{2}=g_{\mu \nu }dx_{\mu }dx_{\nu }$

we have the possibility of getting the relation between naturally measured lengths and times, on the one hand, and the corresponding differences of co-ordinates, on the other hand. As the division into space and time is in agreement with respect to the two systems of co-ordinates, so when we equate the two expressions for $ds^{2}$ we get two relations. If, by (101a), we put

$ds^{2}=-\left(1+{\frac {\kappa }{4\pi }}\int {\frac {\sigma dV_{0}}{r}}\right)\left(dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}\right)+\left(1-{\frac {\kappa }{4\pi }}\int {\frac {\sigma dV_{0}}{r}}\right)dl^{2}$

we obtain, to a sufficiently close approximation,

$\left.{\begin{aligned}{\sqrt {dX_{1}^{2}+dX_{2}^{2}+dX_{3}^{2}}}&=\left(1+{\frac {\kappa }{8\pi }}\int {\frac {\sigma dV_{0}}{r}}\right){\sqrt {dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}}}\\dT&=\left(1-{\frac {\kappa }{8\pi }}\int {\frac {\sigma dV_{0}}{r}}\right)dl.\end{aligned}}\right\}.$

(106)

The unit measuring rod has therefore the length,

$1-{\frac {\kappa }{8\pi }}\int {\frac {\sigma dV_{0}}{r}}$

in respect to the system of co-ordinates we have selected. The particular system of co-ordinates we have selected