Page:The Meaning of Relativity - Albert Einstein (1922).djvu/114

Another important consequence of the theory, which can be tested experimentally, has to do with the path of rays of light. In the general theory of relativity also the velocity of light is everywhere the same, relatively to a local inertial system. This velocity is unity in our natural measure of time. The law of the propagation of light in general co-ordinates is therefore, according to the general theory of relativity, characterized, by the equation

$ds^{2}=0.$

To within the approximation which we are using, and in the system of co-ordinates which we have selected, the velocity of light is characterized, according to (106), by the equation

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

The velocity of light $L$, is therefore expressed in our co-ordinates by

${\frac {\sqrt {dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}}}{dl}}=1-{\frac {\kappa }{4\pi }}\int {\frac {\sigma dV_{0}}{r}}.$

(107)

We can therefore draw the conclusion from this, that a ray of light passing near a large mass is deflected. If we imagine the sun, of mass $M$ concentrated at the origin of our system of co-ordinates, then a ray of light, travelling parallel to the $x_{3}$-axis, in the $x_{1}-x_{3}$ plane, at a distance $\Delta$ from the origin, will be deflected, in all, by an amount