Page:Eddington A. Space Time and Gravitation. 1920.djvu/124

travel along these tracks, and the motion of a light-pulse is always given by the equation $ds=0$.

Using the expression for $ds^{2}$ in a gravitational field, we accordingly have for light $0=-{\frac {1}{\gamma }}\,dr^{2}--r^{2}d\theta ^{2}+\gamma \,dt^{2}$. For radial motion, $d\theta =0$, and therefore $\left({\frac {dr}{dt}}\right)^{2}=\gamma ^{2}$. For transverse motion, $dr=0$, and therefore $\left({\frac {r\,d\theta }{dt}}\right)^{2}=\gamma$. Thus the coordinate velocity of light travelling radially is $\gamma$, and of light travelling transversely is ${\sqrt {\gamma }}$, in the coordinates chosen.

The coordinate velocity must depend on the coordinates chosen; and it is more convenient to use a slightly different system in which the velocity of light is the same in all directions^[1], viz. $\gamma$ or $1-2m/r$. This diminishes as we approach the sun—an illustration of our previous remark that a pulse of light proceeding radially is repelled by the sun.

The wave-motion in a ray of light can be compared to a succession of long straight waves rolling onward in the sea. If the motion of the waves is slower at one end than the other, the whole wave-front must gradually slew round, and the direction in which it is rolling must change. In the sea this happens when one end of the wave reaches shallow water before the other, because the speed in shallow water is slower. It is well known that this causes waves proceeding diagonally across a bay to slew round and come in parallel to the shore; the advanced end

1. This is obtained by writing $r+m$ instead of $r$, or diminishing the nominal distance of the sun by 1 1/2 kilometres. This change of coordinates simplifies the problem, but can, of course, make no difference to anything observable. After we have traced the course of the light ray in the coordinates chosen, we have to connect the results with experimental measures, using the corresponding formula for $ds^{2}$. This final connection of mathematical and experimental results is, however, comparatively simple, because it relates to measuring operations performed in a terrestrial observatory where the difference of $\gamma$ from unity is negligible.