Page:Eddington A. Space Time and Gravitation. 1920.djvu/223

Rh which gives the equation of the orbit in the usual form in particle dynamics. It differs from the equation of the Newtonian orbit by the small term $$3mu^2$$, which is easily shown to give the motion of perihelion.

The track of a ray of light is also obtained from this formula, since by the principle of equivalence it agrees with that of a material particle moving with the speed of light. This case is given by $$ds = 0$$, and therefore $$h = \infty$$. The differentia] equation for the path of a light-ray is thus

An approximate solution is neglecting the very small quantity $$m^2 /R^2$$. Converting to Cartesian coordinates, this becomes

The asymptotes of the light-track are found by taking $$y$$ very large compared with $$x$$, giving so that the angle between them is $$4m/R$$.

Note 10 (p. 126).

Writing the line element in the form the approximate Newtonian attraction fixes $$b$$ equal to + 2; then the observed deflection of light fixes $$a$$ equal to &minus; 2; and with these values the observed motion of Mercury fixes $$c$$ equal to 0.

To insert an arbitrary coefficient of $$r^2d\theta^2$$ would merely vary the coordinate system. We cannot arrive at any intrinsically different kind of space-time in that way. Hence, within the limits of accuracy mentioned, the expression found by Einstein is completely determinable by observation.