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

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 ${\frac {d^{2}u}{d\theta ^{2}}}+u=3mu^{2}$.

An approximate solution is $u={\frac {\cos \theta }{R}}+{\frac {m}{R^{2}}}\left(\cos ^{2}\theta +2\sin ^{2}\theta \right)$, neglecting the very small quantity $m^{2}/R^{2}$. Converting to Cartesian coordinates, this becomes $x=R-{\frac {m}{R}}{\frac {x^{2}+2y^{2}}{\sqrt {\left(x^{2}+y^{2}\right)}}}$.

The asymptotes of the light-track are found by taking $y$ very large compared with $x$, giving $x=R\pm {\frac {2m}{R}}y$ so that the angle between them is $4m/R$.

Note 10 (p. 126).

Writing the line element in the form $ds^{2}=-\left(1+a{\frac {m}{r}}+\ldots \right)dr^{2}-r^{2}\,d\theta ^{2}+\left(1+b{\frac {m}{r}}+c{\frac {m^{2}}{r^{2}}}+\ldots \right)dt^{2}$, the approximate Newtonian attraction fixes $b$ equal to + 2; then the observed deflection of light fixes $a$ equal to − 2; and with these values the observed motion of Mercury fixes $c$ equal to 0.

To insert an arbitrary coefficient of $r^{2}d\theta ^{2}$ 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.