Page:Eddington A. Space Time and Gravitation. 1920.djvu/222

The earth's speed, $v$, is approximately 30 km. per sec., or ⁠+1/10000⁠ in terms of the velocity of light. The radius of its orbit, $r$, is about 1.5 . 10⁸ km. Hence, $m$, the gravitational mass of the sun is approximately 1.5 km.

The radius of the sun is 697,000 kms., so that the quantity $2m/r$ occurring in the formulae is, for the sun's surface, .00000424 or 0″.87.

Note 9 (p. 123).

See Report, §§ 29, 30. The general equations of a geodesic are ${\frac {d^{2}x_{\mu }}{ds^{2}}}+\{a\beta ,\mu \}{\frac {dx_{a}}{ds}}{\frac {dx_{\beta }}{ds}}=0\quad {\text{(}}\mu ={\text{1, 2, 3, 4)}}$.

From the formula for the line-element $ds^{2}=-\gamma ^{-1}dr^{2}-r^{2}\,d\theta ^{2}+\gamma \,dt^{2}\quad {\text{............ (1),}}$ we calculate the three-index symbols and it is found that two of the equations of the geodesic take the rather simple form ${\begin{aligned}{\frac {d^{2}\theta }{ds^{2}}}+{\frac {2}{r}}\cdot {\frac {dr}{ds}}{\frac {d\theta }{ds}}=0{\text{,}}\\{\frac {d^{2}t}{ds^{2}}}+{\frac {d\left(\log \gamma \right)}{dr}}\cdot {\frac {dr}{ds}}{\frac {dt}{ds}}=0{\text{,}}\end{aligned}}$ which can be integrated giving ${\begin{aligned}r^{2}{\frac {d\theta }{ds}}&=h\quad {\text{...........................(2),}}\\{\frac {dt}{ds}}&={\frac {c}{\gamma }}\quad {\text{...........................(3),}}\end{aligned}}$ where $h$ and $c$ are constants of integration.

Eliminating $dt$ and $ds$ from (1), (2) and (3), we have $\left({\frac {h}{r^{2}}}{\frac {dr}{d\theta }}\right)^{2}+{\frac {h^{2}}{r^{2}}}=c^{2}-1+{\frac {2m}{r}}+{\frac {2mh^{2}}{r^{2}}}$, or writing $\left({\frac {du}{d\theta }}\right)^{2}+u^{2}={\frac {c^{2}-1}{h^{2}}}+{\frac {2mu}{h^{2}}}+2mu^{3}$.

Differentiating with respect to $\theta$ ${\frac {d^{2}u}{d\theta ^{2}}}+u={\frac {m}{h^{2}}}+3mu^{2}$,