With respect to the Schwarzschild coordinates $(r, \theta, \phi, t)$, the Schwarzschild geometry is given by

$$d s^{2}=\left(1-\frac{r_{s}}{r}\right)^{-1} d r^{2}+r^{2}\left(d \theta^{2}+\sin ^{2} \theta d \phi^{2}\right)-\left(1-\frac{r_{s}}{r}\right) d t^{2}$$

where $r_{s}=2 M$ is the Schwarzschild radius and $M$ is the Schwarzschild mass. Show that, by a suitable choice of $(\theta, \phi)$, the general geodesic can regarded as moving in the equatorial plane $\theta=\pi / 2$. Obtain the equations governing timelike and null geodesics in terms of $u(\phi)$, where $u=1 / r$.

Discuss light bending and perihelion precession in the solar system.