(i) The Schwarzschild metric is given by

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

Consider a time-like geodesic $x^{a}(\tau)$, where $\tau$ is the proper time, lying in the plane $\theta=\pi / 2$. Use the Lagrangian $L=g_{a b} \dot{x}^{a} \dot{x}^{b}$ to derive the equations governing the geodesic, showing that

$$r^{2} \dot{\phi}=h$$

with $h$ constant, and hence demonstrate that

$$\frac{d^{2} u}{d \phi^{2}}+u=\frac{M}{h^{2}}+3 M u^{2}$$

where $u=1 / r$. State which term in this equation makes it different from an analogous equation in Newtonian theory.

(ii) Now consider Kruskal coordinates, in which the Schwarzschild $t$ and $r$ are replaced by $U$ and $V$, defined for $r>2 M$ by

$$\begin{aligned} &U \equiv\left(\frac{r}{2 M}-1\right)^{1 / 2} e^{r /(4 M)} \cosh \left(\frac{t}{4 M}\right) \\ &V \equiv\left(\frac{r}{2 M}-1\right)^{1 / 2} e^{r /(4 M)} \sinh \left(\frac{t}{4 M}\right) \end{aligned}$$

and for $r<2 M$ by

$$\begin{aligned} &U \equiv\left(1-\frac{r}{2 M}\right)^{1 / 2} e^{r /(4 M)} \sinh \left(\frac{t}{4 M}\right) \\ &V \equiv\left(1-\frac{r}{2 M}\right)^{1 / 2} e^{r /(4 M)} \cosh \left(\frac{t}{4 M}\right) \end{aligned}$$

Given that the metric in these coordinates is

$$d s^{2}=\frac{32 M^{3}}{r} e^{-r /(2 M)}\left(-d V^{2}+d U^{2}\right)+r^{2}\left(d \theta^{2}+\sin ^{2} \theta d \phi^{2}\right),$$

where $r=r(U, V)$ is defined implicitly by

$$\left(\frac{r}{2 M}-1\right) e^{r /(2 M)}=U^{2}-V^{2}$$

sketch the Kruskal diagram, indicating the positions of the singularity at $r=0$, the event horizon at $r=2 M$, and general lines of constant $r$ and of constant $t$.