(i) State and prove Birkhoff's theorem.

(ii) Derive the Schwarzschild metric and discuss its relevance to the problem of gravitational collapse and the formation of black holes.

[Hint: You may assume that the metric takes the form

$$d s^{2}=-e^{\nu(r, t)} d t^{2}+e^{\lambda(r, t)} d r^{2}+r^{2}\left(d \theta^{2}+\sin ^{2} \theta d \phi^{2}\right)$$

and that the non-vanishing components of the Einstein tensor are given by

$$\begin{aligned} & G_{t t}=\frac{e^{2 \nu+\lambda}}{r^{2}}\left(-1+e^{\lambda}+r \lambda^{\prime}\right), \quad G_{r t}=e^{(\nu+\lambda) / 2} \frac{\dot{\lambda}}{r}, \quad G_{r r}=\frac{e^{\lambda}}{r^{2}}\left(1-e^{-\lambda}+r \nu^{\prime}\right), \\ & G_{\theta \theta}=\frac{1}{4} r^{2} e^{-\lambda}\left[2 \nu^{\prime \prime}+\left(\nu^{\prime}\right)^{2}+\frac{2}{r}\left(\nu^{\prime}-\lambda^{\prime}\right)-\nu^{\prime} \lambda^{\prime}\right]-\frac{1}{4} r^{2} e^{-\nu}\left[2 \ddot{\lambda}+(\dot{\lambda})^{2}-\dot{\lambda} \dot{\nu}\right] \\ =&\left.G_{r t} \text { and } G_{\phi \phi}=\sin ^{2} \theta G_{\theta \theta} .\right] \end{aligned}$$