A solution of the Einstein equations is given by the metric

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

For an incoming light ray, with constant $\theta, \phi$, show that

$$t=v-r-2 M \log \left|\frac{r}{2 M}-1\right|,$$

for some fixed $v$ and find a similar solution for an outgoing light ray. For the outgoing case, assuming $r>2 M$, show that in the far past $\frac{r}{2 M}-1 \propto \exp \left(\frac{t}{2 M}\right)$ and in the far future $r \sim t$.

Obtain the transformed metric after the change of variables $(t, r, \theta, \phi) \rightarrow(v, r, \theta, \phi)$. With coordinates $\hat{t}=v-r, r$ sketch, for fixed $\theta, \phi$, the trajectories followed by light rays. What is the significance of the line $r=2 M$ ?

Show that, whatever path an observer with initial $r=r_{0}<2 M$ takes, he must reach $r=0$ in a finite proper time.