The metric of the Schwarzschild solution is

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

Show that, for an incoming radial light ray, the quantity

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

is constant.

Express $d s^{2}$ in terms of $r, v, \theta$ and $\phi$. Determine the light-cone structure in these coordinates, and use this to discuss the nature of the apparent singularity at $r=2 M$.

An observer is falling radially inwards in the region $r<2 M$. Assuming that the metric for $r<2 M$ is again given by $(*)$, obtain a bound for $d \tau$, where $\tau$ is the proper time of the observer, in terms of $d r$. Hence, or otherwise, determine the maximum proper time that can elapse between the events at which the observer crosses $r=2 M$ and is torn apart at $r=0$.