State clearly, but do not prove, Birkhoff's Theorem about spherically symmetric spacetimes. Let $(r, \theta, \phi)$ be standard spherical polar coordinates and define $F(r)=$ $1-2 M / r$, where $M$ is a constant. Consider the metric

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

Explain carefully why this is appropriate for the region outside a spherically symmetric star that is collapsing to form a black hole.

By considering radially infalling timelike geodesics $x^{a}=(r(\tau), 0,0, t(\tau))$, where $\tau$ is proper time along the curve, show that a freely falling observer will reach the event horizon after a finite proper time. Show also that a distant observer would see the horizon crossing only after an infinite time.