In static spherically symmetric coordinates, the metric $g_{a b}$ for de Sitter space is given by

$$d s^{2}=-\left(1-r^{2} / a^{2}\right) d t^{2}+\left(1-r^{2} / a^{2}\right)^{-1} d r^{2}+r^{2} d \Omega^{2}$$

where $d \Omega^{2}=d \theta^{2}+\sin ^{2} \theta d \phi^{2}$ and $a$ is a constant.

(a) Let $u=t-a \tanh ^{-1}(r / a)$ for $r \leqslant a$. Use the $(u, r, \theta, \phi)$ coordinates to show that the surface $r=a$ is non-singular. Is $r=0$ a space-time singularity?

(b) Show that the vector field $g^{a b} u_{, a}$ is null.

(c) Show that the radial null geodesics must obey either

$$\frac{d u}{d r}=0 \quad \text { or } \quad \frac{d u}{d r}=-\frac{2}{1-r^{2} / a^{2}}$$

Which of these families of geodesics is outgoing $(d r / d t>0) ?$

Sketch these geodesics in the $(u, r)$ plane for $0 \leqslant r \leqslant a$, where the $r$-axis is horizontal and lines of constant $u$ are inclined at $45^{\circ}$ to the horizontal.

(d) Show, by giving an explicit example, that an observer moving on a timelike geodesic starting at $r=0$ can cross the surface $r=a$ within a finite proper time.