Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 56 / 1 / b / Solution

For radial null geodesics, the radial metric is $-fd{t}^2+d{r}^2/f$. Define the tortoise coordinate byThen the radial metric becomes $f(-d{t}^2+d{r}_{\ast }^2)=-fdudv$. Its two null directions have $dt=\pm d{r}_{\ast }$, so $\boxed{u=t-r_{\ast }\text{ is constant on outgoing rays},v=t+r_{\ast }\text{ on ingoing rays}}$ in the exterior. The radial null curves are null geodesics up to reparametrization: in the two-dimensional radial metric a null tangent's acceleration is proportional to itself, and spherical symmetry introduces no angular acceleration.