Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 54 / 3 / b / Solution

For radial null geodesics, the metric gives $dt/dr=\pm f^{-1}$ in the exterior. Define the tortoise coordinate byAn outgoing future radial ray has $dt/dr=f^{-1}$, so $u=t-r_{\star }$ is constant; an ingoing ray has $dt/dr=-f^{-1}$, so $v=t+r_{\star }$ is constant. These describe the null trajectories independently of the choice of affine parameter.

For the tortoise coordinate of a five-dimensional charged black hole, with $a=r_{+}$, $b=r_{-}$ and $\Delta =a^2-b^2$, partial fractions giveThus a useful explicit tortoise coordinate isThe logarithms signal failure of the static coordinates at the horizons, rather than a curvature singularity there.