Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 311 / 1 / b / Solution

For radial null geodesics, $d{\Omega }_3=dz=0$ and $d{s}^2=0$, soIntroduce the tortoise coordinate byup to an additive constant. The plus sign gives $d(t-r_{\star })=0$ on outgoing rays, while the minus sign gives $d(t+r_{\star })=0$ on ingoing rays. Henceare respectively constant on the stated radial null families.