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

Write $f(r)=1-r_{+}^2/r^2$ and let a dot denote differentiation with respect to an affine parameter $\tau$. The time-translation and $z$-translation Killing vector fields, together with rotational symmetry of the unit $S^3$, give the conserved quantitiesDefine $\epsilon =-g_{ab}{\dot{x}}^a{\dot{x}}^b$, so $\epsilon =1,0,-1$ for timelike, null and spacelike geodesics respectively. The normalization equation becomesIt therefore has the effective potential form