Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 309 / 1 / c / Solution

The angular Euler-Lagrange equations are homogeneous in $\dot{\theta }$ and $\dot{\varphi }$. Therefore initial data with both angular velocities zero give the unique solution with constant $\theta$ and $\varphi$, so a radial geodesic remains radial.

Time-translation symmetry supplies the geodesic conserved quantity from a Killing vectorMetric compatibility makes the squared tangent norm another constant,For a proper-time parametrized timelike geodesic $k=-1$, for a null geodesic $k=0$, and for a unit-speed spacelike geodesic $k=1$. The constant $e$ is the conserved energy per unit mass associated with the static Killing vector ${\partial }_t$.