Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 1 / 38E / a / Solution

Write $f(r)=1+r^2/a^2$. The geodesic Lagrangian isIts $\theta$ equation isSpherical symmetry lets us rotate the conserved angular-momentum plane into $\theta =\pi /2$. Equivalently, initial data $\theta =\pi /2$, $\dot{\theta }=0$ solve this equation for all $\tau$.

The cyclic coordinates giveUsing the timelike normalization $g_{\mu \nu }{\dot{x}}^{\mu }{\dot{x}}^{\nu }=-1$ givesAfter separating the constant terms,