Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 1 / 38E (General Relativity)

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,

A geodesic passing through $r=0$ must have $h=0$. Its radial equation is thenOn the outgoing and returning portion it is the harmonic solutionIt reacheswhen $\tau =\pi a/2$ and returns to the origin atThe return time is independent of the launch energy, a characteristic focusing property of global anti-de Sitter spacetime.

A circular orbit at $r=r_0$ requires $V^{\prime }(r_0)=0$, henceThis can be satisfied for every $r_0>0$. Moreover,so every such circular orbit is stable. The radial energy equation givesand future direction selects $E=1+r_0^2/a^2=f(r_0)$. Thereforeso proper time and coordinate time differ only by an additive constant on the orbit. Also $\dot{\varphi }=\pm 1/a$.