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

Use proper time $s$, so $k=-1$. The first integral becomesFor an outward geodesic starting at the origin,until it next reaches $r=0$. This occurs atThus every radial timelike geodesic in anti-de Sitter spacetime through the origin returns after the same proper time; restoring anti-de Sitter radius $a$ gives $T=\pi a$.