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

Table of contents
- Solution b

Solution

A geodesic passing through

r = 0

must have

h = 0

. Its radial equation is then

\overset{r}{˙}^{2} + \frac{r ^{2}}{a ^{2}} = E^{2} - 1.

On the outgoing and returning portion it is the harmonic solution

r (τ) = a E^{2} - 1 sin (τ / a), 0 \leq τ \leq πa .

It reaches

r_{m a x} = a E^{2} - 1

when

τ = πa /2

and returns to the origin at

Δ τ = πa .

The return time is independent of the launch energy, a characteristic focusing property of global anti-de Sitter spacetime.

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