Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 2 / 38C / d / Solution

Parametrize the spacelike geodesic by unit length. With $J=r^2\dot{\theta }=b$, its norm equation givesThe total angular change is thereforePutting $r=b\sec u$ evaluates the half-integral as $1/b$, and hence