Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 2 / 38C (General Relativity)

An affine geodesic follows fromIts constant norm may be chosen asfor timelike, null and spacelike geodesics respectively.

The Killing coordinates $t$ and $\theta$ giveTogether with the norm $\epsilon =2\mathcal{L}$, these are the three constants of motion.

For a radial null ray, $d\theta =0$ and the metric gives $dt=dr$ for the outgoing branch. Starting at $r=t=0$, it reaches $r=R$ at

A stationary observer has four-velocity $u=r{\partial }_t$. If $E=-k_t$ is the conserved photon energy, the measured frequency is proportional toHenceThe photon is blueshifted as it moves outward.

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