Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 309 / 2 / c / i / Solution

Since $t=u+r$, the hypersurface ${\Sigma }_t$ is an ordinary constant-Minkowski-time slice. Its future unit normal isin the $(u,r,\theta ,\varphi )$ basis. Lowering the index gives $n_a=(-1,-1,0,0)=-{\nabla }_{a}t$. Restricting the metric to $dt=du+dr=0$ gives the Euclidean spherical metricso its induced Riemannian volume form is