Since $t=u+r$, the hypersurface $\Sigma_t$ is an ordinary constant-Minkowski-time slice. Its future unit normal is
$$
\boxed{n^a=(1,0,0,0)=\partial_u}
$$
in the $(u,r,\theta,\phi)$ basis. Lowering the index gives $n_a=(-1,-1,0,0)=-\nabla_at$. Restricting the metric to $dt=du+dr=0$ gives the Euclidean spherical metric
$$
dr^2+r^2(d\theta^2+\sin^2\theta\,d\phi^2),
$$
so its induced <Riemannian volume form> is
$$
\boxed{d\sigma=r^2\sin\theta\,dr\,d\theta\,d\phi}.
$$

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-309/2/c/i/solution