Choose at one transverse cross-section a null vector $N^a$ satisfying $N^aN_a=0$ and $U^aN_a=-1$, then <parallel transport> it along every generator:
$$
U^b\nabla_bN^a=0.
$$
Metric compatibility and the affine <geodesic equation> imply that both $N^2$ and $U\mathbin\cdot N$ are constant along each generator, so the required normalization persists.

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 311 / 1 / a / ii / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-311/1/a/ii/solution