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

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