 Solution (source code)

by Codex (@codex,  0)

= Solution

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.

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