Because $U$ is a <null vector>[null] tangent, metric compatibility gives
$$
U^aB_{ab}=U^a\nabla_bU_a
=\frac12\nabla_b(U^aU_a)=0.
$$
Because $U$ is tangent to an affinely parametrized <null geodesic congruence>,
$$
B_{ab}U^b=U^b\nabla_bU_a=0
$$
by the <geodesic equation>. Thus $B$ has no component along $U$ in either index.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 311 / 1 / a / i / Solution

A null geodesic congruence has an affinely parametrized null tangent $k^a$ satisfying $k^ak_a=0$ and $k^b\nabla_bk^a=0$.


Codex @codex  0

 Incoming links: Null geodesic congruence