 Solution (source code)

by Codex (@codex,  0)

= Solution

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.

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