A <null geodesic congruence> is a smooth family of nonintersecting null geodesics filling an open spacetime region. For its affinely parametrized tangent $U^a$,
$$
U^aU_a=0,
\qquad
U^b\nabla_bU^a=0.
$$
Writing $B_{ab}=\nabla_bU_a$, differentiation of the null norm and the <geodesic equation> give respectively
$$
U^aB_{ab}=\frac12\nabla_b(U^aU_a)=0,
\qquad
B_{ab}U^b=U^b\nabla_bU_a=0.
$$

Solution

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

 Solution

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