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

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- Solution i

Solution

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^{a} U_{a} = 0, U^{b} \nabla_{b} U^{a} = 0.

B_{ab} = \nabla_{b} U_{a}

, differentiation of the null norm and the geodesic equation give respectively

U^{a} B_{ab} = \frac{1}{2} \nabla_{b} (U^{a} U_{a}) = 0, B_{ab} U^{b} = U^{b} \nabla_{b} U_{a} = 0.

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