Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 311 / 3 / b

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- Solution b

Solution

With

c = - \nabla_{a} t^{a}

, commute covariant derivatives and use the geodesic equation:

t^{b} \nabla_{b} c = - t^{b} \nabla_{b} \nabla_{a} t^{a} = R_{ab} t^{a} t^{b} + (\nabla_{a} t_{b}) (\nabla^{a} t^{b}) .

This is the hypersurface-orthogonal Raychaudhuri equation written in terms of positive convergence rather than expansion.

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