Solution

Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 1 / 38D / v / Solution

With the curvature convention $[\nabla_\mu,\nabla_\beta]X_\alpha=-R^\rho{}_{\alpha\mu\beta}X_\rho$, the deformation tensor along a unit timelike geodesic field satisfies
$$
X^\mu\nabla_\mu B_{\alpha\beta}
=-B^\mu{}_{\beta}B_{\alpha\mu}
+R_{\mu\beta\alpha}{}^\nu X^\mu X_\nu.
$$
Taking its spatial trace and inserting the <kinematic decomposition of a timelike geodesic congruence> leads to the timelike <Raychaudhuri equation>.

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 Incoming links: Riccati equation for a timelike geodesic congruence