With $c=-\nabla_at^a$, commute covariant derivatives and use the geodesic equation:
$$
\begin{aligned}
t^b\nabla_bc
&=-t^b\nabla_b\nabla_at^a\\
&=R_{ab}t^at^b+(\nabla_at_b)(\nabla^at^b).
\end{aligned}
$$
This is the hypersurface-orthogonal <Raychaudhuri equation> written in terms of positive convergence rather than expansion.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-311/3/b/solution