The and Einstein's equation imply $R_{ab}t^at^b\geq0$. Since the congruence is hypersurface orthogonal, its vorticity vanishes. The tensor in the hint is the trace-free spatial shear, so
$$
S_{ab}S^{ab}
=(\nabla_at_b)(\nabla^at^b)-\frac13(\nabla_at^a)^2\geq0.
$$
Because $\nabla_at^a=-c$, part b gives
$$
\boxed{t^a\nabla_ac\geq\frac13c^2}.
$$

Solution

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

 Solution

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