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

The strong energy condition and Einstein's equation imply $R_{ab}{t}^a{t}^b\ge 0$. Since the congruence is hypersurface orthogonal, its vorticity vanishes. The tensor in the hint is the trace-free spatial shear, soBecause ${\nabla }_a{t}^a=-c$, part b gives