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

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- Solution c

Solution

The strong energy condition and Einstein's equation imply

R_{ab} t^{a} t^{b} \geq 0

. 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_{a} t_{b}) (\nabla^{a} t^{b}) - \frac{1}{3} (\nabla_{a} t^{a})^{2} \geq 0.

Because

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

, part b gives

t^{a} \nabla_{a} c \geq \frac{1}{3} c^{2} .

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