Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 64 / 2 / i / Solution

This question uses proper cosmological time $t$. The future-directed unit normal isThe non-null hypersurface projection removes all normal components. The only relevant Christoffel symbols are ${\Gamma }_{ij}^0=a\dot{a}{\delta }_{ij}$. Since the components of $n_{\mu }$ are constant,ConsequentlyWith the printed positive extrinsic curvature convention, this is the expansion of the normal comoving congruence. A fixed coordinate cell has proper volume proportional to $a^3$, so $K=d\log (a^3)/dt={\nabla }_{\mu }{n}^{\mu }$: the fractional proper-volume expansion rate.