Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 311 / 2 / a

Locally write a smooth hypersurface as $\Phi =0$ with $n_a={\nabla }_a\Phi \ne 0$. It is a null hypersurface when $n_a{n}^a=0$ on it. Then $n^a$ is both normal and tangent, and the induced metric is degenerate along $n^a$.

Because covariant derivatives commute on a scalar,The scalar $n^2$ vanishes on the hypersurface, so its derivative annihilates every tangent direction there. Its derivative is consequently proportional to $n_a$: locally $n^2=\Phi q$ for a smooth function $q$, givingThus the normal generates null pregeodesics. Rescale $k^a=\alpha n^a$, choosing $n^b{\nabla }_b\log \alpha =-\kappa$, to get $k^b{\nabla }_b{k}^a=0$. These are the affinely parametrized generators of the null hypersurface. The proportionality need not vanish: setting $n^2=0$ only on the hypersurface does not set its full transverse derivative to zero.