Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 58 / Section I / ii / Solution

Locally write the null hypersurface as $u=0$, $du\ne 0$, with normal ${\ell }_{\mu }=f{\partial }_{\mu }u$, $f\ne 0$. A vector $V$ is tangent exactly when $V(u)=0$. Nullness gives $\ell (u)={\ell }^2/f=0$ on the hypersurface. Hence $\boxed{\ell \in T\mathcal{N}}$: its normal direction lies inside its tangent space.

Put $n=du$ and $q=n_{\mu }{n}^{\mu }$. Torsion freedom gives $n^{\mu }{\nabla }_{\mu }{n}_{\nu }=\frac{1}{2}{\partial }_{\nu }q$. Since $q$ vanishes on $\mathcal{N}$, all tangential derivatives vanish there and $dq=Cdu$ there for a scalar $C$. ConsequentlyThis is the unparametrized null geodesic equation. The integral curves are therefore generators of $\mathcal{N}$. An affine rescaling of a null normal removes the proportionality coefficient locally: if ${\nabla }_{\ell }\ell =\kappa \ell$, set $K=h\ell$ with $\ell (\log h)=-\kappa$.