Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 52 / 2 / a

A null geodesic congruence is a smooth local family of null geodesics, with one generator through each point of the region being described. Choose an affine parameter $\lambda$ on each generator and write $U^a=d{x}^a/d\lambda$. Before a caustic, $U$ is a smooth, nonzero null vector field satisfying $U^a{U}_a=0$ and $U^b{\nabla }_b{U}^a=0$.

With $B^a{}_b={\nabla }_b{U}^a$, compatibility of the Levi-Civita connection with the metric givesThe geodesic equation with affine parameter gives the other contraction:Thus both contractions vanish. Nullness supplies the first identity, and affine parametrization supplies the second; a general nonaffine tangent would instead have $B^a{}_b{U}^b=\kappa U^a$.

Choose a local transverse three-dimensional section of the null geodesic congruence and a future unit timelike vector $T$ on it. Orient $U$ to the future and put $E=-U\cdot T>0$. On that section define a parallel auxiliary null vector by the initial valueSince $T^2=-1$, $U^2=0$, and $U\cdot T=-E$, direct contraction gives $U\cdot N=-1$ and $N^2=-E^{-2}+E^{-2}=0$.

Extend $N$ along each generator by parallel transport, solving $U^b{\nabla }_b{N}^a=0$ with those initial values. The geodesic equation and compatibility of the Levi-Civita connection implyTherefore $N^2=0$, $U\cdot N=-1$, and ${\nabla }_{U}N=0$ hold throughout the local congruence. Smooth initial data and the transport equation give a smooth field up to the breakdown of the congruence at caustics. An arbitrary pointwise choice of $T$ away from the initial section would not automatically have this transport property.

The screen-space projector annihilates both null vectors: $P^a{}_b{U}^b=P^a{}_b{N}^b=0$, and satisfies $P^2=P$. Its image is the two-dimensional spacelike screen orthogonal to $U,N$. The screen metric isIt is positive definite on that screen. Projecting both indices of $B_{ab}={\nabla }_b{U}_a$ gives the optical tensor ${\hat{B}}_{ab}=P_a{}^c{P}_b{}^d{B}_{cd}$.

Define its three parts byThus ${\hat{B}}_{ab}=\frac{1}{2}\theta q_{ab}+{\sigma }_{ab}+{\omega }_{ab}$. The null expansion $\theta$ is its trace and equals $d\log \mathcal{A}/d\lambda$ for the infinitesimal beam area $\mathcal{A}$. The null shear $\sigma$ is symmetric and trace-free, measuring shape change at fixed first-order area. The null twist $\omega$, also called rotation, is antisymmetric and measures failure of screen directions to remain hypersurface-orthogonal. In $D$ dimensions replace $1/2$ by $1/(D-2)$. Here and below expansion means the trace, rather than its average over the screen dimensions.

Locally write the null hypersurface as $F=0$. Its generators are tangent to its raised normal, so on it $U_a=h{\nabla }_{a}F$ for a nonzero scalar $h$. The antisymmetric derivative issince the Hessian of $F$ is symmetric for the torsion-free Levi-Civita connection. Each term contains $\nabla F$, which is proportional to $U$ and is killed by the screen-space projector. HenceThis is the null version of hypersurface orthogonality in the Frobenius theorem. A null hypersurface has no independent normal direction outside its tangent space: its null vector normal also generates it. That is why the same argument applies to the generators' null twist.