Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 311 / 1 / a

A null geodesic congruence is a smooth family of nonintersecting null geodesics filling an open spacetime region. For its affinely parametrized tangent $U^a$,Writing $B_{ab}={\nabla }_b{U}_a$, differentiation of the null norm and the geodesic equation give respectively

Choose at one transverse cross-section a null vector $N^a$ satisfying $N^a{N}_a=0$ and $U^a{N}_a=-1$, then parallel transport it along every generator:Metric compatibility and the affine geodesic equation imply that both $N^2$ and $U\cdot N$ are constant along each generator, so the required normalization persists.

The screen-space projectordefines the optical tensor ${\hat{B}}_{ab}=P_a{}^c{P}_b{}^d{B}_{cd}$. In four spacetime dimensions its irreducible decomposition isThese are the null expansion, null twist, and null shear.

On the null hypersurface, the generator covector is proportional to a normal, locally $U_a=f{\nabla }_{a}u$ for a level-set function $u$. The Frobenius theorem therefore givesContracting once with $N^a$ and projecting the remaining indices with $P^a{}_b$ removes every term containing $U$ or $N$ and leaves $P_b{}^d{P}_c{}^e{\nabla }_{[d}{U}_{e]}=0$. Henceon the hypersurface.