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

Commuting covariant derivatives and using $U^b{\nabla }_b{U}_a=0$ gives the optical evolution equationTaking its screen trace yieldsThe optical decomposition and the symmetry or antisymmetry of its pieces implyThus the four-dimensional Null Raychaudhuri equation is