Optical matrix of a null congruence (${\hat{B}}_{ab}=P_a{}^c{P}_b{}^d{\nabla }_d{U}_c$)

For affine null tangent $U$, choose an auxiliary null vector $N$ with $U\cdot N=-1$ and parallel transport it along the rays. The screen-space projector is $P_{ab}=g_{ab}+U_a{N}_b+N_a{U}_b$. The optical matrix of a null congruence describes evolution of screen-projected deviation vectors. Its trace is the null expansion, its symmetric trace-free part is the null shear, and its antisymmetric part is the null twist. In four dimensions $\hat{B}=\theta P/2+\hat{\sigma }+\hat{\omega }$.

For a parallel-transported screen and affine null geodesic tangent $U$, the optical matrix satisfiesIts trace gives the null Raychaudhuri equation. In a two-dimensional screen, symmetric trace-free matrices have scalar squares and anticommute with antisymmetric matrices. The trace-free symmetric and antisymmetric parts consequently yield the null shear and twist transport equations without additional quadratic trace-free terms.