 Optical matrix of a null congruence

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 $\hat{B}=\theta P/2+\hat{\sigma }+\hat{\omega }$.