= Sachs optical equations
{c}
For a parallel-transported screen and affine null geodesic tangent $U$, the optical matrix satisfies
$$
U^c\nabla_c\widehat B_{ab}=-\widehat B_{ac}\widehat B^c{}_b-P_a{}^eP_b{}^fR_{cedf}U^cU^d.
$$
Its 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.
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