Null-shear propagation equation ($U\cdot \nabla {\hat{\sigma }}_{ab}=-\theta {\hat{\sigma }}_{ab}-P_a{}^e{P}_b{}^f{C}_{cedf}{U}^c{U}^d$)

For an affine null congruence in four spacetime dimensions with a parallel-transported screen, the displayed equation follows by taking the symmetric trace-free part of the Sachs optical equations. The Ricci contribution to the optical tidal tensor is $P_{ab}{R}_{cd}{U}^c{U}^d/2$, so it changes the expansion but contributes no screen-trace-free source. The projected Weyl tensor supplies the tidal source of shear.