= Null-shear propagation equation
{title2=$U\cdot\nabla\widehat\sigma_{ab}=-\theta\widehat\sigma_{ab}-P_a{}^eP_b{}^fC_{cedf}U^cU^d$}
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 $P_{ab}R_{cd}U^cU^d/2$, so it changes the expansion but contributes no screen-trace-free source. The projected Weyl tensor supplies the tidal source of shear.
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