= Solution
A <null geodesic congruence> is a smooth family of null geodesics filling a region, with one curve through each point before caustics form. Choose an affine parameter $\lambda$ and tangent $U^a$, so $U^2=0$ and $\nabla_UU=0$. Choose a null partner $N^a$ with $U\cdot N=-1$, transported parallel to $U$ along each generator. The <screen-space projector>
$$
P^a{}_b=\delta^a{}_b+U^aN_b+N^aU_b
$$
projects onto the positive-definite two-dimensional plane transverse to $U,N$. Define the <optical tensor> and its decomposition by
$$
\widehat B_{ab}=P_a{}^cP_b{}^d\nabla_dU_c
=\frac12\theta P_{ab}+\widehat\sigma_{ab}+\widehat\omega_{ab},
$$
where
$$
\boxed{\theta=P^{ab}\widehat B_{ab},\quad
\widehat\sigma_{ab}=\widehat B_{(ab)}-\tfrac12\theta P_{ab},\quad
\widehat\omega_{ab}=\widehat B_{[ab]}.}
$$
These are respectively the <null expansion>, <null shear> and <null twist>, the latter also called rotation. In the <Null Raychaudhuri equation>, $\widehat\sigma^2=\widehat\sigma_{ab}\widehat\sigma^{ab}$ and $\widehat\omega^2=\widehat\omega_{ab}\widehat\omega^{ab}$ use the screen metric. The trace convention for $\theta$ is important: it is not half the trace. For an affine null tangent it also equals $\nabla_aU^a$.
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