= Null twist vanishes exactly for hypersurface-orthogonal geodesics
{title2=$\widehat\omega_{ab}=0\iff U_{[a}\nabla_bU_{c]}=0$}
For an affine <null geodesic congruence>, $B_{ab}=\nabla_bU_a$ annihilates $U$ in both slots. Relative to a null partner, its antisymmetric part is its screen projection plus a term of the form $U_{[a}q_{b]}$. Hence zero <null twist> implies $U\wedge dU=0$, and <Frobenius theorem> makes $U_a$ locally proportional to a hypersurface gradient. Conversely, if $U_a=h\nabla_aS$, its antisymmetric derivative has a factor of $U$, so the screen projection vanishes. Geodesicity is essential to the converse from vanishing screen twist to the full Frobenius condition.
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