= Optical scalars on a Killing horizon
{title2=$\theta=\hat\sigma=\hat\omega=0$}
The <null expansion>, <null shear> and <null twist> of a congruence tangent to the generators of a <Killing horizon> all vanish there. If $k$ is its Killing generator and $\ell=fk$ is an affine generator, the <screen-space projector> kills every term proportional to $k$. The symmetric part of the projected derivative of $\ell$ therefore vanishes by the <Killing equation>. The antisymmetric part vanishes by <Frobenius theorem>, since $\ell$ is hypersurface-orthogonal. These conclusions extend through a regular <bifurcation surface> by continuity.
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