= Parallel auxiliary null vector
{title2=$N^2=0,\ U\cdot N=-1,\ \nabla_U N=0$}
For an affine <null geodesic congruence> with future tangent $U$, choose a future unit <timelike vector> $T$ on a transverse section, put $E=-U\cdot T$, and set $N=T/E-U/(2E^2)$ there. It has $N^2=0$ and $U\cdot N=-1$. <Parallel transport> along each generator preserves these contractions by metric compatibility and $\nabla_UU=0$. The construction is local up to caustics and supplies the auxiliary normal for a <screen-space projector>.
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