Affine rescaling of a null normal (${\nabla }_{K}K=\kappa K,K(\log h)=-\kappa$)

If a null vector tangent satisfies ${\nabla }_{K}K=\kappa K$, then $U=hK$ is affine when $K(\log h)=-\kappa$. Choose $h=1$ on the initial transverse surface. The screen-space projector kills derivative terms proportional to $K$, so the initial optical tensor and its null expansion agree with those of $K$. In ingoing spherical coordinates, $K={\partial }_v+(f/2){\partial }_r$ has $\kappa =f^{\prime }/2$.