= Spherical null expansion in ingoing coordinates
{title2=$\theta_K=2K(r)/r$}
For a four-dimensional spherical metric $ds^2=-f\,dv^2+2\,dv\,dr+r^2d\Omega^2$, a radial <null vector> normal $K$ to a round sphere has <null expansion> $\theta_K=\tfrac12q^{AB}\mathcal L_Kq_{AB}=2K(r)/r$, where $q_{AB}=r^2\gamma_{AB}$. The angular metric changes only by a common scale, so radial congruences have zero <null shear>; hypersurface orthogonality also gives zero <null twist>. The convention here is the trace, not the screen-averaged expansion.
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