= Spherical null expansions in double-null coordinates
{title2=$\theta_U=\frac{2r_{,U}}{fr},\quad\theta_V=\frac{2r_{,V}}{fr}$}
For a four-dimensional spherically symmetric metric $ds^2=-2f\,dU\,dV+r^2d\Omega^2$, the affine null gradients are $f^{-1}\partial_V$ and $f^{-1}\partial_U$. Their <null expansions> are $2r_{,V}/(fr)$ and $2r_{,U}/(fr)$. Their <null shear> and <null twist> vanish because the transverse metric changes only by a common scale and the covectors are exact gradients.
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