= Parallel null partner for ingoing Schwarzschild rays
{title2=$U=-E\partial_r,\quad N=E^{-1}(\partial_v+\tfrac12f\partial_r)$}
In the <Ingoing Eddington-Finkelstein coordinates> metric $ds^2=-f(r)dv^2+2dv\,dr+r^2d\Omega^2$, where $f=1-2M/r$, the tangent $U=-E\partial_r$ is affine when $E>0$ is constant along each ray. The displayed radial partner obeys $N^2=0$ and $U\cdot N=-1$. Since $\Gamma^v{}_{rv}=\Gamma^a{}_{rr}=0$ and $\Gamma^r{}_{rv}=-f'/2$, its derivative $\partial_rN^r=f'/(2E)$ cancels the connection term, giving $\nabla_UN=0$. Its screen is the angular two-plane, with <optical tensor> $\widehat B_{AB}=-Er\gamma_{AB}$, <null expansion> $-2E/r$, and zero <null shear> and <null twist>.
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