= Solution
Write the <Schwarzschild metric> in <Ingoing Eddington-Finkelstein coordinates> as $ds^2=-f\,dv^2+2dv\,dr+r^2d\Omega^2$, with $f=1-2M/r$. Ingoing radial <null geodesics> have $v,\theta,\phi$ constant. Since $\Gamma^a{}_{rr}=0$, choose their future affine tangent $U=-E\partial_r$, where $E>0$ is constant along a ray; a common normalization sets $E=1$. The <parallel null partner for ingoing Schwarzschild rays> is
$$
\boxed{N=\frac1E\left(\partial_v+\frac f2\partial_r\right).}
$$
The radial conditions give $U\cdot N=-EN^v=-1$ and $N^2=-f/E^2+2(E^{-1})(f/(2E))=0$. Parallel transport can be checked explicitly: $\Gamma^v{}_{rv}=0$ and $\Gamma^r{}_{rv}=-f'/2$, so
$$
\nabla_rN^v=0,\qquad
\nabla_rN^r=\frac{f'}{2E}-\frac{f'}2\frac1E=0,
$$
with angular components also zero. Thus $\nabla_UN=0$. The field is regular at $r=2M$ and throughout the regular advanced chart $r>0$; no vector field can be defined at the curvature-singular endpoint. If the affine normalization differs between rays, the same formula works for $E$ constant along each ray, since $\partial_rE=0$.
Back to article page
