 Solution

Write the Schwarzschild metric in ingoing Eddington-Finkelstein coordinates as $d{s}^2=-fd{v}^2+2dvdr+r^{2}d{\Omega }^2$, with $f=1-2M/r$. Ingoing radial null geodesics have $v,\theta ,\varphi$ 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 isThe radial conditions give $U\cdot N=-E{N}^v=-1$ and $N^2=-f/E^2+2(E^{-1})(f/(2E))=0$. Parallel transport can be checked explicitly: ${\Gamma }^v{}_{rv}=0$ and ${\Gamma }^r{}_{rv}=-f^{\prime }/2$, sowith angular components also zero. Thus ${\nabla }_{U}N=0$. 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 ${\partial }_{r}E=0$.