Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 56 / 3 / d

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 ${\partial }_{r}E=0$.

For the radial null pair above, the screen-space projector selects the angular tangent plane, with metric $q_{AB}=r^2{\gamma }_{AB}$. The ingoing covector is $U_a=-E(dv{)}_a$, and ${\Gamma }^v{}_{AB}=-r{\gamma }_{AB}$. Hence the optical tensor isIt is purely symmetric and proportional to the screen metric, givingFor $E=1$, the null expansion is $-2/r$. The area check $A\propto r^2$ and $dr/d\lambda =-E$ gives $A^{\prime }/A=-2E/r$, in agreement with the spherical null expansion in ingoing coordinates. In vacuum, $d\theta /d\lambda =-2E^2/r^2=-{\theta }^2/2$ also satisfies the Null Raychaudhuri equation; the ingoing beam focuses at the singularity rather than remaining future complete.