Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 56 / 1 / d

For an asymptotically flat spacetime with future null infinity ${\mathcal{I}}^{+}$, its black-hole region iswhere the causal past is understood in the conformal completion. Choose the future orientation in the advanced chart so that $-{\partial }_r$ is future null. For any future causal tangent $X$, the inequality $g(X,-{\partial }_r)=-X(v)\le 0$ gives $X(v)\ge 0$. The causality inequality isIf $X(v)>0$, it implies $X(r)\le fX(v)/2$. Inside $r=2m$, $f<0$, so $r$ strictly decreases. If $X(v)=0$, the angular tangent vanishes and future orientation makes $X$ a positive multiple of $-{\partial }_r$, again decreasing $r$. Thus no future causal curve from $0<r<2m$ can reach the exterior or future null infinity: it is inside the black-hole region.

Conversely, from any point with $r>2m$, the outgoing radial null geodesic has $u$ constant and $dr/dv=f/2>0$. It continues to arbitrarily large $r$, with $r_{\ast }\to +\infty$, and reaches the corresponding future null infinity. Hence $\boxed{\mathcal{B}\cap \{r>2m\}=\varnothing }$ in this advanced exterior chart. This argument concerns the black-hole interior covered by ingoing coordinates; the additional white-hole region of the maximal extension has a different future causal orientation.