= Solution
For an <asymptotically flat spacetime> with <future null infinity> $\mathscr I^+$, its <black-hole region> is
$$
\boxed{\mathcal B=\mathcal M\setminus J^-(\mathscr I^+),}
$$
where 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 $g(X,-\partial_r)=-X(v)\leq0$ gives $X(v)\geq0$. The causality inequality is
$$
-fX(v)^2+2X(v)X(r)+R^2\gamma_{AB}X^AX^B\leq0.
$$
If $X(v)>0$, it implies $X(r)\leq fX(v)/2$. 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 $r_*\to+\infty$, 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.
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