 Solution

First, causal futures of black-hole points remain in the black-hole region. Indeed, if $p\in B$ and $q\in J^{+}(p)$ could send a signal to future null infinity, concatenating the causal curves would put $p$ in $J^{-}({\mathcal{I}}^{+})$, a contradiction. HenceGlobal hyperbolicity supplies the following connectedness lemma: if $C$ is connected on one Cauchy hypersurface, then $J^{+}(C)$ intersects any later Cauchy hypersurface in a connected set. To see the relevant mechanism, flow $C$ to the later surface along a continuous future timelike vector field; the image is connected, and every additional causally reachable point is joined to that image by the endpoint deformation of a causal curve inside the globally hyperbolic diamond. Applying the lemma to the connected component $B$ shows that $J^{+}(B)\cap {\Sigma }_2$ is connected. Since it is contained in $B\cap {\Sigma }_2$, it must lie wholly inside one connected component of that set. This is the black-hole non-splitting theorem: later black-hole components may merge, but one earlier connected black hole cannot split into two future components.