  We study the $\mathsf{SL}(2)$ transformation properties of spherically
symmetric perturbations of the Bertotti-Robinson universe and identify an
invariant $\mu$ that characterizes the backreaction of these linear solutions.
The only backreaction allowed by Birkhoff's theorem is one that destroys the
$AdS_2\times S^2$ boundary and builds the exterior of an asymptotically flat
Reissner-Nordstr\"om black hole with $Q=M\sqrt{1-\mu/4}$. We call such
backreaction with boundary condition change an anabasis. We show that the
addition of linear anabasis perturbations to Bertotti-Robinson may be thought
of as a boundary condition that defines a connected $AdS_2\times S^2$. The
connected $AdS_2$ is a nearly-$AdS_2$ with its $\mathsf{SL}(2)$ broken
appropriately for it to maintain connection to the asymptotically flat region
of Reissner-Nordstr\"om. We perform a backreaction calculation with matter in
the connected $AdS_2\times S^2$ and show that it correctly captures the
dynamics of the asymptotically flat black hole.
