Extreme black hole anabasis

We study the SL(2) transformation properties of spherically symmetric perturbations of the Bertotti-Robinson universe and identify an invariant μ that characterizes the backreaction of these linear solutions. The only backreaction allowed by Birkhoff’s theorem is one that destroys the AdS 2 × S 2 boundary and builds the exterior of an asymptotically flat Reissner-Nordström black hole with Q=M√(1-μ /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 ×S 2 . The connected AdS 2 is a nearly- AdS 2 with its SL(2) broken appropriately for it to maintain connection to the asymptotically flat region of Reissner-Nordström. We perform a backreaction calculation with matter in the connected AdS 2 × S 2 and show that it correctly captures the dynamics of the asymptotically flat black hole.