Extreme Black Hole Anabasis

Understand

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 $AdS_2\times S^2$.