Extreme Black Hole Anabasis

TL;DR

This paper investigates the transformation properties of perturbations in the Bertotti-Robinson universe, identifying an invariant that characterizes backreaction and showing how boundary condition changes relate to black hole geometries.

Contribution

It introduces the concept of anabasis as boundary condition change perturbations and demonstrates how they connect $AdS_2$ to asymptotically flat Reissner-Nordstr"om black holes.

Findings

Identified an invariant $or backreaction characterization.

Showed boundary condition change leads to connected $AdS_2$.

Backreaction in connected $AdS_2$ captures black hole dynamics.

Abstract

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.