Stationary black holes and attractor mechanism

TL;DR

This paper analyzes the near horizon symmetries of extremal stationary black holes in four-dimensional Einstein gravity with gauge fields and scalars, confirming the $SO(2,1) imes U(1)$ symmetry and classifying solutions into two families based on ergoregion presence.

Contribution

It rigorously derives the near horizon symmetry group for extremal black holes and classifies solutions into two distinct families, extending the attractor mechanism understanding.

Findings

Near horizon geometry has $SO(2,1) imes U(1)$ symmetry.

Solutions classified into two families by ergoregion presence.

Comments on the attractor mechanism for both solution branches.

Abstract

We investigate the symmetries of the near horizon geometry of extremal stationary black holes in four dimensional Einstein gravity coupled to abelian gauge fields and neutral scalars. Careful consideration of the equations of motion and the boundary conditions at the horizon imply that the near horizon geometry has $SO(2,1)\times U(1)$ isometry. This complements the rotating attractors proposal of hep-th/0606244 that had assumed the presence of this isometry. The extremal solutions are classified into two families differentiated by the presence or absence of an ergo-region. We also comment on the attractor mechanism of both branches.