Three Theorems on Near Horizon Extremal Vanishing Horizon Geometries

TL;DR

This paper proves three theorems about the near horizon geometries of EVH black holes in various Einstein gravity theories, showing they contain a 3D maximally symmetric space, which is AdS3 or flat, and extends to near-EVH cases with BTZ geometry.

Contribution

It establishes fundamental theorems characterizing the near horizon geometries of EVH and near-EVH black holes across diverse Einstein gravity theories.

Findings

Near horizon geometry contains a 3D maximally symmetric space.

Strong energy condition leads to AdS3 or flat space in near horizon geometry.

Results extend to near-EVH black holes with BTZ geometry.

Abstract

EVH black holes are Extremal black holes with Vanishing Horizon area, where vanishing of horizon area is a result of having a vanishing one-cycle on the horizon. We prove three theorems regarding near horizon geometry of EVH black hole solutions to generic Einstein gravity theories in diverse dimensions. These generic gravity theories are Einstein-Maxwell-dilaton-Lambda theories, and gauged or ungauged supergravity theories with U(1) Maxwell fields. Our three theorems are: (1) The near horizon geometry of any EVH black hole has a three dimensional maximally symmetric subspace. (2) If the energy momentum tensor of the theory satisfies strong energy condition either this 3d part is an AdS3, or the solution is a direct product of a locally 3d flat space and a d-3 dimensional part. (3) These results extend to the near horizon geometry of near-EVH black holes, for which the AdS3 part is replaced with BTZ geometry.