Static near-horizon geometries in five dimensions

TL;DR

This paper classifies static near-horizon geometries of extremal black holes in five-dimensional Einstein-Maxwell theory with a Chern-Simons term, identifying possible geometries and solutions with various field configurations and topologies.

Contribution

It provides a comprehensive classification of near-horizon geometries under specified symmetries, including new solutions with electric and magnetic fields and their topologies.

Findings

Near-horizon geometries are either AdS_3 x S^2 or warped AdS_2 with 3D space.

Two no-magnetic-field solutions: AdS_2 x S^3 and warped S^3.

Existence of solutions with both electric and magnetic fields for certain coupling values.

Abstract

We consider the classification of static near-horizon geometries of stationary extremal (not necessarily BPS) black hole solutions of five dimensional Einstein-Maxwell theory coupled to a Chern-Simons term with coupling xi (with xi=1 corresponding to supergravity). Assuming the black holes have two rotational symmetries, we show that their near-horizon geometries are either the direct product AdS_3 X S^2 or a warped product of AdS_2 and compact 3d space. In the AdS_2 case we are able to classify all possible near-horizon geometries with no magnetic fields. There are two such solutions: the direct product AdS_2 X S^3 as well as a warped product of AdS_2 and an inhomogeneous S^3. The latter solution turns out to be near-horizon limit of an extremal Reissner-Nordstrom black hole in an external electric field. In the AdS_2 case with magnetic fields, we reduce the problem (in all cases) to a single non-linear ODE. We show that if there are any purely magnetic solutions of this kind they must have S^1 X S^2 horizon topology, and for xi^2 <1/4 we find examples of solutions with both electric and magnetic fields.