Double Horizon Limit, AdS Geometry and Entropy Function

TL;DR

This paper demonstrates that the AdS_2 near horizon geometry of four-dimensional stationary black holes arises naturally from the double-horizon limit, with field configurations determined by decoupled equations consistent with the entropy function approach.

Contribution

It shows that the double-horizon limit leads to a decoupled near horizon geometry and field configurations, aligning with the entropy function method, for generic theories of gravity.

Findings

AdS_2 geometry results from the double-horizon limit.

Near horizon field equations decouple from the bulk.

Decoupled equations derive from an action consistent with the original.

Abstract

We start from a generic metric which describes four dimensional stationary black holes in an arbitrary theory of gravity and show that the AdS_2 part of the near horizon geometry is a consequence of the double-horizon limit and finiteness . We also show that the field configurations of the near horizon are determined if the same conditions are applied to the equations of motion. This is done by showing that in the double-horizon limit field equations at the horizon decouple from the bulk of the space. Solving these equations gives the near horizon field configurations. It is shown that these decoupled equations can be obtained from an action derived from the original action by applying the double-horizon condition. Our results agree with the entropy function method.