Non-extremal near-horizon geometries

TL;DR

This paper investigates the structure of non-extremal near-horizon geometries in Einstein gravity, demonstrating that Einstein's equations can be separated into divergent and finite parts, allowing for well-defined near-horizon analysis without extremality.

Contribution

It introduces a method to separate Einstein's equations into divergent and finite parts in non-extremal near-horizon geometries, extending the analysis to include scalar and Maxwell fields.

Findings

Einstein's equations can be separated into divergent and finite parts near non-extremal horizons.

The separation method applies to Einstein gravity coupled with scalar fields.

The approach extends to Maxwell fields with specific potential components.

Abstract

When Gaussian null coordinates are adapted to a Killing horizon, the near-horizon limit is defined by a coordinate rescaling and then by taking the regulator parameter $\varepsilon$ to be small, as a way of zooming into the horizon hypersurface. In this coordinate setting, it is known that the metric of a non-extremal Killing horizon in the near-horizon limit is divergent, and it has been a common practice to impose extremality in order to set the divergent term to zero. Although the metric is divergent, we show for a class of Killing horizons that the vacuum Einstein's equations can be separated into a divergent and a finite part, leading to a well-defined minimal set of Einstein's equations one needs to solve. We extend the result to Einstein gravity minimally coupled to a massless scalar field. We also discuss the case of Einstein gravity coupled to a Maxwell field, in which case the separability holds if the Maxwell potential has non-vanishing components only in the directions of the horizon spatial cross section.