Non-extremal near-horizon geometries

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 $\epsilon$ 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.

v2: 14 pages, comment added regarding the validity of the result to a class of near-horizon geometries where higher order corrections to equation (1.4) are set to zero. Section 1.3.1 added on applications of the result. Version accepted for publication in Classical and Quantum Gravity

References in corpus (4)

• A classification of near-horizon geometries of extremal vacuum black holes
• Uniqueness of near-horizon geometries of rotating extremal AdS(4) black holes
• Black Holes in Higher-Dimensional Gravity
• Classical string solutions in Non-Relativistic AdS${}_5\times$S${}^5$: Closed and Twisted sectors

Cited by in corpus (1)

• Strings near BTZ black holes: A Carrollian Chronicle