Symmetries at the black hole horizon

TL;DR

This paper analyzes the asymptotic symmetry groups of Killing horizons, revealing differences between extremal and subextremal cases, with implications for understanding horizon symmetries and redshift effects.

Contribution

It provides a detailed determination of asymptotic symmetry groups at black hole horizons, including supertranslations and horizon Killing vectors, highlighting differences between extremal and subextremal horizons.

Findings

Symmetry generators involve supertranslations and horizon Killing vectors.

Distinct dependence on horizon coordinate for extremal vs. subextremal cases.

Redshift effect influences symmetry structure in subextremal horizons.

Abstract

We determine the asymptotic symmetry group of Killing horizons by choosing Gaussian null coordinates in the neighbourhood of the horizon and boundary conditions that respect the leading order terms in the metric. The analysis divides naturally into the two cases of subextremal and extremal horizons. In general, we find rather involved asymptotic symmetry generators that nevertheless involve supertranslations and Killing vectors on the compact horizon. The most striking observation is the difference in the dependence on the coordinate along the horizon; we relate this to the redshift effect for subextremal horizons. We consider the spherically symmetric case as a special and illuminating example.