Asymptotic symmetries on Killing horizons

TL;DR

This paper analyzes the asymptotic symmetries of spherically symmetric Killing horizons in Einstein gravity, revealing a group combining rotations and supertranslations, but without central charges, and discusses the implications for symmetry extensions.

Contribution

It derives the form of asymptotic Killing vectors on horizons and clarifies the structure of the symmetry group, including the absence of non-trivial central charges in the conserved charge algebra.

Findings

Asymptotic symmetry group includes O(3) rotations and supertranslations.

Conserved charges do not have non-trivial central extensions.

Extended symmetries require unnatural reductions not justified by spherical symmetry.

Abstract

We investigate asymptotic symmetries regularly defined on spherically symmetric Killing horizons in the Einstein theory with or without the cosmological constant. Those asymptotic symmetries are described by asymptotic Killing vectors, along which the Lie derivatives of perturbed metrics vanish on a Killing horizon. We derive the general form of asymptotic Killing vectors and find that the group of the asymptotic symmetries consists of rigid O(3) rotations of a horizon two-sphere and supertranslations along the null direction on the horizon, which depend arbitrarily on the null coordinate as well as the angular coordinates. By introducing the notion of asymptotic Killing horizons, we also show that local properties of Killing horizons are preserved under not only diffeomorphisms but also non-trivial transformations generated by the asymptotic symmetry group. Although the asymptotic symmetry group contains the $\mathit{Diff}(S^1)$ subgroup, which results from the supertranslations dependent only on the null coordinate, it is shown that the Poisson bracket algebra of the conserved charges conjugate to asymptotic Killing vectors does not acquire non-trivial central charges. Finally, by considering extended symmetries, we discuss that unnatural reduction of the symmetry group is necessary in order to obtain the Virasoro algebra with non-trivial central charges, which will not be justified when we respect the spherical symmetry of Killing horizons.