Asymptotic symmetries on Killing horizons

TL;DR

The paper investigates asymptotic symmetries defined on Killing horizons in static, spherically symmetric spacetimes within Einstein gravity with or without a cosmological constant, aiming to link local horizon data to possible microscopic degrees of freedom.It derives the general form of asymptotic Killing vectors, showing the symmetry group consists of rigid $O(3)$ rotations of the horizon two-sphere and supertranslations along the horizon’s null direction, with the latter arbitrarily dependent on the null coordinate and angles, yielding a Diff($S^1$) subalgebra under certain identifications.Using the covariant phase space formalism, the paper demonstrates that the Poisson bracket algebra of the conserved charges lacks a nontrivial central extension under standard boundary conditions, preventing a Virasoro-like entropy derivation from these symmetries alone.When extending the symmetry group by weakening boundary conditions, a central charge could arise only from angular diffeomorphisms on the horizon two-sphere, which requires an artificial reduction (to Diff($S^1$)) and breaks the assumed spherical symmetry, suggesting the need for alternative approaches to connect horizon symmetries with black hole entropy.

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.