Asymptotic symmetries on Killing horizons

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 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.

Near-horizon conformal symmetry and black hole entropy in any dimension

“…[10]. Thus, this method clearly violates spherical symmetry in dimensions higher than three for instance, and requires unnatural reduction of the symmetry group on the horizon [15]. A framework without choosing an angular direction is required in order to realize this idea in a satisfactory manner.…”

Aspects of the BMS/CFT correspondence

“…More recently, a similar analysis has been used to derive the Bekenstein-Hawking entropy of an extreme 4-dimensional Kerr black hole [13]. One of our hopes is to make progress along these lines in the non extreme case, either directly from an analysis at null infinity or by making a similar analysis at the horizon, as discussed previously for instance in [14,15,16,17,18,19,20,21,22,23,24,25,26].…”

Effective Conformal Descriptions of Black Hole Entropy

“…One can therefore carry out a canonical analysis on this space of solutions, in a covariant manner that does not require a splitting of spacetime into space and time. The covariant canonical derivation of the central term K [ξ, η] was first treated in [14]; the results were corrected and generalized in [36,37], and greatly extended in [38,39].…”