Conformal transformation, near horizon symmetry, Virasoro algebra, and entropy

Abstract

There are certain black hole solutions in general relativity (GR) which are conformally related to the stationary solutions in GR. It is not obvious that the horizon entropy of these spacetimes is also one quarter of the area of horizon, like the stationary ones. Here I study this topic in the context of Virasoro algebra and Cardy formula. Using the fact that the conformal spacetime admits conformal Killing vector and the horizon is determined by the vanishing of the norm of it, the diffemorphisms are obtained which keep the near horizon structure invariant. The Noether charge and a bracket among them corresponding to these vectors are calculated in this region. Finally, they are evaluated for the Sultana-Dyer (SD) black hole, which is conformal to the Schwarzschild metric. It is found that the bracket is identical to the usual Virasoro algebra with the central extension. Identifying the zero mode eigenvalue and the central charge, the entropy of the SD horizon is obtained by using Cardy formula. Interestingly, this is again one quarter of the horizon area. Only difference in this case is that the area is modified by the conformal factor compared to that of the stationary one. The analysis gives a direct proof of the earlier assumption.

A note on entropy of de Sitter black holes

“…We have utilized the formalism developed using the GibbonsHawking-York surface counterterm and near horizon symmetries [23][24][25][26] to derive the total entropy of such two horizon spacetimes. To the best of our knowledge, this has not been done before.…”

“…We shall use below the formalism developed in [23][24][25][26] using the Gibbons-Hawking-York surface counterterm in order to calculate the total entropy of a stationary axisymmetric de Sitter black hole spacetime.…”

“…3, we use this general framework to derive the total entropy of such spacetimes extending the formalism of [23][24][25][26]. We point out there that the mode functions corresponding to the symmetry generating vector fields near the horizons have to be restricted in a certain manner, compared to the single horizon spacetimes, in order to obtain the total entropy.…”

“…A recent approach for deriving the thermal properties of a Killing horizon can be found in [23][24][25][26]. This novel formalism uses the Noether current associated with the variation of the Gibbons-Hawking-York boundary term to obtain the conserved charges associated with the diffeomorphism generating vector fields.…”

“…[2]. Our precise goal in this work is to derive this total entropy using the formalism of [23][24][25][26]. Since such spacetimes have two natural boundaries, i.e.…”