Conformal Transformation, Near Horizon Symmetry, Virasoro Algebra and Entropy

TL;DR

This paper demonstrates that conformally related black hole solutions in general relativity have horizon entropy consistent with the area law, using Virasoro algebra and Cardy formula, exemplified by the Sultana-Dyer black hole.

Contribution

It provides a novel analysis of horizon entropy for conformally related black holes using near horizon symmetries and Virasoro algebra, confirming the area law in these cases.

Findings

Virasoro algebra with central extension is obtained near the horizon.

Horizon entropy matches one quarter of the horizon area, modified by the conformal factor.

The analysis confirms the validity of the Cardy formula for conformal black hole solutions.

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.