Black hole entropy and thermodynamics from symmetries

TL;DR

This work presents a symmetry-based method to derive black hole entropy by extracting the zero mode $Q_{0}$ and central charge $c$ of a Virasoro algebra generated by boundary-preserving diffeomorphisms, and then applying the Cardy formula $S = 2\pi\sqrt{\frac{c}{6} Q_{0}}$. By constructing explicit diffeomorphism algebras on spatial infinity (BTZ) and on the horizon (BTZ and Kerr–AdS$_4$), the authors show that the resulting $Q_{0}$ and $c$ yield the Bekenstein-Hawking entropy $S = \mathcal{A}/(4G)$, provided a horizon-appropriate normalization with $Q_{-1}, Q_{0}, Q_{1}$ forming an $sl(2,\mathbb{R})$ subalgebra is enforced. The approach also derives the first law of black hole thermodynamics from the same charge algebra, and extends to four dimensions, including Kerr–AdS$_4$, reproducing known thermodynamic relations in these spacetimes. The results support a horizon conformal structure contributing to black hole entropy and suggest a quantum horizon CFT description, while highlighting the need for a natural horizon constraint to uniquely select the correct diff$(S^{1})$ algebra.

Abstract

Given a boundary of spacetime preserved by a Diff(S^{1}) sub-algebra, we propose a systematic method to compute the zero mode and the central extension of the associated Virasoro algebra of charges. Using these values in the Cardy formula, we may derive an associated statistical entropy to be compared with the Bekenstein-Hawking result. To illustrate our method, we study in detail the BTZ and the rotating Kerr-adS_{4} black holes (at spatial infinity and on the horizon). In both cases, we are able to reproduce the area law with the correct factor of 1/4 for the entropy. We also recover within our framework the first law of black hole thermodynamics. We compare our results with the analogous derivations proposed by Carlip and others. Although similar, our method differs in the computation of the zero mode. In particular, the normalization of the ground state is automatically fixed by our construction.