The Hamiltonian Dynamics of Bounded Spacetime and Black Hole Entropy: The Canonical Method

TL;DR

This paper derives a consistent conformal field theory description of black hole entropy from first principles, establishing a classical Virasoro algebra at the horizon that leads to the Bekenstein-Hawking entropy through canonical quantization.

Contribution

It provides a new, consistent formulation of the horizon conformal field theory approach to black hole entropy, correcting previous inconsistencies and establishing a universal entropy formula.

Findings

Derives a classical Virasoro algebra at the black hole horizon.

Quantization yields the correct central charge and entropy proportional to horizon area.

Universal applicability to various uncharged black hole solutions.

Abstract

From first principles, I present a concrete realization of Carlip's idea on the black hole entropy from the conformal field theory on the horizon in any dimension. New formulation is free of inconsistencies encountered in Carlip's. By considering a correct gravity action, whose variational principle is well defined at the horizon, I $derive$ a correct $classical$ Virasoro generator for the surface deformations at the horizon through the canonical method. The existence of the classical Virasoro algebra is crucial in obtaining an operator Virasoro algebra, through canonical quantization, which produce the right central charge and conformal weight $\sim A_+/\hbar G$ for the semiclassical black hole entropy. The coefficient of proportionality depends on the choice of ground state, which has to be put in by hand to obtain the correct numerical factor 1/4 of the Bekenstein-Hawking (BH) entropy. The appropriate ground state is different for the rotating and the non-rotating black holes but otherwise it has a $universality$ for a wide variety of black holes. As a byproduct of my results, I am led to conjecture that {\it non-commutativity of taking the limit to go to the horizon and computing variation is proportional to the Hamiltonian and momentum constraints}. It is shown that almost all the known uncharged black hole solutions satisfy the conditions for the universal entropy formula.