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

TL;DR

The work develops a canonical, horizon-based framework to realize Carlip's horizon conformal field theory in any dimension. By formulating a well-defined variational principle with horizon boundary data and imposing horizon fall-off in a grand canonical ensemble, it derives a classical Virasoro generator and, upon canonical quantization, a central charge and conformal weight that scale with the horizon area. Using Cardy’s formula, it reproduces the Bekenstein–Hawking entropy for a wide class of black holes, including BTZ and Kerr-dS, with a universal entropy formula once the ground state is chosen appropriately for rotating versus non-rotating cases. The results imply a universal horizon holography mechanism for black hole entropy and suggest avenues for extending the framework to extremal, non-stationary, or more general horizons and to include gauge/matter fields.

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.