The Hamiltonian dynamics of bounded spacetime and black hole entropy: the canonical method

Abstract

From first principles, I present a concrete realization of Carlip's idea on black hole entropy from conformal field theory on the horizon in any dimension. The new formulation is free of the inconsistencies encountered in Carlip's formulation. By considering a correct gravity action, of which the 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 classical Virasoro algebra is crucial in obtaining an operator Virasoro algebra, through canonical quantization, which produces the right central charge and conformal weight ∼ A +/ℏ 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 rotating and 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 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.