Black hole entropy from Poisson brackets (demystification of some calculations)

TL;DR

The paper critically examines Carlip's derivation of black hole entropy from Virasoro algebra, identifying issues with the original approach and proposing a modified Poisson bracket to resolve these problems.

Contribution

It demonstrates the shortcomings of the original surface deformation approach and introduces a new Poisson bracket definition that enables the derivation of the Virasoro algebra.

Findings

Original functionals are non-differentiable under standard variations.

Standard Poisson brackets for these functionals are zero, preventing Virasoro algebra formation.

A modified Poisson bracket incorporating surface terms allows correct derivation.

Abstract

Recently it has been suggested by S. Carlip that black hole entropy can be derived from a central charge of the Virasoro algebra arising as a subalgebra in the surface deformations of General Relativity in any dimension. Here it is shown that the argumentation given in Section 2 of hep-th/9812013 and based on the Regge-Teitelboim approach is unsatisfactory. The functionals used are really ``non-differentiable'' under required variations and also the standard Poisson brackets for these functionals are exactly zero so being unable to get any Virasoro algebra with a central charge. Nevertheless Carlip's calculations will be correct if we admit another definition for the Poisson bracket. This new Poisson bracket differs from the standard one in surface terms only and allows to work with ``non-differentiable'' functionals.