Black Hole Entropy and the Hamiltonian Formulation of Diffeomorphism Invariant Theories

TL;DR

This paper derives a general formula for black hole entropy in diffeomorphism invariant theories using path integral methods and Hamiltonian formalism, aligning with Noether charge results.

Contribution

It introduces an algorithm to express the action of such theories in Hamiltonian form and compares different path integral approaches for black hole entropy.

Findings

Derived a general entropy expression depending on the Lagrangian's Riemann tensor derivative.

Established equivalence with Noether charge methods by Iyer and Wald.

Presented relationships between various path integral approaches.

Abstract

Path integral methods are used to derive a general expression for the entropy of a black hole in a diffeomorphism invariant theory. The result, which depends on the variational derivative of the Lagrangian with respect to the Riemann tensor, agrees with the result obtained from Noether charge methods by Iyer and Wald. The method used here is based on the direct expression of the density of states as a path integral (the microcanonical functional integral). The analysis makes crucial use of the Hamiltonian form of the action. An algorithm for placing the action of a diffeomorphism invariant theory in Hamiltonian form is presented. Other path integral approaches to the derivation of black hole entropy include the Hilbert action surface term method and the conical deficit angle method. The relationships between these path integral methods are presented.