Black Hole Entropy and the Hamiltonian Formulation of Diffeomorphism Invariant Theories

TL;DR

This paper develops a microcanonical path-integral framework for black-hole entropy in diffeomorphism-invariant theories and shows that, when formulated in a Hamiltonian form, the entropy arises solely from horizon boundary terms. It derives a general expression S_BH = -2π ∫_H d^{D-2}x sqrt{σ} ε_{ab} ε_{cd} U_0^{abcd}, where U_0^{abcd} is the variational derivative of the Lagrangian with respect to the Riemann tensor, and demonstrates its equivalence to the Noether-charge result of Iyer–Wald. An explicit algorithm is provided to place any such theory in Hamiltonian form, clarifying how boundary contributions govern thermodynamics. The work also connects three path-integral approaches—the microcanonical functional integral, the Hilbert action surface term, and the conical deficit angle method—showing their mutual consistency and applicability to general theories, including cases with Maxwell fields and Rindler horizons.

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.