A comparison of Noether charge and Euclidean methods for Computing the Entropy of Stationary Black Holes

TL;DR

This work unifies several Euclidean and semi-classical approaches to black hole entropy by showing that, within their respective domains of applicability, they all reproduce the Noether-charge entropy formula $S = 2\pi \int_{\cal H} \mathbf{Q}[t]$ for stationary black holes. It extends Brown–York’s quasilocal energy to general diffeomorphism-invariant gravities and demonstrates that the microcanonical action, Hilbert action boundary-term, conical deficit, and pair-creation methods each converge to the same entropy expression under appropriate assumptions. The paper also clarifies the domain restrictions of the Euclidean methods (notably the conical deficit’s limitation to linear-in-curvature Lagrangians) and provides an appendix on the off-shell Hamiltonian structure, reinforcing the deep link between boundary terms, constraints, and black hole thermodynamics. Overall, it offers a coherent framework that links Noether charges, quasilocal quantities, and Euclidean path-integral notions of black hole entropy across a broad class of gravitational theories.

Abstract

The entropy of stationary black holes has recently been calculated by a number of different approaches. Here we compare the Noether charge approach (defined for any diffeomorphism invariant Lagrangian theory) with various Euclidean methods, specifically, (i) the microcanonical ensemble approach of Brown and York, (ii) the closely related approach of Bañados, Teitelboim, and Zanelli which ultimately expresses black hole entropy in terms of the Hilbert action surface term, (iii) another formula of Bañados, Teitelboim and Zanelli (also used by Susskind and Uglum) which views black hole entropy as conjugate to a conical deficit angle, and (iv) the pair creation approach of Garfinkle, Giddings, and Strominger. All of these approaches have a more restrictive domain of applicability than the Noether charge approach. Specifically, approaches (i) and (ii) appear to be restricted to a class of theories satisfying certain properties listed in section 2; approach (iii) appears to require the Lagrangian density to be linear in the curvature; and approach (iv) requires the existence of suitable instanton solutions. However, we show that within their domains of applicability, all of these approaches yield results in agreement with the Noether charge approach. In the course of our analysis, we generalize the definition of Brown and York's quasilocal energy to a much more general class of diffeomorphism invariant, Lagrangian theories of gravity. In an appendix, we show that in an arbitrary diffeomorphism invariant theory of gravity, the ``volume term" in the ``off-shell" Hamiltonian associated with a time evolution vector field $t^a$ always can be expressed as the spatial integral of $t^a {\cal C}_a$, where ${\cal C}_a = 0$ are the constraints associated with the diffeomorphism invariance.