Comparison of the Noether charge and Euclidean methods for computing the entropy of stationary black holes

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, Tetelboim, 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 litsed in Sec. II; approach (iii) appears to require the choice of a ‘‘regularizing’’ scheme to deal with curvature singularities (except in the case of Lovelock gravity theories), 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 the Brown-York 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 ${\mathit{t}}^{\mathit{a}}$ always can be expressed as the spatial integral of ${\mathit{t}}^{\mathit{a}}$ ${\mathit{scrC}}_{\mathit{a}}$, where ${\mathit{scrC}}_{\mathit{a}}$=0 are the constraints associated with the diffeomorphism invariance.