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The entropy of stationary black holes has recently been calculated by anumber of different approaches. Here we compare the Noether charge approach(defined for any diffeomorphism invariant Lagrangian theory) with variousEuclidean methods, specifically, (i) the microcanonical ensemble approach ofBrown and York, (ii) the closely related approach of Ba\~nados, Teitelboim, andZanelli which ultimately expresses black hole entropy in terms of the Hilbertaction surface term, (iii) another formula of Ba\~nados, Teitelboim and Zanelli(also used by Susskind and Uglum) which views black hole entropy as conjugateto a conical deficit angle, and (iv) the pair creation approach of Garfinkle,Giddings, and Strominger. All of these approaches have a more restrictivedomain of applicability than the Noether charge approach. Specifically,approaches (i) and (ii) appear to be restricted to a class of theoriessatisfying certain properties listed in section 2; approach (iii) appears torequire the Lagrangian density to be linear in the curvature; and approach (iv)requires the existence of suitable instanton solutions. However, we show thatwithin their domains of applicability, all of these approaches yield results inagreement with the Noether charge approach. In the course of our analysis, wegeneralize the definition of Brown and York's quasilocal energy to a much moregeneral class of diffeomorphism invariant, Lagrangian theories of gravity. Inan appendix, we show that in an arbitrary diffeomorphism invariant theory ofgravity, the ``volume term" in the ``off-shell" Hamiltonian associated with atime evolution vector field $t^a$ always can be expressed as the spatialintegral of $t^a {\cal C}_a$, where ${\cal C}_a = 0$ are the constraintsassociated with the diffeomorphism invariance.Comment: 29 pages (double-spaced) late

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