1995/03/27 by Vivek Iyer, Robert M. Wald · 9 citations
Mathematics · Physics and Astronomy · #Black Holes and Theoretical Physics #Black hole thermodynamics #Cosmology and Gravitation Theories #Curvature #Diffeomorphism #Entropy (arrow of time) #Euclidean geometry #Geometry #Invariant (physics) #Lagrangian #Mathematical physics #Mathematics #Noether's theorem #Noncommutative and Quantum Gravity Theories #Physics #Pure mathematics #Quantum mechanics #gr-qc #hep-th
paper · pdf · doi:10.1103/physrevd.52.4430
published as Phys.Rev. D52 (1995) 4430-4439 · 29 pages (double-spaced) latex
arxiv created 1995/03/27 · openalex publication_date 1995/10/15 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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 Ba\~nados, Teitelboim, and Zanelli which ultimately expresses black hole entropy in terms of the Hilbert action surface term, (iii) another formula of Ba\~nados, 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 ta always can be expressed as the spatial integral of tascrCa, where scrCa=0 are the constraints associated with the diffeomorphism invariance.