Article,
A comparison of Noether charge and Euclidean methods for Computing the Entropy of Stationary Black Holes
Physical Review D, 52 (8): 4430--4439 (Mar 27, 1995)
DOI: 10.1103/physrevd.52.4430
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 C\_a\$, where \$C\_a = 0\$ are the constraints
associated with the diffeomorphism invariance.
