We consider a general, classical theory of gravity with arbitrary matter fields in n dimensions, arising from a diffeomorphism invariant Lagrangian, L. We first show that L always can be written in a "manifestly covariant" form. We then show that the symplectic potential current (n − 1)-form, Θ, and the symplectic current (n − 1)-form, ω, for the theory always can be globally defined in a covariant manner. Associated with any infinitesimal diffeomorphism is a Noether current (n − 1)-form, J, and corresponding Noether charge (n − 2)-form, Q. We derive a general "decomposition formula" for Q. Using this formula for the Noether charge, we prove that the first law of black hole mechanics holds for arbitrary perturbations of a stationary black hole. (For higher derivative theories, previous arguments had established this law only for stationary perturbations.) Finally, we propose a local, geometrical prescription for the entropy, S dyn , of a dynamical black hole. This prescription agrees with the Noether charge formula for stationary black holes and their perturbations, and is independent of all ambiguities associated with the choices of L, Θ, and Q. However, the issue of whether this dynamical entropy in general obeys a "second law" of black hole mechanics remains open. In an appendix, we apply some of our results to theories with a nondynamical metric and also briefly develop the theory of stress-energy pseudotensors. 1Recently, many authors have investigated the validity of the first law of black hole mechanics and the definition of the entropy of a black hole in a wide class of theories derivable from a Hamiltonian or Lagrangian [1]- [10]. In particular, in [6] the first law was proven to hold in an arbitrary theory of gravity derived from a diffeomorphism invariant Lagrangian, and the quantity playing the role of the entropy of the black hole was identified as the integral over the horizon of the Noether charge associated with the horizon Killing vector field. Although some key issues concerning the validity of the first law and the definition of black hole entropy in a general theory of gravity were thereby resolved, the analysis of [6], nevertheless, was deficient in the following ways: (1) It was not recognized that a diffeomorphism covariant choice of the symplectic potential current form always can be made. Consequently, several steps in the arguments were made in an unnecessarily awkward manner.(2) While a completely general proof of the first law of black hole mechanics was given for perturbations to nearby stationary black holes, a proof of the first law for non-stationary perturbations was given only for theories in which the Noether charge takes a particular, simple form. (3) A proposal was made for defining the entropy of a dynamical black hole. However, this proposal made use of a rather arbitrary choice of algorithm for defining the symplectic potential current form, and it turns out to possess the undesireable feature that the addition of an exact form to the Lagrangian (which has no effect upon th...

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 re-1 lated 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.

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The first law of black hole mechanics (in the form derived by Wald), is expressed in terms of integrals over surfaces, at the horizon and spatial infinity, of a stationary, axisymmetric black hole, in a diffeomorphism invariant Lagrangian theory of gravity. The original statement of the first law given by Bardeen, Carter and Hawking for an Einstein-perfect fluid system contained, in addition, volume integrals of the fluid fields, over a spacelike slice stretching between these two surfaces. When applied to the Einstein-perfect fluid system, however, Wald's methods yield restricted results. The reason is that the fluid fields in the Lagrangian of a gravitating perfect fluid are typically nonstationary. We therefore first derive a first law-like relation for an arbitrary Lagrangian metric theory of gravity coupled to arbitrary Lagrangian matter fields, requiring only that the metric field be stationary. This relation includes a volume integral of matter fields over a spacelike slice between the black hole horizon and spatial infinity, and reduces to the first law originally derived by Bardeen, Carter and Hawking when the theory is general relativity coupled to a perfect fluid. We also consider a specific Lagrangian formulation for an isentropic perfect fluid given by Carter, and directly apply Wald's analysis. The resulting first law contains only surface integrals at the black hole horizon and spatial infinity, but this relation is much more restrictive in its allowed fluid configurations and perturbations than that given by Bardeen, Carter and Hawking. In the Appendix, we use the symplectic structure of the Einstein-perfect fluid system to derive a conserved current for perturbations of this system: this current reduces to one derived ab initio for this system by Chandrasekhar and Ferrari.Comment: 26 pages LaTeX-2

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