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      Enthalpy and the Mechanics of AdS Black Holes

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          Abstract

          We present geometric derivations of the Smarr formula for static AdS black holes and an expanded first law that includes variations in the cosmological constant. These two results are further related by a scaling argument based on Euler's theorem. The key new ingredient in the constructions is a two-form potential for the static Killing field. Surface integrals of the Killing potential determine the coefficient of the variation of the cosmological constant in the first law. This coefficient is proportional to a finite, effective volume for the region outside the AdS black hole horizon, which can also be interpreted as minus the volume excluded from a spatial slice by the black hole horizon. This effective volume also contributes to the Smarr formula. Since the cosmological constant is naturally thought of as a pressure, the new term in the first law has the form of effective volume times change in pressure that arises in the variation of the enthalpy in classical thermodynamics. This and related arguments suggest that the mass of an AdS black hole should be interpreted as the enthalpy of the spacetime.

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          Black holes in higher dimensional space-times

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            Some Properties of Noether Charge and a Proposal for Dynamical Black Hole Entropy

            We consider a general, classical theory of gravity with arbitrary matter fields in \(n\) dimensions, arising from a diffeomorphism invariant Lagrangian, \(\bL\). We first show that \(\bL\) always can be written in a ``manifestly covariant" form. We then show that the symplectic potential current \((n-1)\)-form, \(\th\), and the symplectic current \((n-1)\)-form, \(\om\), for the theory always can be globally defined in a covariant manner. Associated with any infinitesimal diffeomorphism is a Noether current \((n-1)\)-form, \(\bJ\), and corresponding Noether charge \((n-2)\)-form, \(\bQ\). We derive a general ``decomposition formula" for \(\bQ\). 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 \(\bL\), \(\th\), and \(\bQ\). 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.
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              Black Hole Entropy is Noether Charge

              We consider a general, classical theory of gravity in \(n\) dimensions, arising from a diffeomorphism invariant Lagrangian. In any such theory, to each vector field, \(\xi^a\), on spacetime one can associate a local symmetry and, hence, a Noether current \((n-1)\)-form, \({\bf j}\), and (for solutions to the field equations) a Noether charge \((n-2)\)-form, \({\bf Q}\). Assuming only that the theory admits stationary black hole solutions with a bifurcate Killing horizon, and that the canonical mass and angular momentum of solutions are well defined at infinity, we show that the first law of black hole mechanics always holds for perturbations to nearby stationary black hole solutions. The quantity playing the role of black hole entropy in this formula is simply \(2 \pi\) times the integral over \(\Sigma\) of the Noether charge \((n-2)\)-form associated with the horizon Killing field, normalized so as to have unit surface gravity. Furthermore, we show that this black hole entropy always is given by a local geometrical expression on the horizon of the black hole. We thereby obtain a natural candidate for the entropy of a dynamical black hole in a general theory of gravity. Our results show that the validity of the ``second law" of black hole mechanics in dynamical evolution from an initially stationary black hole to a final stationary state is equivalent to the positivity of a total Noether flux, and thus may be intimately related to the positive energy properties of the theory. The relationship between the derivation of our formula for black hole entropy and the derivation via ``Euclidean methods" also is explained.
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                Author and article information

                Journal
                17 April 2009
                2009-05-11
                Article
                10.1088/0264-9381/26/19/195011
                0904.2765
                0332040f-2568-4f3b-8b66-d88a11b93d96

                http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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                Custom metadata
                Class.Quant.Grav.26:195011,2009
                21 pages; v2 references added
                hep-th

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