Noether current of the surface term of Einstein-Hilbert action, Virasoro
  algebra and entropy

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Abstract

A derivation of Noether current from the surface term of Einstein-Hilbert action is given. We show that the corresponding charge, calculated on the horizon, is related to the Bekenstein-Hawking entropy. Also using the charge, the same entropy is found based on the Virasoro algebra and Cardy formula approach. In this approach, the relevant diffeomorphisms are found by imposing a very simple physical argument: diffeomorphisms keep the horizon structure invariant. This complements similar earlier results (arXiv:1204.1422) obtained from York-Gibbons-Hawking surface term. Finally we discuss the technical simplicities and improvements over the earlier attempts and also various important physical implications.

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Operator content of two-dimensional conformally invariant theories

John Cardy (1986)

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

Robert Wald, Vivek Iyer (1994)

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

Robert Wald (1993)

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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Publication date Created: 25 October 2012

Publication date Updated: 2012-10-27

Article

DOI: 10.1155/2013/386342

ArXiV ID: 1210.6736

SO-VID: ab2ee980-783b-47fd-b667-0a3752dc6529

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http://creativecommons.org/licenses/by/3.0/

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Journal reference Advances in High Energy Physics 2013 (2013) 386342

Comments V2: 18 pages, no figures, clarifications and references added

Categories gr-qc hep-th

ScienceOpen disciplines: General relativity & Quantum cosmology,High energy & Particle physics

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ScienceOpen disciplines: General relativity & Quantum cosmology, High energy & Particle physics

Noether current of the surface term of Einstein-Hilbert action, Virasoro algebra and entropy

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High Entropy Alloys and Complex Concentrated Alloys

Most cited references 61

Operator content of two-dimensional conformally invariant theories

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

Black Hole Entropy is Noether Charge

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