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      Towards metric-like higher-spin gauge theories in three dimensions

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          Abstract

          We consider the coupling of a symmetric spin-3 gauge field to three-dimensional gravity in a second order metric-like formulation. The action that corresponds to an SL(3,R) x SL(3,R) Chern-Simons theory in the frame-like formulation is identified to quadratic order in the spin-3 field. We apply our result to compute corrections to the area law for higher-spin black holes using Wald's entropy formula.

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          2 + 1 dimensional gravity as an exactly soluble system

          Edward Witten (1988)
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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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              Geometry of the 2+1 Black Hole

              Jorge Zanelli, Claudio Teitelboim, Marc Henneaux … (1993)
              The geometry of the spinning black holes of standard Einstein theory in 2+1 dimensions, with a negative cosmological constant and without couplings to matter, is analyzed in detail. It is shown that the black hole arises from identifications of points of anti-de Sitter space by a discrete subgroup of \(SO(2,2)\). The generic black hole is a smooth manifold in the metric sense. The surface \(r=0\) is not a curvature singularity but, rather, a singularity in the causal structure. Continuing past it would introduce closed timelike lines. However, simple examples show the regularity of the metric at \(r=0\) to be unstable: couplings to matter bring in a curvature singularity there. Kruskal coordinates and Penrose diagrams are exhibited. Special attention is given to the limiting cases of (i) the spinless hole of zero mass, which differs from anti-de Sitter space and plays the role of the vacuum, and (ii) the spinning hole of maximal angular momentum . A thorough classification of the elements of the Lie algebra of \(SO(2,2)\) is given in an Appendix.
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                Author and article information

                Journal
                Publication date Created: 09 August 2012
                Publication date Updated: 2013-09-12
                Article
                DOI: 10.1088/1751-8113/46/21/214017
                ArXiV ID: 1208.1851
                SO-VID: 9c96cc26-27a6-477e-8542-668a6e3708fb
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                http://arxiv.org/licenses/nonexclusive-distrib/1.0/

                History
                Custom metadata
                Report No AEI-2012-081
                Journal reference J.Phys. A46 (2013) 214017
                Comments 29 pages; v2: typos corrected
                Categories hep-th gr-qc

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

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