Abstract
We obtain a black hole solution of minimal massive gravity theory with Maxwell and electromagnetic Chern-Simons terms using the first order formalism. This black hole solution can be translated into the spacelike warped $AdS_3$ black hole solution with some parameters' conditions changing their coordinates system into a Schwarzschild one. Applying the Wald formalism to this theory with the first order formalism, we also find out the entropy, mass, and angular momentum of this black hole solution satisfies the first law of black hole thermodynamics. Under the assumption that minimal massive gravity theory with suitable asymptotically warped $AdS_3$ boundary conditions is holographically dual to a two dimensional boundary conformal field theory, we find appropriate central charges by using the relations between entropy of this black hole and the Cardy formula in dual conformal field theory described by left and right moving central charges and temperatures.
Highlights
There are several three-dimensional gravity theories that are suggested to explain various physical problems, for example, topologically massive gravity (TMG) [1,2], new massive gravity (NMG) [3], minimal massive gravity (MMG) [4], etc
In the viewpoint of anti–de Sitter (AdS)/CFT correspondence, there exists a discrepancy between any three-dimensional gravity theory with asymptotic AdS3 geometry and its dual conformal field theory (CFT) on the boundary
According to Wald’s procedure with the first order formalism, we can define the mass and angular momentum of a black hole as variational forms which are related to the integration of a charge variation form δχξ at spatial infinity (73) [15]
Summary
There are several three-dimensional gravity theories that are suggested to explain various physical problems, for example, topologically massive gravity (TMG) [1,2], new massive gravity (NMG) [3], minimal massive gravity (MMG) [4], etc. We use the Wald’s formalism to obtain the entropy, mass, and angular momentum of black holes with the first order formalism In this method we can define charge variation forms to calculate these physical quantities. The mass and angular momentum of black holes can be obtained by an integration of the charge variation form at spatial infinity, respectively This method becomes a good working tool in case there are some boundaries at infinity such as asymptotically Minkowski and anti–de Sitter (AdS) space-time. Appendices are attached to explain some formulas, and to compute some equations, entropy, mass, and angular momentum of the warped AdS3 black hole in MMG theory
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