The CFT-interpolating black hole in three dimensions

Abstract

We present a new exact black hole solution in three dimensional Einstein gravity coupled to a single scalar field. This is one of the extended solutions of the BTZ black hole and has in fact AdS 3 geometries both at the spatial infinity and at the event horizon. An explicit derivation of Virasoro algebras for CFT 2 at the two boundaries is shown to be possibleà la Brown and Henneaux's calculation. If we regard the scalar field as a running coupling in the dual two dimensional field theory, and its flow in the bulk as the "holographic" renormalization group flow, our black hole should interpolate the two CFT 2 living at the infinity and at the horizon. Following the Hamilton-Jacobi analysis by de Boer, Verlinde and Verlinde, we calculate the central charges c UV and c IR for the CFT 2 on the infinity and the horizon, respectively. We also confirm that the inequality c IR < c UV is satisfied, which is consistent with the Zamolodchikov's c-theorem.It has been believed that three dimensional gravity with negative cosmological constant is an important key to uncover some aspects of the quantum gravity. A vacuum solution is described by three dimensional anti-de Sitter (AdS 3 ) geometry globally [1], and a black hole solution with mass and angular momentum (BTZ black hole) is constructed from locally AdS 3 geometry with appropriate identifications of boundaries [2]. All these geometries become asymptotically AdS 3 , and two dimensional conformal field theory (CFT 2 ) is expected to exist at the boundary of the three dimensional geometry [3]. This is one of important examples of AdS/CFT correspondence [4], which enables us to evaluate physical quantities in CFT from the gravity side [5,6]. Thus many efforts have been directed to understand quantum nature of the three dimensional gravity from the viewpoint of AdS/CFT correspondence.In fact, by using the canonical formalism, Brown and Henneaux showed that general coordinate transformations which preserve the boundary behavior of the geometries form Virasoro algebras for left and right movers [3]. Furthermore they succeeded to evaluate central extensions of these algebras and found that the central charges for left and right movers take the same value. From the CFT 2 viewpoint, the globally AdS 3 corresponds to the ground state, and the BTZ black hole does to excited states. Surprisingly, the macroscopic entropy of the BTZ black hole is explained by counting the number of degenerate states in CFT 2 [7].The Brown-Henneaux's canonical approach has been applied to several interesting three dimensional theories. In ref.[8], the gravity theory with higher derivative corrections was considered. The values of central charges are scaled because of the higher derivative terms. An application to the topologically massive gravity was done in ref. [9], and left-right asymmetric central charges were derived due to the gravitational Chern-Simons term. On the other hand, the canonical approaches to the theories of Einstein gravity coupled to scalar fields were investigated in refs.[10]- ...

Higher-Derivative Corrections to the Asymptotic Virasoro Symmetry of 4d Extremal Black Holes

“…21) In our paper, we deal with Lagrangians which contain arbitrarily high derivatives of the Riemann tensor, but we introduce towers of auxiliary fields so that the derivatives in the Lagrangian is of the second order at most. Then the formulae written above suffice.…”

Scalar hairs and exact vortex solutions in 3D AdS gravity

“…It is interesting to notice that these interpolating features of the vortex solution are more similar to those of the rotating solution of Ref. [5] (both interpolated geometries are AdS 3 ) then to those of the electrically charged solutions of Ref. [6] (the solution interpolates between AdS 3 and AdS 2 × S 1 ).…”

“…Conversely, the black hole-vortex solutions we have derived in this paper share many feature with both rotating [5] and electrically charged [6] 3D black hole solutions. They are characterized by the presence of an inner and outer horizon and the vortex interpolates between two AdS 3 geometries with different AdS lengths.…”

“…The dynamics of scalar fields in the AdS d background gives crucial information about the dynamics of the dual (d − 1)-dimensional field theory. For instance, scalar hair configuration may interpolate between two different AdS geometries (one asymptotic and the other near-horizon) [5] similarly to what happens for charged AdS black holes [6,7,8].…”

“…The study of black holes formed by global vortex is interesting by itself, as a particular case of black hole solutions supported by solitons. This kind of solutions may have some relevance in cosmology (as viable cosmic string candidates [11]), in the context of the AdS/CFT correspondence as nonperturbative 3D bulk solutions interpolating between different AdS vacua [5] and also as possible gravitational duals of condensed matter systems [12].…”