Cosmological New Massive Gravity and Galilean Conformal Algebra in 2 Dimensions

We consider the realization of 2-dimensional Galilean conformal algebraGCA2� on the boundary of cosmological new massive gravity. At first, we consider the contracted BTZ black hole solution. We obtain entropy formula for the GCA2 in terms of contracted scaling dimension Δ and central charge C1. This entropy formula exactly matches with the nonrelativistic limit of Bekenstein- Hawking entropy of BTZ black hole. Then, we extend our study to the contracted warped AdS3 black hole solution of CNMG. We obtain the entropy of dual GCA2 in terms of central charges and finite temperatures, T1, T2. Again, this expression coincides with the nonrelativistic limit of Bekenstein-Hawking entropy formula of warped AdS3 black hole.


Introduction
Recently, there has been some interest in extending the AdS/CFT correspondence to non-relativistic field theories [1], [2]. The Kaluza-Klein type framework for non-relativistic symmetries, used in Refs. [1], [2], is basically identical to the one introduced in [3] (see also [4]). The study of a different non-relativistic limit was initiated in [5], where the non-relativistic conformal symmetry obtained by a parametric contraction of the relativistic conformal group. Galilean conformal algebra (GCA) arises as a contraction relativistic conformal algebras [5], [6], [7], where in 3 + 1 space-time dimensions the Galilean conformal group is a fifteen parameter group which contains the ten parameter Galilean subgroup. Infinite dimensional Galilean conformal group has been reported in [6], [7], the generators of this group are : L n = −(n + 1)t n x i ∂ i − t n+1 ∂ t , M n i = t n+1 ∂ i and J n ij = −t n (x i ∂ j − x j ∂ i ) for an arbitrary integer n, where i and j are specified by the spatial directions. There is a finite dimensional subgroup of the infinite dimensional Galilean conformal group which generated by (J 0 ij , L ±1 , L 0 , M ±1 i , M 0 i ). These generators are obtained by contraction ( t → t, x i → ǫx i , ǫ → 0, v i ∼ ǫ ) of the relativistic conformal generators. Recently the authors of [8] (see also [9]) have shown that the GCA 2 is the asymptotic symmetry of Cosmological Topologically Massive Gravity (CTMG) in the non-relativistic limit. They have obtained the central charges of GCA 2 , and also a non-relativistic generalization of Cardy formula. In the present paper we want to investigate similar problem for Cosmological New Massive Gravity (CNMG). Recently, a new theory of massive gravity in three dimensions was proposed by Bergshoeff, et. al [10]. This theory, referred to as New Massive Gravity (NMG), seems to offer a possibility for formulating a consistent theory of quantum gravity in three dimensions. NMG is defined by adding higher derivative term to the EH term in action, with coupling 1 m 2 . If we add to it the negative cosmological constant we can refer to this three dimensional gravity as Cosmological New Massive Gravity (CNMG). In this theory, the linearized excitations about the anti-de Sitter vacuum describe a propagating massive graviton. One can show that CNMG admits the BTZ black holes as solutions [10], moreover in NMG regular warped black holes have been obtained by Clement [11] (see also [15]). In this paper we propose the contracted BTZ and warped AdS 3 black hole solution of CNMG as a gravity dual of 2d GCA in the context of the nonrelativistic AdS 3 /CF T 2 correspondence. The rest of the paper is organized as: in section 2 we give a brief review of 2d CFT and its contraction, the GCA parameters were realized in term of CFT parameters. In section 3, 4 we study GCA realization of on BTZ, and warped AdS 3 black hole solutions of NMG respectively, in these sections GCA parameters were constructed in term of gravity parameters and finally we obtained finite entropy in nonrelativistic limit. The last section is devoted to the conclusion.

Galilean Conformal Algebra in 2-Dimension
Galilean conformal algebra in 2d can be obtained from contracting 2d conformal symmetry [12]. 2d conformal algebra at the quantum level are described by two copy of Virasoro algebra. In two dimensions space-time (z = t + x, z = t − x), the CFT generators obey the centrally extended Virasoro algebra By taking the non-relativistic limit (t → t, x → ǫx with ǫ → 0), the GCA generators L n and M n are constructed from Virasoro generators by From Eqs. (2) and (3), one obtains centrally extended 2d GCA Note that [M n , M m ] cannot have any central extension. The GCA central charges (C 1 , C 2 ) are related to CFT central charges (c L , c R ) as: Similarly, rapidity ξ and scaling dimensions ∆, which are the eigenvalues of M 0 and L 0 respectively, are given by where h and h are eigenvalues of L 0 and L 0 respectively. So the 2d GCA was obtained by the non-relativistic limit of the Virasoro CF T 2 .

GCA realization of on BTZ black hole solution of NMG
In this section we would like to propose that the contracted BTZ black hole solution of three dimensional new massive gravity (NMG) is gravity dual of 2d GCA in the context of AdS/CF T correspondence. It is notable that the NMG (as a gravity dual) has to yield finite parameters (∆, ξ, C 1 , C 2 and entropy S GCA ) for GCA. The action of the cosmological new massive gravity in three dimension is [10] where the NMG term is The Einstein equation of motion of this action is The parameter λ is dimensionless and characterizes the cosmological constant term, while m has the dimension of mass and provides the coupling to the NMG term. The solution of BTZ black hole is given by The parameters M and J correspond to the mass and angular momentum in the case without the new massive gravity term, but their definitions in the case with NMG term are [11] Due to NMG term, the Bekenstein-Hawking entropy is renormalized by the same factor (1 − 1 2m 2 l 2 ) here r h = 2Gl(lM + J) + 2Gl(lM − J). In the paper [13] Liu and Sun have analyzed the Brown-Henneaux boundary condition for the NMG. They have calculated the conserved charges corresponding to the generators of the asymptotical symmetry under the Brown-Henneaux boundary condition (see the appendix). Then they have obtained the central charges of the Virasoro algebra as [13], [14] c For the BTZ black hole solution in NMG, h and h are calculated as then the microscopic entropy is expressed by Cardy formula which agrees with renormalized Bekenstein-Hawking entropy formula Eq. (14). Now let us consider non-relativistic limit in three dimensional gravity.
Accordingly, the parameters M and J in the BTZ solution must scale like As have been discussed in [8], the black hole metric (11) degenerates and looks singular in the Galilean limits (18), (19). This is similar to the usual Newtonian approximation c → ∞. This situation in the bulk gravity is describe by the Newton-Cartan-like geometry for the geometry with the AdS 2 base [6,8].
Since the Virasoro generators corresponds to L ± n = iξ R,L n , from the gravity side using Eqs.
M n = e inτ (1 + 2e −2ρ n 2 )∂ φ these generators satisfy the center-less version of GCA algebra (4). From Eqs. (5) and (15) the GCA central charges C 1 and C 2 in this case are So, one of the GCA central charges vanishes because of the parity invariance of the NMG. Similarly, from Eqs. (6) and (16), scaling dimensions ∆ and rapidity ξ, which are the eigenvalue of L 0 and M 0 are given by When M are large enough, the last term C 1 2 can be neglected. Now we want to obtain the entropy of the GCA. The scaling limit (M → M , J → ǫJ) requires that the event horizon of the BTZ black hole should scale r h → 2l √ 2GM , so the black hole entropy (14) is given by From the expressions of the central charge C 1 (21) and scaling dimension ∆ (22), we can rewrite the entropy as This expression is the entropy for the GCA in two dimension.

GCA realization on warped AdS 3 black hole solution of NMG
In this section we consider the warped black holes in NMG. The metric of warped NMG can be written as [11,15] The ranges of the coordinates are t ∈ (−∞, +∞), r ∈ [0, +∞) and ϕ ∈ [0, 2π]. There are two horizons which are located at r + and r − . The parameters η, ζ are given by The Bekemstein-Hawking entropy in this case is The microscopic entropy of the dual CFT can be computed by the Cardy formula which matches with the black hole Bekenstein-Hawking entropy [15] The left and right moving temperatures introduced in [15] By the calculation such as we review briefly in appendix, one can obtain the central charges of warped AdS 3 black hole solution of NMG as (see also [15]) From the above central charges and temperatures (29) we have which agrees precisely with the gravity result presented in [15]. Now we consider non-relativistic limit in three dimensional new massive gravity.
GCA central charges C 1 and C 2 are defined in terms of CFT central charges (5) CFT entropy (28) with the limit (ǫ → 0) is converted to Galilean conformal entropy.
We define Galilean conformal temperatures as following the GCA entropy is To make the GCA entropy finite, The parameters r + and r − in the non-relativistic warped NMG must scale like Then, T L + T R = η 2 ζ 2 4πl r + ∼ O(1) and T L − T R = η 2 ζ 2 4πl ǫ ∼ O(ǫ), and the GCA entropy is finite which agrees with Bekenstein-Hawking entropy of warped CNMG in nonrelativistic limit (38).

Conclusion
In the present paper we have considered the BTZ and Warped AdS 3 black hole solution of CNMG in the non-relativistic limit. The BTZ solution of CNMG has been obtained from a BTZ solution of pure Einstein gravity only by redefinition of mass and angular momentum parameter of metric, as Eq.(13), [11]. We have shown that the contracted BTZ solution has a dual description in term of a 2−dimensional GCA. We have obtained the central charges C 1 , C 2 and also the scaling dimension ∆ and rapidity ξ as equations (21), (22) respectively. Using these results and scaling limits (M → M, J → ǫJ), we have obtained the entropy of GCA 2 by Eq.(24), this expression exactly agrees with the non-relativistic limit of Bekenstein-Hawking entropy of the black hole. After that we studied the warped AdS 3 black hole solution in non-relativistic limit. Using the central charges C 1 , C 2 for GCA 2 , we obtained the entropy of non-relativistic CF T 2 , as Eq.(34). Then we have defined Galilean conformal temperatures by Eq.(35). Using these results we have obtained final form of GCA entropy by Eq.(39), which is exactly the non-relativistic limit of entropy formula (27).

Appendix
Here we give a brief review of Brown-Henneaux boundary condition for the NMG (to see more about it refer to [13]). The NMG have an AdS 3 solution The corresponding asymptotic Killing vectors are where τ ± = τ ± φ, ∂ ± = 1 2 (∂ τ ± ∂ φ ). Since φ is periodic, one could choose the basis ξ + m = e imτ + and ξ − n = e inτ − and denote the corresponding Killing vectors as ξ L m and ξ R n . The algebra structure of these vectors is So these asymptotic Killing vectors give two copies of Virasoro algebra. Then by calculating the conserved charges, the authors of [13] have shown that the left moving and right moving conserved charges fulfil two copies of Virasoro algebra with central charges (15).