Cosmological New Massive Gravity and Galilean Conformal Algebra in 2 Dimensions

Abstract

We consider the realization of 2-dimensional Galilean conformal algebra (GCA₂) 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.

1. Introduction

Recently, there has been some interest in extending the AdS/CFT correspondence to nonrelativistic field theories [1, 2]. The Kaluza-Klein-type framework for nonrelativistic symmetries, used in [1, 2], is basically identical to the one introduced in [3, 4] (see also [5, 6]). The study of a different nonrelativistic limit was initiated in [7, 8], where the nonrelativistic conformal symmetry was obtained by a parametric contraction of the relativistic conformal group. Galilean conformal algebra (GCA) arises as contraction relativistic conformal algebras [7–10], 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 [9, 10], the generators of this group are 𝐿^𝑛=−(𝑛+1)𝑡^𝑛𝑥_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 (𝐽⁰_𝑖𝑗,𝐿^±1,𝐿⁰,𝑀_𝑖^±1,𝑀⁰_𝑖). These generators are obtained by contraction (𝑡→𝑡,𝑥_𝑖→𝜖𝑥_𝑖,𝜖→0,𝑣_𝑖∼𝜖) of the relativistic conformal generators. Recently the authors of [11] (see also [12, 13]) have shown that the GCA₂ is the asymptotic symmetry of cosmological topologically massive gravity (CTMG) in the nonrelativistic limit. They have obtained the central charges of GCA₂, and also a nonrelativistic 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. [14, 15]. 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/𝑚². 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 [14, 15], moreover in NMG regular warped black holes have been obtained by Clément [16] (see also [17, 18]).

In this paper we propose the contracted BTZ and warped AdS₃ black hole solution of CNMG as a gravity dual of 2𝑑 GCA in the context of the nonrelativistic AdS₃/CFT₂ correspondence. The rest of the paper is organized as follows: 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 Sections 3 and 4 we study GCA realization of BTZ, and warped AdS₃ black hole solutions of NMG, respectively, in these sections GCA parameters were constructed in terms of gravity parameters, and finally we obtained finite entropy in nonrelativistic limit. The last section is devoted to the conclusion.

2. Galilean Conformal Algebra in 2 Dimensions

Galilean conformal algebra in 2𝑑 can be obtained from contracting 2𝑑 conformal symmetry [19]. 2𝑑 conformal algebra at the quantum level is described by two copy of Virasoro algebra. In two-dimensions space-time (𝑧=𝑡+𝑥, 𝑧=𝑡−𝑥), the CFT generators ℒ_𝑛=−𝑧^𝑛+1𝜕_𝑧,ℒ_𝑛=−𝑧^𝑛+1𝜕_𝑧 (2.1) obey the centrally extended Virasoro algebra ℒ_𝑚,ℒ_𝑛=(𝑚−𝑛)ℒ_𝑚+𝑛+𝑐_𝑅𝑚𝑚12²−1𝛿_𝑚+𝑛,0,ℒ_𝑚,ℒ_𝑛=(𝑚−𝑛)ℒ_𝑚+𝑛+𝑐_𝐿𝑚𝑚12²−1𝛿_𝑚+𝑛,0. (2.2) By taking the nonrelativistic limit (𝑡→𝑡, 𝑥→𝜖𝑥 with 𝜖→0), the GCA generators 𝐿_𝑛 and 𝑀_𝑛 are constructed from Virasoro generators by 𝐿_𝑛=lim_𝜖→0ℒ_𝑛+ℒ_𝑛,𝑀_𝑛=lim_𝜖→0𝜖ℒ_𝑛−ℒ_𝑛. (2.3) From (2.2) and (2.3), one obtains centrally extended 2𝑑 GCA 𝐿_𝑚,𝐿_𝑛=(𝑚−𝑛)𝐿_𝑚+𝑛+𝐶₁𝑚𝑚²−1𝛿_𝑚+𝑛,0,𝐿_𝑚,𝑀_𝑛=(𝑚−𝑛)𝑀_𝑚+𝑛+𝐶₂𝑚𝑚²−1𝛿_𝑚+𝑛,0,𝑀_𝑛,𝑀_𝑚=0. (2.4) Note that [𝑀_𝑛,𝑀_𝑚] cannot have any central extension. The GCA central charges (𝐶₁, 𝐶₂) are related to CFT central charges (𝑐_𝐿, 𝑐_𝑅) as 𝐶₁=lim_𝜖→0𝑐_𝐿+𝑐_𝑅12,𝐶₂=lim_𝜖→0𝜖𝑐_𝐿−𝑐_𝑅12. (2.5) Similarly, rapidity 𝜉 and scaling dimensions Δ, which are the eigenvalues of 𝑀₀ and 𝐿₀ respectively, are given by Δ=lim_𝜖→0ℎ+ℎ,𝜉=lim_𝜖→0𝜖ℎ−ℎ, (2.6) where ℎ and ℎ are eigenvalues of 𝐿₀ and 𝐿₀, respectively. So the 2𝑑 GCA was obtained by the nonrelativistic limit of the Virasoro CFT₂.

3. GCA Realization of 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/CFT correspondence. It is notable that the NMG (as a gravity dual) has to yield finite parameters (Δ,𝜉,𝐶₁,𝐶₂, and entropy 𝑆_GCA) for GCA. The action of the cosmological new massive gravity in three dimensions is [14, 15] 𝑆𝑔_𝜇𝜈=1√16𝜋𝐺1−𝑔𝑅−2𝜆+𝑚²ℒ_NMG, (3.1) where the NMG term is ℒ_NMG=𝑅_𝜇𝜈𝑅^𝜇𝜈−38𝑅². (3.2) The Einstein equation of motion of this action is 𝐺_𝜇𝜈+𝜆𝑚²𝑔_𝜇𝜈−12𝑚²𝐾_𝜇𝜈=0, (3.3) where 𝐾_𝜇𝜈1=−2∇²𝑅𝑔_𝜇𝜈−12∇_𝜇∇_𝜈𝑅+2∇²𝑅_𝜇𝜈+4𝑅_{𝜇𝛼𝜈𝛽}𝑅^𝛼𝛽−32𝑅𝑅_𝜇𝜈−𝑅_𝛼𝛽𝑅^𝛼𝛽𝑔_𝜇𝜈+32𝑅²𝑔_𝜇𝜈, (3.4) where 𝑔^𝜇𝜈𝐾_𝜇𝜈=ℒ_NMG. 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 𝑑𝑠²=−𝑓(𝑟)+16𝐺²𝐽²𝑟²𝑑𝑡²+𝑑𝑟²𝑓(𝑟)+𝑟²𝑑𝜑²+8𝐺𝐽𝑑𝑡𝑑𝜑, (3.5) where 𝑓(𝑟)=𝑟²𝑙²−8𝐺𝑀+16𝐺²𝐽²𝑟²,𝑙⁻²=2𝑚²−1±𝜆1−𝑚². (3.6) 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 [16] 𝑀₁=11−2𝑚²𝑙²𝑀,𝐽₁=11−2𝑚²𝑙²𝐽. (3.7) Due to NMG term, the Bekenstein-Hawking entropy is renormalized by the same factor (1−1/2𝑚²𝑙²)𝑆_BH=𝜋𝑟_ℎ12𝐺1−2𝑚²𝑙², (3.8) here 𝑟_ℎ=2𝐺𝑙(𝑙𝑀+𝐽)+2𝐺𝑙(𝑙𝑀−𝐽). In the paper [20] 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 Appendix). Then they have obtained the central charges of the Virasoro algebra as [20–22] 𝑐_𝐿=𝑐_𝑅=3𝑙12𝐺1−2𝑚²𝑙². (3.9) For the BTZ black hole solution in NMG,ℎ and ℎ are calculated as 1ℎ=2𝑙𝑀₁+𝐽₁+𝑐_𝐿,241ℎ=2𝑙𝑀₁−𝐽₁+𝑐_𝑅24, (3.10) then the microscopic entropy is expressed by Cardy formula 𝑆_CFT=2𝜋𝑐_𝐿ℎ6+𝑐_𝑅ℎ6=𝜋𝑟_ℎ12𝐺1−2𝑚²𝑙², (3.11) which agrees with renormalized Bekenstein-Hawking entropy formula (3.8). Now let us consider nonrelativistic limit in three-dimensional gravity: 𝑡⟶𝑡,𝑟⟶𝑟,𝜑⟶𝜖𝜑. (3.12) Accordingly, the parameters 𝑀 and 𝐽 in the BTZ solution must scale like 𝑀⟶𝑀,𝐽⟶𝜖𝐽. (3.13) As have been discussed in [11], the black hole metric (3.5) degenerates and looks singular in the Galilean limits (3.12), (3.13). 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 AdS₂ base [9, 11].

Since the Virasoro generators corresponds to ℒ^±_𝑛=𝑖𝜉_𝑛^𝑅,𝐿, from the gravity side using (A.3), (3.12), and (2.3) one can define the generators 𝐿_𝑛 and 𝑀_𝑛 as 𝐿_𝑛=𝑒^𝑖𝑛𝜏1−2𝑒^−2𝜌𝑛²𝜕_𝜏+𝑖𝑛𝜙1+2𝑒^−2𝜌𝑛²𝜕_𝜙−𝑖𝑛𝜕_𝜌,𝑀_𝑛=𝑒^𝑖𝑛𝜏1+2𝑒^−2𝜌𝑛²𝜕_𝜙, (3.14) these generators satisfy the centerless version of GCA algebra (2.4). From (2.5) and (3.9) the GCA central charges 𝐶₁ and 𝐶₂ in this case are 𝐶₁=𝑙14𝐺1−2𝑚²𝑙²,𝐶₂=0. (3.15) So, one of the GCA central charges vanishes because of the parity invariance of the NMG.

Similarly, from (2.6) and (3.10), scaling dimensions Δ and rapidity 𝜉, which are the eigenvalue of 𝐿₀ and 𝑀₀ are given by Δ=lim_𝜖→0𝑙𝑀₁+𝑐_𝐿+𝑐_𝑅=1241−2𝑚²𝑙²𝐶𝑙𝑀+¹2,𝜉=lim_𝜖→0𝜖𝑐𝐽+^𝐿−𝑐_𝑅24=0. (3.16) When 𝑀 is large enough, the last term 𝐶₁/2 can be neglected.

Now we want to obtain the entropy of the GCA. The scaling limit (𝑀→𝑀, 𝐽→𝜖𝐽) requires that the event horizon of the BTZ black hole should scale 𝑟_ℎ→2𝑙√2𝐺𝑀, so the black hole entropy (3.8) is given by lim_𝜖→0𝑆_BH=lim_𝜖→0𝑆_CFT1=𝜋1−2𝑚²𝑙²2𝑙²𝑀𝐺. (3.17) From the expressions of the central charge 𝐶₁ (3.15) and scaling dimension Δ (3.16), we can rewrite the entropy as 𝑆_GCA=𝜋8Δ𝐶₁. (3.18) This expression is the entropy for the GCA in two dimensions.

4. GCA Realization on Warped AdS₃ 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 [16–18] 𝑑𝑠²𝑙²=𝑑𝑡²+𝑑𝑟²𝜁²𝜂²𝑟−𝑟₊𝑟−𝑟₋+𝜁𝑟+𝜂√𝑟₊𝑟₋+𝜁𝑑𝑡𝑑𝜑₂4𝑟1−𝜂²𝑟+𝜂²𝑟₊+𝑟₋+2𝜂√𝑟₊𝑟₋𝑑𝜑². (4.1) 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 𝜂²=214+2𝑚²𝑙²𝜁²,𝜁²=4𝑚²𝑙²21−6±3(5−7𝜆). (4.2) The Bekenstein-Hawking entropy in this case is 𝑆_BH=𝜋𝑙𝜁³2𝑚²𝐺𝑟₊+𝜂√𝑟₊𝑟₋. (4.3) The microscopic entropy of the dual CFT can be computed by the Cardy formula which matches with the black hole Bekenstein-Hawking entropy [17, 18] 𝑆_CFT=𝜋²𝑙3𝑐_𝐿𝑇_𝐿+𝑐_𝑅𝑇_𝑅. (4.4) The left and right moving temperatures introduced in [17, 18]: 𝑇_𝐿=𝜂²𝜁²𝑟8𝜋𝑙⁺+𝑟₋+2𝜂√𝑟₊𝑟₋,𝑇_𝑅=𝜂²𝜁²𝑟8𝜋𝑙⁺−𝑟₋. (4.5) By the calculation such as we review briefly in the Appendix, one can obtain the central charges of warped AdS₃ black hole solution of NMG as (see also [17, 18]) 𝑐_𝐿=𝑐_𝑅=6𝑙𝜁𝐺𝜂²𝑚². (4.6) From the above central charges and temperatures (4.5) we have 𝑆_CFT=𝜋𝑙𝜁³2𝑚²𝐺𝑟₊+𝜂√𝑟₊𝑟₋, (4.7) which agrees precisely with the gravity result presented in [17, 18]. Now we consider nonrelativistic limit in three-dimensional new massive gravity: 𝑡⟶𝑡,𝑟⟶𝑟,𝜑⟶𝜖𝜑. (4.8) GCA central charges 𝐶₁ and 𝐶₂ are defined in terms of CFT central charges (2.5) 𝐶₁=𝑙𝜁𝐺𝜂²𝑚²,𝐶₂=0. (4.9) CFT entropy (4.4) with the limit (𝜖→0) is converted to Galilean conformal entropy: 𝑆_GCA=lim_𝜖→0𝜋²36𝐶₁𝑇_𝐿+𝑇_𝑅+6𝐶₂𝑇_𝐿−𝑇_𝑅𝜖. (4.10) We define Galilean conformal temperatures as follows: 𝑇₁=lim_𝜖→06𝑇_𝐿+𝑇_𝑅,𝑇₂=lim_𝜖→06𝑇_𝐿−𝑇_𝑅𝜖, (4.11) the GCA entropy is 𝑆_GCA=𝜋²3𝐶₁𝑇₁+𝐶₂𝑇₂. (4.12) To make the GCA entropy finite, 𝑇_𝐿+𝑇_𝑅∼𝑂(1) and 𝑇_𝐿−𝑇_𝑅∼𝑂(𝜖)𝑇_𝐿+𝑇_𝑅=𝜂²𝜁²𝑟4𝜋𝑙⁺+𝜂√𝑟₊𝑟₋,𝑇_𝐿−𝑇_𝑅=𝜂²𝜁²𝑟4𝜋𝑙⁻+𝜂√𝑟₊𝑟₋. (4.13) The parameters 𝑟₊ and 𝑟₋ in the nonrelativistic warped NMG must scale like 𝑟₊⟶𝑟₊,𝑟₋⟶𝜖²𝑟₋. (4.14) Then, 𝑇_𝐿+𝑇_𝑅=(𝜂²𝜁²/4𝜋𝑙)𝑟₊∼𝑂(1) and 𝑇_𝐿−𝑇_𝑅=(𝜂²𝜁²/4𝜋𝑙)𝜖∼𝑂(𝜖), and the GCA entropy is finite 𝑆_GCA=𝜋𝑙𝜁³2𝑚²𝐺𝑟₊, (4.15) which agrees with Bekenstein-Hawking entropy of warped CNMG in nonrelativistic limit (4.14).

5. Conclusion

In the present paper we have considered the BTZ and warped AdS₃ black hole solution of CNMG in the nonrelativistic 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 (3.7), [16]. 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 𝐶₁, 𝐶₂ and also the scaling dimension Δ and rapidity 𝜉 as (3.15) and (3.16), respectively. Using these results and scaling limits (𝑀→𝑀,𝐽→𝜖𝐽), we have obtained the entropy of GCA₂ by (3.18), this expression exactly agrees with the nonrelativistic limit of Bekenstein-Hawking entropy of the black hole. After that we studied the warped AdS₃ black hole solution in nonrelativistic limit. Using the central charges 𝐶₁, 𝐶₂ for GCA₂, we obtained the entropy of nonrelativistic CFT₂, as (4.10). Then we have defined Galilean conformal temperatures by (4.11). Using these results we have obtained final form of GCA entropy by (4.15), which is exactly the nonrelativistic limit of entropy formula (4.3).

Appendix

Here we give a brief review of Brown-Henneaux boundary condition for the NMG (to see more about it refer to [20]). The NMG have an AdS₃ solution𝑑𝑠²=𝑔_𝜇𝜈𝑑𝑥^𝜇𝑑𝑥^𝜈=𝐿²−cosh²𝜌𝑑𝜏²+sinh²𝜌𝑑𝜙²+𝑑𝜌², (A.1) by expanding 𝑔_𝜇𝜈=𝑔_𝜇𝜈+ℎ_𝜇𝜈 around AdS₃, one could obtain the equation of motion for the linearized excitations ℎ_𝜇𝜈. The Brown-Henneaux boundary condition for the linearized gravitational excitations in asymptotical AdS₃ spacetime in the global coordinate system can be written as ℎ₊₊=𝒪(1)ℎ₊₋=𝒪(1)ℎ_+𝜌=𝒪𝑒^−2𝜌ℎ₋₊=ℎ₊₋ℎ₋₋=𝒪(1)ℎ_−𝜌=𝒪𝑒^−2𝜌ℎ_𝜌+=ℎ_+𝜌ℎ_𝜌−=ℎ_−𝜌ℎ_𝜌𝜌=𝒪𝑒^−2𝜌. (A.2) The corresponding asymptotic Killing vectors are 𝜉=𝜉⁺𝜕₊+𝜉⁻𝜕₋+𝜉^𝜌𝜕_𝜌=𝜖⁺𝜏⁺+2𝑒^−2𝜌𝜕²₋𝜖⁻(𝜏⁻)+𝒪𝑒^−4𝜌𝜕₊+𝜖⁻(𝜏⁻)+2𝑒^−2𝜌𝜕²₊𝜖⁺𝜏⁺+𝒪𝑒^−4𝜌𝜕₋−12𝜕₊𝜖⁺𝜏⁺+𝜕₋𝜖⁻(𝜏⁻)+𝒪𝑒^−2𝜌𝜕_𝜌, (A.3) 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 is𝑖𝜉^𝐿_𝑚,𝜉^𝐿_𝑛=(𝑚−𝑛)𝜉^𝐿_𝑚+𝑛,𝑖𝜉^𝑅_𝑚,𝜉^𝑅_𝑛=(𝑚−𝑛)𝜉^𝑅_𝑚+𝑛,𝜉^𝐿_𝑚,𝜉^𝑅_𝑛=0. (A.4) So these asymptotic Killing vectors give two copies of Virasoro algebra. Then by calculating the conserved charges, the authors of [20] have shown that the left moving and right moving conserved charges fulfil two copies of Virasoro algebra with central charges (3.9).