Asymptotically spacelike warped anti-de Sitter spacetimes in generalized minimal massive gravity
aa r X i v : . [ h e p - t h ] M a y Asymptotically spacelike warped anti-deSitter spacetimes in generalized minimalmassive gravity
M. R. Setare , H. Adami Department of Science, University of Kurdistan, Sanandaj, Iran.
Abstract
In this paper we show that warped AdS black hole spacetime is a solutionof the generalized minimal massive gravity (GMMG) and introducesuitable boundary conditions for asymptotically warped AdS spacetimes.Then we find the Killing vector fields such that transformations generatedby them preserve the considered boundary conditions. We calculate theconserved charges which correspond to the obtained Killing vector fieldsand show that the algebra of the asymptotic conserved charges is given asthe semi direct product of the Virasoro algebra with U (1) current algebra.We use a particular Sugawara construction to reconstruct the conformalalgebra. Thus, we are allowed to use the Cardy formula to calculate theentropy of the warped black hole. We demonstrate that the gravitationalentropy of the warped black hole exactly coincide with what we obtain viaCardy’s formula. As we expect the warped Cardy formula also give usexactly the same result which we obtain from usual Cardy’s formula. Wecalculate mass and angular momentum of the warped black and then checkthat obtained mass, angular momentum and entropy satisfy first law of theblack hole mechanics. According to the results of this paper we belief thatthe dual theory of the warped AdS black hole solution of GMMG is aWarped CFT. In 1992 Ba˜nados, Teitelboim and Zanelli (BTZ) [1, 2] showed that (2 + 1)-dimensional Einstein gravity in the presence of negative cosmological con-stant has a black hole solution. This black hole is described by two (grav-itational) parameters, the mass M and the angular momentum (spin) J .It is locally AdS and thus it differs from Schwarzschild and Kerr solutionssince it is asymptotically anti-de-Sitter instead of flat spacetime. Addition-ally, it has no curvature singularity at the origin. AdS black holes, are E-mail: [email protected] E-mail: [email protected] <
0, thiswill give a negative mass to the BTZ black hole, so the existence of a stableground state is in doubt in this model [8]. However 3-dimensional gravitymodels are rather interesting from the point of view of the AdS/CFT cor-respondence, in particular, the Minimal Massive Gravity (MMG) model [9]has attracted some attention as a theory that circumvents the difficulty ofdefining a bulk theory with positive-energy propagating modes which at thesame time has a CFT dual with positive central charges. This has beencalled the “bulk-boundary unitarity clash” by the authors of [9]. The paper[10] introduces Generalized Minimal Massive Gravity (GMMG), an interest-ing modification of MMG. As has been shown in [10], GMMG also avoidsthe aforementioned “bulk-boundary unitarity clash”. Hamiltonian analysisshow that the GMMG model has no Boulware-Deser ghosts and this modelpropagate only two physical modes. So these models are viable candidatefor semi-classical limit of a unitary quantum 3 D massive gravity.The realization of the existence of three-dimensional (3D) black holes deep-ened our understanding of 3D gravity. In this context an important role isplayed by the notion of asymptotic symmetry. This notion was applied withsuccess some time ago to asymptotically AdS spacetimes, to show that theasymptotic symmetry group (ASG) of AdS is the conformal group in twodimensions [11]. The authors of [12] have introduced spacelike stretched AdS as a new vacuum of TMG, which could be a stable ground state ofthis model [13, 14] (see also [15, 16, 17]). More than this, another rea-son for interest to the warped AdS spacetime, is that they emerge in thenear horizon geometry of extremal Kerr black holes and this fact is impor-tant in the context of Kerr/CFT [18]. In the paper [14], the correctness ofthe hypothesis formulated in [12] have been investigated. The authors of[14] have obtained that the asymptotic symmetry of the spacelike stretched
AdS sector of TMG is a 2-dimensional conformal symmetry with centralcharges. Recently warped AdS black holes have been studied in generic2igher derivatives gravity theories in 2 + 1 dimensions [19]. According toearlier investigations on the warped AdS , the ASG of these spacetimesinstead of the Brown-Henneaux conformal symmetry group, is the semi-direct product of a Virasoro algebra and a U (1) affine Kac-Moody algebra[20, 21, 22, 14, 23]. It is the symmetry of warped conformal field theory(WCFT) in 2-dimension. The authors of [19] have reproduced the matchbetween the Bekenstein-Hawking and WCFT entropies in the case of newmassive gravity (NMG) [24].For systems that admit 2D CFTs as duals, the Cardy formula [25] can beapplied directly. This formula gives the entropy of a CFT in terms of thecentral charge c and the eigenvalue of the Virasoro operator l . However,it should be pointed out that this evaluation is possible as soon as one hasexplicitly shown (e.g using the AdS d /CF T d − correspondence) that the sys-tem under consideration is in correspondence with a 2D CFT [26, 27].In this paper we consider the spacelike warped AdS black hole solutions ofGMMG. The paper is organized as follows. In section 2 we briefly reviewthe quasi-local conserved charges in Chern-Simons-like theories of gravity(CSLTGs). Since GMMG is an example of CSLTGs, we present genericformula for entropy of the black hole solutions of such theories, which previ-ously has obtained in [28]. Then in section 3 we introduce the GMMG modeland present its equations of motion. In section 4 we show that the spacelikewarped AdS black hole is a solution of GMMG model. In section 5 followingpaper [23] we introduce the appropriate boundary conditions for asymptot-ically spacelike warped AdS spacetimes. We show that the algebra amongthe asymptotic Killing vectors is the semi direct product of the Witt algebrawith the U (1) current algebra. In section 6 we find the conserved chargesfor asymptotically spacelike warped AdS spacetime solutions of GMMG.In section 7 we, at first we obtain the conserved charge corresponds to theasymptotic Killing vector, then we find the algebra of conserved charges.After that we show that the algebra of asymptotic conserved charges forthe spacelike warped AdS black hole is semi direct product of the Vira-soro algebra with U (1) current algebra. In section 8 we use a particularSugawara construction [29] to reconstruct the conformal algebra. We thenshow that the Cardy formula reproduces exactly the Bekenstein-Hawkingentropy of the spacelike stretched AdS black hole solution of GMMG. Weshow that this entropy can be reproduce by warped version of the Cardyformula also. In this section we obtain the mass and angular momentum ofblack hole, and show that these quantities satisfy the first law of black holemechanics. In section 9 we obtain the entropy of black hole using our genericentropy formula which we presented in section 2. Last section contain our3onclusions. The Lagrangian 3-form of the Chern-Simons-like theories of gravity is givenby [30] L = 12 ˜ g rs a r · da s + 16 ˜ f rst a r · a s × a t . (1)In the above Lagrangian a ra = a raµ dx µ are Lorentz vector valued one-formswhere, r and a indices refer to flavour and Lorentz indices, respectively.We should mention that, here, the wedge products of Lorentz-vector valuedone-form fields are implicit. Also, ˜ g rs is a symmetric constant metric onthe flavour space and ˜ f rst is a totally symmetric ”flavour tensor” which areinterpreted as the coupling constants. We use a 3D-vector algebra notationfor Lorentz vectors in which contractions with η ab and ε abc are denoted bydots and crosses, respectively . It is worth saying that a ra is a collection ofthe dreibein e a , the dualized spin-connection ω a , the auxiliary field h aµ = e aν h νµ and so on . Also for all interesting CSLTG we have ˜ f ωrs = ˜ g rs [31].The total variation of a ra induced by a diffeomorphism generator ξ is [32] δ ξ a ra = L ξ a ra − δ rω dχ aξ , (2)where χ aξ = ε abc λ bcξ and λ bcξ is generator of the Lorentz gauge transfor-mations SO (2 , δ rs denotes the ordinary Kronecker delta and theLorentz-Lie derivative along a vector field ξ is denoted by L ξ . The Lorentz-Lie derivative of a Lorentz tensor-valued p -form A a ··· b ··· is defined by L ξ A a ··· b ··· = £ ξ A a ··· b ··· + λ aξ c A c ··· b ··· + · · · − λ cξ b A a ··· c ··· − · · · . (3)Dreibein and spin-connection transform like e → Λ e and ω → Λ ω Λ − +Λ d Λ − under Lorentz gauge transformations, where Λ = exp( λ ) ∈ SO (2 , ξ is a combination of variationsdue to a diffeomorphism and an infinitesimal Lorentz gauge transformation[28]. It is obvious that the total variation of a Lorentz tensor-valued p -formis equal to its Lorentz-Lie derivative and the extra term in total variation Here we consider the notation used in [30]. That is a r = { e, ω, h, · · · } , for instance, for r = e and r = ω we have a e = e and a ω = ω . Spin-connection ω ab and dualized spin-connection ω a are related as ω a = ε abc ω bc . a ra comes from the transformation law of spin-connection under Lorentzgauge transformations. Total variation of a ra is covariant under Lorentzgauge transformations as well as diffeomorphism. Hence we are allowed touse covariant phase space method to obtain conserved charges in CSLTG.The arbitrary variation of the Lagrangian (1) is δL = δa r · E r + d Θ( a, δa ) , (4)with E ar = ˜ g rs da sa + 12 ˜ f rst ( a s × a t ) a , Θ( a, δa ) = 12 ˜ g rs δa r · a s . (5)where E ar = 0 are the equations of motion and Θ( a, δa ) is surface term.The total variation of the Lagrangian induced by diffeomorphism generator ξ can be written as δ ξ L = L ξ L + dψ ξ = d ( i ξ L + ψ ξ ) , (6)with ψ ξ = ˜ g ωr dχ ξ · a r . Also, the total variation of the surface term is δ ξ Θ( a, δa ) = L ξ Θ( a, δa ) + Π ξ , (7)with Π ξ = ˜ g ωr dχ ξ · δa r . Now we assume that the variation of Lagrangian(1) is generated by a vector field ξ . For generality, we assume that vectorfield ξ depends on dynamical fields. By using the Bianchi identities, we findoff-shell Noether current J ξ = Θ( a, δ ξ a ) − i ξ L − ψ ξ + i ξ a r · E r − χ ξ · E ω (8)which is conserved off-shell, i.e. we have dJ ξ = 0 off-shell. By virtue of thePoincare lemma, one can obtain off-shell Noether charge density K ξ so that J ξ = dK ξ . By taking variation from Eq.(8) with respect to dynamical fields and by making some calculations one can define extended off-shell ADT current as [33] J ADT ( a, δa, δ ξ a ) =ˆ δa r · i ξ E r + i ξ a r · ˆ δE r − χ ξ · ˆ δE ω + ˜ g rs δ ξ a r · ˆ δa s . (9) i ξ denotes interior product in ξ . We denote variation with respect to dynamical fields by ˆ δ . ADT stands for Abbott, Deser and Tekin.
5f we assume that ξ is a Killing vector field then the last term in Eq.(9)vanishes and extended off-shell ADT current reduces to the generalized off-shell ADT current [32]. Also, if we consider on-shell case, i.e. E r = δE r =0, then extended off-shell ADT current reduces to symplectic current [34].The current (9) is conserved off-shell, i.e. d J ADT = 0, so by virtue of thePoincare lemma, one can find corresponding extended off-shell ADT chargeso that J ADT = d Q ADT . Therefore we can define quasi-local conservedcharge perturbation associated with a field dependent vector field ξ asˆ δQ ( ξ ) = 18 πG Z Σ Q ADT = 18 πG Z Σ (˜ g rs i ξ a r − ˜ g ωs χ ξ ) · ˆ δa s , (10)where G denotes the Newtonian gravitational constant and Σ is a spacelikecodimension two surface. We can take an integration from (10) over theone-parameter path on the solution space [35, 36] and then we find that Q ( ξ ) = 18 πG Z ds Z Σ (˜ g rs i ξ a r − ˜ g ωs χ ξ ) · ˆ δa s , (11)Also, we argued that the quasi-local conserved charge (11) is not only con-served for the Killing vectors which are admitted by spacetime everywherebut also it is conserved for the asymptotically Killing vectors.The entropy of black holes is the conserved charge associated with thehorizon-generating Killing vector field evaluated at the bifurcation surface[34]. Let ζ denotes horizon-generating Killing vector field. The horizon-generating Killing vector field ζ vanishes on the bifurcation surface B . Now,take Σ in Eq.(11) to be the bifurcation surface B then we have Q ( ζ ) = − πG ˜ g ωr Z B χ ζ · a r . (12)To obtain an explicit expression for χ ξ , in an appropriate manner, the au-thors in [37] demand that it must be chosen so that the Lorentz-Lie derivativeof dreibein vanishes when ξ is a Killing vector field and then χ ξ should beprovided as follows [28]: χ aξ = i ξ ω a + 12 ε abc e νb ( i ξ T c ) ν + 12 ε abc e bµ e cν ∇ µ ξ ν . (13)Thus, on the bifurcation surface we will have χ aζ | B = κN a with N a = ε abc n bc , where κ and n ab are respectively surface gravity and bi-normal6o the bifurcation surface. Therefore black hole entropy in the CSLTG canbe defined as S = 2 πκ Q ( ζ ) = 14 G ˜ g ωr Z B N · a r . (14)Since the non-zero components of bi-normal n µν to stationary black holehorizon are n = − n and it is normalized to − S = − G Z B dφ √ g φφ ˜ g ωr a rφφ , (15)where φ is angular coordinate and g φφ denotes the φ - φ component of space-time metric g µν . We use the above generic entropy formula to obtain theBekenstein-Hawking entropy of the spacelike warped AdS black hole solu-tion of GMMG model. Generalized minimal massive gravity (GMMG) is an example of the Chern-Simons-like theories of gravity [10]. In the GMMG, there are four flavoursof one-form, a r = { e, ω, h, f } , and the non-zero components of the flavourmetric and the flavour tensor are˜ g eω = − σ, ˜ g eh = 1 , ˜ g ωf = − m , ˜ g ωω = 1 µ , ˜ f eωω = − σ, ˜ f ehω = 1 , ˜ f fωω = − m , ˜ f ωωω = 1 µ , ˜ f eff = − m , ˜ f eee = Λ , ˜ f ehh = α. (16)where σ , Λ , µ , m and α are a sign, cosmological parameter with dimen-sion of mass squared, mass parameter of Lorentz Chern-Simons term, massparameter of New Massive Gravity term and a dimensionless parameter,respectively. The equations of motion of GMMG are [10] (see also [38]) − σR (Ω)+(1+ σα ) D (Ω) h − α (1+ σα ) h × h + Λ e × e − m f × f = 0 , (17) − e × f + µ (1 + σα ) e × h − µm D (Ω) f + µαm h × f = 0 , (18) R (Ω) − αD (Ω) h + 12 α h × h + e × f = 0 , (19) T (Ω) = 0 , (20)7here Ω = ω − αh (21)is ordinary torsion-free dualized spin-connection. Also, R (Ω) = d Ω + Ω × Ω is curvature 2-form, T (Ω) = D (Ω) e is torsion 2-form, and D (Ω) de-notes exterior covariant derivative with respect to torsion-free dualized spin-connection. Now we consider the stationary black hole metric in ADM form ds l = − N ( r ) dt + dr N ( r ) R ( r ) + R ( r ) (cid:16) dφ + N φ ( r ) dt (cid:17) , (22)where t , r , φ and l are time-coordinate, radial-coordinate, angular-coordinateand AdS space radius, respectively. For the spacelike warped AdS blackhole we have [39] R ( r ) = 14 ζ r (cid:2)(cid:0) − ν (cid:1) r + ν ( r + + r − ) + 2 ν √ r + r − (cid:3) ,N ( r ) = ζ ν ( r − r + ) ( r − r − )4 R ( r ) ,N φ ( r ) = | ζ | r + ν √ r + r − R ( r ) , (23)where r + and r − are outer and inner horizons, respectively. The parametersappeared in Eq.(23), ζ and ν , allow us to keep contact with [8, 40, 41, 42] .The spacetime described by metric (22) with (23) admits SL (2 , R ) × U (1)as isometry group. Therefore, one can write a symmetric-two tensor S µν as[19] S µν = a g µν + a J µ J ν , (24)with J = J µ ∂ µ = ∂ t . (25)It should be noted that J µ J µ = l and ∇ µ J ν = | ζ | l ǫ µνλ J λ (26) For further discussion, see [39]. ∇ µ is covariant derivative with respect to the Christoffel connectionand ǫ µνλ = √− gε µνλ . Hence, the Ricci tensor can be written as R µν = ζ l (cid:0) − ν (cid:1) g µν − ζ l (cid:0) − ν (cid:1) J µ J ν , (27)and the Ricci scalar is given by R = ζ l (cid:0) − ν (cid:1) . (28)It is easy to see that dreibein correspond to the metric (22) can be writtenas e = lN ( r ) dt,e = l R ( r ) N ( r ) dr,e = lR ( r ) (cid:16) dφ + N φ dt (cid:17) . (29)To show that the metric (22) with (23) is a solution of GMMG we considerfollowing ansatz for h and fh aµ = H e aµ + H J a J µ ,f aµ = F e aµ + F J a J µ , (30)where H , H , F , F are constant parameters and J a = e aµ J µ . One can useequations (25)-(30), to show that equations of motion of GMMG (17)-(20)reduce to the following equations ζ l − αl | ζ | H − α H (cid:0) H + l H (cid:1) − (cid:0) F + l F (cid:1) = 0 , (31) − ζ l (cid:0) − ν (cid:1) + 3 α l | ζ | H + α H H + F = 0 , (32)1 µ (cid:0) F + l F (cid:1) − (1 + ασ ) (cid:0) H + l H (cid:1) − l m | ζ | F − αm (cid:2) H F + l ( H F + H F ) (cid:3) = 0 , (33) − µ F + (1 + ασ ) H + 32 lm | ζ | F + αm ( H F + H F ) = 0 , (34) − ζ l σ + 12 (1 + ασ ) l | ζ | H + α (1 + ασ ) H (cid:0) H + l H (cid:1) − Λ + 1 m F (cid:0) F + l F (cid:1) = 0 , (35)9 l (cid:0) − ν (cid:1) σ − l (1 + ασ ) | ζ | H − α (1 + ασ ) H H − m F F = 0 . (36)Thus, the metric (22) with (23) is a solution of GMMG if the introducedparameters satisfy equations (31)-(36). spacetimes In this section we follow the paper [23] to introduce appropriate boundaryconditions. In this way, for asymptotically spacelike warped AdS space-times, we introduce following boundary conditions g tt = l + O ( r − ) , g tr = O ( r − ) , g rφ = O ( r − ) ,g tφ = 12 l | ζ | (cid:20) r + A tφ ( φ ) + 1 r B tφ ( φ ) (cid:21) + O ( r − ) ,g rr = l ζ ν (cid:20) r + 1 r A rr ( φ ) + 1 r B rr ( φ ) (cid:21) + O ( r − ) ,g φφ = 14 l ζ (cid:2)(cid:0) − ν (cid:1) r + rA φφ ( φ ) + B φφ ( φ ) (cid:3) + O ( r − ) , (37)which are consistent with the metric (22). The corresponding componentsof dreibein are e t = lν √ − ν − l (cid:2) (cid:0) ν − (cid:1) A tφ + A φφ (cid:3) rν (1 − ν ) + l r ν (1 − ν ) (cid:20) A tφ (cid:0) ν − (cid:1) + A φφ (cid:0) ν − (cid:1) + 4 A tφ A φφ (cid:0) ν − (cid:1) (cid:0) ν − (cid:1) + 8 B tφ ν (cid:0) ν − (cid:1) + 4 B φφ ν (cid:0) ν − (cid:1)(cid:21) + O ( r − ) (38) e r = lζνr + lA rr ζνr − l ζνr (cid:2) A rr − B rr (cid:3) + O ( r − ) (39) e t = l √ − ν − l (cid:2) (cid:0) ν − (cid:1) A tφ + A φφ (cid:3) r (1 − ν ) + l r (1 − ν ) (cid:20) B tφ (cid:0) ν − (cid:1) + 4 A tφ A φφ (cid:0) ν − (cid:1) + 4 B φφ (cid:0) ν − (cid:1) + 3 A φφ (cid:21) + O ( r − ) (40)10 φ = 12 rl | ζ | p − ν + l | ζ | A φφ √ − ν − l | ζ | r (1 − ν ) (cid:2) B φφ (cid:0) ν − (cid:1) + A φφ (cid:3) + O ( r − ) , (41)and the rest of them are of the order of r − , that is O ( r − ).The metric, under transformation generated by vector field ξ , transforms as δ ξ g µν = £ ξ g µν . The variation generated by the following Killing vectorfield preserves the boundary conditions (37) ξ t ( T, Y ) = T ( φ ) − ∂ φ Y ( φ ) | ζ | ν r + O ( r − ) ,ξ r ( T, Y ) = − r∂ φ Y ( φ ) + O ( r − ) ,ξ φ ( T, Y ) = Y ( φ ) + 2 ∂ φ Y ( φ ) ζ ν r + O ( r − ) , (42)where T ( φ ) and Y ( φ ) are two arbitrary periodic functions. The asymptoticKilling vectors (42) are closed in the Lie bracket and we have[ ξ ( T , Y ) , ξ ( T , Y )] = ξ ( T , Y ) , (43)where T ( φ ) = Y ( φ ) ∂ φ T ( φ ) − Y ( φ ) ∂ φ T ( φ ) ,Y ( φ ) = Y ( φ ) ∂ φ Y ( φ ) − Y ( φ ) ∂ φ Y ( φ ) . (44)By introducing Fourier modes u m = ξ ( e imφ ,
0) and v m = ξ (0 , e imφ ), one canfind that [ u m , u n ] = 0 , [ v m , u n ] = − nu m + n , [ v m , v n ] = ( m − n ) v m + n . (45)Therefore the algebra among the asymptotic Killing vectors is the semi directproduct of the Witt algebra with the U (1) current algebra. Under theaction of a generic asymptotic symmetry generator ξ spanned by (42), thedynamical fields transform as δ ξ A tφ = ∂ φ [ Y ( φ ) A tφ ( φ )] + 2 | ζ | ∂ φ T ( φ ) ,δ ξ A rr = ∂ φ [ Y ( φ ) A rr ( φ )] ,δ ξ A φφ = ∂ φ [ Y ( φ ) A φφ ( φ )] + 4 | ζ | ∂ φ T ( φ ) , (46) Where £ ξ denotes usual Lie derivative along ξ . ξ B tφ = Y ( φ ) ∂ φ B tφ ( φ ) + 2 B tφ ( φ ) ∂ φ Y ( φ ) − ζ ν ∂ φ Y ( φ ) ,δ ξ B rr = Y ( φ ) ∂ φ B rr ( φ ) + 2 B rr ( φ ) ∂ φ Y ( φ ) ,δ ξ B φφ = Y ( φ ) ∂ φ B φφ ( φ ) + 2 B φφ ( φ ) ∂ φ Y ( φ ) + 4 | ζ | A tφ ( φ ) ∂ φ T ( φ ) − (cid:0) ν (cid:1) ζ ν ∂ φ Y ( φ ) . (47)We are interested in solutions which are asymptotically spacelike warpedAdS . Thus, we demand that equations (26) and (27) hold asymptotically,i.e. ∇ µ J ν − | ζ | l ǫ µνλ J λ = O ( r − ) , (48) R µν − ζ l (cid:0) − ν (cid:1) g µν + ζ l (cid:0) − ν (cid:1) J µ J ν = O ( r − ) . (49)By substituting Eq.(37) into the equations (48) and (49), we find that A φφ ( φ ) = ν A rr ( φ ) + 2 A tφ ( φ ) ,B φφ ( φ ) = ν (cid:2) B rr ( φ ) + 2 B tφ ( φ ) − A rr ( φ ) (cid:3) + A tφ ( φ ) + 2 B tφ ( φ ) . (50)Hence, the metric (37) solves equations of motion of GMMG asymptoticallywhen equations (31)-(36) and (50) are satisfied. We emphasis, followingabove analysis, that equations (26), (27) and (30) are held asymptotically. spacetimes in GMMG We first want to simplify the expression for conserved charge perturbation(10) for asymptotically spacelike warped AdS spacetimes (37) in the contextof GMMG. To this end, we use equations (30)-(36), (16), (21) and (27). After12ome calculations we find thatˆ δQ ( ξ ) = 18 πG Z Σ (cid:26) − (cid:18) σ + αH µ + F m (cid:19) h i ξ e · ˆ δ Ω + ( i ξ Ω − χ ξ ) · ˆ δe i + 1 µ ( i ξ Ω − χ ξ ) · ˆ δ Ω + αH (cid:18) αH µ + 2 F m (cid:19) i ξ J · ˆ δ J + (cid:20) − ζ µl (cid:18) − ν (cid:19) + l | ζ | (cid:18) αH µ + F m (cid:19)(cid:21) i ξ e · ˆ δe − (cid:18) αH µ + F m (cid:19) h i ξ J · ˆ δ Ω + ( i ξ Ω − χ ξ ) · ˆ δ J i + (cid:20) ζ µl (cid:0) − ν (cid:1) − | ζ | l (cid:18) αH µ + F m (cid:19)(cid:21) (cid:16) i ξ J · ˆ δe + i ξ e · ˆ δ J (cid:17)(cid:27) . (51)where J aµ = J a J µ for simplicity. One can show that the expression (13) for χ ξ can be rewritten as [43] i ξ Ω − χ ξ = − ε abc e bµ e cν ∇ µ ξ ν . (52)Also we mention that the torsion free spin-connection is given byΩ aµ = 12 ε abc e αb ∇ µ e cα . (53)Now we take spacelike warped AdS spacetime as background which can bedescribed by the following dreibein¯ e = lν √ − ν dt, ¯ e = lζνr dr, ¯ e = l √ − ν dt + 12 rl | ζ | p − ν dφ. (54)The bar sign on the top of a quantity emphasis that the considered quantityhas calculated on background. As we mentioned in section 1, one can takean integration from (51) over the one-parameter path on the solution space13o find the conserved charge corresponds to the Killing vector field ξ , then Q ( ξ ) = 18 πG Z Σ (cid:26) − (cid:18) σ + αH µ + F m (cid:19) (cid:2) i ξ ¯ e · ∆Ω + (cid:0) i ξ ¯Ω − ¯ χ ξ (cid:1) · ∆ e (cid:3) + 1 µ (cid:0) i ξ ¯Ω − ¯ χ ξ (cid:1) · ∆Ω + αH (cid:18) αH µ + 2 F m (cid:19) i ξ ¯ J · ∆ J + (cid:20) − ζ µl (cid:18) − ν (cid:19) + l | ζ | (cid:18) αH µ + F m (cid:19)(cid:21) i ξ ¯ e · ∆ e − (cid:18) αH µ + F m (cid:19) (cid:2) i ξ ¯ J · ∆Ω + (cid:0) i ξ ¯Ω − ¯ χ ξ (cid:1) · ∆ J (cid:3) + (cid:20) ζ µl (cid:0) − ν (cid:1) − | ζ | l (cid:18) αH µ + F m (cid:19)(cid:21) (cid:0) i ξ ¯ J · ∆ e + i ξ ¯ e · ∆ J (cid:1)(cid:27) , (55)with ∆Φ = Φ ( s =1) − Φ ( s =0) , where Φ ( s =1) and Φ ( s =0) are calculated onconsidered spacetime solution and on the background spacetime solution,respectively . As we saw in section 5, the asymptotic Killing vector field is given by Eq.(42).Let’s find out conserved charge corresponds to the asymptotic Killing vectorfield given in Eq.(42). To this end, we need to use (25), (31)-(36), (38)-(42), (50) and (52)-(55). After tedious calculations we find the followingexpression for conserved charge corresponds to the asymptotic Killing vectorfield (42) Q ( T, Y ) = P ( T ) + L ( Y ) , (56)with P ( T ) = − | ζ | π c U Z π T ( φ ) [ A rr ( φ ) + 2 A tφ ( φ )] dφ, (57) L ( Y ) = ζ ν π c V Z π Y ( φ ) (cid:2) − A rr ( φ ) + 4 B rr ( φ ) + 16 B tφ ( φ ) (cid:3) dφ, (58)where c U = 3 l | ζ | ν G (cid:26) σ + αµ (cid:0) H + l H (cid:1) + 1 m (cid:0) F + l F (cid:1) − | ζ | µl (cid:27) , (59) For instance, ∆ e = e − ¯ e , where e and ¯ e are given by (38)-(41) and (54), respectively. V = 3 l | ζ | ν G (cid:26) σ + αµ (cid:0) H + l H (cid:1) + 1 m (cid:0) F + l F (cid:1) − | ζ | µl (cid:0) − ν (cid:1)(cid:27) . (60)It is worth to mention that c U and c V are related as ζ ν c V − c U = 3 ζ ν µG . (61)The algebra of conserved charges can be written as [11, 44] { Q ( ξ ) , Q ( ξ ) } = Q ([ ξ , ξ ]) + C ( ξ , ξ ) (62)where C ( ξ , ξ ) is central extension term. Also, the left hand side of theequation (62) can be defined by { Q ( ξ ) , Q ( ξ ) } = 12 (cid:16) ˆ δ ξ Q ( ξ ) − ˆ δ ξ Q ( ξ ) (cid:17) . (63)Therefore one can obtain the central extension term by using the followingformula C ( ξ , ξ ) = 12 (cid:16) ˆ δ ξ Q ( ξ ) − ˆ δ ξ Q ( ξ ) (cid:17) − Q ([ ξ , ξ ]) . (64)Thus, one can use Eq.(43), Eq.(46), Eq.(47) and Eq.(56) to find that { Q ( T , Y ) , Q ( T , Y ) } = Q ( T , Y )+ | ζ | π c U Z π T ( φ ) [ A rr ( φ ) + 2 A tφ ( φ )] dφ − π c U Z π ( T ∂ φ T − T ∂ φ T ) dφ − π c V Z π (cid:0) Y ∂ φ Y − Y ∂ φ Y (cid:1) dφ. (65)We introduce Fourier modes as P m = Q ( e imφ ,
0) = P ( e imφ ) ,L m = Q (0 , e imφ ) = L ( e imφ ) , (66)so one can read off the algebra of conserved charges as follows: i { P m , P n } = − c U mδ m + n, i { L m , P n } = − nP m + n − | ζ | c U π n Z π e i ( m + n ) φ [ A rr ( φ ) + 2 A tφ ( φ )] dφi { L m , L n } =( m − n ) L m + n + c V m δ m + n, . (67)15ow we consider warped black hole solution as an example. For warpedblack hole (22) with(23), we have A rr = r + + r − , A tφ = ν √ r + r − ,B rr = r + r − + r + r − , B tφ = 0 . (68)In this case, equations (67) will be reduce to i { P m , P n } = − c U mδ m + n, i { L m , P n } = − nP m + n − | ζ | c U n ( r + + r − + 2 ν √ r + r − ) δ m + n, i { L m , L n } =( m − n ) L m + n + c V m δ m + n, . (69)Now we set ˆ P m ≡ P m and ˆ L m ≡ L m , also we replace Dirac brackets by com-mutators i { , } → [ , ], therefore we can rewritten equations (67) as following h ˆ P m , ˆ P n i = − c U mδ m + n, h ˆ L m , ˆ P n i = − n ˆ P m + n + n p δ m + n, h ˆ L m , ˆ L n i =( m − n ) ˆ L m + n + c V m δ m + n, (70)where p is the zero mode eigenvalue of ˆ P m . From Eq.(66) and usingEq.(57), Eq.(58) and Eq.(68), one can easily read off the eigenvalues of ˆ P m and ˆ L m as p m = − | ζ | c U
48 ( r + + r − + 2 ν √ r + r − ) δ m, , (71) l m = ζ ν c V
384 ( r + − r − ) δ m, , (72)respectively. It is clear now that the algebra of asymptotic conserved chargesis given as the semi direct product of the Virasoro algebra with U (1) currentalgebra, with central charges c V and c U . Here, it is reasonable to use the terminology ”eigenvalues” because we improve P m and L m to be operators ˆ P m and ˆ L m , respectively, and we denote eigenvalues of them by p m and l m . Such a terminology have been used by Ba˜nados [45], Strominger [27], Carlip[46] and so on. Mass, angular momentum and entropy of warpedblack hole solution of GMMG
It is clear that the algebra of conserved charges (70) does not describe theconformal symmetry [12]. However, one can use a particular Sugawara con-struction [29] to reconstruct the conformal algebra. Such a method hasbeen used in the topologically massive gravity model [14], also such studyon spacelike warped AdS black hole solutions of NMG has been done in[47]. Therefore, we introduce two new operators as followsˆ L + m = im | ζ | ν ˆ P − m + 6 c U ˆ K − m , (73)ˆ L − m = ˆ L m − p c U ˆ P m + 6 c U ˆ K m , (74)where ˆ K m = X q ∈ Z ˆ P m + q ˆ P − q . (75)One can show that the algebra among ˆ L ± m is given as h ˆ L + m , ˆ L + n i =( m − n ) ˆ L + m + n + c + m δ m + n, h ˆ L − m , ˆ L − n i =( m − n ) ˆ L − m + n + c − m δ m + n, + 3 p c U mδ m + n, h ˆ L + m , ˆ L − n i =0 (76)where c + = c U ζ ν , c − = c V , (77)In this way, Eq.(61) can be rewritten as c − − c + = 3 µG . (78)It is easy to see, from Eq.(73) and Eq.(74), that the values of ˆ L ± are givenas l +0 = 6 p c U , l − = l . (79)Now we can calculate the black hole entropy via Cardy’s formula [25, 48], S CFT = 2 π s c + l +0 π s c − l − S CFT = πl | ζ | G (cid:26)(cid:20) σ + αµ (cid:0) H + l H (cid:1) + 1 m (cid:0) F + l F (cid:1) − | ζ | µl (cid:21) ( r + + ν √ r + r − )+ | ζ | ν µl ( r + − r − ) (cid:27) . (81)Mass and angular momentum can also be obtained by M = − p and J = − (cid:0) l +0 − l − (cid:1) , respectively [14]. Thus, we have M = | ζ | c U
48 ( r + + r − + 2 ν √ r + r − ) , (82) J = − ζ n c U ( r + + r − + 2 ν √ r + r − ) − ζ ν c V ( r + − r − ) o . (83)As we know, angular velocity and surface gravity of horizon are given byΩ H = − N φ ( r + ) = − | ζ | (cid:0) r + + ν √ r + r − (cid:1) , (84) κ H = (cid:20) − ∇ µ k ν ∇ µ k ν (cid:21) r = r + = | ζ | ν ( r + − r − )2 (cid:0) r + + ν √ r + r − (cid:1) , (85)respectively, where k = ∂ t + Ω H ∂ φ is the horizon-generating Killing vectorfield. One can verify that mass (82), angular momentum (83) and entropy(81) of considered black hole satisfy the first law of black hole mechanics, δ M = T H δ S + Ω H δ J , (86)where T H = κ H / π is the Hawking temperature.Let us conclude this section with the following important remark. We knowthat the SL (2 , R ) × SL (2 , R isometry group of AdS space reduce to the SL (2 , R ) × U (1) in warped AdS space due to the presence of warping pa-rameter. So one expect that the asymptotic symmetry of warped AdS spacealso differ from full conformal symmetry. So the dual theory of warped AdS space instead a 2-dimensional CFT, is warped CFT (WCFT), which exhibitspartial conformal symmetry. In another term, for this case the conformalsymmetry is not present, actually not even Lorentz symmetry. Let us define˜ P m = | ζ | − ν − ˆ P m . By that definition the algebra (70) becomes h ˜ P m , ˜ P n i = − ˜ c U mδ m + n, h ˆ L m , ˜ P n i = − n ˜ P m + n + n p δ m + n, h ˆ L m , ˆ L n i =( m − n ) ˆ L m + n + c V m δ m + n, (87)18here ˜ c U = ζ − ν − c U and ˜ p m = | ζ | − ν − p m . The algebra (87) is thesemi direct product of the Virasoro algebra with U (1) current algebra, withcentral charges c V and ˜ c U . This is exactly the symmetry of warped CFT, sothe dual theory of warped black hole solution of GMMG is a WCFT. Thisconformal field theory in the so-called quadratic ensemble has an entropy inthe Cardy regime that looks like usual Cardy’s formula for entropy in a CFT.This warped Cardy’s formula which has been introduced by Detournay et.al [49] present the asymptotic growth of state in the WCFT. Now, we canuse the warped Cardy formula which is given by [49] S WCFT = 24 π ˜ c U ˜ p ( vac )0 ˜ p + 4 π q − l ( vac )0 l , (88)where ˜ p ( vac )0 and l ( vac )0 correspond to the minimum values of ˜ p and l , i.e.the value of the vacuum geometry. Since vacuum corresponds to r ± = 0, sofrom Eq.(87), one can read off˜ p ( vac )0 = − ˜ c U , l ( vac )0 = − c V . (89)By substituting Eq.(71), Eq.(72) and Eq.(89) into Eq.(88), we find that S WCFT = S CFT . The gravitational black hole entropy can be obtained by Eq.(15) in CSLTG.Now we want to simplify Eq.(15) for GMMG model which is an example ofCSLTG. Also, we are interested to find the entropy of Warped black holeentropy (22), with Eq.(23), as a solution of GMMG model. One can useEq.(16) and Eq.(30) to show that, for the considered case, the gravitationalblack hole entropy is simplified as S = − G Z Horizon dφ √ g φφ (cid:26) − (cid:18) σ + αH µ + F m (cid:19) g φφ + 1 µ Ω φφ − (cid:18) αH µ + F m (cid:19) J φ J φ (cid:27) . (90)19lso, for warped black hole solution (22) with (23), we have g φφ | r = r + = 14 l ζ ( r + + ν √ r + r − ) , Ω φφ | r = r + = − ζ ν √ g φφ | r = r + ( r + − r − ) + | ζ | l g φφ | r = r + , ( J φ J φ ) | r = r + = l g φφ | r = r + . (91)By substituting Eq.(91) into Eq.(92), we find the gravitational entropy ofwarped black hole as S = πl | ζ | G (cid:26)(cid:20) σ + αµ (cid:0) H + l H (cid:1) + 1 m (cid:0) F + l F (cid:1) − | ζ | µl (cid:21) ( r + + ν √ r + r − )+ | ζ | ν µl ( r + − r − ) (cid:27) . (92)This result exactly coincide with what we obtained in the previous section(81) via Cardy’s formula.
10 Conclusion
The Chern-Simons-like theories of gravity describe by the Lagrangian (1). Insuch theories, the quasi-local conserved charge perturbation associated witha field-dependent vector field ξ is given by Eq.(10). By integrating fromEq.(10) over the one-parameter path on the solution space, one finds thequasi-local conserved charge (11) associated with a field-dependent vectorfield ξ . The generalized minimal massive gravity (GMMG), is an exam-ple of the Chern-Simons-like theories of gravity, which is described by theequations of motion (17)-(20). Warped black hole spacetime is given by themetric (22) with (23). In section 4, we showed that the warped black holespacetime solves equations of motion of GMMG. In section 5, we have intro-duced boundary conditions (37), which describes asymptotically spacelikewarped AdS spacetimes at spatial infinity. The variation generated by theKilling vector field (42) preserves the considered boundary conditions. Theasymptotic Killing vectors (42) are closed in the Lie bracket ( see Eq.(43)).By introducing Fourier modes, we saw that the algebra among the asymp-totic Killing vectors is the semi direct product of the Witt algebra with the U (1) current algebra. Under the action of a generic asymptotic symmetrygenerator ξ spanned by (42), the dynamical fields appeared in the metric2037) transform as (46) and (47). Also, we have shown that the metric (37)solves equations of motion of GMMG asymptotically when equations (31)-(36) and (50) are satisfied. We found that the conserved charge, correspondsto the asymptotic Killing vector field (42), is given by Eq.(56). Also, in sec-tion 7, we showed that the algebra among the conserved charges is givenby (67). In Eq.(67), we set ˆ P m ≡ P m and ˆ L m ≡ L m , also we replaced theDirac brackets by commutators i { , } → [ , ], then we obtained Eq.(70). Thealgebra of asymptotic conserved charges is given as the semi direct productof the Virasoro algebra with U (1) current algebra, with central charges c V and c U . The algebra of the conserved charges (70) does not describe theconformal symmetry. Therefore, we used a particular Sugawara construc-tion to reconstruct the conformal algebra ( see Eqs.(73)-(77)). Thus, we areallowed to use the Cardy’s formula to calculate the entropy of the warpedblack hole and then we have obtained that the warped black hole entropyis given by Eq.(81). Also, we showed that mass and angular momentumof the warped black hole are given by Eq.(82) and Eq.(83), respectively.The obtained mass, angular momentum and entropy satisfy the first law ofblack hole mechanics. As we mentioned in section 9, the gravitational blackhole entropy can be obtained by Eq.(15) in CSLTG. Hence, in section 9, wehave shown that the gravitational entropy of warped black hole in GMMGmodel is given by Eq.(92). By comparing Eq.(81), Eq.(88) and Eq.(92), wesee that the gravitational entropy of the warped black hole exactly coincidewith what we obtained via Cardy’s formula and the warped version of thisformula. Although we could showed that by Sugawara construction one canobtain an asymptotic symmetry for the warped AdS black hole solution ofGMMG that coincides with the conformal symmetry, described by two in-dependent Virasoro algebra. But we do not claimed that the dual theory ofsuch gravity solutions of GMMG is CFT. It has been argued by El-Showk,and Guica [50] that warped AdS spaces could be dual to certain types ofnon-local CFT. Since the SL (2 , R ) × SL (2 , R isometry group of AdS spacereduce to the SL (2 , R ) × U (1) in warped AdS space due to the presence ofwarping parameter. So one expect that the asymptotic symmetry of warpedAdS space also differ from full conformal symmetry. So the dual theory ofwarped AdS space instead a 2-dimensional CFT, is warped CFT (WCFT),which exhibits partial conformal symmetry.21 M. R. Setare thank Stephane Detournay and Tom Hartman for helpful com-ments and discussions.
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