Phase transitions in BHT Massive Gravity Mahdis Ghodratia, Ali Nasehb aMichigan Center for Theoretical Physics, Randall Laboratory of Physics University of Michigan, Ann Arbor, MI 48109-1040, USA 6 1 0 bSchool of Particles and Accelerators, Institute for Research in Fundamental Sciences (IPM) 2 P.O. Box 19395-5531, Tehran, Iran r p A 7 E-mails: [email protected], [email protected] 1 ] h t - p e h [ Abstract 3 v We present the Hawking-Page phase diagrams in Bergshoeff-Hohm-Townsend (BHT) gravity for 3 the phase transition between AdS and BTZ black hole, warped AdS and warped BTZ black hole 3 3 0 in grand canonical and in non-local/quadratic ensembles, Lifshitz black hole and the new hairy black 4 4 hole solutions. As we expected for all of them except for the quadratic ensemble, the phase diagram 0 is symmetric for the non-chiral theory of BHT. We also examine the laws of inner horizon mechanics . 1 for the warped AdS black holes and proved that they are satisfied. Finally we briefly discuss the 0 3 6 entanglement entropy of an interval in the warped CFT which holographically is dual to the vacuum 2 1 time-like warped AdS or G¨odel geometry. : 3 v i X r a Contents 1 Introduction 1 2 The Bergshoeff-Hohm-Townsend Theory 3 3 Review of calculating conserved charges in BHT 4 3.1 The SL(2,R) reduction method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3.2 Examples of conserved charges of BHT solutions . . . . . . . . . . . . . . . . . . . . . . . 6 4 Phase transitions of AdS solution 9 3 4.1 The stability conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 4.2 Phase diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 5 Phase transitions of warped AdS solution in quadratic ensemble 13 3 5.1 G¨odel space-time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 5.2 Space-like warped BTZ black hole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 5.3 The free energies and phase diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 6 Phase diagram of warped AdS solution in grand canonical ensemble 19 3 6.1 local stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 7 Phase diagram of the hairy black hole 21 8 The inner horizon thermodynamics 23 9 Entanglement entropy of WCFT in BHT 24 10 Discussion 26 References 26 1 Introduction In order to study the quantum field theories with momentum dissipation holographically, the holographic massive gravity theories (HMGs) have been exploited. Using these models one can study different field theory features such as DC resistivity, relaxation rates, or the effect of dissipations or disorders on the confinement-deconfinement phase transitions in strongly correlated systems [1]. Therearedifferentmassivegravitymodelswithmultiplegeometricalsolutionsandtheircorresponding field theory duals. One of these theories is the “Topological Massive Gravity” (TMG), which is the Einstein action, plus a Chern-Simons term breaking the parity. Recently in [2], the Hawking-Page phase transitions between the AdS and BTZ solutions, and warped AdS and warped BTZ black hole of TMG 3 3 were investigated and the Gibbs free energies, local and global stability regions and the phase diagrams were presented. 1 Thereisyetanotherrichtheory,theparitypreservingBergshoeff-Hohm-Townsend(BHT)orthe“New Massive Gravity” (NMG), which has many different solutions as well, in addition to the thermal warped AdS and warped BTZ black hole. The aim of this paper is, similar to [2], we study the Hawking-Page 3 phase transitions between different solutions of NMG and therefore learning more about the properties of the dual CFTs. Particularly we study the phase transitions between the thermal AdS and BTZ black holes, the warped AdS and warped BTZ black holes in two different ensembles, the Lifshitz black hole and the new hairy black hole and their corresponding vacua. The other motivation is to extend the AdS/CFT duality to other more general geometries. One would think that for doing so, the most direct way is to perturbatively deform the AdS manifold to a warped 3 AdS geometry, [3] [4] [5], and then study the dual field theory. The initial works on this extension 3 were done in [6], where the authors studied the magnetic deformation of S3 and the electric/magnetic deformations of AdS which still could remain a solution of string theory vacua. Then in [7] [8] [9], the 3 dual field theories have been studied. In [9] the dual of warped AdS were suggested to be the IR limit 3 of the non-local deformed 2D D-brane gauge theory or the dipole CFT. Constructing this duality could lead to more information about the properties of these new field theories and also some properties of the dual bulk geometries, for instance the nature of the closed time-like curves (CTCs). TheBergshoeff-Hohm-TownsendgravityhavebothwarpedAdSandwarpedBTZblackholesolutions. ThedeformedAdS preserveaSL(2,R)×U(1)subgroupofSL(2,R)×SL(2,R)isometries. Theobtained 3 space-times called null, time-like or space-like warped AdS (WAdS ) corresponding to the norm of U(1) 3 3 killing vectors, where the time-like WAdS is just the Go¨del spacetime [3] [10]. 3 OtherextensionofAdS/CFTincludesAdS/CMT(CondensedMatter),AdS/QCD,dS/CFT,flatspace holography, Kerr/CFT, etc. However, the dual CFT of these theories are not completely known. The advantages of WCFTs are that they posses many properties of CFTs and they can be derived from string theory and low-dimensional gravity theories and hence for studying them the known CFT techniques could be deployed. The specific properties of this new class of WCFTs were studied in [11] and their entanglement entropies were first studied in [12] holographically and in a more recent work in [13] by using the Rindler method of WCFT. To further study this WAdS/WCFT duality, one could study other properties such as the instabilities of the solutions and the Hawking-Page phase transitions [14]. As the phase transitions from the thermal AdS or WAdS, to BTZ or warped BTZ black hole is dual to confining/deconfining phase transitions in the dual field theory, these models could be used in QCD or condensed matter systems with dissipations. The plan of this paper is as follows. First we review two methods of finding the conserved charges for any solution of NMG, the ADT formalism and the SL(2,R) reduction method. Mainly we use the general formulas from SL(2,R) reduction method to calculate the conserved charges for any solution of NMG in different ensembles. Then by finding the free energies, we discuss the phase transitions between the vacuum AdS and BTZ black hole solutions in section 2. We discuss the thermodynamics and local and 3 global stability regions. In section 5 we calculate the free energies of warped AdS vacuum and warped 3 BTZ black hole solutions. We calculate the free energy of the WAdS by three different methods and by 3 doing so we could find a factor in the modular parameter which extends the result of [15] for calculating the free energy of WAdS solutions in NMG. Then we present the phase diagrams of these solutions. In 3 2 section 7 we discuss the free energy and phase transitions of the Lifshitz and the new hairy black hole solution in NMG. We also discuss the inner horizon thermodynamics in section 8. In section 9, we discuss the entanglement entropy of the vacuum solutions corresponding to the WCFT dual of WAdS in NMG and then we conclude with a discussion in section 10. 2 The Bergshoeff-Hohm-Townsend Theory TheBergshoeff-Hohm-Townsend(BHT)orthenewmassivegravity(NMG)isahigher-curvatureextension of the Einstein-Hilbert action in three dimensions which is diffeomorphism and parity invariant. In the linearized level, it is equivalent to the unitary Pauli-Fierz action for a massive spin-2 field [16]. The action of NMG is 1 (cid:90) √ (cid:104) 1 (cid:16) 3 (cid:17)(cid:105) S = d3x −g R−2Λ+ RµνR − R2 , (2.1) 16πG m2 µν 8 N where m is the mass parameter, Λ is a cosmological parameter and G is a three-dimensional Newton N constant. In the case of m → ∞, the theory reduces to the Einstein gravity and in the limit of m → 0, it is just a pure fourth-order gravity. The equation of motion from the action would be derived as 1 1 R − Rg +Λg + K = 0, (2.2) µν 2 µν µν m2 µν with the explicit form of K as in [16], µν (cid:18) (cid:19) K = ∇2R − 1 (cid:0)∇ ∇ R+g ∇2R(cid:1)−4RσR + 9RR + 1g 3RαβR − 13R2 . µν µν 4 µ ν µν µ σν 4 µν 2 µν αβ 8 (2.3) The boundary terms of NMG which make the variational principle well-defined would be 1 (cid:90) √ (cid:18) 1 1 (cid:19) S = d3x −g fµν(R − Rg )− m2(f fµν −f2) , (2.4) Boundary µν µν µν 16πG 2 4 σ where f , the rank two symmetric tensor is µν 2 1 f = (R − Rg ). (2.5) µν m2 µν 4 µν This theory admits different solutions, such as the vacuum AdS , warped AdS , BTZ black hole, 3 3 asymptotic warped AdS black hole, Lifshitz, Schr¨odinger and so on [16], [17]. We construct the phase diagrams between several of these solutions by comparing the on-shell free energies. By constructing the off-shell free energies, one could even find all the states connecting any two solutions and therefore create a picture of continuous evolutions of the phase transitions; similar to the work in [18], who studied the continuous phase transition between the BTZ black hole with M ≥ 0 and the thermal AdS soliton with M = −1 in the new massive gravity. 3 Inthenextsectionwereviewhowonecancalculatetheconservedchargesandwebringseveralgeneral formulas for the solutions of NMG which could be used to find the on-shell Gibbs free energies. In section 4 we study the vacuum AdS and BTZ solutions of NMG, the free energies and the phase diagrams. Then 3 in section 5 we discuss the warped solutions and in section 7 we study the new hairy black hole solution of this theory. 3 Review of calculating conserved charges in BHT In three dimensions, the conserved charges associated to a Killing vector ξ would be 1 (cid:90) 2π√ δQ [δg,g] = −g(cid:15) kµν[δg,g]dϕ. (3.1) ξ 16πG µνϕ ξ 0 As calculated in [19] for BHT, the ADT formalism would result in 1 kµν = Qµν + Qµν, (3.2) ξ R 2m2 K where 3 Qµν = Qµν − Qµν, (3.3) K R2 8 R2 and the term for each charge is 1 Qµν ≡ ξ ∇[µhν]α−ξ[µ∇ hν]α−hα[µ∇ ξν]+ξ[µ∇ν]h+ h∇[µξν], (3.4) R α α α 2 Qµν = 2RQµν +4ξ[µ∇ν]δR+2δR∇[µξν]−2ξ[µhν]α∇ R, (3.5) R2 R α where δR ≡ −Rαβh +∇α∇βh −∇2h, (3.6) αβ αβ and 1 Qµν = ∇2Qµν + Qµν −2Qα[µRν]−2∇αξβ∇ ∇[µhν]−4ξαR ∇[µhν]β −Rh[µ∇ν]ξα R2 R 2 R2 R α α β αβ α +2ξ[µRν]∇ hαβ +2ξ Rα[µ∇ hν]β +2ξαhβ[µ∇ Rν]+2hαβξ[µ∇ Rν] α β α β β α α β −(δR+2Rαβh )∇[µξν]−3ξαR[µ∇ν]h−ξ[µRν]α∇ h. (3.7) αβ α α For the three dimensional case of (t,r,φ), the mass and angular momentum in three dimensions would be [19] 1 (cid:112) (cid:12) 1 (cid:112) (cid:12) M = −det gQrt(ξ )(cid:12) , J = −det gQrt(ξ )(cid:12) , (3.8) 4G T (cid:12)r→∞ 4G R (cid:12)r→∞ 4 where 1 ∂ ∂ ξ = , ξ = . (3.9) T R L∂t ∂φ 3.1 The SL(2,R) reduction method OnecanalsoderivethechargesbySL(2,R)reductionmethodwhichchangesthemetrictoSO(1,2)from. For doing so one should write the metric in the form of [19] dρ2 ds2 = λ (ρ)dxadxb+ , xa = (t,φ). (3.10) ab ζ2U2(ρ) Since there is a reparametrization invariance with respect to the radial coordinate, one can write the √ function U such that detλ = −U2 and this would give −g = 1/ζ. One then applies the equations of motions (EOMs) and the Hamiltonian constraint and then by integrating the EOM one can derive the “super angular momentum” vector. So first one parameterize the matrix λ as (cid:32) (cid:33) X0+X1 X2 λ = , (3.11) ab X2 X0−X1 where X = (X0,X1,X2) would be the SO(1,2) vector. Then one applies the reduced equation of motion and the Hamiltonian constraint as [20] 5 3 9 X∧(X∧X(cid:48)(cid:48)(cid:48)(cid:48))+ X∧(X(cid:48)∧X(cid:48)(cid:48)(cid:48))+ X(cid:48)∧(X∧X(cid:48)(cid:48)(cid:48)(cid:48))+ X(cid:48)∧(X(cid:48)∧X(cid:48)(cid:48)) 2 2 4 1 (cid:20)1 m2(cid:21) − X(cid:48)(cid:48)∧(X∧X(cid:48)(cid:48))− (X(cid:48)2)+ X(cid:48)(cid:48) = 0, (3.12) 2 8 ζ2 1 3 H ≡ (X∧X(cid:48)) . (X∧X(cid:48)(cid:48)(cid:48)(cid:48))− (X∧X(cid:48)(cid:48))2+ (X∧X(cid:48)) . (X(cid:48)∧X(cid:48)(cid:48)) 2 2 1 m2 2m2Λ + (X(cid:48)2)2+ (X(cid:48)2)+ = 0. (3.13) 32 2ζ2 ζ4 From these two equations one can find ζ2 and Λ. Then one can define the vector L ≡ X∧X(cid:48), (3.14) where (cid:48) ≡ d . dρ Finally the super angular momentum of NMG J = (J0,J1,J2) would be J = L+ ζ2 (cid:20)2L∧L(cid:48)+X2L(cid:48)(cid:48)+ 1(cid:0)X(cid:48)2−4X . X(cid:48)(cid:48)(cid:1)L(cid:21), (3.15) m2 8 5 where the products are defined as A.B = η AiBj, (A∧B)i = ηim(cid:15) AjBk, ((cid:15) = 1). (3.16) ij mjk 012 That being so for the case of NMG one would have [19] (cid:20) 1 (cid:21) 1 (cid:20) ζ2 (cid:21) η σQρt+ Qρt = − δJ2+∆ , R m2 K L 2 Cor ζT (cid:20) 1 (cid:21) ζ2 η σQρt+ Qρt = δ(J0−J1), (3.17) R m2 K 2 ζR where η and σ are ±1, depending on the sign in the action. Based on eq. 2.1, both of η and σ would be positive in our case. Also ∆ , the correction term to the mass, for the NMG would be Cor ∆ = ∆ +∆ , (3.18) Cor R K where ∆ = ζ2 (cid:2)−(X . δX(cid:48))(cid:3), R 2 ζ4 (cid:20) U2(cid:104) 1 (cid:105) ∆ = −U2(X(cid:48)(cid:48) . δX(cid:48))+ (UδU)(cid:48)(cid:48)(cid:48)−(X . δX(cid:48))(cid:48)(cid:48)− (X(cid:48) . δX(cid:48))(cid:48) K m2 2 2 UU(cid:48)(cid:104) 5 (cid:105) (cid:104) (cid:105) − (UδU)(cid:48)(cid:48)− (X(cid:48) . δX(cid:48)) + X(cid:48)2−(UU(cid:48))(cid:48) (UδU)(cid:48)+UU(cid:48)(X(cid:48)(cid:48) . δX) 4 2 (cid:21) (cid:104)5 21 (cid:105) (cid:104) 1 9 (cid:105) + (UU(cid:48))(cid:48)− X(cid:48)2 (X . δX(cid:48))+ − (UU(cid:48))(cid:48)(cid:48)+ (X(cid:48) . X(cid:48)(cid:48)) UδU . (3.19) 4 16 2 4 Then the mass and angular momentum in NMG would be (cid:20) (cid:21) 1 (cid:112) 1 M = −det g Qrt+ Qrt , 4G R m2 K ζT,r→∞ (cid:20) (cid:21) 1 (cid:112) 1 J = −det g Qrt+ Qrt . (3.20) 4G R m2 K ζR,r→∞ Also for calculating entropy for any solution of NMG, we can use the following relation from [20] A (cid:18) ζ2 (cid:20) 1 (cid:21)(cid:19) S = h 1+ (X . X(cid:48)(cid:48))− (X(cid:48)2) . (3.21) 4G 2m2 4 Now using these relations one can derive the charges, Gibbs free energies and the phase diagrams of several solutions of NMG. 3.2 Examples of conserved charges of BHT solutions First for the warped AdS black hole in the “grand canonical ensemble” [2], 6   −r2 − H2(−r2−4lJ+8l2M)2 +8M 0 4J − H2(4lJ−r2)(−r2−4lJ+8l2M) l2 4l3(lM−J) 4l2(lM−J)   g =  0 1 0 , (3.22) µν  16J2+r2−8M   r2 l2  4J − H2(4lJ−r2)(−r2−4lJ+8l2M) 0 r2− H2(4Jl−r2)2 4l2(lM−J) 4l(lM−J) by reparametrizing the radial coordinate as r2 → ρ, and then by applying the equation of motion and hamiltonian constraints 3.12 and 3.13 one can find 8l2m2 m2(84H4+60H2−35) ζ2 = , Λ = . (3.23) (1−2H2)(17−42H2) (17−42H2)2 From the above relation one can see that the acceptable region for Λ is −35m2 m2 < Λ < , (3.24) 289 21 and the special case of Λ = m2 corresponds to the AdS ×S1. 21 2 Now for the metric 6.1, the components of the super angular momentum would be H2(cid:0)1+l2(cid:1) H2(cid:0)−1+l2(cid:1) H2 J0 = − , J1 = , J2 = . (3.25) 4l3(−J +lM) 4l3(−J +lM) 2l2(J −lM) Then using 3.26 and 3.20 one can find the charges as 16(cid:0)1−2H2(cid:1)3/2M 16(cid:0)1−2H2(cid:1)3/2J M = , J = . (3.26) GL(17−42H2) G(17−42H2) One should note that ∆ would be zero here. Cor For the above metric using 3.21, the entropy would be 16π(cid:0)1−2H2(cid:1)3/2 (cid:113) (cid:112) S = l2M + l4M2−J2l2. (3.27) G(17−42H2) We can then study this black hole solution in another ensemble. The asymptotically warped AdS black 3 hole in NMG in the ADM form and therefore in the “quadratic/non-local ensemble” would be in the following form ds2 dr2 (cid:112) = dt2+ +(2νr− r r (ν2+3))dtdϕ l2 (ν2+3)(r−r )(r−r ) + − + − (3.28) r(cid:104) (cid:112) (cid:105) + 3(ν2−1)r+(ν2+3)(r +r )−4ν r r (ν2+3) dϕ2. + − + − 4 7 So using 3.12 and 3.13, one would have 8m2 m2(cid:0)9−48ν2+4ν4(cid:1) ζ2 = , Λ = . (3.29) l4(20ν2−3) (3−20ν2)2 The components of the super angular momentum would be (cid:16) (cid:112) (cid:112) (cid:17) l4ν(ν2+3) 4−2r ν r r (ν2+3)−2r ν r r (ν2+3)+r r (5ν2+3) − + − + + − + − J = − , 2(20ν2−3) (cid:16) (cid:112) (cid:112) (cid:17) l4ν(ν2+3) −4−2r ν r r (ν2+3)−2r ν r r (ν2+3)+r r (5ν2+3) − + − + + − + − J = , 2(20ν2−3) (cid:16) (cid:112) (cid:17) 2l4ν(ν2+3) (r +r )ν − r r (ν2+3) + − + − J = − . 20ν2−3 Then by using 3.26 and 3.20 one could find the conserved charges [20] [21] ν(cid:0)ν2+3(cid:1) (cid:16) (cid:112) (cid:17) M = (r +r )ν − r r (ν2+3) , 2G(20ν2−3) + − + − J = ν(cid:0)ν2+3(cid:1) (cid:16)(cid:0)5ν2+3(cid:1)r r −2ν(cid:112)r r (3+ν2)(r +r )(cid:17), 4Gl(20ν2−3) + − + − + − (cid:114) 4πlν2 (cid:16) (cid:112) (cid:17) S = r r (ν2+3)+4r ν r ν − r r (ν2+3) . (3.30) G(20ν2−3) + − + + + − As another example of a practical solution of NMG in condensed matter, one could also study the conserved charges of the Lifshitz geometry r2z l2 r2 ds2 = − dt2+ dr2+ d(cid:126)x2. (3.31) l2z r2 l2 Here ζ2 = −2m2l2+2z and the vector of super angular momentum would be zero. The case of z = 3 and 1+z(z−3) z = 1 could be a solution of the simple NMG with no matter content. For the case of z = 3, one would 3 have ζ2 = −2l8m2. Now considering the Lifshitz black hole solutions [22] r2z (cid:34) (cid:18)l(cid:19)z+21(cid:35) l2 (cid:34) (cid:18)l(cid:19)z+21(cid:35)−1 r2 ds2 = − 1−M dt2+ 1−M dr2+ dϕ2, (3.32) l2z r r2 r l2 (cid:16) (cid:17) 1 √ by taking r → ρ(z+1) 1+z, one would have −g = 1/ζ = l−z which would result in πM2(z+1)2(3z−5) M = − , J = 0, (3.33) 16κ(z−1)(z2−3z+1) 8 in accordance with [22]. This would lead us to the following Gibbs free energy M2πz(z+1)2(3z−5) G = . (3.34) LifshitzBH 16k(z−1)(z(z−3)+1) Comparing this result with the free energy of the Lifshitz metric, one can see that in NMG always the Lifshitz black hole would be the dominant phase. 4 Phase transitions of AdS solution 3 The vacuum AdS solution is 3 ds2 = l2(dρ2−cosh2ρ dt2+sinh2ρ dφ2), (4.1) AdS3 where [23] (cid:114) Λ 1/l2 = 2m2(1± 1+ ), (4.2) m2 and the boundary where the dual CFT is defined is located at ρ → ∞. For this case, we use the relation G(T,Ω) = TS[g ] to find the Gibbs free energy, where g is the c c Euclidean saddle and τ = 1 (−βΩ +iβ) is the modular parameter. We work in the regimes that the 2π E l saddle-point approximation could be used. First we need to find the free energy of the vacuum solution. In [15] [24], the authors derived a general result for deriving the action of the thermal AdS in any theory as, 3 (cid:0) (cid:1) iπ S AdS(τ,τ˜) = (cτ −c˜τ˜). (4.3) E 12l Also the modular transformed version of this equation would give the thermal action of the BTZ black hole. By changing the boundary torus as τ → −1, and then by using the modular invariance, one would τ have (cid:20) (cid:21) 1 ds2 − = ds2 [τ], (4.4) BTZ τ AdS so (cid:0) (cid:1) iπ c c˜ S BTZ(τ,τ˜) = ( − ). (4.5) E 12l τ τ˜ In this equation the contributions of the quantum fluctuations of the massless field is neglected as they are suppressed for large β. One should notice that this equation and its modular transformed version are only true for the AdS 3 and not particularly for the “warped AdS ” or “asymptotically warped AdS black holes”. This equation 3 is correct as in the Lorentzian signature, the thermal AdS has the same form as in the global coordinates 3 and also the global AdS corresponds to NS-NS vacuum with zero Virasoro modes [15]. These statements 3 are not particularly correct for geometries with other asymptotics than AdS, specifically geometries such 9