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Thermodynamics and Statistical Mechanics of Induced 9 Liouville Gravity 9 9 1 n a V. Frolov1, D. Fursaev2, J. Gegenberg3 and G. Kunstatter4 J 0 2 January, 1999 1 v 7 1 Theoretical Physics Institute, Department of Physics, University of Alberta, 8 Edmonton, Canada T6G 2J1 0 1 2Joint Institute for Nuclear Research, 0 Bogoliubov Laboratory of Theoretical Physics, 9 141 980 Dubna, Russia 9 / 3 Dept. of Math. and Statistics, h t University of New Brunswick, Fredericton, NB E3B 5A3 - p 4Physics Department and Winnipeg Institute of Theoretical Physics e h University of Winnipeg, Winnipeg, MB R3B 2E9 : e-mails: [email protected], [email protected], [email protected], v i [email protected] X r a Abstract InthispaperwedescribeaLiouvillegravitywhichisinducedbyasetofquantum fields (constituents) and represents a two-dimensional analog of Sakharov’s induced gravity. The important feature of the considered theory is the presence of massless constituents which are responsible for the appearance of the induced Liouville field. Theroleofthemassive constituents isonly toinducethecosmological constant. We considertheinstanton solutionsoftheEuclideanLiouvillegravity withnegative and zero cosmological constants, some instantons being interpreted as two-dimensional anti-de Sitter AdS black holes. We study thermodynamics of all the solutions and 2 conclude that their entropy is completely determined by the statistical-mechanical entropyofthemasslessconstituents. Thisshows,inparticular,thattheconstituents of the induced gravity are the true degrees of freedom of AdS black holes. Spe- 2 cial attention is also paid to the induced Liouville gravity with zero cosmological constant on a torus. We demonstrate the equivalence of its thermodynamics to the thermodynamics of BTZ black holes and comment on computations of the BTZ black hole entropy. 1 1 Introduction The microscopic explanation of the Bekenstein-Hawking entropy [1],[2] of black holes is one of the most intriguing problems of theoretical physics. Although there are several approaches to its resolution (for a review of some of them see [3],[4]) this problem is a subject of intensive study and continues to inspire new ideas. In particular, it was realized recently that at least some black holes may be macro- scopically equivalent to two-dimensional systems described by a Liouville theory. The statistical-mechanical entropy of these systems can be computed by means of a conformal field theory and coincides with the black hole entropy. This observation was first made for extremal black holes [5] and was based on dualities in string models. However, in the most explicit and simple form it appeared in the work by Strominger [6] concerning BTZ black holes [7] (for a detailed analysis of these computationswiththelargelistofreferencessee[10]). TheBTZblackholeshavethesame thermodynamic characteristics as a dual Liouville theory defined at asymptotic infinity. The reason for such a relation arises from a specific property of three-dimensional gravity with negative cosmological constant. It is equivalent to a Chern-Simons theory which has only boundary degrees of freedom that are described by a Liouville theory [8],[9]. A Liouville theory which is dual to the BTZ black hole can be also defined at the black hole horizon [11]. Moreover, close to the horizon one can find a Liouville-like description of black holes in an arbitrary dimension [12],[13]. This description becomes possible because the gravitational action for spherically symmetric metrics is the action for a two- dimensional dilatonic gravity. In the region near the horizon this gravity is equivalent to a Liouville theory [13] and one can use the conformal theory to calculate the entropy. These results may have the following interpretation [14]. On the level of thermodynamics black holes are equivalent to a Liouville field theory. The Liouville field, which is purely classical, is a collective excitation of some quantum constituents which are described by 2D conformal field theory. To put it in another way, the Liouville theory is an effective theory of the constituents. The constituents are those microscopic degrees of freedom which explain the thermodynamic entropy related to the Liouville field and thus reproduce the black hole entropy in a statistical-mechanical way. Remarkably, this mechanism is basically the same as the mechanism of the generation of the black hole entropy in Sakharov’s induced gravity [17]–[20]. According to Sakharov’s idea [21], the gravitational field is a collective excitation of the matter constituents and the Einstein action is the low-energy effective action of the constituents. The equations for the metric g are µν ˆ T (g) = 0 . (1.1) µν h i HereTˆ isthestress-energy tensorofconstituent fieldsonthebackgroundwiththemetric µν g and its average is taken in some quantum state. µν It should be noted that in Sakharov’s induced gravity the microscopic states of a 2 black hole are related to the constituents which live on the physical space-time. In the Liouville description of black holes the microscopic degrees of freedom live on a dual two- dimensional space-time. For this reason the statistical origin of the Bekenstein-Hawking entropy in the two approaches is different. Nevertheless, the similarity of both approaches suggests that they may be connected. To see whether there is any connection one must first understand better how Sakharov’s mechanism of induced gravity works in the case of the Liouville theory. Studying induced Liouville gravity is the subject of this paper. The important feature of the considered models of two-dimensional induced gravity is the presence of massless constituents which induce the Liouville action. The massive constituents serve only to induce a finite cosmological constant. Thus, as distinct from higher dimensions, all the dynamics in the Liouville induced gravity is due to massless fields. This fact has a crucial consequence for the statistical interpretation of the thermodynamics of the instanton solutions of the theory. We consider models with negative and zero cosmological constants. The theory with negative cosmological constant is a sort of gravity on anti-de Sitter AdS space-time. 2 Studying this theory is motivated by various reasons, one of which is its relation to AdS 2 string theories arising as near-horizon limits of different four and five-dimensional black holes (for a recent discussion of AdS gravity in this context see [15]). The Liouville 2 induced gravity with vanishing cosmological constant is interesting because it is that theory which is thermodynamically equivalent to BTZ black holes. The paper is organized as follows. In Section 2 we use the Sakharov’s mechanism to induce Liouville gravity by massless and massive quantum constituent fields. The considered quantum models are free from ultraviolet divergences and, as a result, the induced theory is finite and well-defined. All the solutions of Liouville gravity with the negative cosmological constant are locally AdS . In the Euclidean theory they can be of three 2 types: elliptic, hyperbolic andparabolic. Weremind thereader theformofthese solutions in Section 3. The instantons of the elliptic type can be interpreted as AdS black holes. 2 Thermodynamics and statistical mechanics of these black holes is studied in Section 4. We demonstrate that in the physical processes the changes of the entropy of a AdS black 2 hole coincide with the corresponding changes of the entanglement (statistical-mechanical) entropy of the massless constituents. Therefore, the constituents of the induced gravity are the true internal degrees of freedom of AdS black hole. In Section 5 we comment on 2 thermodynamicsandstatisticalmechanicsthehyperbolicandparabolicsolutions. Finally, in Section 6 we consider induced Liouville gravity with vanishing cosmological constant on a torus and demonstrate the equivalence of its thermodynamics to the thermodynamics of the BTZ black hole under identifying the central charges of both theories. A brief discussion is given in Section 7. 3 2 Induced Liouville gravity Induced Liouville gravity (ILG) can beconstructed frommodels with different constituent field species, similar to the construction of induced Einstein gravity [18]-[20]. To illustrate the idea we consider here the simplest model. It consists of noninteracting scalar and spinor fields, with some of the scalar fields being massless. The numbers of massive and massless scalars are N and N, respectively, and the number of massive spinor fields is N . s d It is assumed that some massive scalars are non-minimally coupled and the corresponding constants are denoted as ξ . The effective gravitational action of the fields propagating s on a background with the metric g is µν Γ = W + W +NW0 , (2.1) s d s s d X X 1 W = logdet 2 +ξ R+m2 , W = logdet(γµ +m ) , (2.2) s 2 −∇ s s d − ∇µ d (cid:16) (cid:17) 1 W0 = logdet 2 , (2.3) s 2 −∇ (cid:16) (cid:17) Here R is the scalar curvature of the background. It is not difficult to show that Γ is free from ultraviolet divergences if the following constraints are satisfied N +N 2N = 0 , m2 2 m2 = 0 , (2.4) s − d s − d s d X X N +N +N 6 ξ = 0 . (2.5) s d s − s X Constraints (2.4) ensure the finiteness of the induced cosmological constant, while condition (2.5) guarantees the finiteness of the induced Newton constant. Suppose now that masses m have the order of magnitude of a typical mass M. The i low-energy limit of the theory is realized when the curvature of the background geometry is small compared to M2. In this limit contributions of the massive constituents to the induced action can be expanded in powers of the curvature. On the other hand, the contributions of the massless constituents, W0, can be calculated exactly. s It is convenient to represent the induced action in the form Γ = Γm +NW , (2.6) Γm = W + W +NW0 , (2.7) s d s,div s d X X W W0 W0 . (2.8) ≡ s − s,div Here W0 is the divergent part of the action of massless fields, so that W is the ”renor- s,div malized” action. Note that because we are dealing with ultraviolet finite theories the functionals Γm are free from the divergences. The divergences of the massive fields in Γm are canceled by the term NW0 . s,div 4 Let us consider fields given on a manifold with boundary ∂ . The metric on M M M will be denoted g . In the general case, g can have Lorentzian or Euclidean signature. µν µν In what follows we assume that is a Euclidean manifold. In the low energy limit M of the theory the action of the massive constituents Γm can be expanded in powers of the curvature. We keep only the leading (cosmological) term in this decomposition and approximate Γm as Γm = √gd2xλ . (2.9) M Z Here the cosmological constant λ is 1 λ = m2lnm2 2 m2lnm2 . (2.10) −8π s s s − d d d! X X In what follows, we consider only the models where the cosmological constant is negative or zero. The curvature corrections to expression (2.9) are suppressed by powers RM−2. The above approximation is not applicable to the action W of massless fields. In fact, this functional is the well-known Polyakov action [22] which can be computed exactly. Consider the conformal map of onto a space ¯ with the metric g¯ = exp( 2σ)g . µν µν M M − The actions on and ¯ are related as M M 1 W[g] = W[g¯] √gd2x(Rσ ( σ)2)+ h1/2dy(2Kσ+3nµσ ) . (2.11) ,µ − 24π (cid:20)ZM − ∇ Z∂M (cid:21) Here nµ is a unit vector normal to the boundary ∂ of ; K and h are the extrinsic M M curvature and the metric on ∂ . The functional W[g¯] is the effective action computed M on the background ¯ with the metric g¯. It is convenient to assume that ¯ is locally M M flat. As follows from the above analysis, the induced gravitational action after subtracting a boundary term depending on nµσ can be written in the form: ,µ Γ[g] = I [g,φ]+NW[g¯] , (2.12) L 1 2 µ 1 I [g,φ] = √gd2x ( φ)2 + Rφ+ h1/2dyKφ . (2.13) L −8π ZM ∇ γ γ2!− 2πγ Z∂M Here we put 2 φ = σ , (2.14) γ 12 96π γ = , µ = λ . (2.15) sN N | | The action I can be also represented as a functional on the flat space ¯ L M 1 µ β I [φ,g] = I¯ [φ] = d2y (¯φ)2 + eγφ (φ φ ) . (2.16) L L −2π M¯ ∇ γ2 !− 2πγ + − − Z Here β is the circumference length of the boundary, and φ , φ are the values of φ on the + − external and internal parts of the boundary. If the internal boundary is absent φ = 0 in − (2.16). Up to the boundary term, I¯ [φ] is the canonical Euclidean Liouville action 1. The L 1Strictly speaking, I¯ [φ] differs from the standard definition by a sign, see [24]. L 5 Liouville theory is known from the last century as a theory of negatively curved surfaces. A review of some its properties can be found in [23],[24]. The important feature of (2.16) is that it describes a classical conformal theory with the central charge 12 c = , (2.17) γ2 which in our model is just the number of massless constituents c = N , (2.18) see (2.15). The latter fact is not surprising. The massless constituents of our model are conformally invariant in two dimensions. Under quantization the conformal symmetry ac- quires acentral extension due tothe conformalanomaly. The central chargec corresponds to this anomaly. 3 Solutions to the Liouville theory Equation (2.16) shows that the Liouville field is the only dynamical variable of induced Liouville gravity2. By varying I¯ with fixed boundary value of φ, one finds the equation L µ e−γφ∆¯φ = . (3.1) −2γ It follows from this equation that the physical metric g = eγφg¯ corresponds to a space µν µν with constant negative curvature3 R = µ/2. This space is locally a two-dimensional − Lobachevsky space H . The corresponding solution in the Lorentzian space-time is locally 2 anti-de Sitter (AdS ). 2 There is also another approach to the variational problem. One can start with the functional (2.12) and consider the Liouville field and metric g as independent variables. µν Then the equations of motion obtained from (2.12) by varying4 φ and g respectively, µν are: R = γ∆φ , (3.2) G = 0 , (3.3) µν 1 2 µ G φ φ g φ 2 + (g ∆φ φ)+ g . (3.4) µν ≡ ,µ ,ν − 2 µν | ∇ | γ µν −∇µ∇ν 2γ2 µν 2To avoid the confusion, let us note that φ is the dynamical variable only in the classical theory, in the quantum Liouville theory φ appears in the conformal gauge but its contribution is compensated by the contribution of corresponding ghosts. 3ItshouldbenotedthatR= µ/2resultsinfurtherrestrictionsonthe parametersofthe constituent fields of the Polyakov induced g−ravity. Namely, the condition of the large masses, M2 R, becomes M2 48π λ/N where M is a typical scale for masses of the fields and λ is given by Eq≫. (2.10). One ≫ | | can construct models where this condition is satisfied. 4The variational procedure implies that the metric and φ are fixed on the boundary. The boundary termwhichdepends onnµσ wasremovedfrominduced action(2.12)inorderto obeythis requirement. ,µ 6 The trace of Eq.(3.3) results in the relation µ ∆φ = (3.5) −2γ which coincides with the Liouville equation (3.1). This justifies considering the more general variational problem where φ and the metric are independent fields. When φ obeys (3.1) G = 4π Tˆ , (3.6) µν µν − h i ˆ where T is the quantum stress-energy tensor of the constituents computed in the con- µν sidered approximation (3.2). Thus, (3.3) is equivalent to relation (1.1) of the Sakharov’s induced gravity. It should be noted that boundary conditions for φ which are required to solve for it from (3.2) are related to the choice of the quantum state 5. There is three types of solutions of the Euclidean Liouville equations (3.1): elliptic, parabolic and hyperbolic (see, e.g. [24]). These solutions correspond to different coordinate maps on the Lobachevsky space H . 2 1. Elliptic solutions. The metric has the form6 16 1 ds2 = eγφds¯2 = ds¯2 , (3.7) µ (1 ρ2)2 − ds¯2 = ρ2dτ2 +dρ2 , 0 τ 2π . (3.8) ≤ ≤ We specify the flat metric by the boundary condition 0 < ρ ρ . (3.9) + ≤ Solution (3.7) can be also written in the form 1 ds2 = g(x)dτ2 + dx2 , (3.10) g(x) µ 1 g(x) = x2 + x , (3.11) 4 2 8 ρ2 x = . (3.12) µ1 ρ2 − This solution has the topology of a disk and can be interpreted as a black hole instanton. The horizon of the black hole is located at x = 0 (ρ = 0). The normalization of the time coordinate τ is chosen so that the corresponding surface gravity constant is g′(0)/2 = 1. One can make the surface gravity equal to another constant by rescaling τ. The black hole solutions in the Lorentzian metric have the anti-de Sitter geometry and we will call them AdS black holes for the brevity. 2 5A detailed discussion of this problem can be found for instance in Ref. [25]. 6We consider only the solutions which are free from conical singularities 7 2. A hyperbolic solution depends on an integration constant m and has the form 4 m2 ds2 = eγφds¯2 = ds¯2 . (3.13) µρ2sin2(mlnρ) The flat metric ds¯2 is given by (3.8). The hyperbolic solution has the topology of a cylinder. The flat space is defined by the boundary condition ρ ρ ρ where − + M ≤ ≤ e−π/m < ρ < ρ < 1 . (3.14) − + The metric (3.13) can be also written in the form (3.10) with µ 4m2 g = x2 + , (3.15) 4 µ 4m x = cot(mlnρ) . (3.16) − µ 3. Parabolic solutions can be obtained from the hyperbolic ones in the limit m 0 → and look as follows 4 1 ds2 = eγφds¯2 = ds¯2 , (3.17) µ(ρlnρ)2 where ds¯2 is determined by (3.8). The coordinate ρ subjects to the boundary condition ρ ρ ρ , where ρ > 0 and ρ < 1. In the limit when ρ = 0 the curvature of the − + − + − ≤ ≤ solution has a delta-function-like singularity. The metric (3.17) can be also written in the form (3.10) with µ g = x2 , (3.18) 4 4 x = . (3.19) −µlnρ As follows from (3.18) these solutions are analogous to extremal black holes. Note that Liouville solutions of all three types have the same asymptotic behaviour at large x. It is also worth pointing out that each Liouville solution can be interpreted as a boundary theory of a three-dimensional AdS-gravity and corresponds to a particular three dimensional object [14] 7. Elliptic solutions with conical singularities correspond to massive particles in AdS , hyperbolic and parabolic solutions may be related to non- 3 extreme and extreme three-dimensional black holes, respectively. The non-extreme black holes correspond to a particular Liouville theory on a torus. 7It is interesting to note that in a 4D gravity there also exist 3 different families of Schwarzschild- (anti) de Sitter solutions. These solutions are given by metric ds2 = Adt2 +dr2/A+r2dΩ2, where A=K (2m/r)+Λr2/3, dΩ2 =dθ2+sinn2θdφ2. For an elliptic case,−K =1 and sinnθ =sinθ. For a − hyperbolic case, K = 1 and sinnθ = sinhθ. For a parabolic case, K = 0 and sinnθ = θ. The elliptic − solution (with K =1) corresponds to a usual black hole. The hyperbolic solution describes a black hole moving with a superluminar velocity, while the hyperbolic solution describes a black hole moving with the velocity of light. 8 4 Elliptic solutions 4.1 Thermodynamics of AdS black holes 2 Let us now study black holes corresponding to elliptic solutions. It is perhaps necessary at this stage to explain in what sense we expect constant curvature solutions to behave like black holes. The key assumption that we make is that the Liouville field φ is an observable quantity, despite the fact that it is not, strictly speaking, a fundamental field. It is rather introduced for calculational convenience to make the effective action local and defined formally by the relationship φ = 1 R(g). In effect, the observability of φ requires γ∆ matter to couple non-locally to the metric g (i.e. to 1R). This does not occur at the ∆ classical level, but in the full quantum effective action one does expect such terms to appear. It is on this basis that we feel justified in assuming that φ is an observable field, analogous to a dilaton. Henceforth we treat the Liouville action as if it were a dilaton gravity theory of the general form considered extensively in a variety of references[16]8. WewillinterprettheinducedEuclideanaction(2.12)consideredontheellipticsolution (3.7) as T times the free energy of the corresponding black hole. The black hole is completely characterized by the temperature T measured at the boundary x = x + T = (2π g(x ))−1/2 . (4.1) + q This condition defines x in terms of T + 1 x = (4πT)2 +µ 4πT . (4.2) + πµT − (cid:18)q (cid:19) The Liouville action on the elliptic solution is µ I (x ) = x . (4.3) L + −2γ2 + To calculate the induced action (2.12) one also has to know the value of the Polyakov action on the disk of the radius ρ . The dependence of this action on ρ can be found + + out by making a scaling transformation (i.e. a conformal transformation with constant factor) to the disk of unit radius 2 NW[g¯(ρ )] = lnρ +C . (4.4) + −γ2 + 8 It is worth noting that there exists at least one other theory, namely Jackiw-Teitelboim gravity, in which constantcurvature black hole solutions have been analyzedin some detail. In that case, the black holescanbeinterpretedintermsofdimensionallyreduced2+1Einsteingravity(the so-calledBTZblack holes[7]). Interestingly,theJackiw-Teitelboimtheorywasoriginallymotivatedbytheconnectionbetween constant curvature metrics and induced Liouville gravity. The dilaton was considered to be a physically irrelevantlagrangemultiplierfieldneededtoenforcetheconstantcurvatureequation. Oncethedilatonis taken seriously, it becomes clear that Jackiw-Teitelboim gravity and induced Liouville gravity are quite different theories, despite the fact that the solutions to both involve constant curvature metrics. 9 Here C is a constant corresponding to the action on the unit disk. It does not depend on T and can be omitted. From (3.12), (4.3) and (4.4) we obtain for (2.12) 1 µ µx + Γ = x +ln . (4.5) −γ2 2 + µx+ +8! This result can be immediately expressed in terms of the temperature on the boundary by using (4.1) and (4.2). By neglecting in (4.5) a numerical constant we find the free energy of the black hole 1 1 FBH(T) = TΓ(x ) = (4πT)2 +µ+2T ln (4πT)2 +µ 4πT . + |x+=x+(T) −γ2 2π − (cid:20) q (cid:18)q (cid:19)(cid:21) (4.6) Thus, the black hole entropy defined by the standard relation, is ∂FBH 2 SBH(T) = = ln (4πT)2 +µ 4πT . (4.7) − ∂T γ2 − (cid:18)q (cid:19) (Here all numerical constants were omitted.) The energy corresponding to this solution is 1 EBH(T) = (4πT)2 +µ . (4.8) −2πγ2 q The variations of the energy and the entropy are related by the first law dEBH = TdSBH . (4.9) It should be noted that the given thermodynamical system is not stable because its heat capacity is negative for all values of T ∂SBH 1 8πT c(T) = T = . (4.10) ∂T −γ2 (4πT)2 +µ q The energy of this black hole decreases when its temperature grows. Increasing the temperature corresponds to moving the boundary closer to the black hole horizon (x + → 0). As follows from (4.10) the heat capacity is increasing at small temperatures and one may speculate that in this region the black hole may be in a quasi-equilibrium state. Thus at low temperatures our description of the black hole in terms of a canonical ensemble may be justified. On the other hand, at high temperatures T the system is very unstable and one should expect a phase transition. Other possible phases of the theory may be related to other solutions of ILG. 10