Conformal equivalence of 2D dilaton gravity models

arXiv:hep-th/9610201v1 25 Oct 1996 INFNCA-TH9623 October 1996 CONFORMAL EQUIVALENCE OF 2D DILATON GRAVITY MODELS Mariano Cadoni Dipartimento di Scienze Fisiche, Università di Cagliari, Via Ospedale 72, I-09100 Cagliari, Italy. and INFN, Sezione di Cagliari. Abstract We investigate the behavior of generic, matter-coupled, 2D dilaton gravity theories under dilaton-dependent Weyl rescalings of the metric. We show that physical observables associated with 2D black holes, such as the mass, the temperature and the flux of Hawking radiation are invariant under the action of both Weyl transformations and dilaton reparametrizations. The field theoretical and geometrical meaning of these invariances is discussed. E-Mail: [email protected] The recent flurry of activity on two-dimensional (2D) black hole physics [1] even though it has not succeeded in finding a definite answer to challenging questions such as the ultimate fate of black holes or the loss of quantum coherence, has enabled us to gain considerable knowledge on the subject. Among other things, we have got a strong indication that a consistent description of black holes at the semiclassical or even quantum level requires us to treat the matter and the gravitational degrees of freedom on the same footing. Considerable progress has been achieved by considering 2D dilaton gravity models from this purely field theoretical point of view, for example as a non-linear σ-model [2, 3], as a 2D conformal field theory [4] or in the gauge theoretical formulation [5]. One serious problem of this kind of approach is the difficulty in giving a geometrical interpretation to some field theoretical concepts. For example, from a purely field theoretical point of view, performing dilaton-dependent Weyl rescalings of the metric in the 2D dilaton gravity action should give us equivalent models, being these transformations nothing but reparametrizations of the field space. The space-time interpretation of this equivalence presents, however, some problems. Though the causal structure of the 2D space-time does not change under Weyl transformations, geometrical objects such us the scalar curvature of the space-time or the equation for the geodesics do change. This discrepancy has generated a lot a confusion on the subject. Some authors have assumed explicitly or implicitly this equivalence to hold and used it to simplify the description of the general model [6, 7, 8] or even to argue about the existence of Hawking radiation in the context of the Callan-Giddings-HarveyStrominger model (CGHS) [5, 9]. Other authors, focusing on the space-time interpretation of the gravitational degrees of freedom, have pointed out the non-equivalence of 2D dilaton gravity models connected by Weyl rescalings of the metric [9, 10]. In this paper we analyze in detail the role of conformal transformations of the metric in the context of generic, matter-coupled, 2D dilaton gravity theories. We prove a conjecture reported in a previous paper [11], and based on previous results for the CGHS model [12], namely that physical observables for 2D black holes, such us the mass, the temperature and the flux of Hawking radiation, are invariant under dilaton-dependent Weyl rescalings of the metric. Moreover, we show that the same observables are also invariant under reparametrization of the dilaton field. The most general action of 2D dilaton gravity conformally coupled to a set on N matter scalar fields has the form [8, 13] 1 S[g, φ, f ] = 2π Z 2 dx √ −g " N 1X (∇fi )2 , D(φ)R[g] + H(φ)(∇φ) + λ V (φ) − 2 i=1 2 2 # (1) where D, H, V are arbitrary functions of the dilaton φ and λ is a constant. Let us consider the following Weyl transformations of the metric gµν = eP (φ) ĝµν , (2) for the moment we constrain the form of the function P only by requiring the transformation (2) to be non-singular and invertible in the range of variation of the dilaton. Arguments related to the geometrical interpretation of the transformation (2) will impose some additional restrictions on the form of the function P . We will come back to this point later on this paper. Whereas the matter part of the action (1) is invariant under the transformation 1 (2), the gravitational part is not. Nevertheless, the latter maintains its form under these transformations, in fact, modulo a total derivative we have 1 S[g, φ] → S[ĝ, φ] = 2π Z d2 x q ˆ 2 + λ2 V̂ (φ) , −ĝ D̂(φ)R[ĝ] + Ĥ(φ)(∇φ) h i where the new functions D̂, Ĥ, V̂ are related to the old ones through the transformation laws (′ = d/dφ) D̂ = D, Ĥ = H + D ′ P ′, V̂ = eP V. (3) Under dilaton reparametrizations φ = φ(φ̃), V and D behave as scalars, whereas H transforms as !2 dφ . (4) H̃(φ̃) = H(φ) dφ̃ The transformation laws (2), (3) and (4) enable us to find out how the physical parameters characterizing the solutions of the theory transform under Weyl transformations (2) and dilaton reparametrizations. Let us begin with the mass of the solutions. R. B. Mann has shown that for the generic theory defined by the action (1), one can define the conserved quantity [13] F0 M= 2 "Z φ dDV exp − Z ! H(τ ) dτ ′ − (∇D)2 exp − D (τ ) Z H(τ ) dτ ′ D (τ ) !# , (5) where F0 is a constant. M is constant whenever the equation of motion are satisfied and, in this case, it can be interpreted as the mass of the solution. Using Eqs. (2), (3), (4), one can easily demonstrate that the mass M given by the expression (5) is invariant under both Weyl transformation and dilaton reparametrizations. Notice that this invariance means not only that two conformally related solutions have the same mass, but also that the off-shell quantity (5) is Weyl-invariant. The Hawking temperature associated with a generic black hole solution can be defined as the inverse of the periodicity of the Euclidean time necessary to remove the conical singularity at the event horizon. To perform this calculation we need an explicit form for the black hole solutions of 2D dilaton gravity. The generic static solutions in the conformal frame in which Ĥ = 0, have already been found in Ref. [6, 7], ˆ 2 = a2 Jˆ − 2M dt2 + a−2 Jˆ − 2M ds λ2 F0 λ2 F0    −1 dr 2 , D(φ) = λ r, a (6) ˆ D̂ = V̂ , a is an arbitrary integration constant and M is the mass of the solution where dJ/d given by Eq. (5). The static solutions in the generic conformal frame can be easily obtained R from these solutions using Eq. (2) with P = − φ dτ [H(τ )/D ′ (τ )], 2 ds = exp − Z φ ! H(τ ) ˆ 2 dτ ′ ds . D (τ ) (7) A straightforward calculation gives for the Hawking temperature associated with an event horizon of the solution (7), located at φ = φ0 , Z λa V (φ0 ) exp − T = 4π 2 φ0 ! H(τ ) . dτ ′ D (τ ) (8) The temperature is invariant both under Weyl transformations and dilaton reparametrizations. This can be easily checked using Eqs. (3) and (4) in Eq. (8) and taking into account that the transformations (2) do not change the position of the event horizon, because by assumptions they are everywhere non-singular and invertible. In Eq. (6) and in the expressions (5), (8) for the mass and the temperature appear two arbitrary constants a and F0 . As already noted in Ref. [10] for a particular class of 2D dilaton gravity models, their presence is related to the arbitrariness in defining the asymptotical behavior of the metric or, from a physical point of view, to the way an asymptotical observer measures lengths and masses. Our prove does not rely on the way one fixes this arbitrariness, nevertheless, it is useful to have a definite and general prescription to fix it. The proposal of Ref. [10] seems to us too much model-dependent; we will use here a different prescription. First of all, we need a notion of asymptotic region (spatial infinity) for our space-time, which is Weyl-invariant. As already noted in a previous paper [11], the dilaton φ gives a coordinateindependent notion of location and can therefore be used to define the asymptotic region, the singularities and the event horizon of our 2D space-time. Moreover, the natural coupling constant of the theory is D −1/2 so that we have a natural division of our space-time in a strong-coupling region (D = 0) and a weak-coupling region (D = ∞). These considerations limit the range of variation of the function D to 0 ≤ D < ∞ and enable us to identify the weak-coupling region D = ∞ with the asymptotic region of our space-time [11]. Moreover, √ this notion of location is√Weyl-invariant because the term −gDR[g] in the action (1) is transformed by (2) into −ĝDR[ĝ]. In the conformal gauge ds2 = −e2ρ dx+ dx− , (9) using a Weyl transformation (2) one can always put the solution (7) into the form 2ρ e 2 =a 2M 1− , λF0 K   K= Z φ dDV exp − Z ! H(τ ) dτ ′ . D (τ ) (10) This form of the solution can be used to fix the values of the parameters a and F0 . Using arguments similar to those of Ref. [11], one can show that a black hole interpretation of the solution (10) requires K → ∞ for D → ∞. The condition that the metric (10) has asymptotically a Minkowskian form fixes now a = 1. The general solution admits a Killing vector of the form [13] µ µν ζ = ǫ ∇ν F , F = F0 Z φ dD exp − Z ! H(τ ) dτ ′ . D (τ ) The constant F0 can be fixed to F0 = 1/λ by requiring that in the conformal frame in which the metric has the form (10), the norm of the Killing vector approaches, for D → ∞, the value −1. To discuss the Hawking effect we need to be sure that the solutions we are dealing with really represent black holes. Our previous discussion does not rely heavily on the notion of black hole. The mass formula (5) holds for every solution of the theory, whereas the temperature (8) is a local-defined quantity, which does not care if the space-time has the global features of a black hole. In view of the discussion of Ref. [11], one expects that the form of the functions D, H, V has to be constrained in order to be sure that the solution 3 (7) is a black hole. However, the discussion of Ref. [11] cannot be extended trivially to the present context. In Ref. [11] we used the scalar curvature R to define the singularities and the asymptotic behavior of the space-time. R is not Weyl rescaling invariant and cannot be taken as a good quantity for a conformal invariant characterization of black holes. Here, we will not tackle the problem in this general setting, but we will consider the black hole solutions in the particular conformal frame in which the metric is asymptotically Minkowskian and has, therefore, the form (10). In this conformal frame, the ground state solution M = 0 coincides with Minkowsky space. Assuming that the black holes exist in any conformally related frame, we will show that the result for the Hawking radiation is invariant under conformal transformations of the metric. In the conformal frame defined by Eq. (10), the scalar curvature of the black hole spacetime is d2 R = 2MλK ln K. dD 2 We require that the M 6= 0 solutions behave asymptotically as the ground state solution, i.e., R → 0 for D → ∞. This singles out three main classes of 2D dilaton gravity models, according to the asymptotical, D → ∞, behavior of the function K: K ∼ Dα, 0 < α < 2, K ∼ γ ln D, 0 < γ < ∞, βD K ∼ e , 0 < β < ∞. (11) The first class of models has already been found and discussed in Ref. [11]. Our discussion, including the Hawking effect, holds also for models with α = 2. In this case the solutions describe space-times that are asymptotically anti-de Sitter. There are various ways to analyze the Hawking effect. Here, we will use the relationship between Hawking radiation and quantum anomalies [14, 9, 15, 11]. It is well-known that in quantizing the scalar matter fields f in a fixed background geometry the Weyl rescaling or/and part of the diffeomorphism invariance of the classical action for the matter fields has to be explicitly broken. The quantization procedure has two sources of ambiguity. First, one can decide to preserve at the semiclassical level either the diffeomorphism or the Weyl rescaling invariance [16, 17, 18] ( for sake of simplicity we do not consider here the case in which both symmetries are broken). Second, if one decides to preserve diffeomorphism invariance, one has still the freedom of adding local, covariant, dilaton-dependent counterterms to the semiclassical action [19, 4, 3]. The nature of these ambiguities is particularly clear in the path integral formulation. By choosing the diffeomorphism-invariant measure [20] Z  Z Dfi exp i d2 x √ −g fi2  = 1, (12) one breaks explicitly the Weyl invariance of the classical matter action, and introduces an ambiguity related to the choice of the metric to be used in the measure. One is allowed to use in Eq. (12) the metric gµν or a Weyl-rescaled metric ĝµν . The corresponding semiclassical actions differ one from the other for the presence of local, covariant, dilaton-dependent counterterms. On the other hand, by choosing the Weyl-invariant measure [17, 18] Z  Z Dfi exp i 4  d2 xfi2 = 1, (13) one breaks part of the diffeomorphism invariance of the classical action, but there is no ambiguity associated with the choice of the metric to be used in the measure (the measure (13) does not depend on the metric). It has already been shown for a particular 2D dilaton gravity model (the CGHS model) that, though the form of the Weyl anomaly depends on the choice of the measure, the flux of the Hawking radiation does not [15]. Here, we will not only show that this is true for a generic 2D dilaton gravity model but also that the result for the Hawking radiation is independent of the metric used in the measure (12), which is equivalent to prove the invariance of the Hawking radiation rate under the Weyl rescaling (2). To be more precise, in Ref. [15] the measure and the trace anomaly is parametrized by a real parameter k. The two cases we discuss here correspond respectively to k = 1 and k = 0. We expect that our considerations can be trivially extended to arbitrary values of k. Let us first consider a measure defined by Eq. (12). The semiclassical effective action is diffeomorphism-invariant and in the conformal frame where the metric is asymptotically Minkowskian, it is given by Ssc = Scl − Slp , (14) where Scl is the classical action (1) and Slp is the usual non-local Liouville-Polyakov action N Z 2 √ ¯ −2 R[ḡ], Slp = d x −ḡR[ḡ]∇ 96π where the notation ḡ has been used in order to avoid confusion with the metric in the generic conformal frame. The semiclassical action has its ”minimal” Liouville-Polyakov form, with no dilatondependent counterterms present, exactly in the conformal frame where the solutions are asymptotically Minkowskian. This fact follows from very simple physical requirements. Dilaton-dependent counterterms are forbidden if one requires that the expectation value of the stress-energy tensor vanishes when evaluated for Minkowsky space (the M = 0 ground state solution of our models). Under a Weyl transformation (2) the Liouville-Polyakov action acquires local, dilaton-dependent terms that have the same form as those already present in the action (1). These terms depend on the form of the function P (φ) in Eq. (2), so that -as expected- the trace anomaly depends on the particular conformal frame chosen. Using the R equation ḡµν = gµν exp ( dτ [H(τ )/D ′ (τ )] − ln K) in the expression (14), one easily finds the form of the semiclassical action in the generic conformal frame: Ssc [g] = Scl [g] − Slp [g] N − 96π Z d2 x √  −g 2 ln K − Z φ ! K′ H H(τ ) R[g] − − ′ dτ ′ D (τ ) K D !2  (∇φ)2  . (15) The black hole radiation can now be studied along the lines of Ref. [11], working in the conformal gauge (9) and considering a black hole formed by collapse of a f -shock-wave, traveling in the x+ direction and described by a classical stress-energy tensor T++ = Mδ(x+ − + + x+ 0) . The classical solution describing the collapse of the shock-wave, for x ≤ x0 , is given by ! Z φ Z φ dτ λ H(τ ) 2ρ , = (x+ − x− ), (16) e = K exp − dτ ′ D (τ ) K(τ ) 2 5 and, for x+ ≥ x+ 0 , it is given by 2ρ e = exp − Z φ Z dτ K(τ ) − F ′ (x− ) = φ H(τ ) dτ ′ D (τ ) 2M λ = ! 2M K− F ′ (x− ), λ  i λh + − x − x+ − F (x ) , 0 2 dF K = − dx K − 2M λ   x+ =x+ 0 (17) . (18) The next step in our semiclassical calculation is to use the effective action (15) to derive the expression for the quantum contributions of the matter to the stress-energy tensor. The flux of Hawking radiation across spatial infinity is given by < T−− > evaluated on the asymptotical D = ∞ region. For the class of models in Eq. (11) a straightforward calculation, which follows closely that of Ref. [11], leads to < T−− >as = N 1 {F, x− }, 24 (F ′)2 (19) where {F, x− } denotes the Schwarzian derivative of the function F (x− ). This is a Weyl rescaling and dilaton reparametrization invariant result for the Hawking flux. In fact the function F (x− ) is defined entirely in terms of the function K(φ) (see Eq. (18)), which in turn is invariant under both transformations (see Eq. (10). Though the trace anomaly is Weyl rescaling dependent, the Hawking radiation seen by an asymptotic observer is independent of the particular conformal frame chosen. When the horizon φ0 is approached, the Hawking flux reaches the thermal value < T−− >has = Z N λ2 V (φ0 ) exp − 12 16 " φ0 H(τ ) dτ ′ D (τ ) !#2 , (20) which is the result already found in Ref. [11], written in a manifest Weyl rescaling and dilaton reparametrization invariant form. Let us now consider a Weyl-invariant measure defined by Eq. (13). In this case there is no Weyl anomaly, no ambiguity associated with the choice of the conformal frame and therefore there is no need to introduce dilaton-dependent counterterms in the effective action. The Hawking effect is a consequence of the explicit breaking of diffeomorphism invariance. Tµν does not transform as a tensor under general coordinate transformations y µ = y µ (xµ ), but it transforms as follows [15] (x) (y) Tµν = (Tαβ + ∆αβ ) dxα dxβ . dy µ dy ν Applied to the shock-wave solution (16),(17), this transformation law gives the flux of Hawking radiation entirely in terms of the Schwarzian derivative of the conformal map x− → F (x− ), with F given by Eq. (18), i.e. it reproduces our previous result (19). Let us now come to a central question: What is the physical meaning of the equivalence we have found? Does it imply the complete physical equivalence (at least at the semiclassical 6 level) of 2D dilaton gravity models connected by Weyl transformations? The answer to these questions is rather complex. A good starting point for trying to find an answer is to ask ourself the opposite question: What are the features of the models that do change under a Weyl rescaling of the metric? We have already noted that because the scalar curvature of the space-time changes under Weyl transformations, in general the structure of the spacetime singularities is not preserved by such transformations. Also the notion of geodesic motion depends on the choice of the conformal frame. In Fact, the geodesic equation is not invariant under the transformation (2), but acquires terms depending on the derivatives of the function P . As a consequence a space-time that is geodesically complete in the range 0 ≤ D(φ) < ∞ can be mapped by the transformation (2) into a space-time which is not geodesically complete in the same range of variation for the dilaton. The model discussed in Ref. [12] gives a nice example of what happens. In Ref. [12] has been shown that the black hole solutions of the CGHS model can be mapped by a conformal transformation (2) into the black hole solutions of a conformally related model, leaving invariant the values of mass, temperature and Hawking flux. The Weyl rescaled solutions (which describe essentially Rindler space-time) differ from the CGHS ones in two respects. First, whereas the black hole solutions of the CGHS model have a curvature singularity, the Weyl-rescaled ones describe a space-time with no curvature singularities. The structure of the singularities of the CGHS model is however preserved by considering the dilaton on the same footing as the metric, imposing a boundary on the space-time in the strongcoupling region of the theory. Second, whereas the ground state of the CGHS model (the so called linear dilaton vacuum) is a geodesically complete space-time, the ground state of the conformally related model is not geodesically complete in the allowed range of variation for the dilaton. The answer to the previous questions depends on the features of the model we are interested in. After all, 2D dilaton gravity models are just toy models for studying 4D gravity in a simplified context. Differently from the 4D case, where geometrical objects such as the metric or the curvature have a direct physical meaning, in the 2D case these objects acquire a physical significance only through their relation with the 4D problem. There are examples in which 2D space-times with asymptotical anti-de Sitter behavior can be used to model asymptotically flat 4D black holes near extremality [21]. If geometrical features of the 2D model such as the curvature or the geodesic completeness of the space-time are crucial for our problem, we will consider models related by Weyl transformations as non-equivalent. On the other hand, if these geometrical features are irrelevant because we want to treat the gravitational degrees of freedom on the same footing as the matter degrees of freedom or because our problem is focused on physical observables such as masses or temperatures, we can regard the former models as equivalent. In this discussion the form of the function P (φ) in Eq. (2) plays a crucial role. Until now we have assumed, generically, that P is such that the transformation (2) is non-singular and invertible in the range of variation of the field φ. However, one can consider a function P subjected to stronger (or weaker) conditions, restricting (or broadening) in this way the notion of conformal equivalence between 2D dilaton gravity models. For example, one can consider as conformally equivalent models connected by a transformation (2) such that only those functions P , which preserve the structure of the singularities of the model, are allowed. Conversely, one can also allow for functions P leading to transformations that become sin7 gular at some points of the dilaton field space. Again, an example which has been already studied in the literature, can be used to clarify the situation. Let us consider the models investigated in Ref. [22], they are defined by the action S= 1 2π Z d2 x √  2  −g e− n φ R + 4 (∇φ)2 + 4λ2 e−2φ . n   (21) One can easily show that the models with different values of the parameter n are conformally equivalent. In fact, by defining 2 D = e− n φ , ĝµν = D n gµν , (22) the action (21) becomes, modulo a total derivative, S= 1 Z 2 q d x −ĝ (DR[ĝ] + 4λ2 ), 2π (23) which is the model studied in Ref. [12]. This fact explains way the black hole solutions of the models (21) have the same values of mass, temperature and Hawking flux as the solution of the CGHS model: the black hole solutions of the model (21) are related to those of the CGHS model (the model in (21) with n = 1) by a conformal transformation of the metric. However, the existence and the position of curvature singularities depend crucially on the parameter n. In fact, the transformation (22) implies the following on-shell relation R[g] = D n R[ĝ] + 2nMλD n−2 . For 0 ≤ n ≤ 2 the transformation (22) does not change the position of the singularities. The black hole solutions of the models (21) with these values of n have a curvature singularity for D = 0 (the black hole solution of the model (23) have no curvature singularities but we treat the region of strong coupling D = 0 as a singularity). Conversely, for n > 2 the position of the curvature singularity is changed by the transformation (22) from D = 0 to D = ∞. In conclusion, if we use the concept of conformal equivalence in the restricted sense that only those transformations are allowed that preserve the structure of the space-time singularities, then only transformations (22) with 0 ≤ n ≤ 2 are allowed and we have two classes of conformally equivalent models, those with 0 ≤ n ≤ 2 and those with n > 2. Acknowledgments I took benefit from conversations with S. Mignemi and G. Amelino-Camelia. In particular, I am grateful to the latter for having drawn my attention to the papers [15, 18]. 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