We consider the semiclassical dynamics of CGHS black holes with a
Weyl-invariant effective action for conformal matter. The trace anomaly of
Polyakov effective action is converted into the Virasoro anomaly thus leading
to the same flux of Hawking radiation. The covariance of semiclassical
equations can be restored through a non-local redefinition of the
metric-dilaton fields. The resulting theory turns out to be equivalent to the
RST model. This provides a mechanism to solve semiclassical equations of 2D
dilaton gravity coupled to conformal matter for classically soluble models.Comment: LaTeX, needs amssymb, 8 pages. One reference adde

Amekino-Camelia

Amelino-Camelia

Bilal

C.F. Talavera

Callan

Christensen

de Alwis

Hawking

J. Navarro-Salas

Jackiw

Karakhanyan

M. Navarro

Navarro-Salas

Polyakov

Russo

Strominger

Physics Letters B

English

arXiv

We consider the semiclassical dynamics of CGHS black holes with a Weyl-invariant effective action for conformal matter. The trace anomaly of Polyakov effective action is converted into the Virasoro anomaly thus leading to the same flux of Hawking radiation. The covariance of semiclassical equations can be restored through a non-local redefinition of the metric-dilaton fields. The resulting theory turns out to be equivalent to the RST model. This provides a mechanism to solve semiclassical equations of 2D dilaton gravity coupled to conformal matter for classically soluble models

Navarro Salas, José

Navarro Navarro, Miguel

Talavera Usano, César Félix

Repositori d'Objectes Digitals per a l'Ensenyament la Recerca i la Cultura

arXiv:hep-th/9505139v3  29 May 1995FTUV/95-17IFIC/95-17Imperial-TP/94-95/31hep-th/9505139Weyl Invariance and Black Hole Evaporation∗J. Navarro-Salasa†, M. Navarrob‡ and C. F. Talaveraa§a Departamento de F´ısica Teo´rica andIFIC, Centro Mixto Universidad de Valencia-CSIC.Facultad de F´ısica, Universidad de Valencia,Burjassot-46100, Valencia, Spain.b The Blackett Laboratory, Imperial College,London SW7 2BZ, United Kingdom.January 15, 2014AbstractWe consider the semiclassical dynamics of CGHS black holes with a Weyl-invariant effective action for conformal matter. The trace anomaly of Polyakoveffective action is converted into the Virasoro anomaly thus leading to the sameflux of Hawking radiation. The covariance of semiclassical equations can berestored through a non-local redefinition of the metric-dilaton fields. The re-sulting theory turns out to be equivalent to the RST model. This providesa mechanism to solve semiclassical equations of 2D dilaton gravity coupled toconformal matter for classically soluble models.∗ Work partially supported by the Comisio´n Interministerial de Ciencia y Tecnolog´ıa andDGICYT.†jnavarro@evalvx.ific.uv.es‡m.navarro@ic.ac.uk§talavera@evalvx.ific.uv.es1 IntroductionSince the pioneer work of Hawking [1], the formation and subsequent evapora-tion of a black hole has provided an excellent scenario to study the interplaybetween gravity and quantum mechanics. In recent years it has been a revivalof interest in the subject, due to the emergence of simplified (two-dimensional)models sharing basic features with the four-dimensional theory. The modelproposed by Callan, Giddings, Harvey and Strominger [2] (CGHS-model), in-volving gravity coupled to a dilaton and N massless scalar fields fi, i = 1, . . . , N ,describes, at the classical level, the formation of a black hole by incoming con-formal matter. Hawking radiation in the classical background geometry can becomputed from the trace anomaly of the matter fields [3]< T fαα >=N24R , (1)and back-reaction can be incorporated by adding to the classical action thePolyakov effective action [4]SP = − N96pi∫d2x√−gR−1R . (2)The new equations of motion have the quantum stress tensor of matter as thesource for the classical gravity dilaton fields. Although the semiclassical equa-tions have not been solved in closed form, a special modification of the model[5] allows to construct exact solutions and, therefore, to study the evolution ofa quantum black hole analytically.The trace anomaly equation (1) is a direct consequence of the breaking ofWeyl symmetry in the definition of the functional measure for the conformalmatter fields,||δfi||2 =∫d2x√−gδfiδfi ; (3)the above definition respects diffeomorphism invariance of the classical theorybut sacrifices Weyl symmetry. The former symmetry leads to the standardcovariant conservation of the quantum energy-momentum tensor∇µ < T f µν >= 0 , (4)and the effective action capturing the equations (1) and (4) is given by theinduced gravity action (2).Recently, it has been advocated an alternative definition of the measure (3)that preserves Weyl invariance [6, 7]. Diffeomorphism invariance is partiallylost and only area-preserving diffeomorphisms are maintained. In this notewe shall develop further this alternative approach and explore the relationshipbetween the diffeomorphism and Weyl invariant schemes in the context of thesemiclassical theory of 2D gravity.12 The Weyl-invariant effective action and the Vira-soro anomalyThe Weyl-invariant effective action proposed in [6, 7] can be obtained from thePolyakov action replacing gµν by√−ggµν :SW = − N96pi∫d2xR(√−ggµν)(√−g)−1R(√−ggµν) , (5)where  = (√−g)−1∂µ√−ggµν∂ν . To study the relation between the action(5) and (2) in a simple way it is convenient to introduce an auxiliary field Φverifying the equationΦ = R . (6)In terms of gµν and Φ the Polyakov action can be rewritten as (see, for instance,[8])SP = − N96pi∫d2x√−g(−ΦΦ+ 2RΦ) . (7)The equation (6) follows from the action (7) and inserting (6) into (7) we recover(2). The metric-dependent Weyl transformation gµν → √−ggµν induces thefollowing transformation for the scalar curvature and the auxiliary field ΦR → (R+  log√−g)√−g , (8)Φ → Φ+ log√−g . (9)Using (8) and (9) it is easy to find that the Weyl-invariant action SW is givenbySW = − N96pi∫d2x[√−g(−ΦΦ+ 2RΦ)+ ∂µ(√−g(log√−g↔∂µΦ))+√−g(log√−g log√−g + 2R log√−g)]. (10)Therefore, SW and SP differ, up to total derivative terms, by a local actionSW = SP − N96pi∫d2x√−g(log√−g log√−g + 2R log√−g) . (11)The energy-momentum tensor TWµν = − 2pi√−g δSWδgµν coming from (11) can beeasily computed. It admits the following decompositionTWµν = (TPµν −N48gµνR)−N48[∂µ log√−g∂ν log√−g − 12gµν∂α log√−g∂α log√−g]−N24[∇µ∇ν log√−g − 12gµν log√−g], (12)2where TPµν is the energy-momentum associated with the Polyakov effective ac-tionTPµν =N48[∂µΦ∂νΦ− 12gµν∂αΦ∂αΦ]−N24[∇µ∇νΦ− 12gµνΦ]+N48gµνR . (13)Note that TPµµ =N24R is consistent with (1).Due to the breakdown of reparametrization invariance TWµν is not covariantlyconserved, in general. Instead, we have∇µTWµν = −N48[∂ν(R+  log√−g) + (R+  log√−g)∂ν log√−g] , (14)and taking into account thatR+  log√−g = 1√−gR(√−ggµν) , (15)we can rewrite (14) as∇µTWµν = −N481√−g∂νR(√−ggµν) . (16)This formula has been obtained in [7] in a different way. However, in specialgauges the stress tensor TWµν can be conserved. For metrics of the formds2 = −e2ρ (dx+dx− + c (x−)2(dx+)2) , (17)where c is a constant, the r.h.s. of (16) vanishes. In the conformal gauge theconservation equations take the simple form∇+TW++ = 0 = ∇−TW−− , (18)which implies thatTW++ = TW++(x+) , (19)TW−− = TW−−(x−) . (20)Despite of the covariant conservation equations (18) the transformation law ofTW±± is anomalous. Under conformal coordinate transformations x± → y±(x±),TWµν transforms asTWx±x± =(dy±dx±)2TWy±y± −N24{y±, x±}, (21)where{y, x} =∂3y∂x3∂y∂x− 32(∂2y∂x2)2(∂y∂x)2 (22)is the Schwartzian derivative. Note that the expression (21) coincides withthe well-known transformation law of the normal-ordered energy-momentumtensor of a conformal field theory. Therefore we can conclude that the localcounterterm in (11) converts the trace anomaly into the Virasoro anomaly.33 Semiclassical 2D black holes. Field redefinitionsand covarianceThe action of the CGHS model is [2]S =12pi∫d2x√−g[e−2φ(R+ 4(∇φ)2 + 4λ2)− 12N∑i=1(∇fi)2]. (23)The classical solutions describe the formation of a black hole by gravitationalcollapse. In conformal gauge, ds2 = −e2ρdx+dx− (x± = x0 ± x1), and inKruskal coordinates, ρ = φ, the black hole formation from the vacuum by leftmoving incoming matter fi = f+i (x+) is described by the solutione−2φ = e−2ρ = −λ2x+(x− + 1λ2P (x+)) +M(x+)λ, (24)whereM(x+) = λ∫ x+0dx˜+x˜+T f++ , (25)P (x+) =∫ x+0dx˜+T f++ , (26)andT f++ =12N∑i=1(∂+f+i )2 . (27)This solution corresponds to a black hole of mass M = M(x+ → ∞) with anevent horizon located at x− = − 1λ2P (x+ → ∞). In the semiclassical approxi-mation the Hawking radiation at future null infinity can be obtained from thetrace anomaly [2], and the back-reaction is incorporated [2] by adding to theclassical action S the Polyakov term (2).Let us consider an f shock wave travelling in the x−-direction, described bythe stress tensor12∂+f∂+f = a δ(x+ − x+0 ) . (28)For x+ < x+0 the classical solution is the linear dilaton vacuum (LDV). In thisregion the natural coordinate system is the Minkowskian one. We assume, asboundary condition of the gravitational collapse, that the outgoing energy fluxmeasured by the Minkowskian observer σ± vanishes,TWσ−σ− = 0 . (29)After the collapse, x+ > x+0 , the classical solution describes a black hole ofmass M = ax+0 λ with a horizon at x− = − aλ2(= − Pλ2). But then the nat-ural coordinates are the Schwarzschild type coordinates σ˜±. The coordinatetransformationσ˜+ = σ+ , (30)σ˜− = − 1λlog(e−λσ− − aλ), (31)4allows to evaluate the energy flux measured by the Schwarzschild-type observer(σ˜±). It is given byTWσ˜−σ˜− = −N24{σ−, σ˜−}, (32)TWσ˜+σ˜+ = 0 , (33)and henceTWσ˜−σ˜− =λ2N48(1− 1(1 + 1λPeλσ˜−)2), (34)in agreement with the result predicted by the trace anomaly. The reason of thisis that the difference between TWµν and TPµν vanishes at infinity in asymptoti-cally flat coordinates. So both TW and TP yield to the same flux of Hawkingradiation.Let us briefly analyze a more involved model: spherically symmetric gravitycoupled to 2D conformal matter. After appropriate reduction this model isdescribed by the two-dimensional action (see, for instance, [9])S =12pi∫d2x√−g[e−2φ(R + 2(∇φ)2 + 2λ2e2φ)− 12N∑i=1(∇fi)2], (35)where the 4D spherically symmetric metric is related to the 2D metric anddilaton fields by (4)ds2 = ds2 + e−2φ/λ2dΩ2. This model is classically solubleand the solutions are the Vaidya space-times. If we consider the collapse of anull shell of matter, the stress tensor and two-dimensional metric take the formT fvv = mδ(v − v0) ,ds2 = −(1− 2mθ(v − v0)r)dv2 + 2drdv . (36)In conformal coordinates matching the discontinuity across v = v0 we obtainfor the metricds2 = −[θ(v0 − v) + θ(v − v0)(1− 2mr)(1− 4mv0 − u)−1]dvdu , (37)where r is defined implicitly byu− 4m log(v0 − u4m− 1)− v = −2(r + 2m log( r2m− 1)), (38)and for the dilatone−φλ=v − u2θ(v0 − v) + rθ(v − v0) . (39)The coordinates (v, u) are Minkowskian inside the shell (v < v0), whereas theasymptotically flat conformal coordinates (v˜, u˜) are given by the relationsv˜ = v ,u˜ = u− 4m log(v0 − u4m− 1). (40)5Therefore, assuming that TWuu = 0, the stress tensor evaluated by the Schwarzschildobserver is given byTWu˜u˜ = −m(u− v0 + 3m)3(u− v0)4 . (41)As the horizon is approached u → v0 − 4m, TWu˜u˜ builds up to the value (3 ·28m2)−1, that corresponds to the Hawking temperature T = 18pim .Our aim now is to study the semiclassical back-reaction, for the CGHSmodel, defined by the Weyl-invariant effective action (11), in the large N limit.Due to Weyl invariance the ρ, φ classical equations are unmodified,e−2φ(2∂+∂−φ− 4∂+φ∂−φ− λ2e2ρ)= 0 , (42)e−2(φ+ρ)(−4∂+∂−φ+ 4∂+φ∂−φ+ 2∂+∂−ρ+ λ2e2ρ) = 0 . (43)However, the constraint equations are modified according toe−2φ(4∂±ρ∂±φ− 2∂2±φ)+12N∑i=1(∂±fi)2 + TW±± = 0 . (44)The components TW±± of the non-local effective stress tensor are ρ-independentand have to be adjusted by boundary conditions.At this point we should note that, although the Virasoro anomaly of TW±±accounts for the Hawking radiation, it destroys the covariance of the one-loopequations (42-44). Moreover, if we select a particular coordinate system (σ± be-fore the collapse and σ˜± after it) the solution is not continuous at x+ = x+0 ; andcoordinates that match continuously σ± and σ˜± do no adjust to the gauge (17).Then, how can we define, in this scheme, a consistent semiclassical theory?. Togain insights, let us first analyse the question of the vacuum stability.In conformal gauge, the vacuum solution to equations (42-44) ise−2ρ = e−ωe−2φ ,e−2φ = u− h+h− , (45)where ω = ω+(y+) + ω−(y−), u = u+ + u−,u± = −∫ y±eω±∫e−ω±TW±± ,h± = λ∫ y±eω± . (46)According to (21), TWy±y±is given by the Schwartzian derivative with respect tothe Minkowskian coordinates σ±. Hence the general solution to the semiclassi-cal equations in vacuum is found to bee−2ρ = e−2(ρcl−φcl)[e−2φcl +N24(2ρcl − φcl)+N12∫ y+e2(ρcl−φcl)∫e−2(ρcl−φcl)[∂y+(ρcl − φcl)]2+N12∫ y−e2(ρcl−φcl)∫e−2(ρcl−φcl)[∂y−(ρcl − φcl)]2],(47)6where ρcl and φcl represent the classical vacuum solutionsρcl =12logdy+dσ+dy−dσ−,φcl = −λ2(σ+(y+)− σ−(y−)) . (48)In the light of the above expressions we can conclude that, due to the termTW±±(y±), the solutions do no transform covariantly, thus producing some sortof vacuum instability that makes problematic a Weyl-invariant semiclassicaltheory.A way to construct a sensible semiclassical theory is suggested by the non-covariant vacuum solutions themselves (47). The non-local field redefinitionρ→ ρˆ, φ→ φˆ, defined byρ− φ = ρˆ− φˆe−2φ =[e−2φˆ +N24(2ρˆ− φˆ)+N12∫ y+e2(ρˆ−φˆ)∫e−2(ρˆ−φˆ)[∂y+(ρˆ− φˆ)]2+N12∫ y−e2(ρˆ−φˆ)∫e−2(ρˆ−φˆ)[∂y−(ρˆ− φˆ)]2], (49)recovers the classical vacuum solution as a solution of the semiclassical vacuumequations: ρˆ = ρcl, φˆ = φcl, and reestablishes the general covariance as well.After the redefinitions (49), the constraint equations (44) become(e−2φˆ +N48)(4∂±ρˆ∂±φˆ− 2∂2±φˆ)− N12(∂±ρˆ∂±ρˆ− ∂2±ρˆ)+12∂±fi∂±fi + TW±± = 0 , (50)and the equations (42-43) can be rewritten as∂+∂−(ρˆ− φˆ)= 0 , (51)∂+∂−(e−2φˆ +N24ρˆ)= −λ2e2(ρˆ−φˆ) . (52)Remarkably these equations turn out to be equivalent to the equations of motionof the RST model [5]. Note that in Kruskal coordinates ρˆ = φˆ the transforma-tion (49) is local,e−2ρ = e−2ρˆ +N24ρˆ , (53)and coincides with the field redefinitions of [5] that maps (50) into a Liouvilletheory (see also [10]), where the original variable e−2φ is the analogue of the√N12Ω field.This intriguing relation between the RST model and the Weyl-invariantone-loop CGHS model offers a new explanation of the solubility of the former.In the Weyl-invariant scheme the semiclassical equations can be solved in astraightforward way by replacing the classical stress tensor T f±± by Tf±±+TW±± inthe expression of the classical solutions. In other words, the classical solubility ofthe model together with the field redefinitions that ensure the vacuum stabilityallows to solve a related covariant semiclassical model.4 Final commentsWe have exhibit in a simple way the relationship between diffeomorphism andWeyl-invariant effective actions of conformal matter coupled to 2D gravity.They differ in a local term that converts the trace anomaly of the effectivestress tensor into the Virasoro anomaly thus yielding to the same flux of Hawk-ing radiation. In special gauges the semiclassical equations of the Weyl-invarianteffective action are consistent with Bianchi identities although the solutions donot transform covariantly. A consequence of this is the impossibility of defininga covariant vacuum solution. To reestablish the LDV as the semiclassical vac-uum solution of the CGHS model, we are forced to perform specific, non-localfield redefinitions of the metric and dilaton fields. The field redefinitions yieldsto new covariant semiclassical equations, which turn out to be the equations ofthe RST model. The solubility of the classical model and the Weyl invarianceof the semiclassical correction implies exact solubility of the related semiclassi-cal covariant model. This mechanism can be applied to solve the semiclassicaltheory of more realistic models, as 4D-spherically symmetric gravity coupledto 2D conformal matter [11]. Moreover, the insights gained in unravelling therelationship between the Weyl and diffeomorphism invariant schemes could beof great interest for the non-perturbative canonical approach. There are somearguments [7] indicating that the Weyl invariant scheme reflects more appro-priately the exact canonical quantization.AcknowledgementsM. Navarro acknowledges to the MEC for a Postdoctoral fellowship. C. F.Talavera is grateful to the Generalitat Valenciana for a FPI grant.References[1] S. W. Hawking, Commun. Math. Phys. 43 (1975) 199.[2] C. G. Callan, S. B. Giddings, J. A. Harvey and A. Strominger, Phys. Rev.D45 (1992) 1005.[3] S. M. Christensen and S. A. Fulling, Phys. Rev. D15 (1977) 2088.[4] A. M. Polyakov, Phys. Lett. B103 (1981) 207.[5] J. G. Russo, L. Susskind and L. Thorlacius, Phys. Rev. D46 (1993) 3444;Phys. Rev. D47 (1993) 533.8[6] D. R. Karakhanyan, R. P. Manvelyan and R. Mkrtchyan, Phys. Lett. B329(1994) 185.[7] R. Jackiw, Another view on Massless Matter-Gravity Fields in Two Di-mensions, hep-th/9501016.[8] J. Navarro-Salas, M. Navarro and V. Aldaya, Nucl. Phys. B403 (1993) 291.[9] A. Strominger, Les Houches Lectures on Black Holes, hep-th/9501071.[10] A. Bilal and C. Callan, Nucl. Phys. B394 (1993) 73.S. de Alwis, Phys. Lett. B289 (1992) 278.[11] J. Navarro-Salas, M. Navarro and C. F. Talavera, work in progress.9

Weyl Invariance and Black Hole Evaporation

Crossref

Weyl invariance and black hole evaporation

Weyl Invariance and Black Hole Evaporation

Abstract

Similar works

Full text

Available Versions

Repositori d'Objectes Digitals per a l'Ensenyament la Recerca i la Cultura

Crossref