On the Liouville 2D dilaton gravity models with sinh-Gordon matter
OOn the Liouville 2D dilaton gravity models withsinh-Gordon matter
Valeri P. Frolov and Andrei Zelnikov
Theoretical Physics Institute, Department of Physics,University of Alberta, Edmonton, Alberta T6G 2E1, Canada
E-mail: [email protected] , [email protected] Abstract:
We study 1 + 1 dimensional dilaton gravity models which take into accountbackreaction of the sinh-Gordon matter field. We found a wide class of exact solutions whichgeneralizes black hole solutions of the Jackiw-Teitelboim gravity model and its hyperbolicdeformation.
Keywords:
Dilaton gravity, Liouville model, Black Holes a r X i v : . [ h e p - t h ] D ec ontents Q = 1 /b . 44 Vacuum solutions. 65 Non-vacuum solutions. 8 The 1+1 dimensional dilaton gravity models, that on the one hand are exactly solvable and onthe other hand have quite a rich structure, are useful tools to study many different aspects ofclassical and quantum gravity (for comprehensive reviews see [1–3]). These models naturallyappear after dimensional reduction of spherically symmetric gravity and its generalizationsand provide an ideal testing ground for the study of black hole quantum mechanics, thermo-dynamics, and such deep issues in quantum gravity as the endpoint of Hawking radiation andthe nature of black hole entropy. Special 2D dilaton gravity models were extensively usedin the study of classical and quantum properties of Liouville black holes [4], the exact stringand CGHS black holes [5–9]. The Jackiw-Teitelboim model [10, 11] is one of the simplest2D dilaton gravity theories which contains most of the desirable features. Recently it hasbeen used to study backreaction effects in asymptotically AdS spacetimes in the contextof holography [12]. Its deformation has been found [13, 14] using Yang-Baxter deformationtechnique, so that the quadratic potential is replaced by a hyperbolic function of the dilatonfield.In this paper we use a simpler method to derive the hyperbolic deformation [13, 14] ofthe Jackiw-Teitelboim model and further generalize it to include sinh-Gordon type matterfield. The class of the proposed dilaton gravity models is still exactly solvable for a varietyof choices of matter and dilaton distributions.– 1 – The model
We begin by considering a model of a two-dimensional dilaton gravity which interacts with amatter field χ of the sinh-Gordon type and a free massless scalar field f . The action functionalof the system we are interested in reads S = S [ g µν , φ, χ, f ] = S φ + S χ + S f ,S φ = − π (cid:90) d x √− g (cid:104) QφR + (1 − bQ )( ∇ φ ) − λ e − bφ (cid:105) ,S χ = − π (cid:90) d x √− g (cid:104) ( ∇ χ ) + m cosh(2 bχ ) (cid:105) ,S f = − π (cid:90) d x √− g ( ∇ f ) . (2.1)The action S φ is the action of the Liouville dilaton gravity. On a flat background the action S χ describes the matter field χ which obeys the well known sinh-Gordon equation. However,in our case the geometry is curved. The other field f is a free conformal matter useful, e.g.,to study the solutions involving collapsing null shells.We consider the model with positive parameters λ and m . By shifting the dilaton field as φ → φ + const, one can make the coefficient λ in front of the Liouville potential to become anarbitrary (positive) constant. This is possible because in two dimensions (cid:82) R is a topologicalinvariant and this field redefinition does not affect the field equations. Later on we will put λ = m .Using conformal transformations of the metric, the action (2.1) can be rewritten is severaldifferent but equivalent forms. In the literature it is often used the representation when thekinetic term of the dilaton field φ cancels. This choice is given by the following metricredefinition g µν = e (cid:0) b − Q (cid:1) φ ˆ g µν . (2.2)When expressed in terms of the ˆ g µν metric the action functional (2.1) becomesˆ S = − π (cid:90) d x (cid:112) − ˆ g (cid:104) Qφ ˆ R + ( ˆ ∇ χ ) + m e (cid:0) b − Q (cid:1) φ cosh(2 bχ ) − m e − Q φ + 12 ( ˆ ∇ f ) (cid:105) . (2.3)From the computational point of view the following conformal transformation happensto be more convenient g µν = e bφ ˜ g µν , √− g = e bφ (cid:112) − ˜ g, R = e − bφ [ ˜ R − b ˜ (cid:3) φ ] . (2.4)Then the action (2.1) takes the form˜ S = − π (cid:90) d x (cid:112) − ˜ g (cid:104) Qφ ˜ R + ( ˜ ∇ φ ) + ( ˜ ∇ χ ) + m e bφ cosh(2 bχ ) − m + 12 ( ˜ ∇ f ) (cid:105) , (2.5)It is also useful to introduce two other field variables ω = φ + χ, ω = φ − χ. (2.6)– 2 –n terms of these fields the action (2.5) becomes a sum of two decoupled Liouville-type actions and the action of a free scalar field ˜ S = S + S + S f , (2.7)where S k = − π (cid:90) d x (cid:112) − ˜ g (cid:104)(cid:0) Qω k ˜ R + ( ˜ ∇ ω k ) + m e bω k − m (cid:1)(cid:105) , (2.8)and k = (1 , − m . Variation of the action (2.7) over the fields ω k (for k = (1 , f gives the field equations ˜ (cid:3) ω k − Q R − bm e bω k = 0 , ˜ (cid:3) f = 0 . (2.9)Variation of the action (2.7) over the metric leads to the equations˜ T µν = T µν + T µν + T µνf = 0 , (2.10)where ˜ T µν = 2 √− ˜ g δ ˜ Sδ ˜ g µν , T µνk = 2 √− ˜ g δS k δ ˜ g µν . (2.11)Here T µν = − π (cid:104) (cid:0) Qω ; µν − ω ; µ ω ; ν (cid:1) + ˜ g µν (cid:0) − Qω ; αα + ω ; α ω ; α + m ( e bω − (cid:1)(cid:105) (2.12)for ω = ( ω , ω ) correspondingly and T µνf = 14 π (cid:104) f ; µ f ; ν −
12 ˜ g µν f ; α f ; α (cid:105) . (2.13)In the conformal gauge we have d ˜ s = − e ρ dx + dx − , x + = t + z, x − = t − z, (2.14)˜ g ++ = ˜ g −− = 0 , ˜ g + − = ˜ g − + = − e ρ , ˜ g ++ = ˜ g −− = 0 , ˜ g + − = ˜ g − + = − e − ρ , (cid:112) − ˜ g = e ρ . (2.15)Let us denote ω + = ∂ + ω, ω − = ∂ − ω,ω ++ = ∂ + ∂ + ω, ω −− = ∂ − ∂ − ω,ω + − = ∂ − ∂ + ω, ω − + = ∂ + ∂ − ω, (2.16) This representation is similar to that of [14], where the difference of Liouville actions was considered. – 3 –nd introduce similar objects for partial derivatives of ρ , f , and σ . Then˜ R = 4 e − ρ ρ + − , ˜ (cid:3) ρ = − e − ρ ρ + − , (2.17)˜ (cid:3) ω = − e − ρ ω + − , ˜ (cid:3) f = − e − ρ f + − . (2.18)The components of the stress-energy T µν (see (2.12)) take the form T ++ = 14 π (cid:104) − Qω ++ + ω + ω + + 2 Qω + ρ + (cid:105) ,T −− = 14 π (cid:104) − Qω −− + ω − ω − + 2 Qω − ρ − (cid:105) ,T + − = T − + = 18 π (cid:104) Qω + − + m e ρ (cid:0) e bω − (cid:1)(cid:105) . (2.19)Field equations (2.9) become ω k + − + Qρ + − + bm e ρ + bω k ) = 0 ,f + − = 0 . (2.20)Variation of the action (2.7) over the metric leads to (2.10). When written explicitly incomponents these equations take the form2 Q ( ω + − + ω + − ) + m e ρ (cid:0) e bω + e bω − (cid:1) = 0 , (2.21) ω + ω + − Qω ++ + ω + ω + − Qω ++ + 2 Q ( ω + + ω + ) ρ + + f + f + = 0 , (2.22) ω − ω − − Qω −− + ω − ω − − Qω −− + 2 Q ( ω − + ω − ) ρ − + f − f − = 0 . (2.23)From (2.20) and (2.21) we get(1 − bQ )( ω + − + ω + − ) + 2 Qρ + − + mbe ρ = 0 . (2.24)For a specific value of the constant Q = 1 /b the equation for the conformal factor decouplesfrom the matter equations. This case is of a particular interest for us, because the classicalfield equations can be solved exactly. In the next sections we fix Q = 1 /b . Q = 1 /b . In a special case, when Q = 1 /b , (2.24) reduces to the Liouville equation ρ + − + mb e ρ = 0 . (3.1)Taking into account (2.17), one can see that the solution for the metric ˜ g µν describes thespacetime of a constant curvature ˜ R = − mb . (3.2)– 4 –he general solution of the Liouville equation is well known and reads e ρ = 2 mb ∂ + Y + ∂ − Y − ( Y + − Y − ) , (3.3)where Y + = Y + ( x + ) , Y − = Y − ( x − ) (3.4)are arbitrary functions of the advanced and retarded null coordinates.The other field equations are( b ω k + ρ ) + − + mb e b ω k + ρ ) = 0 . (3.5)It is convenient to introduce new fields σ k = b ω k + ρ. (3.6)In terms of these variables (3.5) take the form σ k + − + mb e σ k = 0 . (3.7)Thus, in the conformal gauge the field equations for the metric and two σ -fields reduce tothree identical Liouville equations. The constraint equations (2.22),(2.23) mean that totalfluxes ˜ T ++ = ˜ T −− = 0 and in terms of σ fields read σ ++ − ( σ + ) + σ ++ − ( σ + ) = 2 (cid:2) ρ ++ − ( ρ + ) (cid:3) + b ( f + ) , (3.8) σ −− − ( σ − ) + σ −− − ( σ − ) = 2 (cid:2) ρ −− − ( ρ − ) (cid:3) + b ( f − ) . (3.9)The matter field f satisfies the equation f + − = 0 . (3.10)The general solution of this equation is described by two arbitrary functions V and Uf = V ( x + ) + U ( x − ) . (3.11)The solutions of the (3.7), which have the form of the Liouville equation, are e σ k = 2 mb ∂ + X + k ∂ − X − k ( X + k − X − k ) , k = (1 , , (3.12)where X ± k = X ± k ( x ± ) (3.13)are arbitrary functions of the advanced and retarded null coordinates. For the fields ω k itleads to e bω k = ∂ + X + k ∂ − X − k ( X + k − X − k ) ( Y + − Y − ) ∂ + Y + ∂ − Y − . (3.14)– 5 –sing this solution and (3.3) one can find the original metric (2.4), the dilaton field φ andthe matter fields χ . e bφ = (cid:112) ∂ + X + ∂ − X − ∂ + X + ∂ − X − | X + − X − || X + − X − | ( Y + − Y − ) ∂ + Y + ∂ − Y − , (3.15) e bχ = (cid:115) ∂ + X + ∂ − X − ( X + − X − ) (cid:115) ( X + − X − ) ∂ + X + ∂ − X − . (3.16)The result of substitution of the solutions (3.12) to (3.8)-(3.9) can be written in the form { X + , x + } + { X + , x + } = 2 { Y + , x + } + 2 b ( f + ) , { X − , x − } + { X − , x − } = 2 { Y − , x − } + 2 b ( f − ) . (3.17)Here { A, x } denotes the Schwarzian derivative of the function A ( x ) { A, x } ≡ A (cid:48)(cid:48)(cid:48) A (cid:48) − (cid:16) A (cid:48)(cid:48) A (cid:48) (cid:17) . (3.18)One can always use a coordinate transformation such that Y + = x + , Y − = x − , e ρ = 2 mb x + − x − ) . (3.19)In this gauge the metric becomes d ˜ s = − mb dx + dx − ( x + − x − ) . (3.20)and, evidently, { Y + , x + } = { Y − , x − } = 0 . (3.21)The remaining nontrivial equations (3.17) for the dilaton fields reduce to { X + , x + } + { X + , x + } = 2 b ( f + ) , { X − , x − } + { X − , x − } = 2 b ( f − ) . (3.22)Different choices of the functions X ± k and f ± correspond to different physical setups ofthe problem. Now consider the vacuum solutions of the system (3.22), when the matter field f vanishes.Let the functions X ± k be related according to the rule X ± = X ± , X ± = αX ± + βγ X ± + δ (4.1)– 6 –or some arbitrary coefficients α, β, γ, δ . Substitution of this relation to (3.12),(3.14) gives σ = σ and φ = ω = ω , that corresponds to χ = 0 in the original field variables. Thus thisansatz assumes that the sinh-Gordon matter field χ also vanishes.Thus the choice in question describes the system with the action˜ S = − π (cid:90) d x (cid:112) − ˜ g (cid:104) b φ ˜ R + ( ˜ ∇ φ ) + m (cid:0) e bφ − (cid:1) + 12 ( ˜ ∇ f ) (cid:105) . (4.2)In terms of the metric ˆ g µν (see (2.2)) it readsˆ S = − π (cid:90) d x (cid:112) − ˆ g (cid:104) b φ ˆ R + 2 m sinh( bφ ) + 12 ( ˆ ∇ f ) (cid:105) . (4.3)One can see that the action (4.3) exactly reproduces, up to normalization factors, the hyper-bolic deformation (see [13, 14]) of the Jackiw–Teitelboim gravity model.The vacuum solution assumes that the matter field f = 0 vanishes. Then because of theproperties of the Schwarzian derivative we obtain { X ± , x ± } = { X ± , x ± } = { X ± , x ± } . (4.4)The constraint equations (3.22) reduce to { X ± , x ± } = 0 . (4.5)Their general solutions are X + ( x + ) = α x + + β γ x + + δ , α δ − β γ > , (4.6) X − ( x − ) = α x − + β γ x − + δ , α δ − β γ > . (4.7)In the conformal gauge (3.19) we obtain the vacuum solution for the dilaton e ρ = 2 mb x + − x − ) , (4.8) e bφ = C ( x + − x − ) C x + x − + C x + − C x − + C , (4.9)where C = ( α δ − β γ )( α δ − β γ ) ,C = α γ − α γ , C = α δ − β γ ,C = α δ − β γ , C = β δ − β δ . (4.10)By lifting the solution to a higher dimensional spherical spacetime, the dilaton field gets ameaning of some power of the radial coordinate. In higher dimensions, even for non-staticspacetimes, one can define the apparent horizon using the condition, that a normal to the– 7 –onstant radius surface becomes null on the apparent horizon. This property boils downto the following condition for the analogue of the apparent horizon in the two dimensionaldilaton gravity ( ˜ ∇ φ ) (cid:12)(cid:12) Hor = 0 . (4.11)As an example consider a particular choice of parametrization of the solution (4.6) X + ( x + ) = (1 + 2 ac ) x + + pcc x + + 1 , (4.12) X − ( x − ) = x − . (4.13)where a, c, p are arbitrary constants satisfying the condition 1 + 2 ac > pc . Then the dilatonfield takes the form e bφ = (1 + 2 ac − pc )( x + − x − ) [ x + − x − + c ( p + 2 ax + − x + x − )] . (4.14)The horizon equation (4.11) gives x + (cid:12)(cid:12) Hor = a ± (cid:112) a + p, x − (cid:12)(cid:12) Hor = a ± (cid:112) a + p. (4.15)The obtained solution reproduces the hyperbolic deformation [13, 14] of the Jackiw–Teitelboimgravity model. It can be also generalized to a non-static case [13, 14] that includes the col-lapsing shell of null matter f . Now consider a more general setup of the problem. By choosing a different ansatz for functions F ± X ± = X ± , X ± = F ± ( X ± ) (5.1)and using the chain rule of the Schwarzian derivative, one can write the constraint equations(3.22) in the form { X ± , x ± } + 12 (cid:16) ∂X ± ∂x ± (cid:17) { F ± , X ± } = b ( f ± ) . (5.2)Let us study a few simple examples of deformations of the Jackiw–Teitelboim dilatongravity, that correspond to different choices of the solutions for matter field χ . In this sectionthe matter field f is assumed to vanish.In the conformal gauge (3.19) for any given functions F ± we have (see (3.15),(3.16)) e bφ = | ∂ + X + ∂ − X − | (cid:112) | ( F + ) (cid:48) ( F − ) (cid:48) || X + − X − || F + − F − | ( x + − x − ) , (5.3) e bχ = | F + − F − || X + − X − | (cid:112) | ( F + ) (cid:48) ( F − ) (cid:48) | . (5.4)– 8 –ere F (cid:48) ≡ ∂ X F ( X ) and X ± are the solutions of (5.2). The constraint equations (5.2) definefunctions X ± , and for f = 0 take the form { X, x } + 12 (cid:16) ∂X∂x (cid:17) { F, X } = 0 . (5.5)For any chosen function F ( X ), this equation is a third-order ordinary differential equation.Therefore its general solution X ( x ) is parameterized by three arbitrary constants. In somecases the solution is quite simple. Let us consider a few natural choices of the function F that admit exact solution of (5.5) in terms or elementary functions.It should be noted that if one finds the solution X ( x ) for some function F ( X ), then,because of properties of the Schwarzian derivative, exactly the same function X ( x ) is the so-lution of the problem with F → G , provided the function G is fractional linear transformationof the function F G ( X ) = c F ( X ) + c c F ( X ) + c , { G, X } = { F, X } . (5.6)The dilaton φ and matter field χ depend on choice of functions F ± . Thus, starting from anygiven solution and using this property we can generate a whole class of physically differentsolutions for the dilaton and matter fields.Among all possibilities we single out three simplest types A when { F, X } = α (5.7) B when { F, X } = αX , (5.8) C when { F, X } = αX , (5.9)where α is an arbitrary constant. In this case a particular solution of the problem { F, X } = α reads F = (cid:40) tan( aX ) , for α = +2 a , tanh( aX ) , for α = − a . (5.10)Using the property (5.6) we can derive a most general form of the function F in the case A . F = c tan( aX ) + c c tan( aX ) + c for α = +2 a ,c exp(2 aX ) + c c exp(2 aX ) + c for α = − a . (5.11)One can see that, e.g., the functions tanh( aX ), coth( aX ), and exp( ± aX ) have the sameconstant Schwarzian derivative. – 9 –he general solution of (5.5), after substitution there { F, X } = α , is characterized bythree arbitrary constants q , q , q X = (cid:40) q + √ a arctan[ q ( x + q )] , α = +2 a ,q + √ a arctanh[ q ( x + q )] , α = − a . (5.12)In order to obtain F ± as explicit functions of the coordinates x ± , one has to substitute thesolution (5.12) into the function in question (5.16). A particular solution in the case B is F = (cid:40) X n , for α = − n , ln( X ) , for α = . (5.13)Using the fractional linear transformation (5.6) of (5.13) one can generate the other functionswhich fulfill the condition (5.8).The general solution of the constraint equations (5.5) X = exp (cid:2) q + 2 √ √ n + 1 arctanh[ q ( x + q )] (cid:3) . (5.14)Note that F ( X ) ∼ ln X corresponds to n = 0 case. It it easy to find a function, satisfying the condition (5.9) F = (cid:40) tan aX , for α = +2 a , tanh aX , for α = − a . (5.15)The general form of the function F in the case C reads F = c tan aX + c c tan aX + c for α = +2 a ,c tanh aX + c c tanh aX + c for α = − a . (5.16)Substituting the ansatz { F, X } = α/X to (5.5) we obtain the solution X = (cid:40) q + √ a arctan[ q ( x + q )] , α = +2 a ,q + √ a arctanh[ q ( x + q )] , α = − a . (5.17)– 10 – .4 Some other cases Note that the solutions (5.17) coincide with (5.12). It’s not surprising because { X − , x } = { X, x } and, hence, substitution X → X − transforms constraint equation (5.5) of the case C to that of the case A . In fact, any fractional linear transformation of the function XX ( x ) → (cid:15)X ( x ) + ζγ X ( x ) + δ (5.18)with arbitrary coefficients (cid:15), ζ, γ, δ does not alter the Schwarzian derivative. Therefore thesolution (5.12) is also a solution of the problem { X, x } + 12 (cid:16) ∂X∂x (cid:17) α ( δ(cid:15) − γζ ) ( γX ( x ) + δ ) = 0 , α = ± a . (5.19)It corresponds to F = tan a ( δ(cid:15) − γζ ) γ ( γX + δ ) , for α = +2 a , tanh a ( δ(cid:15) − γζ ) γ ( γX + δ ) , for α = − a (5.20)with { F, X } = α ( δ(cid:15) − γζ ) ( γX ( x ) + δ ) . (5.21)and also to all fractional linear transformations of this function F . We found out the explicit expressions (5.12),(5.14),(5.17) in terms of elementary functions forthe solutions of the constraint equations (5.5) in cases A,B, and C. The same solutions arevalid for the choice of any other function that is the linear fractional (M¨obius) transformationof the functions we have considered. Then one has to substitute these solutions X ± ( x ± ) to F ± ( X ± ) and,using (5.3)-(5.4), derive the value of the dilaton φ and the matter field χ .The metric ˜ g µν (see (2.14)) describes pure AdS spacetime with the constant curvature˜ R = − mb . The metrics g µν and ˆ g µν (see (2.4)-(2.2)) describe conformal deformations ofthe AdS spacetime.The obtained exact classical solutions typically have both horizons and singularities.There are also other solutions we did not present in this paper. Their properties should beanalyzed for every particular choice of the matter fields χ and f . Similar to the cases ofJackiw-Teitelboim gravity [12] or its hyperbolic deformation [13, 14], one can use the derivedsolutions to describe collapsing matter problem, holography, thermodynamics, and Hawkingradiation effects. We will return to these interesting questions in future publications. Acknowledgments
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