Revisit on two-dimensional self-gravitating kinks: superpotential formalism and linear stability
PPrepared for submission to JHEP
Revisit on two-dimensional self-gravitating kinks:superpotential formalism and linear stability
Yuan Zhong School of Physics, Xi’an Jiaotong University, Xi’an 710049, People’s Republic of China
E-mail: [email protected]
Abstract:
Self-gravitating kink solutions of a two-dimensional dilaton gravity are revis-ited in this work. Analytical kink solutions are derived from a concise superpotential for-malism of the dynamical equations. A general analysis on the linear stability is conductedfor an arbitrary static solution of the model. After gauge fixing, a Schr¨odinger-like equa-tion with factorizable Hamiltonian operator is obtained, which ensures the linear stabilityof the solution.
Keywords:
2D gravity; soliton, domain walls and instantons
ArXiv ePrint: Corresponding author. a r X i v : . [ h e p - t h ] J a n ontents During the past decades, two-dimensional (2D) gravitational models continue attractingthe attention of theorists for a variety of reasons. First of all, the field equations obtainedin many 2D gravity models are simple enough to allow a rigorous analysis of some diffi-cult issues of gravitational theory, such as the quantization of gravity [1, 2], gravitationalcollapse [3, 4], black hole evaporation [5–9], see [10–12] for comprehensive reviews on earlyworks. Second, a number of very different approaches of quantum gravity all hint thatat very short distances space-time becomes effectively two dimensional [13–18]. Here, thedimensions that are reduced can be effective, spectral, topological or the usual dimen-sions [19]. Recently, the studies of the Sachdev-Ye-Kitaev (SYK) model [20, 21] also leadto a resurgence of interest in 2D gravity [22–25], see [26–28] for pedagogical introductions.Since the Einstein tensor vanishes identically in two dimensions, the Einstein-Hilbertaction cannot be used to describe 2D gravity. An economical solution to this problem isto introduce a dilaton field. Many different 2D dilaton gravity models have been proposedand studied so far. The simplest action for 2D dilaton gravity is the Jackiw-Teitelboim(JT) action [29, 30] S JT = 1 κ (cid:90) d x √− gϕ ( R + Λ) , (1.1)where the dilaton ϕ plays the role of a Lagrangian multiplier. κ and Λ are the gravitationalcoupling and the cosmological constant, respectively. Two other famous actions for 2Ddilaton gravity are the Mann-Morsink-Sikkema-Steele (MMSS) action, which generalizethe JT action by giving the dilaton a kinetic term [31] S MMSS = 1 κ (cid:90) d x √− g (cid:20) −
12 ( ∇ ϕ ) + ϕR + Λ (cid:21) , (1.2)– 1 –nd the Callan-Giddings-Harvey-Strominger (CGHS) action [5]: S CGHS = 12 π (cid:90) d x √− g (cid:26) e − ϕ (cid:2) R + 4( ∇ ϕ ) + 4Λ (cid:3) −
12 ( ∇ φ ) (cid:27) , (1.3)where φ is a massless scalar matter field. A comprehensive review of 2D dilaton gravitymodels and their applications in black hole physics and quantum gravity can be found inRef. [12].It is a natural idea to extend the discussion on 2D dilaton gravity to other classicalsolutions such as topological solitons, which could be produced by cosmic phase transi-tions [32]. As the simplest topological soliton solution, kink (or domain wall) has beenextensively studied in 4D cosmology [33] and 5D thick brane world models [34, 35]. In thecase of two dimensions, previous works have revealed close connections between kinks and2D black holes [36–39], or naked singularities [3, 4, 40–42].In 1995, an exact 2D self-gravitating sine-Gordon kink solution without curvaturesingularity was found by St¨otzel, in the MMSS gravity model [43]. In addition to the kinkconfiguration of the scalar field, the metric solution [43] describes a 2D asymptotic antide-Sitter (AdS ) geometry. This property remind us the thick brane solution found inasymptotic AdS geometry [44–46]. The aim of the present work is to reveal the intimaterelation between 2D self-gravitating kinks and 5D thick brane worlds.The organization of the paper is as follows. In Sec. 2, we give a brief review of St¨otzel’smodel, and show that for static solutions, the field equations can be written as a groupof first-order differential equations by introducing the so called superpotential. With thesuperpotential formalism, one can easily generate exact self-gravitating kink solutions bychosen proper superpotentials. We will discuss two analytical solutions in Sec. 3. Then, inSec. 4 we give a complete analysis to the linear stability of the solutions. To our knowledge,no such analysis was done before . Finally, we offer in Sec. 5 some concluding remarks. The action of St¨otzel’s model [43] contains an MMSS gravity part along with a canonicalreal scalar φ : S = 1 κ (cid:90) d x √− g (cid:20) − ∂ µ ϕ∂ µ ϕ + ϕR + Λ + κ L m (cid:21) , (2.1)where L m = − ∂ µ φ∂ µ φ − V ( φ ) (2.2)is the Lagrangian density of the scalar field.After variation, one immediately obtains the Einstein equations ∇ µ ϕ ∇ ν ϕ + 2 ∇ µ ∇ ν ϕ − g µν (cid:16) ∇ λ ϕ ∇ λ ϕ + 4 ∇ λ ∇ λ ϕ − (cid:17) = − κT µν , (2.3) In a recent work [47], the authors considered the linear perturbations around self-gravitating kinksolutions in 2D MMSS gravity. However, they expand the metric around the Minkowski metric rather thanthe AdS metric solution. – 2 –he dilaton equation ∇ λ ∇ λ ϕ + R = 0 , (2.4)and the scalar field equation ∇ µ ∇ µ φ − dVdφ = 0 . (2.5)The energy-momentum tensor in Eq. (2.3) is defined as T µν = g µν L m − δ L m δg µν = ∂ µ φ∂ ν φ − g µν ( ∂ α φ∂ α φ + 2 V ) . (2.6)To obtain self-gravitating kink solution, St¨otzel used the following metric ds = − e A ( x ) dt + dx . (2.7)Similar metric ansatz is also used in 5D brane world models with non-factorizable geom-etry [48, 49], therefore, we will follow the terminology of brane world theory and call thefunction A ( x ) as the warp factor. As a convention, the derivative with respect to x willalways be denoted as a subscript, for example, φ x ≡ dφ/dx. Substituting metric (2.7) into the Einstein equations (2.3), one obtains2 A x ϕ x − ϕ xx − ϕ x = κφ x , (2.8) A x ϕ x + ϕ xx = Λ − κV. (2.9)The equations of motion for the dilaton and the scalar fields read − A xx − A x + ϕ xx + A x ϕ x = 0 . (2.10)and A x φ x + φ xx = dVdφ , (2.11)respectively. Note that only three of the above equations are independent. For example,Eq. (2.11) can be derived by using Eqs. (2.8)-(2.10). At a first glance, Eqs. (2.8)-(2.11)constitute a complicate nonlinear differential system, and finding their solutions seems tobe a formidable task. But the study of brane world models has taught us a lesson onhow to solve such system by means of superpotential method, which rewrites second-orderdifferential equations, such as Eqs. (2.8)-(2.11), into some first-order ones [44–46].To construct a superpotential formalism for the present model, we first note that thecombination of Eqs. (2.9) and (2.10) leads to an expression of V in terms of cosmologicalconstant and warp factor: κV = Λ − A xx − A x . (2.12)– 3 –aking the derivative of the above equation and eliminating dV /dφ by using Eq. (2.11),one obtains a relation between A and φ : A xxx + 2 A x A xx = − κ ( A x φ x + φ xx φ x ) . (2.13)The superpotential method starts with an assumption that the first-order derivative of φ equals to a function of φ itself, namely, the superpotential W ( φ ) via the following equation: φ x = dWdφ . (2.14)Under this assumption, one can show that Eq. (2.13) supports a very simple specialsolution: A x = − κW. (2.15)Then, Eq. (2.12) enables us to write V in terms of superpotential: V = 12 (cid:18) dWdφ (cid:19) − κW + Λ κ . (2.16)Finally, the general solution of Eq. (2.10) gives a simple relation between dilaton and warpfactor: ϕ = 2 A + β (cid:90) e − A dx + ϕ , where β and ϕ are just two integral constants. Since the field equations only contain thederivatives of the dilaton, the value of ϕ is unimportant to the solution of other variables,and can be taken as ϕ = 0. Besides, to consist with Eq. (2.8), β must be set as zero, so ϕ = 2 A. (2.17)Eqs. (2.14)-(2.17) constitute the first-order superpotential formalism of the presentmodel. Exact kink solutions can be derived by choosing proper superpotentials. Thefreedom of choosing a superpotential comes from the fact that there are four unknownvariables ( A, φ, ϕ and V ) but only three independent equations. Taking a superpotentialamounts to specifying one of the four unknown variables. In this section, we show how to use the superpotential formalism to derive exact self-gravitating kink solutions. We first reproduce St¨otzel’s solution and then report a newsolution. – 4 – .1 Reproducing St¨otzel’s solution
In fact, the superpotential formalism presented in last section has been derived and used,although unconsciously, by St¨otzel [43]. Instead of choosing a superpotential W ( φ ), St¨otzelstarted with the Sine-Gordon potential V ( φ ) = 2 m sin φ . (3.1)He observed that when κ = λ m − λ , Eq. (2.16) surports two solutions of the superpotential: W ± = ± (cid:112) m − λ cos (cid:18) φ (cid:19) , (3.2)where 0 < λ ≡ κ < m . The solutions of φ ( x ) corresponds to W − could be obtained byintegrating Eqs. (2.14), and the result turns out to be the sine-Gordon kink [43]: φ K ( x ) = 4 arctan (cid:16) e M ( x − x ) (cid:17) . (3.3)Here x is an integral constant that represents the position of the kink, and will be set tozero from now on. The constant M is defined as M ≡ √ m − λ . Obviously, M ∈ (0 , m ).The solution corresponds to W + is an antikink φ ¯ K ( x ) = 4 arctan (cid:0) e − Mx (cid:1) , (3.4)which is similar as the kink in many aspects. Thus, we will focus on the kink solution only,and eliminate the subscript K from now on.Plugging the solutions of W ( φ ) and φ ( x ) into Eq. (2.15), one immediately obtains theexpression of the warp factor: A ( x ) = A − λ M ln(2 cosh( M x )) , (3.5)which further reduces to [43] A ( x ) = − λ M ln cosh( M x )= − κ ln cosh( M x ) (3.6)after taking integral constant to A = λ M ln 2. Obviously, this warp factor describes anasymptotic AdS geometry. Finally, the dilaton field reads ϕ ( x ) = 2 A ( x ) = − κ ln cosh( M x ) . (3.7)The profiles of φ , A and ϕ are plotted in Fig. 1.– 5 – s c a l a r f i e l d (a) kinkantikink 8 6 4 2 0 2 4 6 8x15.012.510.07.55.02.50.0 w a r p f a c t o r a n d d il a t o n (b) A(x)( x ) Figure 1 . The shapes of some important variables in St¨otzel’s solution, incluting (a) scalar field,(b) warp factor and the dilaton field. The parameters are taken as κ = 1, m = √ λ = 4,therefore M = 1 and Λ = 2. It seems that St¨otzel did not realize the power of the superpotential formalism, as headmitted that he can only find a special soliton solution by trail and error [43]. However,as shown repeatedly in the study of 5D thick brane models, it is quite easy to constructexact self-gravitating kink solutions once the superpotential formalism is established. Inthe following discussions, we will take Λ = 0 for simplicity, as it can be absorbed into thedefinition of V ( φ ).Consider a simple polynomial potential with parameter c [50–52] W = c + φ (cid:18) − φ (cid:19) . (3.8)It has two minima at φ ± = ±
1, where W ( φ ± ) = ± + c . With this superpotential, oneobtains [52] φ ( x ) = tanh( x ) , (3.9) ϕ ( x ) = 2 A ( x ) , (3.10) A ( x ) = 124 κ (cid:2) − cx + sech ( x ) − x )) − (cid:3) , (3.11) V ( φ ) = − κ (cid:0) − c + φ − φ (cid:1) + 12 (cid:0) φ − (cid:1) . (3.12)The asymptotic behaviors of the warp factor and the scalar potential are A ± ( x ) = − κW ( φ ± ) x = − κ ( 23 ± c ) | x | , (3.13) V ± = −
172 (3 c ± κ. (3.14)Depending on the value of c , there are four different situations [52]:1. c = 0: In this case, the kink connects two equivalent AdS spaces symmetrically, and V + = V − = − κ . – 6 –. 0 < | c | < : The kink connects two distinct AdS spaces.3. | c | = : The kink connects an AdS space and a 2D Minkowski space (M ) asym-metrically. This situation is of particular interesting when considering kink collisionin asymptotical AdS space-time [51, 53].4. | c | > : The warp factor diverges at one side of the kink.The behavior of e A for different values of c has been plotted in Fig. 2. e A c = 0 c = 1/3 c = 2/3 c = 1 Figure 2 . Plots of warp factor e A ( x ) of the polynomial model with κ = 1. In this section, we discuss the linear stability of the self-gravitating kink solutions. Thisissue has been studied extensively in 5D brane world models [45, 54–57], but remainsuntouched in the case of 2D. The reducing of dimensions and the introducing of dilatonfield make it impossible to analyze linear stability of 2D self-gravitating kinks by simplycopying the stability analysis of 5D thick branes. For example, there are no vector andtensor perturbation in 2D, so the traditional scalar-vector-tensor decomposition [55, 57]is no longer needed. Beside, in 2D there is no way to eliminate the non-minimal gravity-dilaton coupling by using conformal transformation.It is convenient to discuss the linear stability in the conformal flat coordinates ds = e A ( r ) η µν dx µ dx ν , (4.1)where r is defined through dr ≡ e − A ( x ) dx . For simplicity, we use a prime and an overdotto represent the derivatives with respect to r and t , respectively.In this coordinates, the Einstein equations take the following form: κφ (cid:48) = 4 A (cid:48) ϕ (cid:48) − ϕ (cid:48)(cid:48) − ϕ (cid:48) , (4.2) ϕ (cid:48)(cid:48) = e A (Λ − κV ) . (4.3)The equation of motion for the scalar and dilaton fields are φ (cid:48)(cid:48) = e A dVdφ , (4.4)– 7 –nd ϕ (cid:48)(cid:48) = 2 A (cid:48)(cid:48) , (4.5)respectively. Obviously, the general solution of Eq. (4.5) is ϕ = 2 A + βr + ϕ , but as statedbefore, we will take β = 0 = ϕ .Equation (2.13) becomes 2 A (cid:48)(cid:48)(cid:48) − A (cid:48) A (cid:48)(cid:48) + κφ (cid:48) φ (cid:48)(cid:48) = 0 , (4.6)which, after integration, gives A (cid:48)(cid:48) − A (cid:48) + 14 κφ (cid:48) = 0 , (4.7)where the integral constant has been taken as zero.Now, let us consider small field perturbations around an arbitrary static backgroundsolution: ϕ ( r ) + δϕ ( r, t ) , φ ( r ) + δφ ( r, t ) , g µν ( r ) + δg µν ( r, t ) . (4.8)We also define δg µν ( r, t ) ≡ e A ( r ) h µν ( r, t ) , (4.9)for convenience.In the linear perturbation analysis of cosmological or brane world models, one usu-ally decompose h µν into scalar, vector and tensor sectors [58, 59]. Each sector can bediscussed independently. In the present case, we have only one spatial dimension and nosuch decomposition is needed. So we will directly deal with the components of the metricperturbation h µν = (cid:32) h ( r, t ) Φ( r, t )Φ( r, t ) h rr ( r, t ) (cid:33) , (4.10)where we have renamed h = h as Φ, and h as h rr .To the first order, the perturbation of the metric inverse is given by δg µν = − e − A h µν . (4.11)Note that the indices of h are always raised or lowered with η µν , thus, h µν ≡ η µρ η νσ h ρσ = (cid:32) h − Φ − Φ h rr (cid:33) . (4.12)After linearization, the Einstein equations (2.3) lead to three nontrivial perturbationequations, namely, the (0 ,
0) component:2 A (cid:48) δϕ (cid:48) − A (cid:48) ϕ (cid:48) h rr − δϕ (cid:48)(cid:48) − δϕ (cid:48) ϕ (cid:48) + h (cid:48) rr ϕ (cid:48) + 2 h rr ϕ (cid:48)(cid:48) + 12 h rr ϕ (cid:48) = κ (cid:18) φ (cid:48) δφ (cid:48) + φ (cid:48)(cid:48) δφ − φ (cid:48) h rr (cid:19) , (4.13)– 8 –he (0 ,
1) or (1 ,
0) components:2 A (cid:48) δϕ − δϕ (cid:48) − ϕ (cid:48) δϕ + ϕ (cid:48) h rr = κφ (cid:48) δφ, (4.14)and the (1 ,
1) component:2 A (cid:48) δϕ (cid:48) − A (cid:48) ϕ (cid:48) h rr − δϕ (cid:48) ϕ (cid:48) − δϕ + 12 h rr ϕ (cid:48) + Ξ ϕ (cid:48) = κ (cid:18) φ (cid:48) δφ (cid:48) − φ (cid:48)(cid:48) δφ − φ (cid:48) h rr (cid:19) . (4.15)Here we have defined a new variable Ξ ≡ − h (cid:48) . One can testify that after usingbackground equations (4.2)-(4.5), Eq. (4.13) reduces to Eq. (4.14).Another independent equation comes from the perturbation of the scalar equation ofmotion: − ¨ δφ + δφ (cid:48)(cid:48) + 2 A (cid:48) φ (cid:48)(cid:48) φ (cid:48) δφ − φ (cid:48)(cid:48)(cid:48) φ (cid:48) δφ − φ (cid:48) h (cid:48) rr − φ (cid:48)(cid:48) h rr + 12 φ (cid:48) Ξ = 0 . (4.16)One can also linearize the dilaton equation (2.4), but it does not offer new informationfurther.Therefore, we have three independent perturbation equations, i.e., (4.14)-(4.16). Butone should note that the perturbation variables are not all independent. The invariance ofthe dynamical equations under coordinate transformations x µ → ˜ x µ = x µ + ξ µ ( r, t ) (4.17)induces an invariance of the linear perturbation equations (4.14)-(4.16) under the followinggauge transformations: ∆ h µν ≡ (cid:101) h µν − h µν = − ξ ( µ,ν ) − η µ,ν A (cid:48) ξ , (4.18)∆ δφ ≡ (cid:102) δφ − δφ = − φ (cid:48) ξ , (4.19)∆ δϕ ≡ (cid:102) δϕ − δϕ = − ϕ (cid:48) ξ . (4.20)The components of h µν transform as∆ h = 2 ∂ t ξ + 2 A (cid:48) ξ , (4.21)∆Φ = − ∂ t ξ + ∂ r ξ , (4.22)∆ h rr = − ∂ r ξ − A (cid:48) ξ , (4.23)which means that the variable Ξ = 2 ˙Φ − h (cid:48) should transforms as∆Ξ = − (cid:104) ¨ ξ + (cid:0) A (cid:48) ξ (cid:1) (cid:48) (cid:105) . (4.24)We see that the gauge degree of freedom ξ has been canceled.The residual gauge degree of freedom in ξ allows us to eliminate one of the perturba-tion variables. Here we simply take δϕ = 0, with which Eq. (4.14) reduces to ϕ (cid:48) h rr = κφ (cid:48) δφ, (4.25)– 9 –nd Eq. (4.15) becomes − A (cid:48) ϕ (cid:48) h rr + 12 h rr ϕ (cid:48) + Ξ ϕ (cid:48) = κ (cid:18) φ (cid:48) δφ (cid:48) − φ (cid:48)(cid:48) δφ − φ (cid:48) h rr (cid:19) . (4.26)After eliminating h rr and Ξ, equation (4.16) can be written as a wave equation of δφ :¨ δφ − δφ (cid:48)(cid:48) + V eff ( r ) δφ = 0 , (4.27)where the effective potential reads V eff ( r ) = 4 A (cid:48)(cid:48) − A (cid:48) φ (cid:48)(cid:48) φ (cid:48) − ϕ (cid:48)(cid:48) + 2 (cid:18) ϕ (cid:48)(cid:48) ϕ (cid:48) (cid:19) − ϕ (cid:48)(cid:48)(cid:48) ϕ (cid:48) + φ (cid:48)(cid:48)(cid:48) φ (cid:48) . (4.28)Using Eqs. (4.5)-(4.7), one can obtain an useful identity: ϕ (cid:48)(cid:48) = ϕ (cid:48)(cid:48)(cid:48) ϕ (cid:48) + φ (cid:48)(cid:48) φ (cid:48) ϕ (cid:48) − φ (cid:48)(cid:48) φ (cid:48) ϕ (cid:48)(cid:48) ϕ (cid:48) , (4.29)which enable us to rewrite the effective potential as V eff = φ (cid:48)(cid:48)(cid:48) φ (cid:48) − φ (cid:48)(cid:48) φ (cid:48) ϕ (cid:48)(cid:48) ϕ (cid:48) + 2 (cid:18) ϕ (cid:48)(cid:48) ϕ (cid:48) (cid:19) − ϕ (cid:48)(cid:48)(cid:48) ϕ (cid:48) , (4.30)or, in a more compact form V eff = f (cid:48)(cid:48) f , with f ≡ φ (cid:48) ϕ (cid:48) . (4.31)If we take δφ = ψ ( r ) e iwt , Eq. (4.27) becomes a Schr¨odinger-like equation of ψ ( r ): − ψ (cid:48)(cid:48) + V eff ψ = w ψ. (4.32)It is interesting to note that the Hamiltonian operator are factorizable:ˆ H = − d dr + V eff = ˆ A ˆ A † , (4.33)with A = ddr + f (cid:48) f , A † = − ddr + f (cid:48) f . (4.34)According to the theory of supersymmetric quantum mechanics [60], the eigenvalues of afactorizable Hamiltonian operator are semipositive definite, namely, w ≥
0. Therefore,static kink solutions are stable against linear perturbations. The zero mode ( w = 0)satisfies A † ψ ( r ) = 0, and the solution reads ψ ( r ) ∝ f = φ (cid:48) ϕ (cid:48) = φ (cid:48) A (cid:48) . (4.35)– 10 –bviously, for any solution with a non-monotonic warp factor, ψ ( r ) diverges at the extremaof A , and would be unnormalizable. Since it is not always possible to obtain the explicitexpression of x ( r ), it is useful to transform V eff back to the x -coordinates: V eff ( x ) = e A (cid:18) A x f x f + f xx f (cid:19) , (4.36)with f ( x ) = φ x /ϕ x .It should be note that the discussions presented so far are rather general and does notdepend on the specific form of the solution, but only on the general form of the metric(4.1) and of the action (2.1).Now, we move on to the specific solutions. For St¨otzel’s sine-Gordon model and thepolynomial model, the effective potentials read V eff ( x ) = M cosh − κ ( M x ) (cid:2) κ + 2csch ( M x ) + 1 (cid:3) , (4.37)and V eff ( x ) = exp (cid:2) (cid:0) − cx + sech ( x ) − (cid:1)(cid:3) (cid:112) cosh( x ) (cid:2) c + tanh( x ) (cid:0) sech ( x ) + 2 (cid:1)(cid:3) (cid:8) − sech ( x ) [296+ 702 c + (cid:0) c − (cid:1) sech ( x ) + 118sech ( x ) + sech ( x ) + sech ( x ) (cid:3) + 18 c tanh( x ) (cid:2) c + 23sech ( x ) − ( x ) + 36 (cid:3) + 540 c + 208 (cid:9) , (4.38)respectively. For the later case, we have taken κ = 1, for simplicity.The profiles of the V eff ( x ) are depicted in Fig. 3. For St¨otzel’s model, we take m = √ κ such that M ≡ (cid:113) m − κ = 1, while keep κ as a free parameter. We see that V eff is positive and divergent at x = 0 for κ = 0 .
2, 1 and 3.For the polynomial model, we take c = 0, 1/3, 2/3 and 1 as examples. We see that V eff ( x ) diverges at x = 0 for both c = 0 and 1/3, while blows up at x → −∞ if c = 1, butbecomes finite when c = 2 / V e ff ( x ) Stotzel's model = 0.2= 1= 3 4 3 2 1 0 1 2 3 4x505101520 V e ff ( x ) Polynomial model c = 0 c = 1/3 c = 2/3 c = 1 Figure 3 . Plots of V eff ( x ). For polynomial model with c = 2 / V eff ( x ) becomes finite, andapproaches to 4 √ e − ≈ .
637 as x → −∞ . It is worth to mention that in many 5D thick brane models the effective potentials ofthe scalar perturbation also have singularities, and the corresponding scalar zero modes are– 11 –sually unnormalizable. Without normalizable scalar zero modes, these models are free ofthe problem of long range scalar fifth force [54, 55, 57]. For the 2D self-gravitating kinksolutions considered in this paper, however, we find an unusual situation where the zeromode might be normalizable, namely, the polynomial model with c > /
3. In this case,the zero mode reads ψ ( x ) = N φ x A x = −N ( x )3 c + tanh( x ) (cid:0) sech ( x ) + 2 (cid:1) , (4.39)where N is the normalization constant, and we have taken κ = 1. The normalization ofzero mode requires 1 = (cid:90) + ∞−∞ drψ ( r ) = N (cid:90) + ∞−∞ dxe − A (cid:18) φ x A x (cid:19) . (4.40)The integration can be done numerically, for instance, taking c = 1, 1.2 and 1.5 we obtain |N | ≈ ψ ( x ) is depicted in Fig. 4. ( x ) c = 1 c = 1.2 c = 1.5 Figure 4 . Plots of ψ ( x ) for the polynomial model with κ = 1, c = 1, 1.2 and 1.5. In this work, we revisited smooth self-gravitating kink solutions of a type of 2D dilatongravity proposed by Mann et al. [31]. We first showed that exact kink solutions canbe constructed with the aid of a first-order superpotential formalism (2.14)-(2.17) of thedynamical equations. This formalism has already been derived and used by St¨otzel in1995, for 2D self-gravitating sine-Gordon model [43], but its virtue was not completelyappreciated until the advent of 5D thick brane world models. After reproducing St¨otzel’ssolution [43], we reported another kink solution generated by a polynomial superpotentialused in some 5D brane world models [50–52].The main contribution of the present work, however, is a general analysis on thestability of static kink solutions under small linear perturbations. After eliminating theredundant gauge degrees of freedom, we derived a Schr¨odinger-like equation for the physicalperturbation. We found that the Hamiltonian operator can be factorized as ˆ H = ˆ A ˆ A † ,which implies the stability of the solutions. Besides, the zero mode takes the form ψ ( r ) ∝ – 12 – ≡ φ (cid:48) ϕ (cid:48) = φ (cid:48) A (cid:48) , which diverges at the extrema of A . For St¨otzel’s model, the zero modeis not normalizable, because the symmetric solution of the warp factor corresponds toa singularity of ψ ( r ) at r = 0. For the polynomial model, however, the zero mode isnormalizable provides c > / Acknowledgements
This work was supported by the National Natural Science Foundation of China (GrantNos. 11847211, 11605127), Fundamental Research Funds for the Central Universities (GrantNo. xzy012019052), and China Postdoctoral Science Foundation (Grant No. 2016M592770).
References [1] M. Henneaux,
Quantum Gravity in Two-Dimensions: Exact Solution of The Jackiw Model , Phys. Rev. Lett. (1985) 959.[2] S. de Alwis, Quantization of a theory of 2-d dilaton gravity , Phys. Lett. B (1992) 278[ hep-th/9205069 ].[3] C. Vaz and L. Witten,
Formation and evaporation of a naked singularity in 2-d gravity , Phys. Lett. B (1994) 27 [ hep-th/9311133 ].[4] C. Vaz and L. Witten,
Do naked singularities form? , Class. Quant. Grav. (1996) L59[ gr-qc/9511018 ].[5] C. G. Callan, Jr., S. B. Giddings, J. A. Harvey and A. Strominger, Evanescent black holes , Phys. Rev. D (1992) R1005 [ hep-th/9111056 ].[6] A. Bilal and J. Callan, Curtis G., Liouville models of black hole evaporation , Nucl. Phys. B (1993) 73 [ hep-th/9205089 ].[7] J. G. Russo, L. Susskind and L. Thorlacius,
Black hole evaporation in (1+1)-dimensions , Phys. Lett. B (1992) 13 [ hep-th/9201074 ].[8] J. G. Russo, L. Susskind and L. Thorlacius,
The Endpoint of Hawking radiation , Phys. Rev.D (1992) 3444 [ hep-th/9206070 ].[9] J. G. Russo, L. Susskind and L. Thorlacius, Cosmic censorship in two-dimensional gravity , Phys. Rev. D (1993) 533 [ hep-th/9209012 ].[10] J. Brown, Lower Dimensional Gravity . World Scientific Publishing Co. Pte. Ltd., 1988.[11] L. Thorlacius,
Black hole evolution , Nucl. Phys. B Proc. Suppl. (1995) 245[ hep-th/9411020 ]. – 13 –
12] D. Grumiller, W. Kummer and D. Vassilevich,
Dilaton gravity in two-dimensions , Phys.Rept. (2002) 327 [ hep-th/0204253 ].[13] J. Ambjorn, J. Jurkiewicz and R. Loll,
The Spectral Dimension of the Universe is ScaleDependent , Phys. Rev. Lett. (2005) 171301 [ hep-th/0505113 ].[14] P. Horava, Spectral dimension of the universe in quantum gravity at a lifshitz point , Phys.Rev. Lett. (2009) 161301 [ ].[15] J. R. Mureika and D. Stojkovic,
Detecting Vanishing Dimensions Via PrimordialGravitational Wave Astronomy , Phys. Rev. Lett. (2011) 101101 [ ].[16] L. Anchordoqui, D. C. Dai, M. Fairbairn, G. Landsberg and D. Stojkovic,
VanishingDimensions and Planar Events at the LHC , Mod. Phys. Lett. A (2012) 1250021[ ].[17] D. Stojkovic, Vanishing dimensions: A review , Mod. Phys. Lett. A (2013) 1330034[ ].[18] R. Loll, Quantum Gravity from Causal Dynamical Triangulations: A Review , Class. Quant.Grav. (2020) 013002 [ ].[19] S. Carlip, Dimension and Dimensional Reduction in Quantum Gravity , Class. Quant. Grav. (2017) 193001 [ ].[20] S. Sachdev and J. Ye, Gapless spin fluid ground state in a random, quantum Heisenbergmagnet , Phys. Rev. Lett. (1993) 3339 [ cond-mat/9212030 ].[21] A. Kitaev, A simple model of quantum holography , 2015,http://online.kitp.ucsb.edu/online/entangled15/.[22] A. Almheiri and J. Polchinski,
Models of AdS backreaction and holography , JHEP (2015)014 [ ].[23] J. Maldacena, D. Stanford and Z. Yang, Conformal symmetry and its breaking in twodimensional Nearly Anti-de-Sitter space , PTEP (2016) 12C104 [ ].[24] J. Maldacena and D. Stanford,
Remarks on the Sachdev-Ye-Kitaev model , Phys. Rev. D (2016) 106002 [ ].[25] K. Jensen, Chaos in AdS Holography , Phys. Rev. Lett. (2016) 111601 [ ].[26] V. Rosenhaus,
An introduction to the SYK model , .[27] G. S´arosi, AdS holography and the SYK model , PoS
Modave2017 (2018) 001 [ ].[28] D. A. Trunin,
Pedagogical introduction to SYK model and 2D Dilaton Gravity , .[29] R. Jackiw, Lower Dimensional Gravity , Nucl. Phys. B (1985) 343.[30] C. Teitelboim,
Gravitation and Hamiltonian Structure in Two Space-Time Dimensions , Phys. Lett. B (1983) 41.[31] R. B. Mann, S. Morsink, A. Sikkema and T. Steele,
Semiclassical gravity in(1+1)-dimensions , Phys. Rev. D (1991) 3948.[32] A. Vilenkin and E. P. S. Shellard, Cosmic Strings and Other Topological Defects . CambridgeUniversity Press, 2000.[33] T. Vachaspati,
Kinks And Domain Walls . Cambridge University Press, 2006. – 14 –
34] V. Dzhunushaliev, V. Folomeev and M. Minamitsuji,
Thick brane solutions , Rept. Prog.Phys. (2010) 066901 [ ].[35] Y.-X. Liu, Introduction to Extra Dimensions and Thick Braneworlds , .[36] H.-S. Shin and K.-S. Soh, Black hole formation by Sine-Gordon solitons in two-dimensionaldilaton gravity , Phys. Rev. D (1995) 981 [ hep-th/9501045 ].[37] H.-M. Johng, H.-S. Shin and K.-S. Soh, Sine-gordon solitons coupled with dilaton gravity intwo-dimensional spacetime , Phys. Rev. D (1996) 801.[38] J. Gegenberg and G. Kunstatter, Geometrodynamics of sine-gordon solitons , Phys. Rev. D (1998) 124010.[39] M. Cadoni, , Phys. Rev. D (1998) 104001[ hep-th/9803257 ].[40] C. Vaz and L. Witten, Soliton induced singularities in 2-d gravity and their evaporation , Class. Quant. Grav. (1995) 2607 [ gr-qc/9504037 ].[41] J. Yan and X. Qiu, Sinh-Gordon matter field and a solvable model in two-dimensionalgravity , Gen. Rel. Grav. (1998) 1319.[42] J. Yan, S.-J. Wang and B.-Y. Tao, A solvable model in two-dimensional gravity coupled to anonlinear matter field , Commun. Theor. Phys. (2001) 19.[43] B. St¨otzel, Two-dimensional gravitation and Sine-Gordon solitons , Phys. Rev. D (1995)2192 [ gr-qc/9501033 ].[44] K. Skenderis and P. K. Townsend, Gravitational stability and renormalization-group flow , Phys. Lett. B (1999) 46 [ hep-th/9909070 ].[45] O. DeWolfe, D. Z. Freedman, S. S. Gubser and A. Karch,
Modeling the fifth dimension withscalars and gravity , Phys. Rev. D (2000) 046008 [ hep-th/9909134 ].[46] M. Gremm, Four-dimensional gravity on a thick domain wall , Phys. Lett. B (2000) 434[ hep-th/9912060 ].[47] A. A. Izquierdo, W. G. Fuertes and J. M. Guilarte,
Self-gravitating kinks in two-dimensionalpseudo-riemannian universes , Phys. Rev. D (2020) 036020.[48] L. Randall and R. Sundrum,
An alternative to compactification , Phys. Rev. Lett. (1999)4690 [ hep-th/9906064 ].[49] L. Randall and R. Sundrum, A large mass hierarchy from a small extra dimension , Phys.Rev. Lett. (1999) 3370 [ hep-ph/9905221 ].[50] M. Eto and N. Sakai, Solvable models of domain walls in N = 1 supergravity , Phys. Rev. D (2003) 125001 [ hep-th/0307276 ].[51] Y.-i. Takamizu and K.-i. Maeda, Collision of domain walls in asymptotically anti de Sitterspacetime , Phys. Rev. D (2006) 103508 [ hep-th/0603076 ].[52] D. Bazeia, R. Menezes and R. da Rocha, A Note on Asymmetric Thick Branes , Adv. HighEnergy Phys. (2014) 276729 [ ].[53] J. Omotani, P. M. Saffin and J. Louko,
Colliding branes and big crunches , Phys. Rev. D (2011) 063526 [ ].[54] M. Giovannini, Gauge invariant fluctuations of scalar branes , Phys. Rev. D (2001) 064023[ hep-th/0106041 ]. – 15 –
55] M. Giovannini,
Localization of metric fluctuations on scalar branes , Phys. Rev. D (2002)064008 [ hep-th/0106131 ].[56] M. Giovannini, Scalar normal modes of higher dimensional gravitating kinks , ClassicalQuantum Gravity (2003) 1063 [ gr-qc/0207116 ].[57] Y. Zhong and Y.-X. Liu, Linearization of thick K-branes , Phys. Rev. D (2013) 024017[ ].[58] V. F. Mukhanov, H. A. Feldman and R. H. Brandenberger, Theory of cosmologicalperturbations , Phys. Rep. (1992) 203.[59] H. Kodama and M. Sasaki,
Cosmological perturbation theory , Progr. Theoret. Phys. Suppl. (1984) 1.[60] F. Cooper, A. Khare and U. Sukhatme, Supersymmetry and quantum mechanics , Phys. Rep. (1995) 267 [ hep-th/9405029 ].[61] N. Ikeda and K. I. Izawa,
General form of dilaton gravity and nonlinear gauge theory , Prog.Theor. Phys. (1993) 237 [ hep-th/9304012 ].[62] K. Takahashi and T. Kobayashi, Generalized 2D dilaton gravity and kinetic gravity braiding , Class. Quant. Grav. (2019) 095003 [ ].[63] M. Bianchi, D. Z. Freedman and K. Skenderis, How to go with an RG flow , J. High EnergyPhys. (2001) 041 [ hep-th/0105276 ].[64] E. Kiritsis, F. Nitti and L. Silva Pimenta,
Exotic RG Flows from Holography , Fortsch. Phys. (2017) 1600120 [ ].].