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A Solvable Model of Two-Dimensional Dilaton-Gravity Coupled to a Massless Scalar Field

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a r X i v :h e p -t h /9802013v 1 3 F eb 1998A Solvable Model of Two-Dimensional Dilaton-Gravity Coupled to a Massless ScalarFieldMarco Cavagli`a ∗Max-Planck-Institut f¨u r GravitationsphysikAlbert-Einstein-InstitutSchlaatzweg 1,D-14473Potsdam,Germany(February 1,2008)We present a solvable model of two-dimensional dilaton-gravity coupled to a massless scalarfield.We locally integrate the field equations and briefly discuss the properties of the solutions.For a particular choice of the coupling between the dilaton and the scalar field the model can beinterpreted as the two-dimensional effective theory of 2+1cylindrical gravity minimally coupled toa massless scalar field.PACS number(s):04.60.Kz,04.20.Jb I.INTRODUCTION The investigation of lower-dimensional gravity is receiving a lot of attention because of its connection with string theory,dimensionally reduced models (minisuperspaces and midisuperspaces)and black hole physics [1].Lower dimensional models may further provide some insight into the difficult challenge of quantizing gravity theories in the (more physical)four dimensional case.Hence,many 0+1,1+1and 2+1integrable models have been analyzed and solved in the literature,both from the classical and quantum points of view [2].In this context many papers have been devoted to the discussion of integrable and non-integrable two-dimensional models [2].It is well known that two-dimensional dilaton-gravity with an arbitrary potential is (classically)completely integrable [3].A remarkable feature of this model is that any solution possesses a Killing vector [4],i.e.the metric tensor and the dilaton φcan be cast in the form [3]ds 2=4ρ(u,v )dudv ,(1.1)ρ(u,v )=h (ψ)∂u ψ∂v ψ,φ≡φ(ψ),(1.2)where ψis a harmonic function,i.e.∂u ∂v ψ=0.(With a somewhat improper terminology we will call these solutions “static”,even though the Killing vector is not timelike and hypersurface orthogonal.)This is the content of the generalized Birkhofftheorem [3,4].When a scalar field is added to the model,the Birkhofftheorem is no longer valid and non-static solutions appear.It is then worthwile to investigate how the presence of a scalar field is related to issues like integrability and absence of Killing vectors.This perhaps can help shed light on some unsolved problems in classical and quantum gravity as,for instance,the origin of the black hole entropy.Having this in mind,the purpose of this brief report is the discussion of the general solution of the model describedby the actionS = d 2x√R ×S 1cylindrical gravity minimally coupled to a massless scalar field [5].In thiscase the dilaton plays the role of the scale factor of S 1.II.FIELD EQUATIONS AND GENERAL SOLUTIONSFrom the action Eq.(1.3)it is straightforward to obtain the fieldequations.They can be cast in a useful and simple form writing the two-dimensional line element in the “conformal gauge”(1.1).The result is∂u ∂v φ=0,(2.1)∂u ∂v (ln ρ)=γV ′(φ)∂u χ∂v χ,(2.2)∂u [V (φ)∂v χ]+∂v [V (φ)∂u χ]=0,(2.3)ρ∂u∂u φρ =γV (φ)(∂v χ)2,(2.5)where the prime represents the derivative w.r.t.φ.It is surprising that the field equations can be locally integrated.The equation for the field φis the key for solving the system.From Eq.(2.1),it follows that φis indeed a harmonic function.Therefore,the general solution of Eq.(2.1)is φ(u,v )=a (u )+b (v ),where a (u )and b (v )are arbitrary functions.The general solution can be classified in three distinct classes:i)φ(u,v )=φ0,i.e.φconstant;ii)∂u φ(u,v )=0or ∂v φ(u,v )=0,i.e.φdepending on a single variable;iii)φdepending on both u and v variables.The first two cases identify degenerate solutions of the model.Let us now discuss separately the above cases.A.Constant φThis is the simplest (degenerate)case.When ∂u φ(u,v )=0and ∂v φ(u,v )=0Eqs.(2.2-2.5)reduce to (we assume V (φ0)=0)∂u ∂v (ln ρ)=0,∂u χ=0,∂v χ=0.(2.6)The general solution of Eqs.(2.6)is obviousχ=χ0,ln(ρ)=c (u )+d (v ),(2.7)where χ0is a constant and c (u )and d (v )are two arbitrary functions.Recalling Eq.(1.1)and using the reparametriza-tion invariance of the metric in the u and v variables,the three-dimensional line element can be cast in the form ds 2=4dζdυ,where ζ= du e c (u )and υ= dv e d (v ).The solution is then static and the spacetime is flat,as expected since this case corresponds to a pure dilaton-gravity model with vanishing potential.Since φis constant,the above solution represents a flat (du 2−∂u ln(ρ)dφdu 2.(2.9)(Equation (2.3)is identically satisfied.)The general solution of Eqs.(2.8,2.9)isρ=dφdφ′ 2 ,(2.10)where d (v )is an arbitrary function of v ,φ0is an integration constant,and χ≡χ[φ(u )].The two-dimensional line element then represents a flat spacetime,even though the scalar field is not constant.Note that all fields depend on a single variable (φ),however the solution (2.10)is not strictly a “static”solution in the sense (1.1,1.2).(Asimilar solution arises in two-dimensional dilaton-gravity when the dilaton has vanishing potential.)The nature of the solution becomes particularly evident when V(φ′)=φ′and the model is interpreted in2+1dimensions.In this case the three-dimensional line element corresponding to the solution(2.10)reads(V(φ′)=φ′)ds2=4exp −γ φφ0dφ′φ′ dχR2×S1,and the scale factorφis a function of the light cone variable orthogonal toυ.For instance, choosingχ=χ0+arctan(φ/k),Eq.(2.11)reads(x=υ+φand t=υ−φ)ds2=f0eγ/24dθ2.(2.12)plete CaseIn this case,settingρ(u,v)=f(u,v)da(u)dv,(2.13)and using the new coordinates(a,b)(note that the coordinate transformation is never degenerate,the degenerate cases being included in the previous sections),equations(2.2-2.5)reduce to∂a∂b(ln f)=γV′(a+b)∂aχ∂bχ,(2.14)V′(a+b)(∂aχ+∂bχ)+2V(a+b)∂a∂bχ=0,(2.15)∂a(ln f)=−γV(a+b)(∂aχ)2,(2.16)∂b(ln f)=−γV(a+b)(∂bχ)2.(2.17) Now the system of second order partial differential equations(2.14-2.17)can be integrated,solvingfirst Eq.(2.15), and then using the solutionχin Eqs.(2.16,2.17).This program can be easily completed using the new variables z=a+b and w=a−b.Since Eq.(2.15)and the constraints(2.16,2.17)imply Eq.(2.14),we can forget the latter and write∂2zχ+V′(z)dz .Equation(2.18)corresponds to Eq.(2.15)and Eqs.(2.19-2.20)are the sum and the difference ofEqs.(2.16,2.17)respectively.Finally,given a solution of Eq.(2.18),the(logarithm of the)physical conformal factor of the two-dimensional metric f can be written as a functional ofχand locally cast in the formln f(z,w;f0)=ln f0+−2γV(z) w w0dw′∂w′χ(w′,z)∂zχ(w′,z)+(2.21)−γ z dz′V(z′) [∂wχ(w,z′)]2+[∂zχ(w,z′)]2 w=w0,where f0and w0are two constants.Let us now focus attention on Eq.(2.18).Since the latter is a linear separable partial differential equation in the variables z and w,the solution can be written in the form[7]χ(z,w)=χ0+ ∞−∞dλC(λ)η(z,λ)ξ(w,λ),(2.22) whereηandξare the solutions of the linear second order ordinary differential equationsd2ηV(z)dηdw2=λξ.(2.23)It can be proved that the solution(2.22)is the most general solution of Eq.(2.18)provided that the completeness theorem for both Eqs.(2.23)holds.(See for instance[7].)This rather weak assumption is satisfied for a wide class of physical,well-behaved,functions V(z)in Eqs.(2.23).As a concrete example of the formalism,let us now consider the case V(z)=z corresponding to the2+1-dimensional model.The(real)solution(2.22)reads(in this case it is straightforward to verify that the completeness theorem holds because thefirst equation in(2.23)coincides with the Bessel/modified Bessel equation depending on the sign ofλ)χ(z,w)=χ0+ ∞0dα χ(1)α(z,w)+χ(2)α(z,w) ,(2.24) whereχ(1)α(z,w)= A1(α)sin(αw)+B1(α)cos(αw)· C1(α)J0(αz)+D1(α)Y0(αz) ,(2.25)χ(2)α(z,w)= A2(α)eαw+B2(α)e−αw· C2(α)I0(αz)+D2(α)K0(αz) ,(2.26) where J0and Y0are the zero order Bessel functions offirst and second kind,I0and K0are the zero order modified Bessel functions,and the coefficients A,..,D are real functions.Starting from Eqs.(2.24-2.26),or directly from Eqs.(2.18-2.20),some interesting particular solutions can be calcu-lated.Let us look,for example,for a solution of Eq.(2.18)of the formχ(z,w)=ζ(z)+υ(w).Inserting the previous ansatz in Eq.(2.18),we haveχ(z,w)=χ0+k1ln z+k2w+k3 w2+z24(2k22+4k1k3+k23z2).(2.29) The solution(2.27-2.29)is generally non-static in the sense(1.1,1.2)because thefields depend on both variables.(A particular case is given by the choice k1=k3=0.In this caseχ≡χ(w)and f=exp(−γk22z2/2)depends only on z.Hence,according to(1.1,1.2)we have aflat spacetime but a non-static solution.)However,when k2=k3=0,i.e. g(z,w)=1,Eqs.(2.27-2.29)reduce to the(well-known)static solution[6].(Alternatively,the latter can be obtained directly from Eqs.(2.24-2.26)setting,for instance,A i=B i=δ(α).)Let us see this in detail.Recalling Eq.(1.1)and Eq.(2.13)the three-dimensional line element reads(w=τ,z=R)ds2=f0R−γk21(−dτ2+dR2)+R2dθ2,(2.30)χ=k1ln(R/R0).(2.31) The above solution can be cast in a more familiar form with a redefinition of the integration constants and a change of coordinates.Let us set R=βr N/2andτ=tM,whereβis a parameter with dimensions of length and N,M are related to k1and f0by N=2/(1+4k21),M=(2/Nf0)β1−2/N.With these redefinitions the line element(2.30) becomes(γ=−4,low-energy string case)ds2=−U(r)dt2+β2U−1(r)dr2+β2r N dθ2,(2.32) where U(r)=(2M/N)r1−N/2.As expected,the model does not admit black hole solutions and the metric becomesχ(z,w)=χ0+Kz2−w2,(2.33)where K is a constant.From Eq.(2.21)it is straightforward to obtain the metricds2=f0eγK2z2√2(w2−z2)2(−dw2+dz2)+z2dθ2.(2.36) The discussion of the global properties of Eqs.(2.33-2.36)is beyond the scope of this brief report and will be discussed elsewhere,so here we will not enter into details.Let us stress,however,that the above solutions are asymptotically flat and singular when w±z=0,i.e.when the scalarfield diverges.With different choices of the coefficients in Eqs.(2.25-2.26)is it possible to construct non singular solutions.III.CONCLUSIONSIn this note we have briefly discussed the two-dimensional dilaton-gravity-matter theory described by the action (1.3).When V(φ)=φthe model can be interpreted as the two-dimensional effective theory of2+1cylindrical gravity minimally coupled to a massless scalarfield.The model of Eq.(1.3)has the remarkable property of being completely solvable and we have derived and classified its solutions for a large class of functions V(φ).It is well-known that any two-dimensional pure dilaton-gravity theory satisfies the generalized Birkhofftheorem,i.e.any solution of the system can be reduced to the form(1.1-1.2).Here, due to the presence of the scalarfield,the Birkhofftheorem is no longer valid.It is then interesting to examine how the presence of the scalarfield modifies the equations of motion w.r.t.to the pure dilaton-gravity case.Let us consider for simplicity∂uφ=0and∂vφ=0.In this case Eqs.(2.1)and(2.3-2.5)imply Eq.(2.2),so we can neglect the latter. Equation(2.3)simply defines the scalarfield and can be solved setting V(φ)∂vχ=∂vϕand V(φ)∂uχ=−∂uϕ.Using ϕthe remaining equations read∂u∂vφ+ρ¯V(φ)=0,(3.1)ρ∂u ∂uφρ =γV−1(φ)(∂vϕ)2,(3.3) where we allow for the presence of a dilatonic potential¯V(φ).Equation(3.1)does not depend onϕ.Hence,only the constraints(3.2,3.3)are modified by the presence of the scalarfield.The validity of the Birkhofftheorem is thus related to the r.h.s.of the Eqs.(3.2,3.3).Further,a static solution is obtained only when the r.h.s.of Eq.(3.2)is equal to the r.h.s.of Eq.(3.3).In that caseϕ≡ϕ(u+v)orϕ≡ϕ(u−v)and the equations of motion can be reduced to a system of ordinary differential equations.Clearly,the above condition is satisfied in the case of pure dilaton-gravity whenϕis identically zero but cannot be satisfied by any dilaton-gravity theory coupled to a scalarfield.Finally,the integrability property of the system does not seem to be related to the existence of non-static solutions.ACKNOWLEDGMENTSWe are indebted to Vittorio de Alfaro,Alexandre T.Filippov,Larry Ford,and Alexander Vilenkin for interesting discussions and useful suggestions on various questions connected to the subject of this paper.This work has been partially supported by the grant No.3229/96of the University of Torino and by a Human Capital and Mobility grant of the European Union,contract number ERBFMRX-CT96-0012.[1]See for instance:E.Witten,Phys.Rev.D44,314(1991);H.Kastrup and T.Thiemann,Nucl.Phys.B425,665(1994);D.Marolf,Phys.Rev.D53,6979(1996);K.V.Kuchaˇr,Phys.Rev.D50,3961(1994);C.Callan,S.Giddings,J.Harvey,andA.Strominger,Phys.Rev.D45,1005(1992);J.Russo,L.Susskind,and L.Thorlacius,Phys.Rev.D46,3444(1992);S.Carlip,J.Gegenberg,and R.B.Mann,Phys.Rev.D51,6854(1995);A.Barvinskii and G.Kunstatter,Phys.Lett.B389, 231(1996);M.Banados,M.Henneaux,C.Teitelboin,and J.Zanelli,Phys.Rev.D48,1506(1993);J.D.Barrow,A.B.Burd, and ncaster,Class.Quantum Grav.3,551(1996)and references therein.[2]See for instance:H.Nicolai,D.Korotkin,and H.Samtleben,“Integrable Classical and Quantum Gravity”,Lectures givenat NATO Advanced Study Institute on Quantum Fields and Quantum Space Time,Carg`e se,France,22July-3August 1996,e-Print Archive:hep-th/9612065;E.Benedict,R.Jackiw,and H.-J.Lee,Phys.Rev.D54,6213(1996);D.Cangemi, R.Jackiw,and B.Zwiebach,Ann.Physics(N.Y.)245,408(1995);K.V.Kuchaˇr,J.D.Romano,and M.Varadarajan,Phys.Rev.D55,795(1997);D.Louis-Martinez,J.Gegenberg,and G.Kunstatter,Phys.Lett.B321,193(1994)and references therein.[3]A.T.Filippov,Mod.Phys.Lett.A11,1691(1996);Int.J.Mod.Phys.A12,13(1997).[4]T.Kloesch and T.Strobl,Class.Quantum Grav.13,965(1996);Class.Quantum Grav.14,2395(1997);Class.QuantumGrav.14,1689(1997).[5]For a deep discussion of the asymptotic structure of2+1cylindrical gravity coupled to a massless scalarfield see A.Ashtekar,J.Biˇc´a k,and B.G.Schmidt,Phys.Rev.D55,687(1997).[6]K.C.K.Chan and R.B.Mann,Phys.Rev.D50,6385(1994)and Erratum,Phys.Rev.D52,2600(1995);Phys.Lett.B371,199(1996).[7]D.Zwillinger,Handbook of Differential Equations(Boston,Academic Press,1989),pp.59and370.[8]See for instance:Bateman Manuscript Project,Tables of Integral Transforms,ed.A.Erd´e lyi et al.(Mc Graw-Hill BookCompany Inc.,1954)Vol.I,p.43,formula(1).。

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Definition of polymers A simple understanding of polymers can be gained by imaging them to be like a chain or, perhaps, a string of pearls, where the individual pearl represent small molecules that are chemically bonded together. Therefore, a polymer is a molecule made up of smaller molecules that are joined together by chemical bonds. The word polymer means „many parts or units.‟ The parts or units are the small molecules that combine. The result of the combination is, of course, a chainlike molecule (polymer). Usually the polymer chains are long, often consisting of hundreds of units, but polymers consisting of only a few units linked together are also known and can be commercially valuable.
Figure 1.1 Diagram illustrating the definition of plastics.
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2维声学黑洞与1维流体的对应研究

2维声学黑洞与1维流体的对应研究

2维声学黑洞与1维流体的对应研究杨晓焕;颜骏;陈海霖;余毅【摘要】The correspondence relationship between two-dimensional black holes and one-dimensional fluid is studied in this paper.The expression of acoustic metric is derived according to the hydromechanics equations,and we obtain some exact black hole solutions in two-dimensional dilaton gravity model.Moreover,the energy density ρ,speed ν and drive potential f in one-dimension fluid are calculated and the physical properties of these fluid parameters are also analyzed and discussed.%研究2维声学黑洞与1维流体的对应关系,在流体力学方程的基础上推导声学度规的表达式,获得了2维dilaton引力模型中的一些精确黑洞解.计算了1维流体中的能量密度ρ,速度ν和驱动势f,还分析和讨论了这些流体参量的物理性质.【期刊名称】《四川师范大学学报（自然科学版）》【年(卷),期】2016(039)006【总页数】7页(P875-881)【关键词】2维引力;声学黑洞;引力-流体对应【作者】杨晓焕;颜骏;陈海霖;余毅【作者单位】四川师范大学物理与电子工程学院, 四川成都 610066;四川师范大学物理与电子工程学院, 四川成都 610066;四川师范大学物理与电子工程学院, 四川成都 610066;四川师范大学物理与电子工程学院, 四川成都 610066【正文语种】中文【中图分类】O351.2高维时空中的引力方程难以求解，2维引力中的场方程相对简单，因此2维引力可以为研究高维空间中的广义相对论提供理想实验室.2维引力与理论物理中的弦理论和共形场论有密切关系,还与纯数学理论中的几何拓扑学和调和映照有一定的联系.近年来,2维引力作为一种Toy模型,有助于人们对4维引力模型及其量子化深人理解,因此对它的研究具有积极的理论意义.在20世纪80年代初,物理学家已开始着手研究2维引力及其相关的量子Liouville理论,分别在光锥规范、共形规范下深入研究了2维引力问题,并在矩阵模型的框架下获得了2维量子引力模型的一些精确解.这些解极大地丰富了人们对弦理论、共形场论甚至临界现象的进一步理解,因此,从各个不同侧面深入研究2维引力模型就显得非常必要了.2维引力的物理性质已经得到了充分研究[1-12].描述2维时空中黑洞的精确的共形场论是在WZW模型中发展起来的，2维dilation引力理论已经被广泛地应用于研究黑洞的蒸发问题.此外，2维高阶引力模型、2维引力的可积与可解性质已分别在共形规范和光锥规范下得到分析.另一方面，在平坦的2维时空中存在一种非线性标量场的作用模型,即sine-Gordon模型,这一模型中存在孤子解.因此,人们自然希望研究2维引力和sine-Gordon物质场的相互作用，通过CGHS模型的框架研究sine-Gordon物质场作用下的黑洞解.文献[13-16]发现了2维引力模型中的sine-Gordon孤子解和sinh-Gordon时空带解.由于天体物理中黑洞表面温度极低，其霍金辐射非常微弱，因此目前尚未观察到这一物理效应.另外，无论在早期宇宙残留物中寻找小型黑洞或者是在粒子物理对撞机中制造出微黑洞，在短期内这些探索的成功机率都很小.由于声波在不均匀流体中的传播性质和光波在弯曲空间中的传播性质非常相似，所以在流体力学实验中可较容易模拟黑洞的物理性质.流体力学的基础方程是连续性方程和Euler方程[17])ν]=F,式中，ρ为流体密度，ν是流体速度，F=-P,P是压力，F是压力P产生的力密度，这时流体假定没有粘滞性.根据速度矢量的关系式(1/2ν2)=(ν·)ν，引入速度式ν=-φ,那么Euler方程变为p)-).再定义h=p/ρ,那么Euler方程约化为在流体中当振动很小和速度很小时，那么流体中的压强和密度相对变化也很小，这时p和ρ可以表示为p=p0+p1,ρ=ρ0+ρ1,p0和ρ0分别代表流体中平衡密度和平衡压强，p1和ρ1表示围绕平衡的微小涨落.当涨落的二阶小量忽略后，那么线性化处理后的速度势所满足的波动方程为∂t2φ=c22φ,这里c表示声速.根据连续性方程(1)式得到∂t ρ1+·(ρ1ν0+ρ0ν1)=0.并且h(p)可展开为如果忽略流体的牛顿引力势和外力的驱动，那么只剩下流体压强产生的作用力，这时利用(7)式对Euler方程(4)进行线性化处理后得到对方程φ0)2=0,φ1=0.方程(9)式可以重新表示为这时有φ1+ν0·φ1),现将(11)式代入(6)式可以得到如下波动方程φ1+ν0·φ1))+·(-ρ0φ1+φ1+ν0·φ1))=0,这一二阶偏微分方程可以描述线性化标量势φ1的传播规律，即这一方程确定了声学扰动的传播形式.为了将流体方程和引力理论联系起来首先定义如下的局域声速再构造一个4×4矩阵根据(13)式和(14)式那么波动方程(12)式可以重新写成这时定义弯曲时空上的达朗贝尔算符为式中这里g=det(gμν)是度规的行列式，并且有根据矩阵(14)式的行列式可以得到因此有所以得到了如下形式的逆声学度规那么声学度规应为这时声学度规的间隔形式可以表示为].2维dilaton引力模型的作用量[18-20]为2b(φ)2-8πG(-V(φ)Λ)},式中，ψ是辅助场，φ是dilaton场，V(φ)是势函数，G是牛顿常数，b、Λ为常数作用量(24)式对应的辅助场方程为引力场方程为ψ)2)+gμν2ψ-μνψ=8πGTμν,φ)-2b(μφνφφ)2),dilaton场方程为这时2维静态度规选择为[21-29]其中，α(x)是度规因子.此时引力物质系统方程组化为α″=-8πGΛV(φ),(αφ命题 1 dilaton场φ和度规α有如下关系：式中，X0是积分常数，下面证明这一关系式.用αφ′乘以(32)式两边得对(31)式两边求导得将(34)式与(35)式联立消去dV/dx后得即φ又因为并且‴-2α′α″)=αα‴.由(37)～(39)式可得φ′)2]=即所以命题1证毕.命题 2 当标量场势能V(φ)=e-2aφ时有式中a是势能常数，下面证明这一关系式成立.当标量场势能取为V(φ)=e-2aφ时,(31)式变为将上式整理得又因为并且φ′)2]=所以即那么(47)式变为式中β=b/a2.化简上式得所以命题2证毕.命题 3 当β=p/(p+2)=1(p→∞)，场方程有如下的黑洞度规和dilaton场解φ(51)式中A、C、E是积分常数，下面证明这一命题成立.由(50)式得α″=-8πGΛe-2aEe-Cx,α‴=8πGΛCe-2aEe-Cx.将α′、α″、α‴带入式命题2中的(41)式的左端得同理，将α″带入(43)式得所以命题3证毕.下面说明命题3得到的2维度规可以描述一种黑洞，取B=8GΛπe-2aE/C2则度规(50)式化为当C>0,x→-∞或C<0,x→+∞时，可知黑洞度规出现奇异性质；当xC=-ln(A/B)/C，同样可知黑洞度规也出现奇异性质，根据曲率R=-α″可计算出不同时空奇点处的曲率分别为所以xC表示黑洞的视界位置，可以为正值或负值.标量场φ(x)在奇点处的性质为φ因此，这个解可以描写2维黑洞，此黑洞的真正奇点位于x→±∞处.黑洞的ADM 质量定义为φ′)2].K是积分常数，可以证明解析解(50)式和(51)式对应的2维黑洞质量为这里，A、C>0,当a>0,由(51)式知系数C越大，φ(x)越强，那么对应的黑洞质量越大.取C=1，E=0，8πG=1，Λ=1，则(50)和(51)式化为命题 4 对(29)式中黑洞度规和时空坐标做如下变换，则黑洞度规可化为声学度规的形式,下面证明这一命题成立.如果使用如下变换[30]则有dx2=ρ02dx2,于是有这时声学度规的表达式为2v0dxdt+dx2].这一度规恰好对应于(23)式中i=j=1的特殊情况.由(65)式可以看出当2M=J时，黑洞的视界为于xc=-ln2M处，这时其对应的声学视界位于c=v0处.命题 5 1维流体力学中的Euler方程和连续方程为ρ0(x)v0(x)A(x)=常数,式中，ρ0是流体密度，p是流体压强，v0是流体速度，f是驱动外力的势，A(x)通量截面，下面证明这一命题.引入如下变换则(70)和(71)式等价于如下方程组这时讨论一种特殊情况，当声速c=常数时有如下关系式并且另外以及将(75)～(78)式代入(73)式命题即可得这一等式成立.另外有(70)式容易得到(71)式，所以命题5成立,再根据Euler方程(70)式和连续性方程(71)式可求出1维流体中的能量密度ρ0(x)、速度v0(x)和驱动势f(x)的如下表达式式中，s是与流体通量A有关的常数，f0是积分常数.当黑洞质量取为M=1/2,声速取为c=1，常数取为s=1，f0=1时，那么可以作出黑洞视界外流体参量，如图1～3所示.当黑洞质量取定时，计算结果表明流体密度随空间坐标增大而增大，而流体速度随空间坐标增大而减小.另外，流体驱动势也随空间坐标增大而减小.根据流体方程组解可以直接看出，当空间坐标不变时，随着黑洞质量的增大，流体密度变大，对应的流体速度变小.本文首先根据流体力学中的连续方程和Euler方程分析了流体中的微小振动，这种振动所对应的速度势满足的方程可以描述声波现象，对流体方程组进行线性化处理后得到密度、压强和速度势涨落满足的波动方程，这一方程也可描述标量势的传播规律.当定义适当的度规张量后，那么波动方程就可化为一个弯曲时空下的标量场方程，由此可以导出声学度规的表达式.其次，本文推导了2维dilaon引力模型中的场方程组，根据3个命题进一步获得了2维度规的解析解，通过物理分析后表明这一解可以描述2维时空中的黑洞，其时空奇点为无穷远处，而视界的位置由势能强度Λ和a所决定.经过适当的坐标变换后发现2维黑洞度规可以变化为对应的声学度规，这时2维引力场方程可与1维流体中的Euler方程发生联系,因此可以求出声学黑洞中的流体密度、压强和驱动势的解析解.当黑洞质量取为定值时，本文对流体速度和驱动势作出了数值图形，并讨论了这些流体参量在空间中的变化规律.另外，2维定态时空中霍金温度定义为4πTH=(dα/dx)|x=xc,代入视界坐标xc=-ln 2M便可求出霍金温度为TH=M/2π,因此2维时空中黑洞质量越大，霍金温度越高，最近，W. 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A Model of Two-Dimensional Quantum Gravity in the Strong Coupling Regime

A Model of Two-Dimensional Quantum Gravity in the Strong Coupling Regime


The tensionless limit at the level of the action is a problem that was studied by Schild many years ago [3] and after considered by several other authors [4–8] in differents contexts. In essence the procedure delineated above is equivalent to make the hamiltonian formulation of (1) and to take directly the tensionless limit in the constraints 1 ′2 H⊥ = (p2 + T 2 x ), 2 because in such limit one replace (2) by 1 H⊥ = p2 , 2 and the algebra of constraints becomes [H⊥ (σ ), H⊥ (σ )] = 0, [H⊥ (σ ), H1 (σ )] = (H⊥ (σ ) + H⊥ (σ ))δ (σ − σ ), [H1 (σ ), H1 (σ )] = (H1 (σ ) + H1 (σ ))δ (σ − σ ). A careful analysis of the action in the tensionless limit reveal that g = det(gαβ ) is vanishes and, as a consequence, all the points of the string moves with the velocity of the light. Thus, one see the constraints of the tensionless string as describing a set of infinite massless relativistic particles (one for every point along the string) suplemented by the transversality condition px = 0. Now, we would like to construct an analog model for 2D gravity. Two-dimensional gravity is described by the following action S=− 1 2 √ 1 1 d2 σ −g g αβ ∂α φ∂β φ + Rφ + √ µ2 eφ , k M 4 k (5)
Quasi-Exactly Solvable Models with Spin-Orbital Interaction

Quasi-Exactly Solvable Models with Spin-Orbital Interaction

a r X i v :h e p -t h /9810045v 1 7 O c t 1998TPI-MINN -98/04Quasi-Exactly Solvable Models with Spin-Orbital Interaction 1Alexander UshveridzeTheoretical Physics Institute,University of MinnesotaandDepartment of Theoretical Physics,University of Lodz,Pomorska 149/153,90-236Lodz,Poland 2Abstract First examples of quasi-exactly solvable models describing spin-orbital interaction are constructed.In contrast with other examples of matrix quasi-exactly solvable models discussed in the literature up to now,our models admit infinite (but still incomplete)sets of exact (algebraic)solutions.The hamiltonians of these models are hermitian operators of the form H =−∆2+V 1(r )+(s ·l )V 2(r )+(s +l )·h V 3(r )where V 1(r ),V 2(r )and V 3(r )are scalar functions,l is a vector of the angular momentum operator,s is a matrix-valued vector spin-operator and h is an external (constant)vector magnetic field.1Introduction Quasi-exactly solvable (QES)problems are distinguished by the fact that only some of their energy levels and corresponding wavefunctions admit explicit construction.In the last decade a big progress has been acieved in elaborating the concepts of the phenomenon of quasi-exact solvability and formulating methods of constructing and solving QES models of a variety oftypes (for more detail see e.g.the reviews [[4],[7],[3],[5]]and book [[8]]).In this paper we undertake a new step in this direction and construct new classes of QES models which (up to now)have never been discussed in the QES-literature.These are QES models with a spin-orbital interaction.The potentials of such models have the following general formV (x,y,z )=V 1(r )+(s ·l )V 2(r )+(j ·h )V 3(r )where r =1This work was supported by the grant No DOE/DE-FG02-94ER408232E-mail address:alexush@mvii.uni.lodz.pl and alexush@krysia.uni.lodz.plenergy levels and corresponding eigenvalues.They are however quasi-exactly solvable because the set of their exact solutions is still incomplete and does notfill all the spectrum of a model. First examples of such models were presented in our recent work[[1]]where they have been called”infinite QES models”.Another unusual feature of these models is that they(in contrast with models discussed in paper[[1]])are matrix models with physically realistic hermitian hamiltonians.Everybody who has some experience with quasi-exact solvability in the matrix(multi-channel)case knows how difficult is to satisfy the condition of hermiticity when constructing such models.It is hardly neccessary to remind the reader that up to now only a couple of hermitian matrix QES models have been constructed(see e.g.[[4],[2]]).2Starting pointTo demonstrate how does our construction procedure work we start with the simplest one-dimensional QES model with hamiltonianH=−∂2r2+[b2−2a(2m+c+1)]r2+2abr4+a2r6(1)acting in Hilbert space of functions defined on the positive half axis r∈[0,∞]and vanishing sufficiently fast at its ends r=0and r=∞.Here a,b,c are real parameters satisfying the conditions(a>0,c>0)and m is a non-negative integer.As it was demonstrated in[[8]], for anyfixed m the Schroedinger equationHψ(r)=Eψ(r)(2) for model1admits algebraic solutions whose general form is given by the formulasψ(r)=r c−1/2mi=1 r2/2−ξi exp −ar42 (3)E=2b(2m+c)+8ami=1ξi(4)The m complex numbersξi in expressions3and4satisfy the system of m algebraic equationsmk=1,k=i12ξi−b−2aξi=0,i=1,...,m.(5)It turns out that system5has only m+1permutationally invariant solutions for any given m which are represented by the sets of real pointsξi.Each solution is completely characterized by a(quantum)number k=0,1,...,m which indicates the number of positiveξi-points. According to formula3,the number of positiveξi-points determines the number of(real) wavefunction zeros,which,in turn,determines the ordinal number of an excitation(oscillation theorem).This means that model1has m+1exactly constructable solutions describing the ground state and mfirst excited states.A more detailed exposition of properties of model1 and its algebraic solutions can be found in the book[[8]].3The modified equationIt is not difficult to see that the transformationψ(r)=rϕ(r)(6)reduces the equation2to the form−∂2r∂r2+V(r,l,m) ϕ(r)=Eϕ(r)(7) in which we used the notationV(r,l,m)=[b2−2a(2m+5/2+l)]r2+2abr4+a2r6(8) andl=c−3/2(9)Hereafter we shall consider l as a new independent parameter taking(by agreement)only non-negative integer values.The form of thefirst three terms in the equation7coincides with the form of the radial part of a tree-dimensional Laplace operator.For this reason it seems quite natural to interpret l as the3-dimensional angular momentum and try to relate the equation7to a certain3-dimensional quantum problem.In the following three sections we show that there are three such possibilities leading to three different kinds of quasi-exactly solvable problems in the3-dimensional space.4Thefirst possibilityOne of the simplest possibilities of interpreting equation7is based on the assumption that function8entering into7is l independent:V(r,l,m)=V0(r,N)=[b2−2a(N+5/2)]r2+2abr4+a2r6(10) For this the numberN=l+2m(11)must befixed.In this case,equation7takes the form of a typical radial Schroedinger equation for a spherically symmetric3-dimensional equation(−∆+V0(r,N))Ψ(x,y,z)=EΨ(x,y,z).(12) Since both m and l are assumed to be positive,the condition11leads to afinite number of possibilities with m=0,1,...,[N/2],and l=N,N−2,...,N−2[N/2],respectively.For this reason,for any given N,the model12is quasi-exactly solvable and has(as usually)only a finite([N/2]([N/2]+1)/2)number of explicit solutions.The model of such a form and even its more complicated spherically non-symmetric versions were considered many years ago in papers[[6],[7]].5The second possibilityAnother possibility of interpreting equation7is to consider m as afixed number not restricting the value of l.In this case the function8becomes linearly dependent on l and can be represented in the formV(r,l,m)=V1(r,m)−l·V2(r)={[b2−2a(2m+5/2)]r2+2abr4+a2r6}−l·{2ar2}(13)It is quite obvious that,in order to associate the equation13with a certain3-dimensional Schroedinger equation,we mustfind a proper3-dimensional source for the term which is linear in the momentum l.Thefirst think which comes in ones head is to look for the3-dimensional scalar operators O which wuld commute with both the Laplace operator and r and would have the eigenvalues linear in l.In this case we could consider7as a reduction of a3-dimensional problem(−∆+V1(r,m)−O·V2(r))Ψ(x,y,z)=EΨ(x,y,z)(14)The linearity in l means that the operator must be proportional to the operator of the angular momentuml=(l x,l y,l z)=(i(y∂z−z∂y),i(z∂x−x∂z),i(x∂y−y∂x)).(15) But this is a3-dimensional vector while the operator we are looking for must be a scalar. The only possibility to construct a scalar from15is to take a scalar product of l with another vector operator.It is quite obvious that there is no such operator if we restrict ourselves to the single-channel problems.However,if we admit the consideration of multi-channel problems, then a good candidate for the second operator can immediately be found.This is obviously the spin operator s!Rerstricting ourselves(for the sake of simplicity)to the1/2-spin case(2 by2matrices),we can easily check that the spectrum of the operatorO=2·s·l(16)(which,obviously commutes with both∆and r)is linear in l.Indeed,representing operator 16in the formO=j2−l2−s2(17) (where j=l+s is a total momentum)and taking for concreteness a particular case with j=l+1/2we easilyfind the corresponding branch of the spectrumo=j(j+1)−l(l+1)−s(s+1)=(l+1/2)(l+3/2)−l(l+1)−3/4=l.(18)Thisfinally leads us to a3-dimensional matrix QES models−∆+{[b2−2a(2m+5/2)]r2+2abr4+a2r6}−2(s·l)·{2ar2} Ψ(x,y,z)=EΨ(x,y,z)(19) describing spin-orbital interaction.It is a time to ask ourselves of what kind of models did we obtain?First of all,one should stress again that these models are actually quasi-exactly solvable.This follows from the fact that for any given m and l they have an infinite number of normalizable solutions,but only m+1of them are exactly(algebraically)constructable.Second,and this is may be the most important thing,despite the fact that the set of exactly constructable solutions is incomplete,this set is infinitely large.This is so because the number l is notfixed by the3-dimensional model19.It appears as a solution of the eigenvalue problem for operators O and may take arbitrary non-negative integer values.In conclusion of this section note that the hamiltonians of models we obtained are hermi-tian by construction.6The third possibilityThe last interesting possibility of reducing the equation7to a3-dimensional form appears when the function8depends on both parameters l and m.In this case the function8becomes linearly dependent on both l and m and can be represented in the formV(r,l,m)=V1(r)−l·V2(r)−m·V3(r)={(b2−5a)r2+2abr4+a2r6}−l·{2ar2}−m·{4ar2}(20) By analogy with the previous section we can consider the numbers l as the eigenvalues of the operator of spin-orbital interaction,and the only thing which remains to do is to interpret m as an independent quantum number appearing in equation7as an eigenvalue of a certain operator M commuting with the variable r,Laplasian∆and the spin-orbital operator s·l.A good candidate for such an operator is the z-projection of the total momentum s+l.In fact,it should not neccessarily be a z-projection.Because of the spherical symmetry,it could be equally weel a x-or y-projection,or any other projection.We can therefore represent this operator in a covariant formM=2(s+l)·h(21) where h is a unit magneticfield.We introduced an additional factor2to make the eigenvalues of operator M integer rather than half integer.Of course,the negative integers are not interesting for us,because,as we remember,only for non-negative integer values of m the system admits algebraic solution.Summarizing,we can consider7as a reduction of a3-dimensional problem(−∆+V1(r)−(s·l)·V2(r)−2(s+l)·h·V3(r))Ψ(x,y,z)=EΨ(x,y,z)(22) which can be treated as a spectral problem for a matrix quantum model describing spin-orbital interaction together with the interaction of a total momentum with an external magneticfield. It is remarkable,that the model22does not contain any integer parameters anylonger.All these parameters appear dynamically as solutions of the eigenvalue problems for additionally introduced symmetry operators.At the same time,the model22remains quasi-exactly solvable,because for any particular values of these eigenvalues the equation7has only a certain incomplete set of solutions.7ConclusionThe method of construction infinite(matrix)QES models exposed in this paper is,obviously, quite general and can easily be used for building other spin-orbital models with more com-plicated potentials and higher matrix dimensions.For this it is sufficient to start with other known one-dimensional QES modelsfirst rewritting them in the form of a radial Schroedinger equation and then interpreting the l-dependent terms appearing in their potential as the eigenvalues of a spin-orbital operators.8AcknowledgementsI would like to express my sincere gratitude to my colleagues from the Theoretical Physics Institute of the University of Minnesota(where this work has been written)for their kind hos-pitality.I am especially grateful to Professor M.Shifman for very intersting and fruitful dis-cussions during my visit.This work was supported by the grant DOE/DE-FG02-94ER40823. 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