Black Holes and Two-Dimensional Dilaton Gravity

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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The Physics of Black Holes and theirPropertiesBlack holes have always been a subject of great curiosity and fascination for scientists and laymen alike. They are mysterious and intriguing, defying our understanding of the physical laws that govern the universe. But what exactly are black holes? How are they formed? What are their properties? These are the questions that physicists have been grappling with for decades, and although some answers have been found, much of their nature remains a mystery.One of the key features of black holes is their immense gravitational pull. This is because they are formed from the gravitational collapse of a massive object, such as a star, and their density is so high that their gravitational field becomes incredibly strong. In fact, the gravitational pull of a black hole is so powerful that even light cannot escape it, hence the name "black hole". This phenomenon is known as the event horizon, which marks the point of no return for any object that ventures too close to the black hole.But it's not just the gravitational pull of black holes that make them unique. They also have a number of other unusual properties, such as their temperature, their spin, and their size. Let's take a closer look at each of these characteristics.TemperatureIn the 1970s, physicist Stephen Hawking proposed that black holes emit radiation, which came to be known as Hawking radiation. This was a groundbreaking discovery, as it suggested that black holes are not entirely black after all, but rather emit a small amount of energy in the form of radiation. This radiation is due to quantum fluctuations near the event horizon, which cause particle-antiparticle pairs to be created. If one of the particles falls into the black hole, the other can escape, carrying energy away with it.The temperature of a black hole is proportional to its gravitational pull, so the more massive the black hole, the colder it is. This might sound counterintuitive, but it's becausea larger black hole has a larger event horizon, so it emits less radiation. Conversely, a smaller black hole has a smaller event horizon, so it emits more radiation and is therefore hotter.SpinAnother important property of black holes is their spin, which is determined by the angular momentum of the object that formed them. Like a spinning top, a black hole can spin either clockwise or counterclockwise, and its spin affects how it interacts with the surrounding matter. For example, a spinning black hole can drag nearby matter along with it, creating a swirling disk of gas and dust around itself.The spin of a black hole can also affect the characteristics of its event horizon. A non-spinning black hole has a spherical event horizon, whereas a spinning black hole has a flattened, donut-shaped event horizon known as an ergosphere. This region is where matter can enter the black hole, but also where energy and momentum can be extracted from it.SizeFinally, black holes come in a range of sizes, from the smallest "stellar" black holes, which have the mass of a few suns, to the largest "supermassive" black holes, which can have a mass billions of times greater than the sun. But size is not the only factor that determines the properties of a black hole - its spin and temperature also play a role.One interesting feature of black holes is that they have a maximum density, known as the Planck density. This is the point at which the laws of physics as we know them break down, and quantum effects become dominant. Beyond this density, we simply don't know what happens, and it's possible that new physical laws may need to be discovered to explain the behavior of matter in this extreme environment.ConclusionIn conclusion, black holes are fascinating objects that challenge our understanding of the universe. Their immense gravitational pull, temperature, spin, and size make themunique, and studying them can help us learn more about the fundamental laws of physics. However, much of their nature remains a mystery, and it will likely take many more years of research to unlock their secrets.。

高中英语世界著名科学家单选题50题1. Albert Einstein was born in ____.A. the United StatesB. GermanyC. FranceD. England答案：B。

解析：Albert Einstein（阿尔伯特·爱因斯坦）出生于德国。

本题主要考查对著名科学家爱因斯坦国籍相关的词汇知识。

在这几个选项中，the United States是美国，France是法国，England是英国，而爱因斯坦出生于德国，所以选B。

2. Isaac Newton is famous for his discovery of ____.A. electricityB. gravityC. radioactivityD. relativity答案：B。

解析：Isaac Newton 艾萨克·牛顿）以发现万有引力gravity）而闻名。

electricity是电，radioactivity是放射性，relativity 是相对论，这些都不是牛顿的主要发现，所以根据对牛顿主要成就的了解，选择B。

3. Marie Curie was the first woman to win ____ Nobel Prizes.A. oneB. twoC. threeD. four答案：B。

解析：Marie Curie 居里夫人）是第一位获得两项诺贝尔奖的女性。

这题主要考查数字相关的词汇以及对居里夫人成就的了解，她在放射性研究等方面的贡献使她两次获得诺贝尔奖，所以选B。

4. Thomas Edison is well - known for his invention of ____.A. the telephoneB. the light bulbC. the steam engineD. the computer答案：B。

解析：Thomas Edison（ 托马斯·爱迪生）以发明电灯（the light bulb）而闻名。

Black Hole Hair in Higher Dimensions
曹超;陈一新;李剑龙
【期刊名称】《理论物理通讯：英文版》
【年(卷),期】2010(000)002
【摘要】我们在 D 维的 spacetime 与一个静态的、球状地对称的黑洞在平衡学
习物质的性质。

它要求这种物质有州的 =pr/= 的一个方程 ?n/(n + 2k ) ， k， n 吗？(在哪儿n > 1 对应于真空物质和“头发”的混合物物质) ，它似乎独立于D。

然而，当我们把这结果与特定的模型联系时，我们发现这些毛乎乎的黑洞能仅仅住在某特殊维的 spacetime：(i) D = 2 + 2k/n 当黑洞被宇宙的绳包围时，它要求 D 是平的或 D ?，取决于 n 的价值，这与在超弦理论的一些重要结果一致，它可能
在另一个方面揭示在宇宙的绳和超弦之间的关系；(ii ) 黑洞能仅仅在 4-dimensional spacetime 由线性 dilaton 地被包围。

在两个盒子中， D = 4 是特
殊的。

我们也在更高的尺寸举如此的毛乎乎的黑洞的一些例子，包括有否定精力密度的一个玩具模型。

【总页数】6页(P285-290)
【关键词】高维;头发;超弦理论;玩具模型;静态平衡;PR方程;真空问题;能量密度
【作者】曹超;陈一新;李剑龙
【作者单位】
Zhejiang;Institute;of;Modern;Physics,;Zhejiang;University,;Hangzhou;31002 7,;China
【正文语种】中文
【中图分类】O175.29
因版权原因，仅展示原文概要，查看原文内容请购买。

a rXiv:h ep-th/0611194v117N ov26A CFT description of the BTZ black hole:topology versus geometry (or thermodynamics versus statistical mechanics)G.Maiella 1,2and C.Stornaiolo 1,21Istituto Nazionale di Fisica Nucleare,Sezione di Napoli,Complesso Universitario di Monte S.Angelo Edificio N’via Cinthia,45–80126Napoli 2Dipartimento di Scienze Fisiche,Universit`a “Federico II”di Napoli,Complesso Universitario di Monte S.Angelo Edificio N’via Cinthia,45–80126Napoli February 2,2008Abstract In this paper we review the properties of the black hole entropy in the light of a general conformal field theory treatment.We find that the properties of horizons of the BTZ black holes in ADS 3,can be described in terms of an effective unitary CFT 2with central charge c =1realized in terms of the Fubini-Veneziano vertex operators.It is found a relationship between the topological properties of the black hole solution and the infinite algebra extension of the conformalgroup in 2D,SU (2,2),i.e.the Virasoro Algebra,and its subgroup SL (2,Z )which generates the modular symmetry.Such a symmetry induces a duality for the black hole solution with angular momentum J =0.On the light of such a global symmetry we reanalyze the Cardy formula for CFT 2and its possible generalization to D >2proposed by E.Verlinde.11IntroductionGravity in2+1dimensions is a very simple model where it is possible to derive exactly the correspondence between quantum black holes properties and thermodynamical quantities.In presence of a negative cosmological constantΛ=−1/ℓ2,the Gen-eral Relativity in2+1dimensions admits the analogues of the3+1 Schwarzschild and the Kerr black hole known as the BTZ black hole[1].In fact these solutions exhibit all the usual thermodynamic properties of black holes[2].Their entropy is found to be1/4of the horizon area divided by G3the gravitational constant in three dimensions2πr+S=.(1.3)2G3The paper is organized as follows.In sect.2we discuss the geometrical diffeomorphisms which preserve the conformal boundary and its possible description by a Chern-Simons topologicalfield theory.The classical black hole general solution(which we shall call in the following as BTZ)is given in section3together with its properties.The explicit relation of the mass M and the angular momentum J in terms of the generators of the classical Virasoro algebra is found.In section4the quantum version of the BTZ so-lution in the Euclidean space is briefly analyzed.Moreover it is argued that the relevant unitary representation of CFT2is the c=1Fubini-Veneziano vertex operator one[5],following the Cardy argument[6].In section5we2give a short summary of the mathematical properties of the unitary repre-sentation of CFT2which are used in this paper.In particular the Hilbert space of the quantized“momenta”ˆp and the winding“numbers”ˆw for theFubini-Veneziano scalarfield compactified on a circle S1is analyzed usingthe SL(2,Z)modular symmetry.That is reflected in a“duality”relation between(ℓ,J)or(r+,r−).In section6we derive the Brown-Henneaux relation by evaluating the quantum“anomaly”for the locally conformal (Weyl)transformations which is a classical symmetry in both cases,for thegravitational equations in the bulk and for CFT2defined on the boundaryand identifying the results[7].That is also a proof of the validity of the ADS3-CFT2duality guessed by Maldacena[8].The analogue of the Hawking temperature and entropy S H is derivedin section7from the topology of spacetime,identified to be R2×S1.then as for the duality between(r+,r−),the topology of spacetime is strictly related to the physical quantities of BTZ.In section8we stress how the modular invariance,SL(2,Z),of CFT2,which implies the Cardy equation (see sect.4)is also crucial for the validity of the cosmological Bekenstein bound[9]in2D.We present a derivation of the Casimir energy E C and entropy S C which should be relevant for the possible extension to higher dimensions.Some hints in such a direction are briefly discussed in section9where general comments and conclusions are also given.2The2+1anti-de Sitter spacetime(ADS3)Before discussing the BTZ black hole,i.e.the black hole in2+1dimensions, let us introduce the ADS3spacetime.The reason is that a black hole solution in2+1dimension,differently from the Schwarzschild solution,has not a Newtonian asymptotic limit but an ADS3one.The ADS3is a vacuum solution of Einstein equations in3dimensions with a negative cosmological constantΛ.Its metric in polar coordinates isds2= rℓ 2+1 −1dr2+r2dϕ2.(2.1)whereℓ=|Λ|−1/2.An ADS3is characterized by being a manifold with a(conformal) boundary.In this case not all the diffeomorphisms are allowed,but only those which preserve the conformal boundary.These can be identified with the conformal group SL(2,C),which is the covering group of O(2,2)[10].3From the geometrical point of view,these diffeomorphisms are gener-ated by the vectorfieldsξ(+)t=ℓT++ℓ3r4 ξ(−)t=ℓT−+ℓ3r4(2.2)ξ(+)ϕ=ℓT+−ℓ3r4 ξ(−)ϕ=−ℓT−+ℓ3r4 (2.3)ξ(+)r=−r∂u T++O 1r (2.4) where T±are functions of u=t/ℓ+ϕand v=t/ℓ−ϕ.The commutators[ξ±1,ξ±2]=ξ±3define new vectorfields of the form given above and induce two Virasoro algebras with vanishing central charges on the functions T±.It is possible to show that the presence of a boundary modifies these algebras by introducing a central charge c related toℓas in eq.(1.3).Some properties of the ADS3spacetime are better described by observ-ing that the General Relativity action in2+1dimensions is equivalent to the Chern-Simons theory[3][11].Then by doing appropriate identifications the general relativistic action with a negative cosmological constant can be expressed by the following actionI CS=k3A∧A∧A (2.5)which can be split in the spatial and the time partsI CS=kr001r g(u)(2.8)4˜G=˜g(v) 1r00√2T r(σ3∂u A u−A u A u),(2.10) T vv= n L n e−inv=kℓ2+J2Lorℓ2+J2Lor2rdt 2(3.1)We can note the following features in this solution.Differently from the3+1solution the length scale is given by the “curvature”radiusℓ,because the mass M Lor is a dimensionless quantity.The lapse functionN=−M Lor+r24r2(3.2)vanishes forr2±=ℓ2M Lor1−J2LorIt followsr2++r2−M Lor=.(3.5)ℓg00vanishes atr=r erg≡ℓM1/2Lor,(3.6) which as for Kerr solution called ergodic radius,which defines the infinite red-shift surface of the black hole.Finally at r=0there is a singularity on the causal structure but not in the curvature,because the curvature is everywherefinite and constant.For large r the BTZ metric(3.1)approaches the ADS3metric(2.1),then the asymptotic symmetry group for this metric is the conformal group in two dimensions SO(2,2)or its covering SL(2,C).It is noteworthy that one canfind some dual relations in the BTZ black hole in presence of an angular momentum.Indeed the second degree algebraic equation N=0can be solved in terms of the unknown r2or its of inverse1/r2.The solutions in terms of r2are related to the solutions of their inverse through(3.5)according to whichJ2Lorℓ2r2−=only asymptotically,but it generally has a lesser number of Killing vectors. Therefore a black hole can be defined by its symmetries.Given the Killing vectorsξ,one can construct a parameter subgroup such that to a given point P we have P→e tξP.When t is an integral multiple of a step(conventionally one can take it as2π)we define an identification subgroup of SO(2,2).Correspondingly we can take the quotient space,which preserves the properties of the ADS3.This quotient space is still a solution of Einstein’s equations.If we label the coordinates by x a=(u,v,x,y),then the six Killing vectors of the ADS3areJ ab=x b ∂∂x b(3.10)In[12]it is proved that the black hole solutions are obtained by making the identifications defined previously by the discrete group generated by the Killing vectorξ=1ℓJ12−r−ℓ2(r2++r2−)=−2M(3.12)I2=1ℓ2=−2|J|ℓ2−J2ℓ2−J22rdτ 2.(4.1) 7The singularities of this metric are in r=0,r=∞and inr2±=ℓ2M1+J2ℓ2 dτ2+ −M+r2Mℓand r−=0,which is the equivalent in2+1dimensions of the Schwarzschild radius in3+1dimensions[12].By appropriate changes of coordinates we can put in evidence the prop-erties of the metric(4.3)in this limiting case,it is possible to show the existing of the periodicity conditionθ∼θ+2πand the identifications of the coordinates lead to quotient the upper semispace by identifying R∼e2π√M, where the singular points are identified through the radial lines,the mani-fold so defined is a solid torus.The extension of this procedure to the case J=0is straightforward.Going back to the CFT2for the boundary dynamics,we stress a very relevant result.It has been proven in[6]that the density of states is given byS=2π cL06 1/2 .(4.4)In[14]the Cardy equation was used to derive an effective CFT2with c=1described by the Coulomb gas(vertex)operator(for details see section5)as a consequence of modular invariance,SL(2,Z),and other very general properties of CFT2.Then the“highest weight states”of such CFT2 can be considered as the microstates of the gravity in2+1dimensions,while the global states are described by the boundary dynamics.More precisely the gravity is determined only by the global geometric data and does not have“local excitations”,however eq.(4.4)sets for ADS3spacetime the correspondence with local states of the(conformal)field theory which are the“microscopic”excitations for2+1gravity.This point of view resembles the one advocated by Martinec in[14]and it will be made precise in the following section.85Short summary of unitary representations of CFT2To analyze the properties of the unitary representations of CFT2,it is customary to use the Euclidean spacetime.In complex coordinates the metric isds2E=dzd¯z(5.1)where z=x+iy and¯z=x−iy.Being thefield theory conformal invariant one can split all thefield in an analytic and an antianalytic part,i.e.for Φ(x,y)one can writeΦ(x,y)=ΦL(z)+¯ΦR(¯z).(5.2)Henceforth we discuss only the analytic part,i.e.the left sector.Relevant conformalfields called highest weight states area)The energy-momentum tensor T ab(x,y)which is an operator of con-formal dimension2and it is written asT(x,y)=T zz(z)+¯T¯z¯z(¯z)=T L(z)+¯T R(¯z).(5.3) b)The currents J a(z)which have conformal dimension1and are gener-ators of symmetries.They are a necessary ingredient for the solvability of CFT2,for details see[10]and[16].Then one can define the Virasoro algebra for the left sector as follows [L n,L m]=(n−m)L n+m+cn(n2−1)δn+m;0(5.4) whereL n=1(z−w)2+∂w JA CF T2is completely and exactly known if one can derive all the highest weight states and the associated operators O a(z)a=1,...k and their3point functions exactlyO c(w)O a(z)O b(w)=limz→w(5.10)uu being an infrared cut-off.A very interesting case is when the scalarfieldΦ(z)is compactified on a circle S1with radius R Then the highest weight states are defined by the Hilbert space aslˆp|l>=+R2ˆw2(5.13)R210∆−¯∆=2ˆp·ˆw.(5.14) from which∆= ˆp R−Rˆw 2(5.15)If R2=m and m∈Z+then(l,k)≤m.In other terms the high-est weight states arefinite and all the correlation functions are given by products of binomials as(z−w)α2withα2a positive integer.Furthermore ˆp andˆw can be interpreted as charged in the so-called2D Coulomb gas interpretation of the vertex operators[16][17].Eqs.(5.13)and(5.14)for R2=1reproduce respectively eqs.(3.4)and(3.5)if we identifyˆp=r+ℓ(5.16)from which∆+¯∆=ˆp2+ˆw2=r2++r2−ℓ2(5.19) and¯∆=(ˆp−ˆw)2=(r+−r−)2ℓ2r2+,¯∆=0.(5.21)We notice that for such extremal case the CFT2contains one sector or in other terms it is a chiral theory.In such a case it looks similar to the one used in[17]to describe the Laughlin anyons for a Quantum Hallfluid[18]. Such a connection has previoulsy been noticed,in a different context,in [19].6Cosmological constant and central chargeAs it is well known there have been different ways to analyze the gravita-tional properties of the anti de Sitter space in2+1dimensions.Here we shall start from a very interesting fact derived in the work by Brown and Henneaux,i.e.the relation between the cosmological constant11Λ=−1/ℓ2in AdS3and the central charge of the boundary CF T2,equation (1.3).We shall briefly summarize the derivation as given in[7].In fact starting from the actionS=−1g R−d(d−1)8πG ∂M√8πG S ct(γµν)(6.1)whereΘis the trace of the extrinsic curvature of the boundary.M is AdS and∂M is its boundary.From(6.1)one getsTµν=1√δγµν (6.2)where S ct has the role of canceling the divergences whenδ¯M goes to the AdS boundaryδM.By a careful analysis onefindsTµν=1ℓγµν (6.3)where all the quantities refer to the boundary metric andGµν=Rµν−1r 2dr2+ rℓ2 .(6.6) Following such line of thought one reproduces general results usually de-rived with conventional technique see ref[]for details.In particular when M=−1/8πG;J=0the BT Z metric approaches the global AdS3,while M=0and J=0it is similare to the Poincar´e metric.Let us derive the Weyl anomaly,in a covariant way.As it is well-known for Euclidean CF T2with metric is ds2=dzd¯z the diffeomorphisms are defined as byz→z−f(z)¯z→¯z−g(¯z)(6.7)12andT zz →T zz +(2∂z f (z )T zz +z∂z T zz −c24π∂+¯z g (6.8)where z =x +iy and ¯z=x −iy .Really equation (6.7)is a symmetry of the CF T 2only at a classical level.At the quantum level.In fact we have to introduce an ultraviolet cut-offu in quantum time-like convergent loops and show that a field theory in ADS 3remains invariant if we rescale z →z ′=e λz (λ>0).Equivalently the metric should be Weyl rescaled to preserve ds 2=−dzd ¯z .Starting from eq.(6.5),if we consider the diffeomorphism eq.(6.7)there is a Weyl scaling of the boundary metricThen we require that the asymptotic form (for r 2→∞)remains in-variant.One can prove that it is so if for r 2→∞g zz =−r 2r 2+O 1r 3(6.10)From now on the boundary CF T 2is analyzed in the Euclidean metric as usually one does for the relativistic quantum field theory.With these diffeomorphisms the metric changes asds 2→ℓ22(∂3z ξ+)dz 2−ℓ216πG ∂3z ξz T ¯z ¯z =−ℓ2G (6.13)according to (1.3).Naturally the analysis above is in agreement with the ADS 3/CFT 2du-ality on which we have not much to say here.We stress that r plays the rˆo le of the ultraviolet cut-offin the general relativity analysis of the BTZ metric as the usual cut-offa does on the quantum field theory (CFT 2)side.137Thermodynamics and topology of a BTZ black holeWe will show that the BTZ black hole is a thermodynamic object with “effective”temperatureT0=12πℓr+.(7.1)as one would guess by analogy to the Schwarzschild case.To do so we can apply the Euclidean path integral method[20].In complete(and straightforward)analogy to the3+1dimensional case one finds the Euclidean metric(4.1)(with t=iτ)introduced in section3.Such metric is singular atr2+= Mℓ2Mℓ2 1/2 (7.2) andr2−≡[−i|r−|]2= Mℓ2Mℓ2 1/2 .(7.3)As shown in[13]the metric(4.1)is positive definite of constant negative curvature;then it is isometric to the hyperbolic three-space H3.With a coordinate change one obtains the metric of the standard half-space of H3, i.e.ds2=ℓ2sin2χ dR2r2+−r2−,β0=2πr+ℓ2r+.(7.8)14For a temperature T0=β−10and a rotational chemical potentialΩthenI E=4πr+−β0(M−ΩJ)(7.9) then one has2πr+S≡S E=c12 .(8.1) The term c/12is the vacuum energy of the system.Moreover the central charge c,for a system offinite volume,is strictly related to the boundary(surface)energy,i.e.the Casimir energy E c.Such a quantity is not an extensive term,i.e.proportional to the volume V,but a subextensive one.All the previous statements have been proved to be true for a large class of CFT2[6].Finally it has been argued recently by Verlinde[21]that eq.(8.1)can be generalized for any dimension D if the central charge c/12is replaced by the Casimir energy[see also ref3].The main support of such an assumption consists in relating the Cardy formula to the cosmological Beckenstein bound[9]S≤S BH.(8.2) Here S is the entropy of the entire system,while S BH is the Beckenstein entropy,which for any D can be defined as2πS BH=d S BH.By looking more closely to the role of Casimir energy in the cosmological bounds(see[21]and[23]for details)we will see that eq.(8.2)has a very surprising physical interpretation.15To this aim we shall repeat a general argument.The entropy of a thermodynamical system S V and the associated energy E V are extensive quantities,i.e.proportional to the volume V in D dimensions.That simply implies the relationρ+p=T s.(8.4) whereρ,p and s are respectively the energy density,the pressure and the entropy density.Now the extensiveness of E means that E(λV,λE)=λE(S,V).By differentiating onefindsE=V ∂E∂S V(8.5)forλ=1.Thefirst law of thermodynamics tells that∂ENotice that the exponent ofλdepends on the dimensions D in contrast to the extensive quantities.Finally we can separate the two terms in EE=E ext.+14πR S1+1/(D−1);E C=b√E C(2E−E C)=2πR ab6→E C R(8.13) Notice that we have used only conformal invariance and scaling argu-ments to derive eq.(8.12)then it is very tempting to assume that it is true for any D>2.Forfixed E eq.(8.12)has a maximum when E=E C,i.e.S≤2πabER.(8.14)That is the Beckenstein bound up to a constant term.Many of our results are nice exemplifications of Maldacena ADS k/CFT k−1 correspondence for ADS3.More precisely the thermodynamics of CFT2is identified with the thermodynamics of the BTZ black hole as argued in [26].179Comments and conclusionsIn this paper we have emphasized the role and the strict relations between the different properties,1)the origin of the Brown-Henneaux relation(1.3)is found to be the Weyl(trace)anomaly which for CFT2is proportional to the central charge of the Virasoro algebra(sect.6).A related(dual)derivation of the Weyl anomaly(6.9)gives a contri-bution proportional toℓ/G.The eq.(1.3)relates the(quantum)vacuum energy in CFT2with the(classical)gravity vacuum energy in ADS3pa-rameterized byℓ.2)The metric of the space-time ADS3and its symmetries(i.e.the ge-ometry of space-time)implies the symmetries(diffeomorphisms)i.e.the classical conformal group SO(2,2)at the boundary3)The quantum extension of this symmetry,the Virasoro algebra,is assumed to be true for the boundary of ADS3(assumption needed in order to reproduce the Weyl anomaly).Therefore the role of the global symmetry SL(2,Z)(the modular in-variance of CFT2)is to generate a“duality”relation between r+and r−, eq.(5.16)or equivalently betweenℓand J very reminiscent of the electro-magnetic duality in the description of the Quantum Hall effect by CFT2(see ref.[17]).This fact needs a deeper understanding.4)The thermodynamics of CFT2now gives the results of sect.7where the Bekenstein-Hawking temperature T BH,and the related“entropy”eq.(8.3)are evaluated to be the correct ones.Surprisingly those results are consequence of the topology of the space-time metric when a BTZ black hole of mass M and angular momentum J is present.Then topology implies thermodynamical properties of CFT2and viceversa.Moreover the black-hole physical quantities eqs.(3.4)and(3.5) are all expressed in terms ofℓand become zero when c→0(E C→0) or equivalently whenℓ→0.That seems to be a further support for the validity of the Cardy formula for D>2or of a related more general formula.These results emphasize the relevance of the Casimir energy E C for the cosmological implication of the Cardy formula eq.(8.1)analyzed in papers [21]and[23].Our result shows that in2D the ADS3entropy S≡S BH as derived in section7does saturate the Bekenstein bound(8.2)implying that E=E C which,of course,is the maximum value of the Casimir energy.Then for D>2one shouldfind black hole solutions which do not saturate the bound.Therefore it seems quinte important to deepen the study of black hole phase 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a rXiv:g r-qc/1157v 211Feb23SOGANG-HEP 285/01Global Embeddings of Two-dimensional Dilatonic Black Holes Soon-Tae Hong,∗Won Tae Kim,†John J.Oh,‡and Young-Jai Park §Department of Physics and Basic Science Research Institute,Sogang University,C.P.O.Box 1142,Seoul 100-611,Korea (Dated:February 7,2008)Abstract We obtain minimal (2+1)-and (2+2)-dimensional global flat embeddings of uncharged and charged dilatonic black holes in (1+1)-dimensions.Moreover,we obtain the Hawking temperatures and the black hole temperatures of these dilatonic black holes.However,even though the minimal flat embedding structures are mathematically meaningful,through these minimal embeddings,the proper entropies are shown to be unattainable,in contrast to the cases of other black holes in (2+1)or much higher dimensions.PACS numbers:PACS number(s):04.70.Dy,04.62.+v,04.20.Jb,11.25.-w Keywords:superstring theory,global flat embeddingI.INTRODUCTIONThere has been tremendous progress in the study of two-dimensional black holes[1]and string theory[2].It is also well known that in the string theory,a U-duality exists between two-dimensional dilatonic black holes[3,4,5,6]andfive-dimensional ones.On the other hand,it is well-known that a thermal Hawking effect on a curved manifold[7]can be viewed at as an Unruh effect[8]in a higher-dimensionalflat spacetime.Following the global embedding Minkowski space(GEMS)approach[9],several authors[10,11,12,13]recently have shown that this approach could yield a unified derivation of temperature for various curved man-ifolds,such as the rotating Banados-Teitelboim-Zanelli(BTZ)manifold[14,15,16,17,18], Schwarzschild manifold[19]together with its anti-de Sitter(AdS)extension,the Reissner-Nordstr¨o m(RN)manifold[20]and the RN-AdS manifold[12].Historically,the higher-dimensional globalflat embeddings of black-hole solutions have been subjects of great interest to mathematicians,as well as physicists.In differential geometry,it is well-known that the four-dimensional Schwarzschild metric is not embedded in R5[21].Recently,Deser and Levinfirst obtained(5+1)-dimensional globalflat embeddings of the(3+1)Schwarzschild black-hole solution[10].The(3+1)-dimensional RN-AdS,RN, and Schwarzschild-AdS black holes are also shown to be embedded in(5+2)-dimensional GEMS manifolds[12].On the other hand,very recently,the brane metric has also been embedded in six dimensions.[22]Moreover,static,rotating,and charged(2+1)-dimensional BTZ AdS black holes are shown to have(2+2),(2+2),and(3+3)GEMS structures[10,13],respectively,while the static,rotating,and charged(2+1)-dimensional dS black holes are shown to have(3+1), (3+1)and(3+2)GEMS structures,respectively[13].Very recently,we have obtained(3+1) and(3+2)GEMS of uncharged and charged(2+1)black strings,respectively[23].Until now,we have analyzed the GEMS structure of the black hole and black strings in(2+1) and(3+1)dimensions.It is now interesting to study the geometry of(1+1)-dimensional dilatonic black-hole solutions in the GEMS approach to directly yield their minimalflat embeddings.In this paper,we will analyze the Hawking and Unruh effects of(1+1)-dimensional dila-tonic black holes[3,4,5]in the framework of the GEMS scheme.In Section II,we will briefly recapitulate two-dimensional dilatonic black holes[3,4,5]associated with type IIA stringsto yield asymptoticallyflat two-dimensional dilatonic black holes.In Section III,we will obtain the minimal GEMS structures of uncharged and charged two-dimensional dilatonic black holes and their corresponding Hawking temperatures.In Section IV,we will discuss the entropies of dilatonic black holes and the embedding constraints in the framework of the GEMS scheme.II.TYPE IIA STRING THEORY AND TWO-DIMENSIONAL BLACK HOLESIn this section,we briefly recapitulate two-dimensional dilatonic black holes[3,4,5]as-sociated with type IIA string theories and their compactification tofive dimensions whose metric is the product of a three-sphere and an asymptoticallyflat two-dimensional geometry. The ten-dimensional type IIA superstring solution consists of a solitonic NS5-brane wrap-ping around the compact coordinates,say,x5,x i(i=6,7,8,9)and a fundamental string wrapping around x5,and a gravitational wave propagating along x5.In the string frame, the10-metric,dilaton and2-formfield B are given as[24,25,26]ds2=−(H1K)−1fdt2+H−11K(dx5−(K′−1−1)dt)2+H5(f−1dr2+r2dΩ23)+dx i dx i, e−2φ=H1H−15,B05=H′−11−1+tanhα,B056789=H′−15−1+tanhβ,(1) where r2=x21+···+x24andH1=1+r20sinh2αr2,K=1+r20sinh2γr2H1,K′−1=1−r20sinhγcoshγr2.(2)Here,the B05component of the Neveu-Schwarz2-form B is the electricfield of the funda-mental strings,and B056789is the electricfield dual to the magneticfield of the5-brane with components B ij.Exploiting the dimensional reduction in x5,x i(i=6,7,8,9)directions in the Einstein frame,one can obtain thefive-dimensional black hole metric[24,25]ds2=−(H1H5K)−2/3fdt2+(H1H5K)1/3(f−1dr2+r2dΩ23).(3)On the other hand,performing an T5ST6789ST5transformation with the T-duality T ij... along the ij...directions and the S-duality S of type IIB string[27]and then performing an SL(2,R)coordinate transformation associated with the O(2,2)T-duality group,one canalso obtain the5-metricds2=−(H1¯H5)−1fdt2+H−11¯H5(dx5−(¯H−15−1)dt)2+K(f−1dr2+r2dΩ23)(f−1dr2+r2dΩ23)(4) with¯H 5=r20r20+sinh2α,(7) which,after exploiting dimensional reduction in the x5,x i(i=6,7,8,9)directions with α=γ,yield thefive-dimensional black hole metric[28]ds2=− 1−r20r2 −2dt2+ r2r20+sinh2α (m2−q2)1/2,Q=2where the lapse function is given asN 2=1−2me −Qx +q 2e −2Qx ,(12)from which one can obtain the horizons x H and x −in terms of the mass m and the charge q ,e Qx H =m +(m 2−q 2)1/2,e Qx −=m −(m 2−q 2)1/2.(13)By using these relations,one can rewrite the lapse function asN 2= 1−e −Q (x −x H ) 1−e −Q (x −x −) .(14)In order to construct the GEMS structures of the two-dimensional dilatonic black hole,we first consider the uncharged dilatonic black-hole 2-metricds 2=− 1−2me −Qx dt 2+ 1−2me −Qx −1dx 2.(15)Making an ansatz of two coordinates (z 0,z 1)in Eq.(17)to yield−(dz 0)2+(dz 1)2=−(1−e −Q (x −x H ))dt 2+e −2Q (x −x H )Q(1+e −Q (x −x H ))1/2+1(1+e −Q (x −x H ))1/2+1 ,(17)where k H =Q 2(1−e −2Q (x −x H ))1/2,(18)from the definition of the acceleration in n -dimensional spacetimes,a n =√2π=Q(1−e −Q (x −x H ))1/2,T =(−g 00)1/2T H =Q−g tt .Here,one notes that the above Hawking temperature is also given by therelation[7]T H =1(−g 00)1/2.(20)Next,we consider the charged dilatonic black hole whose 2-metric is given by Eqs.(11)and (12).Similarly to the uncharged case,after some algebra,we arrive at the (2+2)GEMS metric ds 2=−(dz 0)2+(dz 1)2+(dz 2)2−(dz 3)2for the charged two-dimensional dilatonicblack hole given by the coordinate transformations z 0=k −1H 1−e −Q (x −x H ) 1/2 1−e −Q (x −x −) 1/2sinh k H t,z 1=k −1H 1−e −Q (x −x H )1/2 1−e −Q (x −x −) 1/2cosh k H t,z 2=22ln F 1/2−1Q (e −Q (x −x H )−e −Q (x −x −)),(21)where the surface gravity k H and F are given ask H=Q1−e −Q (x −x −).(23)Here,one can easily check that,in the uncharged limit (q →0or e Qx −→0),the above coordinate transformations reduce exactly to the previous ones in Eq.(17)for the uncharged dilatonic black hole case.[31].For the trajectories,which follow the Killing vector ξ=∂t on the charged dilatonic black-hole manifold described by (t,x ),one can obtain the constant 2-acceleration,a 2=Qme −Q (x −x H )and the Hawking temperature and the black-hole temperature,T H=a44π1−e−Q(x H−x−)4π(1−e−Q(x H−x−))1/2,(25)where a4is an acceleration in embedded four-dimensional spacetimes.IV.ENTROPIES OF TWO-DIMENSIONAL DILATONIC BLACK HOLESIn this section,we consider the entropies of the dilatonic black holes in the framework of the GEMS scheme.For the uncharged case,the Rindler horizon condition(z1)2−(z0)2=0implies r=r H,and the remaining embedding constraints yield z1=f(r)where f(r)can be read offfrom Eq.(17).The area of the Rindler horizon[29]then seems to yield the entropy of the uncharged dilatonic black hole:S=14G3,(26)where we have explicitly included the constant of proportionality1/4G3[30].However,the entropy in Eq.(26)is not equivalent to the previous result in Refs.[4]and[28]since we still have the Newton constant G3instead of G2.Moreover,in defining the entropy(26), we have used an improper constraint condition,δ(z2−f(r)),since one needs at least onenontrivial constraint describing a relation among the GEMS coordinates and the constraint in Eq.(26)cannot yield any relation between the coordinates in Eq.(17).In that sense,one cannot obtain the desired proper entropy in the minimal higher-dimensional embeddings of the dilatonic black hole.In order to avoid these difficulties,as a plausible candidate,one could consider other higher-dimensional embeddings such as(3+1)-dimensional GEMS:z0=k−1H 1−e−Q(x−x H) 1/2sinh k H t,z1=k−1H 1−e−Q(x−x H) 1/2cosh k H t,z2=x,z3=2entropy:S=1Qe−Q(z2−x H)/2)=1Qd z3=1Q(1+e Q(x H−x−))1/2sin−1e−Q(x−x−)/2, z4=2e−3Q(x−x H)/2e−Q(x−x−)/2functions),regardless of the GEMS coordinate transformations.However,even though the minimal GEMS structures are mathematically meaningful,one has difficulties in deriving the GEMS coordinate transformations to yield the proper entropies since the entropies of the dilatonic black holes have nontrivial G n factors which are associated with the U-duality structure involved in type IIA string theory and depend on the dimensionalities of the GEMS structures.In fact,we have succeeded in obtaining a(3+1)GEMS structure for the uncharged dilatonic black hole that yielded an entropy consistent with previous results with-out appealing to the somehow complicated U-duality transformations.However,the entropy calculation in the charged case remains unsolved and suggests an open problem.Through further investigation,it will be interesting to study the entropy in the GEMS approach to charged(1+1)dilatonic black holes and to investigate the relations between the GEMS and the U-duality schemes.AcknowledgmentsSTH would like 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