Rotating Kaluza-Klein Black Holes
Wavy Strings Black or Bright
McGill 96–44 WATPHYS-TH-96/17
Wavy Strings: Black or Bright?
arXiv:hep-th/9612248v1 30 Dec 1996
Nemanja Kaloper,a,1 Robert C. Myersb,2 and Harold Roussedevelopments in string theory have brought forth a considerable interest in time-dependent hair on extended objects. This novel new hair is typically characterized by a wave profile along the horizon and angular momentum quantum numbers l, m in the transverse space. In this work, we present an extensive treatment of such oscillating black objects, focusing on their geometric properties. We first give a theorem of purely geometric nature, stating that such wavy hair cannot be detected by any scalar invariant built out of the curvature and/or matter fields. However, we show that the tidal forces detected by an infalling observer diverge at the ‘horizon’ of a black string superposed with a vibration in any mode with l ≥ 1. The same argument applied to longitudinal (l = 0) waves detects only finite tidal forces. We also provide an example with a manifestly smooth metric, proving that at least a certain class of these longitudinal waves have regular horizons.
关于太阳黑洞的英文作文
关于太阳黑洞的英文作文Title: Exploring the Mysteries of Solar Black Holes。
Introduction:Solar black holes, also known as stellar black holes, are enigmatic celestial objects that continue to fascinate astronomers and physicists alike. These cosmic phenomenaare born from the collapse of massive stars, leading to the formation of incredibly dense regions in space with gravitational fields so strong that not even light can escape from them. In this essay, we delve into theintricacies of solar black holes, exploring their formation, characteristics, and the profound impact they have on the universe.Formation of Solar Black Holes:Solar black holes originate from the explosive deathsof massive stars, known as supernovae. When a star exhaustsits nuclear fuel, it can no longer withstand its own gravitational force, causing it to collapse under its own weight. This collapse results in a supernova explosion, where the outer layers of the star are expelled into space, leaving behind the core.If the core of the star is sufficiently massive –typically several times the mass of the Sun – it undergoes further collapse, forming a black hole. This process compresses the mass of the core into an infinitesimally small point known as a singularity, surrounded by an event horizon beyond which no information or matter can escape.Characteristics of Solar Black Holes:Solar black holes possess several uniquecharacteristics that distinguish them from other celestial objects:1. Gravitational Singularity: At the center of a black hole lies a gravitational singularity, where matter is infinitely dense and spacetime curvature is infinitelysteep. The laws of physics, as we understand them, break down at this point, leading to a theoretical realm where our current understanding fails.2. Event Horizon: Surrounding the singularity is the event horizon, the boundary beyond which escape is impossible. Once an object crosses the event horizon, it is inexorably drawn towards the singularity, never to return.3. No Hair Theorem: According to the no-hair theorem, black holes are characterized by only three properties: mass, electric charge, and angular momentum. All other information about the material that formed the black hole is lost beyond the event horizon.Impact on the Universe:Solar black holes play a significant role in shaping the cosmos, influencing the dynamics of galaxies, stars, and even spacetime itself:1. Galactic Evolution: Black holes, including solarblack holes, are thought to reside at the centers of most galaxies. Their immense gravitational pull can influence the motion of stars and gas within galaxies, affectingtheir evolution over cosmic timescales.2. Stellar Dynamics: In binary star systems, a solar black hole can gravitationally interact with its companion star, leading to phenomena such as accretion disk formation and the emission of X-rays. These interactions can provide valuable insights into the properties of black holes and their surrounding environments.3. Time Dilation: The extreme gravitational fields near black holes cause time to dilate significantly. This phenomenon, predicted by Einstein's theory of general relativity, has practical implications for space exploration and our understanding of the nature of spacetime.Conclusion:Solar black holes stand as some of the most enigmaticand intriguing objects in the universe. From their mysterious formation to their profound impact on the cosmos, these cosmic behemoths continue to captivate theimagination of scientists and laypeople alike. As we probe deeper into the nature of black holes, we unlock newinsights into the fundamental laws that govern the universe and our place within it.。
量子引力和弦论在中国—李淼
取体积是可观测宇宙的大小，有
更多的理由：
1、 宇宙的开始有奇点，黑洞塌缩有奇点。 2、 黑洞熵的存在隐含引力的全息原理，这个原 理包含引力和所有其它相互作用。 3、在暴涨宇宙论中，密度涨落由曲率涨落所引 起，而后者与标量场一起作量子涨落才能实现。 ……
中国对量子引力的研究，开始于文革 末期。 引力规范理论的研究： 北京：陆启铿，郭汉英、吴咏时，张 元仲、安英、陈时、邹振隆、黄鹏、 兰州：段一士、 合肥：闫沐霖（Veltman-’t Hooft）
Canonical Quantization Of Gauge Fields. Iii. Gravitational Field. (In Chinese). High Energy Phys.Nucl. Phys.3:594-603,1979
其他尝试，如包括引力的大统一， 吴岳良、周光召：
A Possible unification model for all basic forces.
5、更加具体地说，我们应该重视可 能出现的新物理，暗能量和暗物质的 理论研究 6、弦论目前已经出现二级学科之间 的交叉，特别是凝聚态物理中的一些 问题已经成为弦论的研究对象
7、过去10年我一直相信，弦论以及 量子引力的发展已经不可能依赖内 部的逻辑发展，今后粒子物理和宇宙 学实验对发展弦论、量子引力将起决 定性的作用
弦论开始于68年Veneziano amplitude, Nambu (68)、Susskind (69)等人的弦 论的解释，但局限于强相互作用。 从74年开始，Scherk-Schwarz， Yoneya等人建议弦论是量子引力理 论。
引力为什么要量子化？
可以找到一些理由，下面我给出一个简单 的理由。如果引力场是经典场，就会发生 紫外灾难。将引力场放在一个空腔里，给 一个经典波长截断，能量是 。为了 避免黑洞，得
庞加莱猜想应用篇
（一）庞加莱是法国数学家，1904年他在一组论文中提出有关空间几何结构的猜想，但1905年发现提法中有错误，并对之进行了修改，这就是“庞加莱猜想”：在一个三维空间中，假如每一条封闭的曲线都能收缩成一点，那么这个空间一定是一个三维的圆球。
后来，这个猜想被推广至三维以上空间，被称为“高维庞加莱猜想”。
丘成桐院士认为，庞加莱猜想和三维空间几何化的问题是几何领域的主流，它的证明将会对数学界流形性质的认识，甚至用数学语言描述宇宙空间产生重要影响。
庞加莱猜想证明对用数学语言描述宇宙空间产生重要影响，我们可举在超弦理论上的应用来说明。
首先我们要对庞加莱猜想的“点”作一个约定：庞加莱猜想中的“点”可以指数轴、坐标、直线、曲线、平面、曲面等等数学空间的数值点、标点、原点、奇点、焦点、鞍点、结点、中心点......而不能指我们说的“曲点”和“点内空间”的点，不然就会产生矛盾。
因为我们说的“曲点”，是指环圈面、圆环面收缩成的一点，以及“环绕数”收缩成的一点---如圈是“绳”一致分布中间没有打结的封闭线；在这种纽结理论定义中，两个圈套圈的纽结，有一个交点；如果这种圈套圈有两次纽合，圈套圈的纽结“点”就包含了“环绕数”，把有一个以上“环绕数”的圈套圈，紧致化到一个交点，就是一个“曲点”。
即“曲点”最直观的数学模型，是指包含“环绕数”的点。
而我们说的“点内空间”的点，是指虚数一类虚拟空间内的“点”。
如果把“在一个三维空间中，假如每一条封闭的曲线都能收缩成一点，那么这个空间一定是一个三维的圆球”称为“庞加莱猜想正定理”，那么“曲点”和“点内空间”正是来源于庞加莱猜想之外还有的一个庞加莱猜想：在一个三维空间中，假如每一条封闭的曲线都能收缩成类似一点，其中只要有一点是曲点，那么这个空间就不一定是一个三维的圆球，而可能是一个三维的环面---我们称为“庞加莱猜想逆定理”。
庞加莱猜想至少有两个来源---一个是函数论，一个是代数拓扑学。
弦理论探索：穿越维度
弦理论探索：穿越维度弦理论的基本介绍弦理论是一种寻求解释自然界基本粒子与力的理论，该理论认为所有基本粒子并非是零维度的，而是细小且具有能量的弦状对象。
这些弦可以振动，不同模式的振动对应不同的基本粒子，包括了我们所熟知的质子、中子、电子等。
弦理论的一个重要假设是空间不止存在三个维度，还可能存在额外的紧缩维度。
弦理论的基本思想是将所有基本粒子和力都看作是弦的振动模式。
不同的振动模式会产生不同的粒子，而粒子之间的相互作用则可以通过弦之间的相互作用来解释。
这在某种程度上是对现有粒子物理学中的点粒子模型的一种拓展和统一。
根据弦理论，空间不仅仅是我们所感知到的三个维度，而是一个更高维度的多元空间。
其中，一些额外的维度被假设为紧缩的、微小的，以至于我们无法直接观测到它们。
这些额外维度的存在可以提供对一些基本物理问题的全新解释，例如引力的来源和统一力量的起源。
弦理论的一个重要特点是它具有一种无量纲的固定精细结构常数，称为弦耦合常数。
这个常数决定了弦的相互作用强度，并且与自然界中的其他基本常数，如引力常数和元电荷，存在一定的关联。
这意味着弦理论具有一种内在的统一性，可以为自然现象提供全局性的解释。
弦理论是一种革命性的物理学理论，试图通过将基本粒子和力归结为振动的弦。
它提供了一种对空间维度的重新解释，探索了更高维度的存在，并追求建立一个全局统一的物理学模型。
然而，弦理论仍然面临许多未解决的问题，需要进一步研究和实验验证来验证其有效性和准确性。
维度的概念和重要性在弦理论与多维度的研究中，维度扮演着关键的角色，不仅对理解空间的结构起着重要作用，还为解释基本粒子和力的性质提供了新的视角。
维度可以被看作是物理空间中独立的方向，它描述了物体或者空间在各个方向上的自由度。
在传统的物理学中，我们认为自然界是三维的，即存在长度、宽度和高度。
然而，弦理论提出了更高维度的概念，认为除了我们熟知的三个维度之外，还可能存在额外的维度。
这些额外维度被假设为微小且紧凑的，因此无法直接观测到。
物理本原分析：超越完备元子及其脉动球体模型
Gλ = Bt+ G + + (1 − Bt元子应有：
(3 )
Gλ = (1 − Bt+ )G − + Bt+ ⋅ iG + 。
(4)
3. 正负能级系统交叠作用与零质量元子脉动球体模型
+ t + t
初宇宙耗散算子 Bt 可如下给出：
+
Bt+ =
因为总有
+ + ( ρ εt ) − SU ( ρ ε0 ) SU 。 + + SU ( ρ εt ) − SU (ρ t )
(1 )
+ + SU ( ρ ε ) − SU ( ρ ) = ln(∫ | ϕ ( z ) | 2 dz
∫ exp{−ε
+
3
在本书建立的理论描述形式中，实轴、实变量、实流形、实物理方程等用于反映直接实 在的正能级系统作用，而虚轴、虚变量、虚流形、虚物理方程等用于反映间接实在的负能级 系统作用。联系到正能级宇宙的不可逆性演变过程，我们可以通过正能级宇宙的耗散算子
Bt+ ，建立四维超越完备时空连续体，给出超越完备物理量、超越完备流形和超越完备物理
1
两部分的超越完备量子系统， 或者说， 本书讨论的超越完备量子系统是正能级量子系统与负 能级量子系统在极端条件和特殊方式下的交叠联结和统一， 是直接实在量子系统与间接实在 量子系统在极端条件和特殊方式下的交叠联结和统一。 ——在超越完备量子系统之内， 本书讨论的终极基元是包含正能级基元和负能级基元这 两部分的超越完备元子， 或者说， 本书讨论的超越完备元子是正能级元子与负能级元子在极 端条件和特殊方式下的交叠联结和统一， 是直接实在元子与间接实在元子在极端条件和特殊 方式下的交叠联结和统一。 本文及后续文将处于大爆炸初始阶段的太初宇宙所包含的基本实体称为元子。 作为超越 完备宇宙所包含的终极基元， 超越完备元子是正负能级系统交叠作用的基本载体， 是基本相 互作用统一的基础。
《一名物理学家的教育历程》有关资料
《一名物理学家的教育历程》有关资料【作者介绍】加来道雄，美籍日裔物理学家，毕业于美国哈佛大学，获加利福尼亚大学伯克利分校哲学博士学位，后任纽约市立大学都市学院理论物理学教授。
要紧著作有《超越爱因斯坦》(与特雷纳合著)、《量子场论》《超弦导论》。
【相关知识】为了更好地明白得课文，我们需要了解一些相关的理论物理学知识：1、统一场论：依照现代物理学知识，将我们的宇宙结合起来的力有四种：引力、电磁力、强核力和弱核力。
物理学家运用量子力学，差不多把后三种力统一起来(美籍华裔物理学家杨振宁和他的学生米尔斯提出杨—米尔斯场理论，解决了这一问题，被称为“标准模型”。
然而这一理论因为运算繁复无比而让人头疼)，然而引力仍旧游离在外。
爱因斯坦毕终生之力想寻求四种力的统一，建立一个大一统的理论，最终也没有实现。
2、高维空间：现代理论物理学认为，统一四种力的前景，在于高维空间(如十维或更高)理论的确立。
比如关于古人来讲，风暴是如何样产生的，风暴会突击什么地点，什么时候袭来，什么时候终止，他们是一无所知的，因为他们生活在平坦的大地上，只能靠肉眼从近似于二维平面的角度来观看，即使有简单的预报，也差不多上靠体会来估量的。
现在有了气象卫星，从太空如此三维角度观看地球，在地面上看来奇异莫测的风暴被看得一清二楚，能够精准地预报风暴的动向。
同样，理论物理学家认为，传统的四维(空间三维加上时刻)理论太“小”，不能说明宇宙中的四种力。
当他们超越四维而在更高维(如十维或更高)中寻求统一这四种力时，就能得到一种简单、漂亮的解决模型(科学家认为宇宙应该是简单、和谐的)。
高维空间理论认为，宇宙大爆炸后10-43秒，十维宇宙分解成四维宇宙和六维宇宙，四维宇宙暴胀，通过近150亿年，演变成今天我们生活的宇宙。
大爆炸后10-35秒，大统一力分开。
然而高维空间理论专门难在实验室中得到证实，因为要模拟当时的环境，需要的能量太大，全然无法做到，因此现在高维空间理论只能是“理论”。
黑洞的准正模式(quasinormal modes)
Quasi-Normal Modes of Stars and Black HolesKostas D.KokkotasDepartment of Physics,Aristotle University of Thessaloniki,Thessaloniki54006,Greece.kokkotas@astro.auth.grhttp://www.astro.auth.gr/˜kokkotasandBernd G.SchmidtMax Planck Institute for Gravitational Physics,Albert Einstein Institute,D-14476Golm,Germany.bernd@aei-potsdam.mpg.dePublished16September1999/Articles/Volume2/1999-2kokkotasLiving Reviews in RelativityPublished by the Max Planck Institute for Gravitational PhysicsAlbert Einstein Institute,GermanyAbstractPerturbations of stars and black holes have been one of the main topics of relativistic astrophysics for the last few decades.They are of partic-ular importance today,because of their relevance to gravitational waveastronomy.In this review we present the theory of quasi-normal modes ofcompact objects from both the mathematical and astrophysical points ofview.The discussion includes perturbations of black holes(Schwarzschild,Reissner-Nordstr¨o m,Kerr and Kerr-Newman)and relativistic stars(non-rotating and slowly-rotating).The properties of the various families ofquasi-normal modes are described,and numerical techniques for calculat-ing quasi-normal modes reviewed.The successes,as well as the limits,of perturbation theory are presented,and its role in the emerging era ofnumerical relativity and supercomputers is discussed.c 1999Max-Planck-Gesellschaft and the authors.Further information on copyright is given at /Info/Copyright/.For permission to reproduce the article please contact livrev@aei-potsdam.mpg.de.Article AmendmentsOn author request a Living Reviews article can be amended to include errata and small additions to ensure that the most accurate and up-to-date infor-mation possible is provided.For detailed documentation of amendments, please go to the article’s online version at/Articles/Volume2/1999-2kokkotas/. Owing to the fact that a Living Reviews article can evolve over time,we recommend to cite the article as follows:Kokkotas,K.D.,and Schmidt,B.G.,“Quasi-Normal Modes of Stars and Black Holes”,Living Rev.Relativity,2,(1999),2.[Online Article]:cited on<date>, /Articles/Volume2/1999-2kokkotas/. The date in’cited on<date>’then uniquely identifies the version of the article you are referring to.3Quasi-Normal Modes of Stars and Black HolesContents1Introduction4 2Normal Modes–Quasi-Normal Modes–Resonances7 3Quasi-Normal Modes of Black Holes123.1Schwarzschild Black Holes (12)3.2Kerr Black Holes (17)3.3Stability and Completeness of Quasi-Normal Modes (20)4Quasi-Normal Modes of Relativistic Stars234.1Stellar Pulsations:The Theoretical Minimum (23)4.2Mode Analysis (26)4.2.1Families of Fluid Modes (26)4.2.2Families of Spacetime or w-Modes (30)4.3Stability (31)5Excitation and Detection of QNMs325.1Studies of Black Hole QNM Excitation (33)5.2Studies of Stellar QNM Excitation (34)5.3Detection of the QNM Ringing (37)5.4Parameter Estimation (39)6Numerical Techniques426.1Black Holes (42)6.1.1Evolving the Time Dependent Wave Equation (42)6.1.2Integration of the Time Independent Wave Equation (43)6.1.3WKB Methods (44)6.1.4The Method of Continued Fractions (44)6.2Relativistic Stars (45)7Where Are We Going?487.1Synergism Between Perturbation Theory and Numerical Relativity487.2Second Order Perturbations (48)7.3Mode Calculations (49)7.4The Detectors (49)8Acknowledgments50 9Appendix:Schr¨o dinger Equation Versus Wave Equation51Living Reviews in Relativity(1999-2)K.D.Kokkotas and B.G.Schmidt41IntroductionHelioseismology and asteroseismology are well known terms in classical astro-physics.From the beginning of the century the variability of Cepheids has been used for the accurate measurement of cosmic distances,while the variability of a number of stellar objects(RR Lyrae,Mira)has been associated with stel-lar oscillations.Observations of solar oscillations(with thousands of nonradial modes)have also revealed a wealth of information about the internal structure of the Sun[204].Practically every stellar object oscillates radially or nonradi-ally,and although there is great difficulty in observing such oscillations there are already results for various types of stars(O,B,...).All these types of pulsations of normal main sequence stars can be studied via Newtonian theory and they are of no importance for the forthcoming era of gravitational wave astronomy.The gravitational waves emitted by these stars are extremely weak and have very low frequencies(cf.for a discussion of the sun[70],and an im-portant new measurement of the sun’s quadrupole moment and its application in the measurement of the anomalous precession of Mercury’s perihelion[163]). This is not the case when we consider very compact stellar objects i.e.neutron stars and black holes.Their oscillations,produced mainly during the formation phase,can be strong enough to be detected by the gravitational wave detectors (LIGO,VIRGO,GEO600,SPHERE)which are under construction.In the framework of general relativity(GR)quasi-normal modes(QNM) arise,as perturbations(electromagnetic or gravitational)of stellar or black hole spacetimes.Due to the emission of gravitational waves there are no normal mode oscillations but instead the frequencies become“quasi-normal”(complex), with the real part representing the actual frequency of the oscillation and the imaginary part representing the damping.In this review we shall discuss the oscillations of neutron stars and black holes.The natural way to study these oscillations is by considering the linearized Einstein equations.Nevertheless,there has been recent work on nonlinear black hole perturbations[101,102,103,104,100]while,as yet nothing is known for nonlinear stellar oscillations in general relativity.The study of black hole perturbations was initiated by the pioneering work of Regge and Wheeler[173]in the late50s and was continued by Zerilli[212]. The perturbations of relativistic stars in GR werefirst studied in the late60s by Kip Thorne and his collaborators[202,198,199,200].The initial aim of Regge and Wheeler was to study the stability of a black hole to small perturbations and they did not try to connect these perturbations to astrophysics.In con-trast,for the case of relativistic stars,Thorne’s aim was to extend the known properties of Newtonian oscillation theory to general relativity,and to estimate the frequencies and the energy radiated as gravitational waves.QNMs werefirst pointed out by Vishveshwara[207]in calculations of the scattering of gravitational waves by a Schwarzschild black hole,while Press[164] coined the term quasi-normal frequencies.QNM oscillations have been found in perturbation calculations of particles falling into Schwarzschild[73]and Kerr black holes[76,80]and in the collapse of a star to form a black hole[66,67,68]. Living Reviews in Relativity(1999-2)5Quasi-Normal Modes of Stars and Black Holes Numerical investigations of the fully nonlinear equations of general relativity have provided results which agree with the results of perturbation calculations;in particular numerical studies of the head-on collision of two black holes [30,29](cf.Figure 1)and gravitational collapse to a Kerr hole [191].Recently,Price,Pullin and collaborators [170,31,101,28]have pushed forward the agreement between full nonlinear numerical results and results from perturbation theory for the collision of two black holes.This proves the power of the perturbation approach even in highly nonlinear problems while at the same time indicating its limits.In the concluding remarks of their pioneering paper on nonradial oscillations of neutron stars Thorne and Campollataro [202]described it as “just a modest introduction to a story which promises to be long,complicated and fascinating ”.The story has undoubtedly proved to be intriguing,and many authors have contributed to our present understanding of the pulsations of both black holes and neutron stars.Thirty years after these prophetic words by Thorne and Campollataro hundreds of papers have been written in an attempt to understand the stability,the characteristic frequencies and the mechanisms of excitation of these oscillations.Their relevance to the emission of gravitational waves was always the basic underlying reason of each study.An account of all this work will be attempted in the next sections hoping that the interested reader will find this review useful both as a guide to the literature and as an inspiration for future work on the open problems of the field.020406080100Time (M ADM )-0.3-0.2-0.10.00.10.20.3(l =2) Z e r i l l i F u n c t i o n Numerical solutionQNM fit Figure 1:QNM ringing after the head-on collision of two unequal mass black holes [29].The continuous line corresponds to the full nonlinear numerical calculation while the dotted line is a fit to the fundamental and first overtone QNM.In the next section we attempt to give a mathematical definition of QNMs.Living Reviews in Relativity (1999-2)K.D.Kokkotas and B.G.Schmidt6 The third and fourth section will be devoted to the study of the black hole and stellar QNMs.In thefifth section we discuss the excitation and observation of QNMs andfinally in the sixth section we will mention the more significant numerical techniques used in the study of QNMs.Living Reviews in Relativity(1999-2)7Quasi-Normal Modes of Stars and Black Holes 2Normal Modes–Quasi-Normal Modes–Res-onancesBefore discussing quasi-normal modes it is useful to remember what normal modes are!Compact classical linear oscillating systems such asfinite strings,mem-branes,or cavitiesfilled with electromagnetic radiation have preferred time harmonic states of motion(ωis real):χn(t,x)=e iωn tχn(x),n=1,2,3...,(1) if dissipation is neglected.(We assumeχto be some complex valuedfield.) There is generally an infinite collection of such periodic solutions,and the“gen-eral solution”can be expressed as a superposition,χ(t,x)=∞n=1a n e iωn tχn(x),(2)of such normal modes.The simplest example is a string of length L which isfixed at its ends.All such systems can be described by systems of partial differential equations of the type(χmay be a vector)∂χ∂t=Aχ,(3)where A is a linear operator acting only on the spatial variables.Because of thefiniteness of the system the time evolution is only determined if some boundary conditions are prescribed.The search for solutions periodic in time leads to a boundary value problem in the spatial variables.In simple cases it is of the Sturm-Liouville type.The treatment of such boundary value problems for differential equations played an important role in the development of Hilbert space techniques.A Hilbert space is chosen such that the differential operator becomes sym-metric.Due to the boundary conditions dictated by the physical problem,A becomes a self-adjoint operator on the appropriate Hilbert space and has a pure point spectrum.The eigenfunctions and eigenvalues determine the periodic solutions(1).The definition of self-adjointness is rather subtle from a physicist’s point of view since fairly complicated“domain issues”play an essential role.(See[43] where a mathematical exposition for physicists is given.)The wave equation modeling thefinite string has solutions of various degrees of differentiability. To describe all“realistic situations”,clearly C∞functions should be sufficient. Sometimes it may,however,also be convenient to consider more general solu-tions.From the mathematical point of view the collection of all smooth functions is not a natural setting to study the wave equation because sequences of solutionsLiving Reviews in Relativity(1999-2)K.D.Kokkotas and B.G.Schmidt8 exist which converge to non-smooth solutions.To establish such powerful state-ments like(2)one has to study the equation on certain subsets of the Hilbert space of square integrable functions.For“nice”equations it usually happens that the eigenfunctions are in fact analytic.They can then be used to gen-erate,for example,all smooth solutions by a pointwise converging series(2). The key point is that we need some mathematical sophistication to obtain the “completeness property”of the eigenfunctions.This picture of“normal modes”changes when we consider“open systems”which can lose energy to infinity.The simplest case are waves on an infinite string.The general solution of this problem isχ(t,x)=A(t−x)+B(t+x)(4) with“arbitrary”functions A and B.Which solutions should we study?Since we have all solutions,this is not a serious question.In more general cases, however,in which the general solution is not known,we have to select a certain class of solutions which we consider as relevant for the physical problem.Let us consider for the following discussion,as an example,a wave equation with a potential on the real line,∂2∂t2χ+ −∂2∂x2+V(x)χ=0.(5)Cauchy dataχ(0,x),∂tχ(0,x)which have two derivatives determine a unique twice differentiable solution.No boundary condition is needed at infinity to determine the time evolution of the data!This can be established by fairly simple PDE theory[116].There exist solutions for which the support of thefields are spatially compact, or–the other extreme–solutions with infinite total energy for which thefields grow at spatial infinity in a quite arbitrary way!From the point of view of physics smooth solutions with spatially compact support should be the relevant class–who cares what happens near infinity! Again it turns out that mathematically it is more convenient to study all solu-tions offinite total energy.Then the relevant operator is again self-adjoint,but now its spectrum is purely“continuous”.There are no eigenfunctions which are square integrable.Only“improper eigenfunctions”like plane waves exist.This expresses the fact that wefind a solution of the form(1)for any realωand by forming appropriate superpositions one can construct solutions which are “almost eigenfunctions”.(In the case V(x)≡0these are wave packets formed from plane waves.)These solutions are the analogs of normal modes for infinite systems.Let us now turn to the discussion of“quasi-normal modes”which are concep-tually different to normal modes.To define quasi-normal modes let us consider the wave equation(5)for potentials with V≥0which vanish for|x|>x0.Then in this case all solutions determined by data of compact support are bounded: |χ(t,x)|<C.We can use Laplace transformation techniques to represent such Living Reviews in Relativity(1999-2)9Quasi-Normal Modes of Stars and Black Holes solutions.The Laplace transformˆχ(s,x)(s>0real)of a solutionχ(t,x)isˆχ(s,x)= ∞0e−stχ(t,x)dt,(6) and satisfies the ordinary differential equations2ˆχ−ˆχ +Vˆχ=+sχ(0,x)+∂tχ(0,x),(7) wheres2ˆχ−ˆχ +Vˆχ=0(8) is the homogeneous equation.The boundedness ofχimplies thatˆχis analytic for positive,real s,and has an analytic continuation onto the complex half plane Re(s)>0.Which solutionˆχof this inhomogeneous equation gives the unique solution in spacetime determined by the data?There is no arbitrariness;only one of the Green functions for the inhomogeneous equation is correct!All Green functions can be constructed by the following well known method. Choose any two linearly independent solutions of the homogeneous equation f−(s,x)and f+(s,x),and defineG(s,x,x )=1W(s)f−(s,x )f+(s,x)(x <x),f−(s,x)f+(s,x )(x >x),(9)where W(s)is the Wronskian of f−and f+.If we denote the inhomogeneity of(7)by j,a solution of(7)isˆχ(s,x)= ∞−∞G(s,x,x )j(s,x )dx .(10) We still have to select a unique pair of solutions f−,f+.Here the information that the solution in spacetime is bounded can be used.The definition of the Laplace transform implies thatˆχis bounded as a function of x.Because the potential V vanishes for|x|>x0,the solutions of the homogeneous equation(8) for|x|>x0aref=e±sx.(11) The following pair of solutionsf+=e−sx for x>x0,f−=e+sx for x<−x0,(12) which is linearly independent for Re(s)>0,gives the unique Green function which defines a bounded solution for j of compact support.Note that for Re(s)>0the solution f+is exponentially decaying for large x and f−is expo-nentially decaying for small x.For small x however,f+will be a linear com-bination a(s)e−sx+b(s)e sx which will in general grow exponentially.Similar behavior is found for f−.Living Reviews in Relativity(1999-2)K.D.Kokkotas and B.G.Schmidt 10Quasi-Normal mode frequencies s n can be defined as those complex numbers for whichf +(s n ,x )=c (s n )f −(s n ,x ),(13)that is the two functions become linearly dependent,the Wronskian vanishes and the Green function is singular!The corresponding solutions f +(s n ,x )are called quasi eigenfunctions.Are there such numbers s n ?From the boundedness of the solution in space-time we know that the unique Green function must exist for Re (s )>0.Hence f +,f −are linearly independent for those values of s .However,as solutions f +,f −of the homogeneous equation (8)they have a unique continuation to the complex s plane.In [35]it is shown that for positive potentials with compact support there is always a countable number of zeros of the Wronskian with Re (s )<0.What is the mathematical and physical significance of the quasi-normal fre-quencies s n and the corresponding quasi-normal functions f +?First of all we should note that because of Re (s )<0the function f +grows exponentially for small and large x !The corresponding spacetime solution e s n t f +(s n ,x )is therefore not a physically relevant solution,unlike the normal modes.If one studies the inverse Laplace transformation and expresses χas a com-plex line integral (a >0),χ(t,x )=12πi +∞−∞e (a +is )t ˆχ(a +is,x )ds,(14)one can deform the path of the complex integration and show that the late time behavior of solutions can be approximated in finite parts of the space by a finite sum of the form χ(t,x )∼N n =1a n e (αn +iβn )t f +(s n ,x ).(15)Here we assume that Re (s n +1)<Re (s n )<0,s n =αn +iβn .The approxi-mation ∼means that if we choose x 0,x 1, and t 0then there exists a constant C (t 0,x 0,x 1, )such that χ(t,x )−N n =1a n e (αn +iβn )t f +(s n ,x ) ≤Ce (−|αN +1|+ )t (16)holds for t >t 0,x 0<x <x 1, >0with C (t 0,x 0,x 1, )independent of t .The constants a n depend only on the data [35]!This implies in particular that all solutions defined by data of compact support decay exponentially in time on spatially bounded regions.The generic leading order decay is determined by the quasi-normal mode frequency with the largest real part s 1,i.e.slowest damping.On finite intervals and for late times the solution is approximated by a finite sum of quasi eigenfunctions (15).It is presently unclear whether one can strengthen (16)to a statement like (2),a pointwise expansion of the late time solution in terms of quasi-normal Living Reviews in Relativity (1999-2)11Quasi-Normal Modes of Stars and Black Holes modes.For one particular potential(P¨o schl-Teller)this has been shown by Beyer[42].Let us now consider the case where the potential is positive for all x,but decays near infinity as happens for example for the wave equation on the static Schwarzschild spacetime.Data of compact support determine again solutions which are bounded[117].Hence we can proceed as before.Thefirst new point concerns the definitions of f±.It can be shown that the homogeneous equation(8)has for each real positive s a unique solution f+(s,x)such that lim x→∞(e sx f+(s,x))=1holds and correspondingly for f−.These functions are uniquely determined,define the correct Green function and have analytic continuations onto the complex half plane Re(s)>0.It is however quite complicated to get a good representation of these func-tions.If the point at infinity is not a regular singular point,we do not even get converging series expansions for f±.(This is particularly serious for values of s with negative real part because we expect exponential growth in x).The next new feature is that the analyticity properties of f±in the complex s plane depend on the decay of the potential.To obtain information about analytic continuation,even use of analyticity properties of the potential in x is made!Branch cuts may occur.Nevertheless in a lot of cases an infinite number of quasi-normal mode frequencies exists.The fact that the potential never vanishes may,however,destroy the expo-nential decay in time of the solutions and therefore the essential properties of the quasi-normal modes.This probably happens if the potential decays slower than exponentially.There is,however,the following way out:Suppose you want to study a solution determined by data of compact support from t=0to some largefinite time t=T.Up to this time the solution is–because of domain of dependence properties–completely independent of the potential for sufficiently large x.Hence we may see an exponential decay of the form(15)in a time range t1<t<T.This is the behavior seen in numerical calculations.The situation is similar in the case ofα-decay in quantum mechanics.A comparison of quasi-normal modes of wave equations and resonances in quantum theory can be found in the appendix,see section9.Living Reviews in Relativity(1999-2)K.D.Kokkotas and B.G.Schmidt123Quasi-Normal Modes of Black HolesOne of the most interesting aspects of gravitational wave detection will be the connection with the existence of black holes[201].Although there are presently several indirect ways of identifying black holes in the universe,gravitational waves emitted by an oscillating black hole will carry a uniquefingerprint which would lead to the direct identification of their existence.As we mentioned earlier,gravitational radiation from black hole oscillations exhibits certain characteristic frequencies which are independent of the pro-cesses giving rise to these oscillations.These“quasi-normal”frequencies are directly connected to the parameters of the black hole(mass,charge and angu-lar momentum)and for stellar mass black holes are expected to be inside the bandwidth of the constructed gravitational wave detectors.The perturbations of a Schwarzschild black hole reduce to a simple wave equation which has been studied extensively.The wave equation for the case of a Reissner-Nordstr¨o m black hole is more or less similar to the Schwarzschild case,but for Kerr one has to solve a system of coupled wave equations(one for the radial part and one for the angular part).For this reason the Kerr case has been studied less thoroughly.Finally,in the case of Kerr-Newman black holes we face the problem that the perturbations cannot be separated in their angular and radial parts and thus apart from special cases[124]the problem has not been studied at all.3.1Schwarzschild Black HolesThe study of perturbations of Schwarzschild black holes assumes a small per-turbation hµνon a static spherically symmetric background metricds2=g0µνdxµdxν=−e v(r)dt2+eλ(r)dr2+r2 dθ2+sin2θdφ2 ,(17) with the perturbed metric having the formgµν=g0µν+hµν,(18) which leads to a variation of the Einstein equations i.e.δGµν=4πδTµν.(19) By assuming a decomposition into tensor spherical harmonics for each hµνof the formχ(t,r,θ,φ)= mχ m(r,t)r Y m(θ,φ),(20)the perturbation problem is reduced to a single wave equation,for the func-tionχ m(r,t)(which is a combination of the various components of hµν).It should be pointed out that equation(20)is an expansion for scalar quantities only.From the10independent components of the hµνonly h tt,h tr,and h rr transform as scalars under rotations.The h tθ,h tφ,h rθ,and h rφtransform asLiving Reviews in Relativity(1999-2)13Quasi-Normal Modes of Stars and Black Holes components of two-vectors under rotations and can be expanded in a series of vector spherical harmonics while the components hθθ,hθφ,and hφφtransform as components of a2×2tensor and can be expanded in a series of tensor spher-ical harmonics(see[202,212,152]for details).There are two classes of vector spherical harmonics(polar and axial)which are build out of combinations of the Levi-Civita volume form and the gradient operator acting on the scalar spherical harmonics.The difference between the two families is their parity. Under the parity operatorπa spherical harmonic with index transforms as (−1) ,the polar class of perturbations transform under parity in the same way, as(−1) ,and the axial perturbations as(−1) +11.Finally,since we are dealing with spherically symmetric spacetimes the solution will be independent of m, thus this subscript can be omitted.The radial component of a perturbation outside the event horizon satisfies the following wave equation,∂2∂t χ + −∂2∂r∗+V (r)χ =0,(21)where r∗is the“tortoise”radial coordinate defined byr∗=r+2M log(r/2M−1),(22) and M is the mass of the black hole.For“axial”perturbationsV (r)= 1−2M r ( +1)r+2σMr(23)is the effective potential or(as it is known in the literature)Regge-Wheeler potential[173],which is a single potential barrier with a peak around r=3M, which is the location of the unstable photon orbit.The form(23)is true even if we consider scalar or electromagnetic testfields as perturbations.The parameter σtakes the values1for scalar perturbations,0for electromagnetic perturbations, and−3for gravitational perturbations and can be expressed asσ=1−s2,where s=0,1,2is the spin of the perturbingfield.For“polar”perturbations the effective potential was derived by Zerilli[212]and has the form V (r)= 1−2M r 2n2(n+1)r3+6n2Mr2+18nM2r+18M3r3(nr+3M)2,(24)1In the literature the polar perturbations are also called even-parity because they are characterized by their behavior under parity operations as discussed earlier,and in the same way the axial perturbations are called odd-parity.We will stick to the polar/axial terminology since there is a confusion with the definition of the parity operation,the reason is that to most people,the words“even”and“odd”imply that a mode transforms underπas(−1)2n or(−1)2n+1respectively(for n some integer).However only the polar modes with even have even parity and only axial modes with even have odd parity.If is odd,then polar modes have odd parity and axial modes have even parity.Another terminology is to call the polar perturbations spheroidal and the axial ones toroidal.This definition is coming from the study of stellar pulsations in Newtonian theory and represents the type offluid motions that each type of perturbation induces.Since we are dealing both with stars and black holes we will stick to the polar/axial terminology.Living Reviews in Relativity(1999-2)K.D.Kokkotas and B.G.Schmidt14where2n=( −1)( +2).(25) Chandrasekhar[54]has shown that one can transform the equation(21)for “axial”modes to the corresponding one for“polar”modes via a transforma-tion involving differential operations.It can also be shown that both forms are connected to the Bardeen-Press[38]perturbation equation derived via the Newman-Penrose formalism.The potential V (r∗)decays exponentially near the horizon,r∗→−∞,and as r−2∗for r∗→+∞.From the form of equation(21)it is evident that the study of black hole perturbations will follow the footsteps of the theory outlined in section2.Kay and Wald[117]have shown that solutions with data of compact sup-port are bounded.Hence we know that the time independent Green function G(s,r∗,r ∗)is analytic for Re(s)>0.The essential difficulty is now to obtain the solutions f±(cf.equation(10))of the equations2ˆχ−ˆχ +Vˆχ=0,(26) (prime denotes differentiation with respect to r∗)which satisfy for real,positives:f+∼e−sr∗for r∗→∞,f−∼e+r∗x for r∗→−∞.(27) To determine the quasi-normal modes we need the analytic continuations of these functions.As the horizon(r∗→∞)is a regular singular point of(26),a representation of f−(r∗,s)as a converging series exists.For M=12it reads:f−(r,s)=(r−1)s∞n=0a n(s)(r−1)n.(28)The series converges for all complex s and|r−1|<1[162].(The analytic extension of f−is investigated in[115].)The result is that f−has an extension to the complex s plane with poles only at negative real integers.The representation of f+is more complicated:Because infinity is a singular point no power series expansion like(28)exists.A representation coming from the iteration of the defining integral equation is given by Jensen and Candelas[115],see also[159]. It turns out that the continuation of f+has a branch cut Re(s)≤0due to the decay r−2for large r[115].The most extensive mathematical investigation of quasi-normal modes of the Schwarzschild solution is contained in the paper by Bachelot and Motet-Bachelot[35].Here the existence of an infinite number of quasi-normal modes is demonstrated.Truncating the potential(23)to make it of compact support leads to the estimate(16).The decay of solutions in time is not exponential because of the weak decay of the potential for large r.At late times,the quasi-normal oscillations are swamped by the radiative tail[166,167].This tail radiation is of interest in its Living Reviews in Relativity(1999-2)。
De Sitter时空中黑洞视界面积的量子化
De Sitter时空中黑洞视界面积的量子化陈德友;杨树政【摘要】最近的研究表明黑洞的面积谱具有8πnl2p的表达式.在本文中,我们采用玻尔-索末菲量子化条件对Schwarzschild de Sitter黑洞的事件视界和宇宙视界进行量子化,得到它们的面积谱表达式为8πl2p(n+1/2).黑洞基态熵的出现意味着黑洞不能完全蒸发,有残余物质存在.【期刊名称】《西华师范大学学报（自然科学版）》【年(卷),期】2013(034)003【总页数】4页(P203-206)【关键词】熵谱;基态熵;欧氏化【作者】陈德友;杨树政【作者单位】西华师范大学理论物理研究所,四川南充637009;西华师范大学理论物理研究所,四川南充637009【正文语种】中文【中图分类】P145.61 引言黑洞视界面积化是黑洞的一个重要量子特征[1]，这里有几种方法量子化视界面积. 最常见的方法是准正则模方法：利用准正则模与黑洞参数的关系以及热力学第一定律对黑洞视界面积进行量子化[2，3].Hod提出该方法的模型，并将准正则模的频率与其实部联系[2]，得到面积谱为其中，n=0，1，2，3……，Maggiore考虑到准正则模与谐振子的类比性[3]，认为准正则模的频率应该是它的物理频率，它同时与实部和虚部有关，即因此，模的物理频率的渐进值决定着黑洞的面积谱，得到谱的表达式为该结果与Bekenstein得到的结果完全一致.利用粒子隧穿黑洞对视界面积进行量子化的工作在文献[4，5]中提出.从粒子的出射波函数出发，该方法得到面积谱的最小间隔为利用绝热不变性和玻尔-索末菲量子化条件，Majhi、Vegenas通过系统的哈密顿量对球对称黑洞的面积进行了研究[6]，得到黑洞的熵谱和面积谱.熵谱是等间隔的，而面积谱是否等间隔依赖于所选取的理论框架：在爱因斯坦理论中等间隔，在圈量子引力理论中不等间隔．该方法中黑洞的准正则模不再出现.本文将利用此方法对de Sitter时空中黑洞的面积进行量子化.我们采用朗道和栗弗席兹从粒子一维周期运动得到的量子化条件[7]，发现黑洞的基态熵不为零，这意味着黑洞不能完全蒸发，有残余物质存在.该结果与广义测不准关系得到的结果一致.在相关文献中，黑洞的残余物质被视为暗物质，因此，我们的结果揭示了基态黑洞可视为暗物质.2 Schwarzschild de Sitter 黑洞面积量子化Schwarzschild de Sitter黑洞度规给出如下：ds2=-f(r)dt2+f-1(r)d2+r2(dθ2+sin2θdφ2)(1)其中，参数l与宇宙常数(Λ)的关系式为为黑洞的物理质量.f(r)=0的三个根分别对应于宇宙视界(rc)，事件视界(r+)及无意义的负根(r-)：其中，在以前的工作中，人们通常采用玻尔-索末菲量子化条件，对黑洞面积进行量子化.而本文将采用由朗道和栗弗席兹得到的一维周期运动粒子的量子化条件[7]：(2)其中，n=1，2，3…，pi和pi分别是广义动量和广义坐标.为了对黑洞面积进行量子化，我们首先把广义动量和广义坐标与黑洞的相关参数联系起来：∮pidqi(3)τ是欧氏时间，它具有的周期性，κ是黑洞的表面引力，Η是系统的哈密顿函数.由于欧氏时间的出现，黑洞度规也应该对应于欧氏的空间.将度规(1)欧氏化后得到[8] ds2=f(r)dτ2+f-1(r)dr2+r2(dθ2+sin2θdφ2).(4)该黑洞系统的哈密顿量是系统的ADM质量.在事件视界处，黑洞的ADM质量为我们只考虑出射路径，这意味着的取值为因此，方程(3)变为ħS.(5)利用温度与表面引力的关系以及黑洞热力学第一定律dM=TdS，我们得到上面第二个等号.结合方程(2)和(5)可以得到熵的表达式为(6)该结果表明熵谱与黑洞的质量、宇宙常数等参数无关.当n=0时，基态熵出现，Sbase=π；这意味着黑洞不可能完全蒸发，有残余物存在[9].该结果与广义测不准关系得到的结果一致.在相关文献中，人们把黑洞的残余物视为暗物质.因此，这意味着基态黑洞与暗物质有一定的联系.此外，黑洞熵的最小间隔为ΔS=Sn-Sn-1=2π.(7)该结果与以前得到熵间隔的结果一致[10，11].但是，由于基态熵的出现，该结果与Majhi和Vagenas以及其他人得到的结果有一定差别，这是由于量子化条件的选取不同产生的.考虑到熵与面积的关系事件视界的面积谱得到为在宇宙视界处，黑洞的哈密顿量为黑洞温度和熵分别为和同样采用方程(3)和(4)以及宇宙视界处的热力学第一定律我们可以得到ħ(8)利用量子化条件(2)，同样可以得到熵谱的表达式为这与方程(6)一致.所以，事件视界和宇宙视界处，黑洞的熵谱与面积谱具有相同的性质.在最近的研究中，人们发现黑洞的视界面积之积与黑洞质量无关[12].然而，Matt Visser的研究表明，de Sitter黑洞的视界面积直接与黑洞的质量有关.对于Schwarzschild de Sitter黑洞，视界面积之积为(9)因此，黑洞视界面积之积它依赖于黑洞的质量.在极端情况下，黑洞事件视界与宇宙视界相互重合，即rc=r+，这意味着φ=(3m+1)π，m是正整数.如果黑洞在极端和非极端情况下的面积谱具有相同的表达式那么，我们可以得到宇宙常数为这表明宇宙常数也是量子化的.3 结论采用由朗道-栗弗席兹从粒子的一维周期运动中得到的量子化条件对Schwarzschild de Sitter黑洞的熵和面积进行量子化.结果表明：黑洞事件视界和宇宙视界处的熵谱是等间隔的，间隔最小值为2π；当时n=0，黑洞存在Sbase=π的基态熵，这意味着黑洞不能完全蒸发，有残余物质存在，该残余物与暗物质有关.当黑洞在极端情况和非极端下具有相同的面积谱表达式时，我们可以得到量子化的宇宙常数.参考文献：[1] BEKENSTEIN J D．The Quantum Mass Spectrum of the Kerr Black Hole [J]．Lett Nuovo Cimento，1974，11(9): 467-470.[2] HOD S．Bohr’s Correspondence Principle and the Area Spectrum of Quantum Black Holes [J]．Phys Rev Lett，1998，81:4293-4296.[3] MAGGIORE M．The Physical Interpretation of the Spectrum of Black Hole Quasinormal Modes [J]．Phys Rev Lett，2008，100:141301．[4] BANERJEE R，MAJHI B R，VAGENAS E C．Quantum Tunneling and Black Hole spectroscopy [J]．Phys Lett B，2010，686: 279-282.[5] BANERJEE R，MAJHI B R，VAGENAS E C． A Note on the Lower Bound of Black Hole Area Change in Tunneling Formalism [J]．Euro phys Lett，2010，92: 20001.[6] MAJHI B R，VAGENAS E C．Black Hole Spectroscopy via Adiabatic Invariance [J]． Phys Lett B，2011，701:623-625.[7] LANDAU L D，LIFSHITZ E M．Mechanics，translated by J.B．Sykes and J.S．Bell．Pergamon Press，New York (1969).[8] MA Z Z．Euler Numbers of Four-dimensional Rotating Black Holes with the Euclidean Signature [J]． Phys Rev D，2003，67: 024027．[9] CHEN D，ZENG X X．The Schwarzschild Black Hole’s Remnant via the Bohr-Sommerfeld Quantization Rule [J]．Gen Relativ Gravit，2013，45:631-641.[10] ZENG X X，LIU X M，LIU W B．Periodicity and Area Spectrum of Black Holes [J]．Eur Phys J C，2012，72:1967.[11] CHEN D，YANG H，ZU X T．Area Spectra of Near Extremal Black Holes [J]．Eur Phys J C，2010，69:289-292.[12] CVETIC M，GIBBONS G W，POPE C N．Universal Area Product Formulae for Rotating and Charged Black Holes in Four and Higher Dimensions [J]．Phys Rev Lett，2011，106:121301.。
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