分类号 密级

Transcription

1 分类号 U C D 密级 编号 中国科技大学 博士后研究工作报告 黑洞熵的起源 吴双清 工作完成日期 报告提交日期 2002 年 7 月 ~ 2004 年 6 月 2004 年 6 月 中国科学技术大学 ( 安徽合肥 ) 二〇〇四年六月

2 黑洞熵的起源 The Origin of Black Hole Entropy 博士后姓名 : 吴双清 流动站 ( 一级学科 ) 名称 : 物理学 专业 ( 二级学科 ) 名称 : 理论物理 研究工作起始时间 : 2002 年 7 月 研究工作期满时间 : 2004 年 6 月 中国科学技术大学近代物理系交叉学科理论研究中心 2004 年 6 月

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4 Abstract Black hole physics especially black hole thermodynamics provides a unique big stage to synthesize several disciplines such as General Relativity, Quantum Field Theory, and Statistical Physics, etc. Researches on black holes have become the intersecting point of multi-disciplines. It is generally expected that a thorough understanding of the quantum mechanical nature of black holes will help pave the way towards a fully consistent theory of quantum gravity. As such, black hole physics will be a good test ground for quantum gravity. Many topics in black hole physics, such as the origin problem of black hole entropy, information loss and the endpoint of Hawking radiation, are the front subjects in theoretical physics and are one of the most hot spots in current researches. Amongst these topics, studies of the microscopic origin of black hole entropy have been achieved a lot of progress during the past decade. The aim of this report mainly investigates black hole thermodynamics and the origin problem of black hole entropy. After a brief introduction of some basic knowledge in black hole physics, we review in Chapter 2 the usual thermodynamic laws of Kerr-Newman black holes and setup a set of similar laws of thermodynamics on the inner horizon by extending the four laws of black hole thermodynamics on the outer event horizon to those on the inner horizon. The main content of Chapter 3 is to discuss the statistical, quantum and dynamical origin problem of black hole entropy. In this chapter, we review progress made on this subject during the last decade, focusing on some representative viewpoints. In Chapter 4, we return to discuss black hole thermodynamics again. A characteristic feature of this chapter is that we have established a set of new first laws of black hole thermodynamics. Distinct from the first laws ( horizon version) of thermodynamics in Chapter 2, this set of first laws is related to the conformal properties of black holes in the near-horizon geometric region, thus it can be called as a conformal version of first laws of black hole thermodynamics. This version may play a key role in our understanding of many important issues such as the microscopic origin of black hole entropy, area spectrum (area quantization), dynamic processes (for example, Hawking radiation iii

5 and graybody factor), quasinormal and nonquasinormal modes, near-horizon conformal symmetry and geometric conformal field theory as well as holography principle, etc. In Chapter 5 we investigate the effect of particle spin and the cosmological constant on the black hole entropy. Specifically we use the improved thin-layer brick wall model to study quantum correction to the entropies of a Kerr-de Sitter black hole due to arbitrary spin fields. Several previously obtained results are included as special cases in ours. Finally in the concluding section, we summarize some problems existing in the current researches about the origin of black hole entropy and outline what deserves to be done in the next step. Key Words: Black Hole Thermodynamics, Statistical Mechanics of Black Holes, Black Hole Entropy, Origin Problem and Quantum Correction, Conformal Field Theory iv

6 Contents! i Abstract iii Chapter 1 Introduction Chapter 2 Black Hole Thermodynamics Chapter 3 The Origin of Black Hole Entropy Classical and Semi-classical Theory Entanglement Entropy Brick Wall Model Induced Gravity Theory String/M-Theory and D-brane Physics Loop QG and Quantum Geometry Conformal Field Theory Summary Chapter 4 Black Hole Thermodynamics Revisited Chapter 5 Effect of Spin on Black Hole Entropy Introduction Perturbations of Spin Fields in the Kerr-de Sitter Space Entropy of Kerr-de Sitter Black Holes due to Spin Fields Discussions and Conclusions Chapter 6 Concluding Remarks v

7 Appendix A Teukolsky-Starobinsky Identities B Integrals in terms of Thin-layer Brick Wall Model References List of Publications Publications during Doctoral Study Publications during Postdoctoral Study Submission for Publication First Pages of 4 Papers \ 85 ;» 86 vi

8 Chapter 1 Introduction Black holes are specific solutions of the Einstein equations which describe regions of a space-time where the gravitational field is so strong that nothing, including light signals, can escape them. The interior of a black hole is hidden from an external observer. The boundary of the unobservable region is called the horizon. A black hole can appear as a result of the gravitational collapse of a star. In this case it quickly reaches a stationary state characterized by a certain mass and an angular momentum as well as an electric charge. These are the only parameters a black hole in the Einstein-Maxwell theory can have. Its metric in the most general case is the Kerr- Newmann metric. This statement is known as the no-hair theorem [1] which allows us to describe a stationary black hole by a small number of parameters. On the other hand, thermodynamics describes behavior of averaged quantities of a system with a large number of physical degrees of freedom. The behavior is traced by a small number of parameters. Mathematically, the microscopic description of thermodynamics, statistical mechanics, is grounded by the ergodic hypothesis. Since the cosmic censorship conjecture [2] combined with the singularity theorem [3] predicts inevitable occurrence of black holes, it seems that the no hair theorem plays the same role as the ergodic hypothesis plays in thermodynamics. In fact, it is well known that black holes have many properties analogous to those of thermodynamics. Those are as a whole called black hole thermodynamics. In particular, four laws of black holes combined with the generalized second law make up a main framework of the black hole thermodynamics. In these laws, black hole entropy defined as follows plays an important role. S = 1 4 A [ = k Bc 3 4G A ], (1. 1) where A is the surface area of black hole horizon. Moreover, it was suggested that the black hole entropy, or the horizon area, is an adiabatic invariant [4]. The formula (1. 1) for black hole entropy is often called Bekenstein-Hawking formula since the concept of black hole entropy was first introduced by Bekenstein [5] as a quantity proportional to the horizon area and the proportionality coefficient was fixed by Hawking s discovery of thermal radiation from a black hole [6]. Hawking showed that a black hole emits thermal radiation of a quantum matter field with temperature given by 1

9 2 Chapter 1 Introduction T = κ 2π [ = κ 2πck B ], (1. 2) where κ is the surface gravity of a background black hole, it characterizes the strength of the gravitational field near the horizon. This thermal radiation and its temperature are called Hawking radiation and Hawking temperature, respectively. Moreover if a matter field in a thermal-equilibrium state at some temperature is scattered by a black hole, then it always becomes closer to the thermal-equilibrium state at the Hawking temperature (1. 2) [7, 8]. Let us recall basic properties of the black-hole thermodynamics. We assume that a relation analogous to the first law of thermodynamics holds for a black-hole system. If Ω is the angular velocity of the black hole at the horizon, Φ is the difference of the electric potential at the horizon and at infinity, then by using purely classical equations one arrives at the following variational formula [9] dm = κ A + ΩdJ + ΦdQ 8π = T ds + ΩdJ + ΦdQ. (1. 3) Relation (1. 3) has the form of the first law of thermodynamic where S has the meaning of an entropy, T is a temperature, and M is an internal energy. Strictly speaking (1. 3) defines the entropy and the temperature up to a multiplier. This multiplier is fixed from another considerations: T is defined as the temperature of the Hawking radiation from a black hole [6]. One can also find an analogy with other laws of thermodynamics. For instance, by considering classical processes with black holes one can conclude that the area of the event horizon does not decreases in time (the area law [10] or the second law of black hole) just as the ordinary thermodynamical entropy. The Bekenstein-Hawking formula (1. 1) looks reasonable in this sense. Indeed this observation which is reminiscent to the second law was the original motivation for the introduction of the black hole entropy [5]. Moreover, when quantum effects are taken into account, it is believed that a sum of the black hole entropy and the entropy of a matter outside the horizon does not decrease (the generalized second law). Black hole must have an intrinsic entropy proportional to the horizon area. Otherwise processes like a gravitational collapse would be at odds with the second law. The zeroth law of black hole thermodynamics states that surface gravity of a Killing horizon is constant throughout the horizon. This supports our choice of the black hole temperature. (For a proof of the zeroth law, see Refs. [9, 11].)

10 Chapter 1 Introduction 3 At this stage we would like to point out that for a black hole the third law does not hold in the sense of Planck: S as T 0 for the family of Schwarzschild black holes. Rather, the third law does hold in the sense of Nernst: it is impossible by any process, no matter how idealized, to reduce κ to zero in a finite sequence of operations [9, 12]. Thermodynamics and statistical mechanics of black holes is one of the most interesting and rapidly developing branches of black hole physics. In the Einstein theory S is a pure geometrical quantity. It is known that quantum statistical mechanics is a well-established microscopic description of thermodynamics. In the thermodynamical description of real thermodynamical systems, information on each microscopic degree of freedom is lost, and only macroscopic variables are concerned. However, the number of all microscopic degrees of freedom is implemented in a macroscopic variable: entropy is the logarithm of the number of all consistent microscopic states corresponding to a given set of macroscopic parameters. In analogy, it is expected that there might be a microscopic description of the black hole thermodynamics, too. In particular, it is widely believed that black holes should have microscopic degrees of freedom whose number is consistent with the Bekenstein- Hawking entropy. Since the microscopic description seems to require a quantum theory of gravity, detailed investigations of the black hole entropy should contribute a lot toward construction of the theory of quantum gravity. This is one among the several reasons why the origin of the black hole entropy needs to be understood at the fundamental level. Another strong motivation to investigate the black hole entropy is the so-called information loss problem. Hawking argued that, if a black hole is formed by gravitational collapse, then evolution of quantum fields becomes non-unitary because of evaporation of the black hole due to the Hawking radiation [13]. This means that some information is lost in the process of the black hole evaporation. Moreover, this suggests that the conventional field-theoretical approach may be useless for the purpose of construction of the theory of quantum gravity since the field theory is based on the unitarity. Hence, the evaporation of a black hole makes people, who wish to construct a unitary theory of quantum gravity, be in difficulties: if the evaporation of a black hole would actually occur and lead to the information loss, then they would be obliged to give up the unitarity. Thus, we have to clarify whether information is lost or not due to the Hawking radiation in order to take a step forward. This problem is called the information loss problem. On the other hand, since entropy is strongly connected with information in the theory of information, it is natural to expect that the black hole entropy might be related to

11 4 Chapter 1 Introduction some information. Therefore, investigations of the origin of the black hole entropy seem to provide important insight toward the information loss problem. In these senses, the origin of the black hole entropy is one of the most important issues at the present stage of black hole physics. There were many attempts to explain the origin of the black hole entropy. Related researches can be classified as [14]: classical and semi-classical quantum field theory (for example, entanglement entropy and brick wall model), induced gravity theory, string/m-theory and D-brane physics, loop quantum gravity and quantum geometry as well as conformal field theory, etc. We leave them to be subject of our review in Chapter 3. The aim of this report mainly investigates black hole thermodynamics and the origin problem of black hole entropy. After a brief introduction of some basic knowledge in black hole physics, we review in Chapter 2 the usual thermodynamic laws of Kerr-Newman black holes and setup a set of similar laws of thermodynamics on the inner horizon by extending the four laws of black hole thermodynamics on the outer event horizon to those on the inner horizon. The main content of Chapter 3 is to discuss the statistical, quantum and dynamical origin problem of black hole entropy. In this chapter, we review progress made on this subject during the last decade, focusing on some representative viewpoints. In Chapter 4, we return to discuss black hole thermodynamics again. A characteristic feature of this chapter is that we have established a set of new first laws of black hole thermodynamics. Distinct from the first laws ( horizon version) of thermodynamics in Chapter 2, this set of first laws is related to the conformal properties of black holes in the near-horizon geometric region, thus it can be called as a conformal version of first laws of black hole thermodynamics. This version may play a key role in our understanding of many important issues such as the microscopic origin of black hole entropy, area spectrum (area quantization), dynamic processes (for example, Hawking radiation and greybody factor), quasinormal and nonquasinormal modes, near-horizon conformal symmetry and geometric conformal field theory as well as holography principle, etc. In Chapter 5 we investigate the effect of particle spin and the cosmological constant on the black hole entropy. Specifically we use the improved thin-layer brick wall model to study quantum correction to the entropies of a Kerr-de Sitter black hole due to arbitrary spin fields. Several previously obtained results are included as special cases in ours. Finally in the concluding section, we summarize some problems existing in the current researches about the origin of black hole entropy and outline what deserves to be done in the next step.

12 Chapter 2 Black Hole Thermodynamics Studies of black hole physics during the late 1960 s and early 1970 s yielded laws of black hole mechanics [9] that bear a striking resemblance to the laws of thermodynamics. Law Thermodynamics Black Hole Mechanics Zeroth Temperature T constant over Surface gravity κ constant over body in thermal equilibrium horizon of stationary black hole First de = T ds+ work terms dm = κda/(8π)+ rotation, charge terms Second δs 0 in any process δa 0 in any process Third Impossible to achieve T = 0 Impossible to achieve κ = 0 via physical processes via physical processes Motivated by these parallels, and the fact that the irreducible mass [15] of a black hole is related to the area of the event horizon, Bekenstein [5] argued that the entropy of the black hole should be directly proportional to the area of the event horizon measured in Planck units. A starting point was the no-hair theorem for gravity coupled to a Maxwell field, which states that the only information available to observers outside a black hole is a set of conserved quantities: the mass M, charge Q, and angular momentum J. The presence of the black hole event horizon thus motivated Bekenstein s definition of the black hole entropy as a measure of information about the black hole interior that is inaccessible to outside observers. Bekenstein then argued, on the basis of gedanken experiments involving infalling particles and then merging black holes, that the entropyarea relation should be linear. Bekenstein also proposed a generalized second law in which the total entropy, given by the sum of the entropy of the black hole and the common entropy of outside matter fields, is nondecreasing. This generalized second law can be used to argue that the entropy of a self-gravitating system of a given spatial extent and given (M, Q, J) can never exceed that of a black hole. Alternatively phrased in terms of information theory, this says that the maximum amount of information in a self-gravitating system is, up to a constant of order unity, a single bit per unit Planck area. Although Bekenstein s arguments were made in four dimensions, they carry over to other dimensions as well. However the concept of black hole entropy is in conflict with classical physics. In the light of Einstein s general relativity, a black hole does not radiate anything including light. But according to standard thermodynamics, entropy must accompany with the 5

13 6 Chapter 2 Black Hole Thermodynamics presence of temperature. So if a black hole has an entropy, it should have a temperature and must have thermal radiation. This superficial conflict is resolved with the Hawking s famous discovery [6] of quantum radiation by black hole event horizon, which in the same time also uncover the deeper connection among General Relativity, Quantum Mechanics and Statistical Mechanics, etc. Hawking discovered that black holes emit radiation, due to quantum pair production in their gravitational potential gradient and the presence of the event horizon. The emitted radiation has a thermal spectrum, with deviations from a perfect black body spectrum, the greybody factors, determined by the frequency dependence of the gravitational potential barriers outside the event horizon. The thermodynamic temperature is given in terms of the geometrical surface gravity at the event horizon, T = κ/(2π). An immediate consequence of this identification of the temperature is that the proportional constant between the entropy and the area is fixed: S = A/4. In what follows we consider in details the laws of black hole mechanics. The validity the laws of black hole mechanics does not appear to depend upon the details of the underlying dynamical theory. It should be emphasized that there is a number of unresolved issues such as the interpretation of the third law arising in ordinary thermodynamics in the context of general relativity. Thus, a deeper understanding of the relationship between black holes and thermodynamics may provide us with an opportunity not only to gain a better understanding of the nature of black holes in quantum gravity, but also to better understand some aspects of the fundamental nature of thermodynamics itself. To begin with let us discuss thermodynamic laws of black holes in the classical Einstein-Maxwell theory and in low energy effective heterotic string theory. The most general three-parameters rotating charged solutions are represented by the Kerr-Newman [16] and Kerr-Sen [17] black holes, respectively. We then extend the conventional four laws of black hole thermodynamics on the exterior horizon to those on the interior horizon [18]. In the Boyer-Lindquist coordinates [19], the metric of a rotating charged black hole and the vector potential of an electromagnetic field can be rewritten as [20]: ds 2 = ( ) 2 dt a sin 2 sin 2 θ [ ] 2 ( dr θdϕ + adt (Σ + a 2 sin 2 2 ) θ)dϕ + Σ Σ Σ + dθ2 A (em) = Qr Σ (dt a sin2 θdϕ), (2. 1) where for the Kerr-Newman metric, = r 2 2Mr + a 2 + Q 2, Σ = r 2 + a 2 cos 2 θ, and the horizons are located at r ± = M ± ɛ with ɛ = M 2 a 2 Q 2 ; while for the Kerr-Sen black hole, = r 2 + 2(b M)r + a 2, Σ = r 2 + 2br + a 2 cos 2 θ, and the horizons are

14 Chapter 2 Black Hole Thermodynamics 7 denoted by r ± = M b ± ε where ε = (M b) 2 a 2. In both cases, the specific angular momentum is a = J/M. Note that in the latter case, the twist parameter b is related to the Sen s parameter α via b = Q 2 /2M = M tanh 2 (α/2). Because M b 0, r = r is a new singularity in the region r 0, the event horizon of the Kerr-Sen black hole is located at r = r +. For a Kerr-Newman black hole, the reduced horizon area, surface gravity, angular velocity, and electric potential corresponding to the outer and inner horizons are given by A ± = r 2 ± + a 2 = 2Mr ± Q 2, Ω ± = a A ± = κ ± = r ± M = A ± a 2Mr ± Q, Φ 2 ± = Qr ± = A ± respectively. Similarly for the twisted Kerr(-Sen) solution, we have A ± = r 2 ± + 2br ± + a 2 = 2Mr ±, Ω ± = a A ± = ±ɛ 2Mr ± Q, 2 Qr ±, (2. 2) 2Mr ± Q2 κ ± = r ± M + b A ± = ±ε 2Mr ±, a, Φ ± = Qr ± = Q 2Mr ± A ± 2M = b. (2. 3) Q Note that it is more convenient to use the reduced area rather than the horizon area A ± = 4πA ±. We summarize the chief points of four laws of conventional black hole thermodynamics: 0th Law: The surface gravity κ + or temperature T + = κ + /(2π) of a stationary black hole (at equilibrium) is constant on the event horizon. 1st Law: In an isolated system containing a stationary black hole, the total energy, total angular momentum, and total charge are conserved. In the evolving process of a black hole, the mass, angular momentum, and charge vary as certain functions of its area A + and its surface gravity κ +. The first law can be formulated as the Bekenstein- Smarr (B-S) differential mass formula [21] dm = 1 2 κ +da + + Ω + dj + Φ + dq, (2. 4) and integral relation M = κ + A + + 2JΩ + + QΦ +. (2. 5) A few remarks should be made about the derivation of the first law of black hole mechanics. Its rigorous proof was done using a covariant technique or Hamiltonian formalism

15 8 Chapter 2 Black Hole Thermodynamics [22]. However, there exist other possible derivations also. For example, the B-S differential and integral mass formulas can be derived from Christodoulou mass-squared formula by using thermodynamical relations. It is also possible to deduce the mass formulas directly by a simple algebra manipulation [18]. 2nd Law: Known as area nondecreasing theorem, it asserts that the area A + of black hole event horizon can never decrease for any processes satisfying the weak energy condition [5], δa + 0. (2. 6) Generalized 2nd law: In any classical process, the total entropy S T = S + + S M of a system composed of the outer horizon and the outside matter fields, where S + = πa + is the entropy of event horizon and S M is the total entropy of ordinary matter outside the event horizon, never decreases in any physical process: δs T 0. 3rd Law: It is also known as Nernst theorem, which states that it is not possible by any physical process to reduce the surface gravity κ + of a black hole to zero (0 + ) through any physical process and any finite transformations. This law says that one can not obtain a zero-temperature extremal black hole from from the nonextremal one by a finite sequence of operations. This is just the conventional thermodynamic laws on the outer event horizon. As a Kerr-Newman black hole has two horizons: the outer and inner horizon, we can also establish a set of thermodynamic laws on the inner horizon. In Ref. [18], we have extended Bardeen-Carter-Hawking s four laws of black hole thermodynamics on the exterior horizon to those on the interior horizon. Here we quote them as follows: (1) Zeroth Law: The surface gravity κ or temperature T = κ /(2π) of a stationary black hole is constant on the inner horizon. (2) First Law: In an isolated system including the inner (Cauchy) horizon, the total energy of the system is conserved. The Bekenstein-Smarr differential and integral mass formulae are respectively. dm = 1 2 κ da + Ω dj + Φ dq, (2. 7) M = κ A + 2JΩ + QΦ, (2. 8) (3) Second Law: The area A + of inner horizon can never increase for any processes [23], δa 0. (2. 9)

16 Chapter 2 Black Hole Thermodynamics 9 We suggest by conjecture a Generalized 2nd law: In any classical process, the total entropy S T = S + S M of a system composed of the inner horizon and the matter fields inside the inner horizon, where S = πa is the entropy of inner horizon and S M is the total entropy of ordinary matter in the intrinsic singular region, still never decreases in any physical process: δs T 0. (4) Third Law: It is impossible by any physical process to reduce κ to zero (0 ) by a finite sequence of operations. Now we turn to the thermodynamic laws of Kerr-Sen black hole [17] arising in the low energy effective string theory. This solution is a rotating charged black hole generated from the Kerr solution by the twist technique, its geometric character resembles that of a Kerr black hole. The thermodynamic property of this twisted Kerr black hole was discussed in Ref. [24] by using separation of the Hamilton-Jacobi equation of a test particle. In Ref. [20], we have investigated quantum thermal effect of the twisted Kerr- Sen black hole and shown that its thermal property has the same character as that of the Kerr-Newman black hole though its geometry character is unlike that of Kerr-Newman black hole. In addition we have pointed out that the Kerr-Sen solution shares similar four black hole thermodynamical laws as the Kerr-Newman space-time does. In the following we list the four thermodynamical laws of the Kerr-Sen black hole and demonstrate they are similar to those of Kerr-Newman black hole thermodynamics. The four laws of black hole thermodynamics are generalized as follows: Zeroth Law: The surface gravity κ ± of a stationary black hole are two constants on the entire surface of its corresponding horizons. First Law: In an isolated system including black holes, the total energy of the system is conserved. The Bekenstein-Smarr differential and integral relations are dm = 1 2 κ ±da ± + Ω ± dj + Φ ± dq, M = κ ± A ± + 2JΩ ± + QΦ ±, (2. 10) where Φ ± = Q/(2M) for the Kerr-Sen black hole. Second Law: The total entropy S T = S ± + S M, where S M is the total entropy of ordinary matter outside the event horizon or inside the inner horizon, never decreases in any physical process: δs T 0. Third Law: It is impossible by any physical process to reduce κ ± to zero by a finite sequence of operations. We conclude this chapter by making a few remarks in order:

17 10 Chapter 2 Black Hole Thermodynamics (A) The inner horizon is quantum mechanically unstable, while the event horizon has been shown to be stable under small external perturbations. For the Kerr-Sen black hole, the interior horizon is in the region r 0. Nevertheless we regard they are irrespective to the discussion made here. (B) There is Hawking radiation from the outer horizon, and there should exist Hawking absorption on the inner horizon from the intrinsic singular region [18, 25]. (C) If we accept the temperature interpretation of surface gravity κ ±, then the temperature T + is positive while T is negative. The definition of negative temperature has no contradict with black hole having negative specific heat. Each horizon can be viewed as a single thermodynamic system, so a Kerr-Newman black hole have a pair of entropies S ± = πa ± accompanied by a pair of temperatures T ± = κ ± /(2π). Finally if the Kerr-Newman (or Kerr-Sen) black hole is being treated as a compound thermodynamic system composed of its outer and inn horizons, then it is possible to supply a routine to settle down the contradiction inside the third law (Nernst theorem) [25, 26]. Clearly the origin problem of black hole entropy, information loss problem and the interpretation of the third laws constitute three important but still unsolved subjects in black hole physics. Thus one should keep an open eye on this issue. To summarize, we have considered not only the thermodynamic laws on the event horizon but also those on the inner horizon, which is generally called as horizon thermodynamics. They relate thermodynamic quantities on the horizon with the global charges defined at the infinity. In Chapter 4, we shall encounter another conformal version of thermodynamics which relates the same global charges but with thermodynamic quantities reflecting the conformal symmetry of the near-horizon geometry.

18 Chapter 3 The Origin of Black Hole Entropy The thermodynamics and statistical mechanics of black holes is one of the most exciting and rapidly developing areas of black hole physics. Black holes are known to possess the properties similar to the properties of thermodynamical systems. According to this analogy, a black hole has an entropy equal to one quarter of its surface area and a temperature proportional to its surface gravity of the event horizon. Despite considerable effort [27] on the quantum [28], dynamic [29], and statistical [30] origin of black hole thermodynamics in the past thirty years, the exact source and mechanism of the Bekenstein-Hawking black hole entropy remain unclear [31]. The origin of black hole entropy is one of the most important and long-standing unsolved problems at the present stage of black hole physics. From study of Bekenstein- Hawking entropy, it gradually became clear that its resolution may be found only in a fully unified quantum theory of gravitation and matter. It is generally believed now that the explanation of the microscopic origin of the Bekenstein-Hawking entropy of black holes should be available in quantum gravity theory, whatever this theory will finally look like. Thus the statistical-mechanical derivation of the black hole entropy is a highly non-trivial test for a fundamental theory of quantum gravity. (See Ref. [32] for an early review.) In the last few years there also appeared a hope that an understanding of black hole entropy may be possible even without knowing the details of quantum gravity. The thermodynamics of black holes is a low energy phenomenon, so only a few general features of the fundamental theory may be really important [33]. Our aim of this chapter is to present a brief review on the current status of black hole entropy. There were many attempts to explain the origin of the black hole entropy in the past thirty years. Related researches can be classified as: classical and semi-classical quantum field theory (for example, entanglement entropy and brick wall model), induced gravity theory, string/m-theory and D-brane physics, loop quantum gravity (Loop QG) and quantum geometry as well as conformal field theory, etc. Among them, string/mtheory, D-brane statistical mechanics and loop QG are the theory-dependent approaches to explain the of the black hole entropy. In what follows, we shall briefly review these methods. 11

19 12 Chapter 3 The Origin of Black Hole Entropy 3.1 Classical and Semi-classical Theory In the last thirty years many authors tried to give answers to the origin problem of black hole entropy. To answer the problem of a dynamical origin for black hole entropy is to achieve a statistical mechanics explanation of it. This means to give an interpretation of horizon thermodynamics in a familiar way, in the sense that it could be seen as related to dynamical degrees of freedom associated with black hole entropy. In order to understand the real nature of horizon thermodynamics, one still has to answer another important question: How does General Relativity know about black hole thermodynamics? As we have seen, the four laws of black hole thermodynamics have been formulated some years before Hawking s discovery of quantum radiation. Most of these appear in fact as General Relativity theorems of black hole dynamics and so are already encoded at a classical level. But in spite of this, the consistence of this frame is achieved only at the quantum stage by the introduction of Hawking radiation. How can geometry know about quantum matter behavior is still an unsolved question. During the past thirty years, there were many attempts to provide a microscopic statistical explanation of the Bekenstein-Hawking entropy in classical and smi-classical theory. Here I shall only quote some of major proposals: 1. Bekenstein s Proposal. Bekenstein [34] initially proposed that the black hole entropy can be seen as a S = ln W, where W is the number of possible microscopical configurations of astrophysical body that generate the same black hole (relation to the no hair theorem). 2. York s Opinion. York [35] advocated that the dynamical degrees of freedom at the origin of black hole entropy are identified with black hole quasi normal modes. 3. Topological Origin. Gibbons and Hawking [36] took the view that black hole entropy has a topological origin, depending crucially on the presence of a horizon. Liberati and Pollifrone [37] argued that the topological structure of space-time determines the presence of gravitational entropy, and the Bekenstein-Hawking entropy formula should be extended as S = χa/8, where χ is the Euler number of the spacetimes. As we have known, in the Einstein theory of general relativity the Bekenstein- Hawking entropy is a pure geometrical quantity. In Euclidean gravity the black hole entropy is associated with the topology of an instanton which corresponds to a black hole [36, 38 41]. Instantons are non singular solutions of the classical equations in 4- dimensional Euclidean space. For a wide class of black hole space-times, metrics that extremize the Euclidean action are gravitational instantons if one removes the conical

20 Chapter 3 The Origin of Black Hole Entropy 13 singularity at the horizon. This forces to fix a period for the imaginary time. It is well known that Euclidean quantum field theory with periodic imaginary time is equivalent to a finite temperature quantum field theory in pseudoeuclidean space-time, the temperature being the inverse of imaginary time period. Thermodynamical aspect of black holes appear more apparent in Euclidean pathintegral approach. Considering the generating functional of the Euclidean theory with the action equal to the Einstein-Hilbert one plus the matter contribution, in a semiclassical approach, a calculation of the partition function shows that the tree level contribution is due only to gravitational part and S emerges as a boundary contribution to the geometrical part of the Euclidean action. (A non-extremal horizon is represented by a regular point in the Euclidean sector, so the presence of a horizon corresponds to the absence of an inner boundary in this sector.) Thermodynamics appears in this way as a request of consistence of quantum field theory on black hole space-times with Killing horizon. 4. Noether Charge Interpretation. Wald [42] defined the black hole entropy as a Noether charge associated with a bifurcating Killing horizon. A Killing horizon is null hypersurface whose null generators are orbits of a Killing vector field. In General Relativity it has been demonstrated that the event horizon of a stationary black hole is always a Killing horizon. The existence of a Killing vector reflects the symmetry of the spacetimes. In this symmetry-based method, black hole entropy is identified with the Noether charge, associated to a diffeomorphism invariant theory, in presence of bifurcate Killing horizons. 5. Entanglement Entropy Explanation. Srednicki [43], Bombelli et al. [28], Frolov and Novikov [29] argued that the black hole entropy can be obtained by identifying the dynamical degrees of freedom of a black hole with the states of all fields which are located inside the black hole. Using the brick wall model, t Hooft [44] identified the black hole entropy with the statistical entropy of a thermal gas of quantum quantum field excitations with a mirror-like boundary just outside the event horizon. It should be noted that the matter entropy calculated by the brick wall model is also a kind of entanglement entropy, it is one loop correction to the tree level (classical) gravitational contribution [45]. 6. Shell Entropy. Pretorius et al. [46] identified the black hole entropy with the thermodynamical entropy of a shell in thermal equilibrium with acceleration radiation due to the shell s gravity in the limit that the shell forms a black hole. The first two proposals appear at the moment not successful due to their appeal

21 14 Chapter 3 The Origin of Black Hole Entropy to a count over the entire life of black hole. This point has been stigmatized by a Gedanken experiment by Frolov and Novikov [47]. In this work a wormhole is used as a device to explore the backside of the horizon, the behavior found seems to show a deep relation between some sort of dynamical degrees of freedom living behind the horizon and black hole entropy. In the topological interpretation of black hole, it was difficult to justify ignoring quantum corrections. In addition, the continuation from Lorentzian to Euclidean signature in quantum gravity is not well understood. The nature of the microscopic degrees of freedom giving rise to the black hole entropy was therefore still obscured from view. The fourth proposal (that of Noether charge) shows a deep link between rescaling properties of the (gravitational plus matter) action and the presence of thermodynamical behaviour in presence of bifurcate Killing horizons. Unfortunately, although very impressive, it lacks in giving a properly statistical interpretation of black hole entropy. In this sense it is more a way of recovering Bekenstein-Hawking results that casts a new light on the nature of the problem than an interpretative frame. On the other hand, entanglement entropy is often speculated as a strong candidate for the statistical origin of the black hole entropy. So in the rest of this section we are focusing on this proposal Entanglement Entropy The idea of relating the black hole entropy to its quantum excitations was formulated first in [30, 44, 48] and it has stimulated the study of statistical mechanics of quantum fields in the presence of a Killing horizon. The properties of physical vacuum, especially in the presence of gravity, are nontrivial. In the state of vacuum there always exist zero-point fluctuations of physical fields. An observer at rest near a horizon would register these zero-point fluctuations in the form of a thermal atmosphere of a black hole [48, 49]. Historically the first suggestions to relate the entropy of a black hole to the entropy of its thermal atmosphere were made in Refs. [30, 44]. t Hooft [44] estimated the thermal entropy by assuming that the red-shifted temperature of the atmosphere is Hawking temperature and showed that the entropy is proportional to the horizon area. To avoid the divergences t Hooft assumed that fields vanish within some distance near the horizon (the corresponding model was called the brick wall model). If this distance is of the order of a Planck lengths the thermal entropy turns out to be comparable to S. The reason why the static observer near the black hole sees the vacuum as a mixed state is explained by the loss of the information about a part of the quantum system

22 Chapter 3 The Origin of Black Hole Entropy 15 located inside a the black hole horizon. It has been shown [28, 43] that even in a flat space when observations in vacuum are restricted by a part of the system, the entropy is not zero and is proportional to the surface area of the restricted region Ω. A similar result for the entropy was also established for non-zero spin fields [50] and for the pure sates different from the vacuum [51]. The non-vanishing entropy appears because the observable and non-observable vacuum fluctuations are entangled (correlated) on the boundary of Ω. By taking into account these properties the authors of Refs. [28, 43] suggested to explain S as the entropy of entanglement between quantum fluctuations propagating on the different sides of the horizon. Frolov and Novikov [29] proposed to relate the black hole entropy to the degrees of freedom corresponding to quantum states in the black hole interior. The density matrix of these degrees of freedom can be obtained by averaging the quantum state of the complete system over the states of fields located outside the black hole. For modes in the close vicinity to the horizon this density matrix is thermal. The particles are created in pairs, and only one of the components can be created outside the horizon. A pair inside the black hole is in a pure state and does not contribute to the entropy. For this reason, the statistical mechanical entropy is connected with entanglement, and it can be written in the form of summation over the modes in the black hole exterior. In other words, this approach incorporates main features of the earlier approaches [28, 30, 44]. A remarkable property of a black hole is that its entaglement entropy, and entropy connected with its thermal atmosphere coincide [27, 52 55]. In the literature this quantity is often referred to statistical-mechanical entropy. Small fluctuations of fields (including the gravitational one) propagating in the black hole background can be related to small deformations of the black hole geometry. This can be explicitly demonstrated in the approach using the no-boundary wave function of the black hole [55]. For this reason, counting states of quantum fields is connected with the counting the states of quantum excitations of the black hole. Now let us consider a stationary black hole and define ˆρ init the density matrix describing, in Heisenberg representation, the initial state of quantum matter propagating on its background. For an external observer the system consist of two parts: black hole and radiation outside of it. By defining a spacelike hypersurface we can consider quantum modes of radiation at a given time so that they can be separated in external and internal to the black hole. The state for external radiation is obtainable from ˆρ init by tracing on all the state of matter inside the event horizon and so inaccessible to the external observer ˆρ rad = T r inv ˆρ init.

23 16 Chapter 3 The Origin of Black Hole Entropy For a black hole alone this density matrix would describe its Hawking radiation at infinity. We can also define the density matrix for the black hole state ˆρ bh = T r vis ˆρ init, where one now performs the trace on external degrees of freedom. From this matrix is it possible to find the related von Neumann entropy S = T r inv (ˆρ bh ln ˆρ bh ). (3. 1) This is exactly what is called entanglement entropy. It is important to note that this definition is invariant in the sense that independent changes of definitions of vacuum for external and internal states do not change the value of S. According to the entanglement entropy interpretation, black hole entropy is generated by dynamical degree of freedom, excited at a certain time, associated to the matter in black hole interior near the horizon through non-causal correlation with external matter. The ignorance of an observer outside the black hole about the modes inside the horizon is associated with a so called entanglement entropy which is identified with Bekenstein-Hawking one. Entanglement entropy is a statistical entropy measuring the information loss due to a spatial division of a system [28]. The entanglement entropy is based only on the spatial division, and can be defined independently of the theory, although explicit calculations are dependent on the model employed. Moreover it is expected independently of the details of the theory that the entanglement entropy is proportional to the area of the boundary of the spatial division. In this sense, the entanglement entropy is considered to be a strong candidate for the statistical origin of the black hole entropy. Nevertheless entanglement entropy is always divergent on division plane due to the presence of modes of arbitrary high modes near the horizon. The divergence form is general and independent on the kind of field. There are two different approaches for the divergence problem resolution: regularization and renormalization. The main idea of regularization [29, 55] approach is to apply a physical cut-off to entropy by justifying it through the quantum fluctuations of the event horizon. This cut-off has been estimated to be of the order of the Planck length. Remarkably the introduction of such a cutoff gives a value of entanglement entropy of the same order of the Bekenstein-Hawking one. The second approach is instead based on elimination of divergences through a renormalization of gravitational coupling constant [56] and of constants related to second order curvature terms [57, 58]. This way appears very interesting for its relation to elementary particle physics but is penalized by its necessity of a renormalizable theory of quantum gravity in order to give exact results.

24 Chapter 3 The Origin of Black Hole Entropy 17 The regularization approach has been considered in Ref. [59] in order to probe its consistency. The first idea of regularization of entanglement entropy was implicitly proposed by t Hooft [44] as applied to his brick wall model. One of the problems t Hooft proposed in his seminal work was the divergence of not only entropy but also of quantum matter contribute to internal energy of the black hole, which has to be regularized by using the same cut-off one has to introduce for entropy. He found that, fixing the cut-off in order to obtain S ent = S = A/4, one obtains U = 3M/8. So matter contribution to internal energy appeared to be a very consistent fraction of the black hole mass M. As t Hooft underlined, this is a signal for a strong back-reaction effect, not a good aim for a model based on semiclassical (negligible back-reaction) approximation. The same problem is also present in Barvinsky-Frolov-Zelnikov model [55] Brick Wall Model The entanglement interpretation seems to be implicit in, and is certainly closely related to a pioneering calculation done by t Hooft [44] in his brick wall model. He considered the statistical thermodynamics of quantum fields in the Hartle-Hawking state (i.e. having the Hawking temperature at large radii) propagating on a fixed Schwarzschild background of mass M. To control divergences, t Hooft introduced a brick wall actually a static spherical mirror at which the fields are required to satisfy Dirichlet or Neumann boundary conditions with radius a little larger than the gravitational radius 2M. t Hooft found, in addition to the expected volume-dependent thermodynamical quantities describing hot fields in a nearly flat space, additional wall contributions proportional to the area. These contributions are, however, also proportional to α 2, where α is the proper altitude of the wall above the gravitational radius, and thus diverge in the limit α 0. For a specific choice of α (which depends on the number of fields, etc., but is generally of order l pl ), t Hooft was able to recover the Bekenstein-Hawking formula with the correct coefficient. However, this calculation raises a number of questions which have caused many, including t Hooft himself, to have reservations about its validity and consistency. (a) S is here obtained as a one-loop effect, originating from thermal excitations of the quantum fields. Does this material contribution to S have to be added to the zero-loop Gibbons-Hawking contribution which arises from the gravitational part of the action and already by itself accounts for the full value of S? [59] (b) The ambient quantum fields were assumed to be in the Hartle-Hawking state. Their stress-energy should therefore be bounded (of order M 4 in Planck units)

25 18 Chapter 3 The Origin of Black Hole Entropy near the gravitational radius, and negligibly small for large masses. However, t Hooft s calculation assigns to them enormous (Planck-level) energy densities near the wall. (c) The integrated field energy gives a wall contribution to the mass M = 3M/8 when α is adjusted to give the correct value of S. This suggests a substantial gravitational back-reaction [44] and that the assumption of a fixed geometrical background may be inconsistent [56, 59, 60]. S. Mukohyama [61] pointed out that these difficulties are only apparent and easy to resolve. The basic remark is that the brick-wall model strictly interpreted does not represent a black hole. It represents the exterior of a starlike object with a reflecting surface, compressed to nearly (but not quite) its gravitational radius. The ground state for quantum fields propagating around this star is not the Hartle-Hawking state [38] but the Boulware state [62], corresponding to zero temperature, which has a quite different behavior near the gravitational radius. We may conclude that many earlier concerns [44, 59, 60] were unnecessary: t Hooft s brick wall model does provide a perfectly self-consistent description of a configuration which is indistinguishable from a black hole to outside observers, and which accounts for the Bekenstein-Hawking entropy purely as thermal entropy of quantum fields at the Hawking temperature (i.e. in the Hartle-Hawking state), providing one accepts the ad hoc but plausible ansatz for a Planck-length cutoff near the horizon. The model does, however, present us with a feature which is theoretically possible but appears strange and counterintuitive from a gravitational theorist s point of view. Although the wall is insubstantial (just like a horizon) i.e., space there is practically a vacuum and the local curvature low it is nevertheless the repository of all of the Bekenstein-Hawking entropy in the model. It has been argued [29] that this is just what might be expected of black hole entropy in the entanglement picture. Entanglement will arise from virtual pair-creation in which one partner is invisible and the other visible (although only temporarily nearly all get reflected back off the potential barrier). Such virtual pairs are all created very near the horizon. Thus, on this picture, the entanglement entropy (and its divergence) arises almost entirely from the strong correlation between nearby field variables on the two sides of the partition, an effect already present in flat space [52]. Complementarity. t Hooft [44] sought the origin of S in the thermal entropy of ambient quantum fields raised to the Hawking temperature. He derived an expression which is indeed proportional to the area, but with a diverging coefficient which has to be