Black Holes Physics

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Advances in High Energy Physics

Black Holes Physics

Guest Editors: Xiaoxiong Zeng, Christian Corda, and Deyou Chen Black Holes Physics Advances in High Energy Physics

Black Holes Physics

This is a special issue published in “Advances in High Energy Physics.” All articles are open access articles distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Editorial Board

Luis A. Anchordoqui, USA Maria Giller, Poland Reinhard Schlickeiser, Germany Marco Battaglia, USA Xiaochun He, USA Frederik G. Scholtz, South Africa Botio Betev, Switzerland Hong-Jian He, China Sally Seidel, USA Rong-Gen Cai, China Ian Jack, UK George Siopsis, USA Duncan L. Carlsmith, USA Filipe R. Joaquim, Portugal Terry Sloan, UK Kingman Cheung, Taiwan Kyung Joo, Republic of Korea Luca Stanco, Italy Ashot Chilingarian, Armenia Michal Kreps, UK EliasC.Vagenas,Kuwait Sridhara Dasu, USA Ming Liu, USA Nikos Varelas, USA Shi-Hai Dong, Mexico Piero Nicolini, Germany K. S. Viswanathan, Canada Edmond C. Dukes, USA Seog H. Oh, USA Yau W. Wah, USA Amir H. Fatollahi, Iran Stephen Pate, USA Moran Wang, China Frank Filthaut, The Netherlands Anastasios Petkou, Greece Gongnan Xie, China Chao-Qiang Geng, Taiwan Alexey A. Petrov, USA Contents

Black Holes Physics, Xiaoxiong Zeng, Christian Corda, and Deyou Chen Volume 2014, Article ID 453586, 2 pages

State-Space Geometry, Statistical Fluctuations, and Black Holes in String Theory, Stefano Bellucci and Bhupendra Nath Tiwari Volume 2014, Article ID 589031, 17 pages

Quantum Tunnelling for Hawking Radiation from Both Static and Dynamic Black Holes, Subenoy Chakraborty and Subhajit Saha Volume 2014, Article ID 168487, 9 pages

Analyzing Black Hole Super-Radiance Emission of Particles/Energy from a Black Hole as a Gedankenexperiment to Get Bounds on the Mass of a Graviton,A.Beckwith Volume 2014, Article ID 230713, 7 pages

Magnetic String with a Nonlinear 𝑈(1) Source, S. H. Hendi Volume 2014, Article ID 697914, 10 pages

HolographicBrownianMotioninThree-DimensionalGodel¨ Black Hole, J. Sadeghi, F. Pourasadollah, and H. Vaez Volume2014,ArticleID762151,9pages

Holographic Screens in Ultraviolet Self-Complete Quantum Gravity, Piero Nicolini and Euro Spallucci Volume 2014, Article ID 805684, 9 pages

Black Holes and Quantum Mechanics,B.G.Sidharth Volume 2014, Article ID 606439, 4 pages

Particle Collisions in the Lower Dimensional Rotating Black Hole Space-Time with the Cosmological Constant,JieYang,Yun-LiangLi,YangLi,Shao-WenWei,andYu-XiaoLiu Volume 2014, Article ID 204016, 7 pages

Entropy Spectrum of a KS Black Hole in IR Modified Horava-Lifshitzˇ Gravity,ShiweiZhou, Ge-Rui Chen, and Yong-Chang Huang Volume2014,ArticleID396453,6pages

The Critical Phenomena and Thermodynamics of the Reissner-Nordstrom-de Sitter Black, Hole Ren Zhao, Mengsen Ma, Huihua Zhao, and Lichun Zhang Volume2014,ArticleID124854,6pages

ResearchingonHawkingEffectinaKerrSpaceTimeviaOpenQuantumSystemApproach, Xian-Ming Liu and Wen-Biao Liu Volume 2014, Article ID 794626, 8 pages

The Geometry of Black Hole Singularities,OvidiuCristinelStoica Volume2014,ArticleID907518,14pages

Intermediate Mass Black Holes: Their Motion and Associated Energetics, C. Sivaram and Kenath Arun Volume 2014, Article ID 924848, 8 pages Electrostatics in the Surroundings of a Topologically Charged Black Hole in the Brane, Alexis Larranaga,˜ Natalia Herrera, and Sara Ramirez Volume 2014, Article ID 146094, 6 pages

Energy Loss of a Heavy Particle Near 3D Rotating Hairy Black Hole, Jalil Naji and Hassan Saadat Volume 2014, Article ID 720713, 5 pages

Hawking Radiation-Quasi-Normal Modes Correspondence and Effective States for Nonextremal Reissner-Nordstrom¨ Black Holes,C.Corda,S.H.Hendi,R.Katebi,andN.O.Schmidt Volume 2014, Article ID 527874, 9 pages

A Little Quantum Help for Cosmic Censorship and a Step Beyond All That, Nikolaos Pappas Volume 2013, Article ID 236974, 4 pages Hindawi Publishing Corporation Advances in High Energy Physics Volume 2014, Article ID 453586, 2 pages http://dx.doi.org/10.1155/2014/453586

Editorial Black Holes Physics

Xiaoxiong Zeng,1 Christian Corda,2,3,4 and Deyou Chen5

1 School of Science, Chongqing Jiaotong University, Nanan 400074, China 2 Dipartimento di Fisica e Chimica, Istituto Universitario di Ricerca Scientifica “Santa Rita,”Centro di Scienze Naturali, ViadiGalceti,74,59100Prato,Italy 3 Institute for Theoretical Physics and Advanced MathematicsM)Einstein-Galilei,ViaSantaGonda14,59100Prato,Italy (IF 4 International Institute for Applicable Mathematics & Information Sciences (IIAMIS), B.M. Birla Science Centre, Adarsh Nagar, Hyderabad 500 463, India 5 Institute of Theoretical Physics, China West Normal University, Nanchong, Sichuan 637009, China

Correspondence should be addressed to Xiaoxiong Zeng; [email protected]

Received 15 May 2014; Accepted 15 May 2014; Published 28 May 2014

Copyright © 2014 Xiaoxiong Zeng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP3.

Black hole (BH) is perhaps the most fascinating object in The paper “Intermediate mass black holes: their motion the research fields of astrophysics and gravitational physics. and associated energetics,” by C. Sivaram and A. Kenath, This mysterious object is predicted by the general theory of is devoted to exploring the astrophysical signatures and relativity (GTR), but such a classical theory cannot explain all evidences for the intermediate mass BHs. Especially authors its properties. It is believed that only a more general, definitive describe the specific features of their motion and energetics, theory of quantum gravity, which should unify GTR with related with the Bondi accretion. quantum mechanics, should clarify all the mysteries of BH The paper “Energy Loss of a heavy particle near 3d rotating physics, starting from the unsolved problem of the singularity hairy black hole,” by J. Naji and H. Saadat, considers rotating in the BH’s core to arrive to the BH information puzzle and BH in 3 dimensions with a scalar charge and discusses energy to the last stages of the BH evaporation, where very high loss of heavy particle moving near the BH horizon. energies are involved. In fact, it is general conviction that The paper “Holographic screens in ultraviolet self-complete black holes result in highly excited states representing both quantum gravity,” by P. Nicolini and E. Spallucci, investigates the “hydrogen atom” and the “quasi-thermal emission” in the idea of a short distance fundamental scale below which quantum gravity. On this issue we recall that an exciting con- it is not possible to probe and analyzes the geometry and sequence of TeV-scale quantum gravity could be the potential thermodynamics of a holographic screen in the framework production of mini-BHs in high-energy experiments, like the of the ultraviolet self-complete quantum gravity. LHC and beyond. The paper “Black holes and quantum mechanics,”byB.G. This special issue on BH physics consists of 17 interesting Sidharth, reconsiders BHs in the context of general relativity and well written papers. critically reviewing all problems relating these phenomena. The paper “Alittlequantumhelpforcosmiccensorshipand The paper “Magnetic string with a nonlinear U(1) source,” astepbeyondallthat”, by N. Pappas, discusses the weak and by Seyed H. Hendi, deals with the study of magnetic string strong versions of the cosmic censorship conjecture and also solutions in Einstein gravity in the presence of nonlinear deals with the well-known problem of naked singularities. electrodynamics. Also the effects of these nonlinear fields The paper “On the critical phenomena and Thermody- as well as other properties of the solutions are investigated, namics of the Reissner-Nordstrom-de Sitter black hole,”byR. precisely. Zhao et al., deals with the effective thermodynamic quantities The paper “ResearchingonHawkingeffectinaKerrspace in Reissner-Nordstrom-de Sitter BH by also discussing its time via open quantum system approach,” by X.-M. Liu and thermodynamic stability. W.-B.Liu,investigatestheHawkingeffectinaKerrspacetime 2 Advances in High Energy Physics in the framework of open quantum systems, showing that the correspondence between Hawking radiation and BH Hawking effect of the Kerr space time can also be understood quasi-normal modes and defines the concept of “effective as the manifestation of thermalization phenomena via open state” for the non-extremal Reissner-Nordstrom black holes. quantum system approach. The work is the extension of earlier works of the same The paper “Holograghic Brownian motion in three dimen- research group and contributes to the understanding of the sional Godel¨ black hole,” by J. Sadeghi et al., uses the AdS/CFT quantum properties of BHs. correspondence and Godel¨ BH background to study the dynamics of heavy quark under a rotating plasma. Xiaoxiong Zeng The paper “EntropyspectrumofaKSblackholeinIR Christian Corda modified Hovrava-Lifshitz gravity,”byS.Zhouetal.,discusses Deyou Chen theentropyspectrumandareaspectrumofaKSBHbased on the proposal of adiabatic invariant quantity. It is found that that the entropy spectrum is discrete and equidistant spaced and the area spectrum is not equidistant spaced, which depends on the parameter of gravity theory. The paper “Particle collisions in the lower dimensional rotating black hole space-time with the cosmological constant,” by J. Yang et al., deals with the effect of ultrahigh energy collisions of two particles with different energies near the horizon of a 2 + 1 dimensional BTZ BH (BSW effect), finding that the particle with the critical angular momentum could exist inside the outer horizon of the BTZ BH regardless of the particle energy. The paper “The geometry of black hole singularities,”byO. C. Stoica, is a review about singularity problem in general and BH singularity in particular. The importance of dimensional reduction effects is also stressed. The paper “State-space geometry, statistical fluctuations and black holes in string theory,”byS.BellucciandB.N. Tiwari, considers statistical properties of the charged and anticharged BH configurations by using the notion of the thermodynamic geometry. The authors highlight the utility of thermodynamic geometry in understanding the state space correlations and fluctuations in BHs in string theory while treating them thermodynamically. The paper “Quantum tunnelling for Hawking radiation from both static and dynamic black hole,”byS.Chakraborty and S. Saha, deals with the well-known Hawking radiation andquantumcorrectionsinordertofurtherimprovethe theory from the semiclassical approach. The authors specially study the Hawking radiation from both static and nonstatic spherically symmetric BHs. The paper “Electrostatics in the surroundings of a topologically charged black hole in the brane,”byA.Larranaga˜ et al., studies the EM properties of 4D BH due to brane contributions, determining the electrostatic potential generated by a static point-like charge in the brane-world space-time of a BH with topological (or tidal) charge. The paper “Analyzing black hole super-radiance emission of particles/energy from a black hole as a Gedankenexper- iment to get bounds on the mass of a graviton,”byA. Beckwith, discusses the process of particles emission and adopted a standard approach proposed by Padmanabhan. In the author’s point of view “super-radiance allows massive gravity to be consistent with BH physics and general relativity.” The paper “Hawking radiation-quasi-normal modes correspondence and effective states for non-extremal Reissner- Nordstrom black holes,” by C. Corda et al., investigates Hindawi Publishing Corporation Advances in High Energy Physics Volume 2014, Article ID 589031, 17 pages http://dx.doi.org/10.1155/2014/589031

Research Article State-Space Geometry, Statistical Fluctuations, and Black Holes in String Theory

Stefano Bellucci and Bhupendra Nath Tiwari

INFN-Laboratori Nazionali di Frascati, Via E. Fermi 40, 00044 Frascati, Italy

Correspondence should be addressed to Stefano Bellucci; [email protected]

Received 5 February 2014; Accepted 8 April 2014; Published 7 May 2014

Academic Editor: Christian Corda

Copyright © 2014 S. Bellucci and B. N. Tiwari. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP3.

We study the state-space geometry of various extremal and nonextremal black holes in string theory. From the notion of the intrinsic geometry, we offer a state-space perspective to the black hole vacuum fluctuations. For a given black hole entropy, we explicate the intrinsic geometric meaning of the statistical fluctuations, local and global stability conditions, and long range statistical correlations. We provide a set of physical motivations pertaining to the extremal and nonextremal black holes, namely, the meaning of the chemical geometry and physics of correlation. We illustrate the state-space configurations for general charge extremal black holes. In sequel, we extend our analysis for various possible charge and anticharge nonextremal black holes. From the perspective of statistical fluctuation theory, we offer general remarks, future directions, and open issues towards the intrinsic geometric understanding of the vacuum fluctuations and black holes in string theory.

1. Introduction nonextremal black holes in string theory, in general. String theory [30], as the most promising framework to understand In this paper, we study statistical properties of the charged all possible fundamental interactions, celebrates the physics and anticharged black hole configurations in string theory. of black holes, in both the zero and the nonzero temperature Specifically, we illustrate that the components of the vacuum domains. Our consideration hereby plays a crucial role fluctuations define a set of local pair correlations against in understanding the possible phases and stability of the the parameters, for example, charges, anticharges, mass, string theory vacua. A further motivation follows from the and angular momenta. Our consideration follows from the consideration of the string theory black holes; namely, N =2 notion of the thermodynamic geometry, mainly introduced supergravity arises as a low energy limit of the Type II string by Weinhold [1, 2] and Ruppeiner [3–9]. Importantly, this theory solution, admitting extremal black holes with the zero framework provides a simple platform to geometrically Hawking temperature and a nonzero macroscopic attractor understand the statistical nature of local pair correlations entropy. and underlying structures pertaining to the vacuum phase A priori, the entropy depends on a large number of transitions. In diverse contexts, the state-space geometric scalar moduli arising from the compactification of the 10- perspective offers an understanding of the phase structures dimensional theory down to the 4-dimensional physical of mixtures of gases, black hole configurations [10–26], spacetime. This involves a 6-dimensional compactifying generalized uncertainty principle [27], strong interactions, manifold. Interesting string theory compactifications involve 6 2 for example, hot QCD [28], quarkonium configurations29 [ ], 𝑇 , 𝐾3 ×𝑇, and Calabi-Yau manifolds. The macroscopic and some other systems, as well. entropy exhibits a fixed point behavior under the radial The main purpose of the present paper is to consider flow of the scalar fields. In such cases, the near horizon the state-space properties of various possible extremal and geometry of an extremal black hole turns out to be an 2 Advances in High Energy Physics

2 𝐴𝑑𝑆2 ×𝑆 manifold which describes the Bertotti-Robinson From the perspective of the intrinsic state-space geometry, vacuum associated with the black hole. The area of the black if one pretends that the notion of statistical fluctuations hole horizon is 𝐴 and thus the macroscopic entropy [31– applies to intermediate regimes of the moduli space, then 𝑆 =𝜋|𝑍|2 42]isgivenas macro ∞ . This is known as the the attractor horizon configurations require an embedding Ferrara-Kallosh-Strominger attractor mechanism, which, as to the higher dimensional intrinsic Riemanian manifold. the macroscopic consideration, requires a validity from the Physically, such a higher dimensional manifold can be viewed microscopic or statistical basis of the entropy. In this concern, as a possible blow-up of the attractor fixed point phase- there have been various investigations on the physics of black space to a nontrivial moduli space. From the perspective holes, for example, horizon properties [43, 44], counting of thermodynamic Ruppenier geometry, we have offered of black hole microstates [45–47], spectrum of half-BPS future directions and open issues in the conclusion. We leave states in N =4supersymmetric string theory [48], and the explicit consideration of these matters open for further fractionation of branes [49]. From the perspective of the research. fluctuation theory, our analysis is intended to provide the In Section 2, we define the general notion of vacuum nature of the statistical structures of the extremal and nonex- fluctuations. This offers the physical meaning of the state- tremal black hole configurations. The attractor configurations space geometry. In Section 3, we provide a brief review of exist for the extremal black holes, in general. However, the statistical fluctuations. In particular, for a given black hole corresponding nonextremal configurations exist in the throat entropy, we firstly explicate the statistical meaning of state- approximation. In this direction, it is worth mentioning that space surface and then offer the general meaning of the there exists an extension of Sen entropy function formalism local and global stability conditions and long range statistical for 𝐷1𝐷5 and 𝐷2𝐷6𝑁𝑆5 nonextremal configurations [50– correlations. In Section 4, we provide a set of physical 52]. In the throat approximation, these solutions, respectively, motivations pertaining to the extremal and nonextremal 3 4 correspond to Schwarzschild black holes in 𝐴𝑑𝑆3 ×𝑆 ×𝑇 black holes, the meaning of Wienhold chemical geometry, 2 1 4 and 𝐴𝑑𝑆3 ×𝑆 ×𝑆 ×𝑇 . In relation with the intrinsic state- and the physics of correlation. In Section 5, we consider space geometry, we will explore the statistical understanding state-space configurations pertaining to the extremal black of the attractor mechanism and the moduli space geometry holes and explicate our analysis for the two and three charge and explain the vacuum fluctuations of the black brane configurations. In Section 6, we extend the above analysis for configurations. the four, six, and eight charge-anticharge nonextremal black In this paper, we consider the state-space geometry of holes. Finally, Section 7 provides general remarks, conclusion the spherical horizon topology black holes in four spacetime and outlook, and future directions and open issues towards dimensions. These configurations carry a set of electric mag- the application of string theory. netic charges (𝑞𝑖,𝑝𝑖).DuetotheconsiderationofStrominger and Vafa [53],thesechargesareassociatedwithanensemble 2. Definition of State-Space Geometry of weakly interacting D-branes. Following [53–59], it turns outthatthecharges(𝑞𝑖,𝑝𝑖) are proportional to the number of Considering the fact that the black hole configurations in electric and magnetic branes, which constitute the underlying string theory introduce the notion of vacuum, it turns out ensemble of the chosen black hole. In the large charge limit, for any thermodynamic system, that there exist equilibrium namely, when the number of such branes becomes large, thermodynamic states given by the maxima of the entropy. we have treated the logarithm of the degeneracy of states These states may be represented by points on the state-space. of the statistical configuration as the Bekenstein-Hawking Along with the laws of the equilibrium thermodynamics, entropy of the associated string theory black holes. For the the theory of fluctuations leads to the intrinsic Riemannian extremal black holes, the entropy is described in terms of geometric structure on the space of equilibrium states [8, 9]. the number of the constituent D-branes. For example, the The invariant distance between two arbitrary equilibrium two charge extremal configurations can be examined in terms states is inversely proportional to the fluctuations connecting of the winding modes and the momentum modes of an the two states. In particular, a less probable fluctuation means {𝑋 } excited string carrying 𝑛1 winding modes and 𝑛𝑝 momentum that the states are far apart. For a given set of states 𝑖 ,the modes. Correspondingly, the state-space geometry of the state-space metric tensor is defined by nonextremal black holes is described by adding energy to 𝑔𝑖𝑗 (𝑋) =−𝜕𝑖𝜕𝑗𝑆(𝑋1,𝑋2,...,𝑋𝑛). (1) the extremal D-branes configurations. This renders as the contribution of the clockwise and anticlockwise momenta in Aphysicalmotivationof(1) can be given as follows. Up to the Kaluza-Klein scenarios and that of the antibrane charges the second order approximation, the Taylor expansion of the in general to the black hole entropy. entropy 𝑆(𝑋1,𝑋2,...,𝑋𝑛) yields From the perspective of black hole thermodynamics, we 1 𝑛 describe the structure of the state-space geometry of four- 𝑆−𝑆 =− ∑ 𝑔 Δ𝑋𝑖Δ𝑋𝑗, 0 2 𝑖𝑗 (2) dimensional extremal and nonextremal black holes in a given 𝑖=1 duality frame. Thus, when we take arbitrary variations over where the charges (𝑞𝑖,𝑝𝑖) on the electric and magnetic branes, the underlying statistical fluctuations are described by only the 𝜕2𝑆(𝑋 ,𝑋 ,...,𝑋 ) 𝑔 := − 1 2 𝑛 =𝑔 (3) numbers of the constituent electric and magnetic branes. 𝑖𝑗 𝜕𝑋𝑖𝜕𝑋𝑗 𝑗𝑖 Advances in High Energy Physics 3 is called the state-space metric tensor. In the present investi- 3.2. State-Space Surface. The Ruppenier metric on the state- gation, we consider the state-space variables {𝑋1,𝑋2,...,𝑋𝑛} space (𝑀2,𝑔)oftwochargeblackholesisdefinedby as the parameters of the ensemble of the microstates of the underlying microscopic configuration (e.g., conformal 𝜕2𝑆(𝑞,𝑝) 𝜕2𝑆(𝑞,𝑝) 𝑔 =− ,𝑔=− , field theory [60], black hole conformal field theory61 [ ], and 𝑞𝑞 𝜕𝑞2 𝑞𝑝 𝜕𝑞𝜕𝑝 hidden conformal field theory [62, 63]), which defines the (8) corresponding macroscopic thermodynamic configuration. 𝜕2𝑆(𝑞,𝑝) 𝑔 =− . Physically, the state-space geometry can be understood as the 𝑝𝑝 𝜕𝑝2 intrinsic Riemannian geometry involving the parameters of the underlying microscopic statistical theory. In practice, we Subsequently, the components of the state-space metric will consider the variables {𝑋1,𝑋2,...,𝑋𝑛} as the parameters, tensor are associated with the respective statistical pair namely, charges, anticharges, and others if any, of the corre- correlation functions. It is worth mentioning that the coor- sponding low energy limit of the string theory, for example, dinates on the state-space manifold are the parameters of N =2supergravity. In the limit, when all the variables, the microscopic boundary conformal field theory which is namely, {𝑋1,𝑋2,...,𝑋𝑛}, are thermodynamic, the state-space dual the black hole space-time solution. This is because metric tensor equation (1) reduces to the corresponding the underlying state-space metric tensor comprises of the Ruppenier metric tensor. In the discrete limit, the relative 𝑖 𝑖 𝑖 𝑖 Gaussian fluctuations of the entropy which is the function coordinates Δ𝑋 are defined as Δ𝑋 := 𝑋 −𝑋0,forgiven {𝑋𝑖 }∈𝑀 of the number of the branes and antibranes. For the chosen 0 𝑛. In the Gaussian approximation, the probability black hole configuration, the local stability of the underly- distribution has the following form: ing statistical system requires both principle minors to be 1 𝑖 𝑗 positive. In this setup, the diagonal components of the state- 𝑃(𝑋1,𝑋2,...,𝑋𝑛)=𝐴exp (− 𝑔𝑖𝑗 Δ𝑋 Δ𝑋 ). (4) {𝑔 |𝑥 =(𝑛,𝑚)} 2 space metric tensor, namely, 𝑥𝑖𝑥𝑖 𝑖 ,signifythe heat capacities of the system. This requires that the diagonal With the normalization components of the state-space metric tensor ∫∏𝑑𝑋 𝑃(𝑋 ,𝑋 ,...,𝑋 )=1, 𝑖 1 2 𝑛 (5) 𝑔 >0, 𝑖= 𝑛,𝑚, 𝑖 𝑥𝑖𝑥𝑖 (9) we have the following probability distribution: be positive definite. In this investigation, we discuss the √𝑔 (𝑋) 1 𝑖 𝑗 significance of the above observation for the eight parameter 𝑃(𝑋1,𝑋2,...,𝑋𝑛)= exp (− 𝑔𝑖𝑗 𝑑𝑋 ⊗𝑑𝑋 ), (6) (2𝜋)𝑛/2 2 nonextremal black brane configurations in string theory. From the notion of the relative scaling property, we will where 𝑔𝑖𝑗 now, in a strict mathematical sense, is properly demonstrate the nature of the brane-brane pair correlations; 𝑖 𝑗 defined as the inner product 𝑔(𝜕/𝜕𝑋 ,𝜕/𝜕𝑋 ) on the corre- namely, from the perspective of the intrinsic Riemannian geometry, the stability properties of the eight parameter black sponding tangent space 𝑇(𝑀𝑛)×𝑇(𝑀𝑛). In this connotation, the determinant of the state-space metric tensor, branes are examined from the positivity of the principle 󵄩 󵄩 minors of the space-state metric tensor. For the Gaussian 󵄩 󵄩 𝑔 (𝑋) := 󵄩𝑔𝑖𝑗 󵄩 , (7) fluctuationsofthetwochargeequilibriumstatisticalconfigu- rations, the existence of a positive definite volume form on can be understood as the determinant of the corresponding the state-space manifold (𝑀2(𝑅), 𝑔) imposes such a global [𝑔 ] (𝑀 ,𝑔) matrix 𝑖𝑗 𝑛×𝑛. For a given state-space manifold 𝑛 ,we stability condition. In particular, the above configuration 𝑖 ⋆ will think of {𝑑𝑋 } as the basis of the cotangent space 𝑇 (𝑀𝑛). leads to a stable statistical basis if the determinant of the state- In the subsequent analysis, by taking an account of the fact space metric tensor, 𝑖 that the physical vacuum is neutral, we will choose 𝑋0 =0. 󵄩 󵄩 2 󵄩𝑔󵄩 =𝑆𝑛𝑛𝑆𝑚𝑚 −𝑆𝑛𝑚, (10) 3. Statistical Fluctuations remains positive. Indeed, for the two charge black brane 3.1. Black Hole Entropy. As a first exercise, we have illustrated configurations, the geometric quantities corresponding to the thermodynamic state-space geometry for the two charge underlying state-space manifold elucidate typical features of extremalblackholeswithelectriccharge𝑞 and magnetic the Gaussian fluctuations about an ensemble of equilibrium charge 𝑝. The next step has thence been to examine the branemicrostates.Inthiscase,weseethattheChristoffel thermodynamic geometry at an attractor fixed point(s) for connections on the (𝑀2,𝑔)are defined by the extremal black holes as the maxima of their macroscopic entropy 𝑆(𝑞, 𝑝). Later on, the state-space geometry Γ𝑖𝑗𝑘 =𝑔𝑖𝑗,𝑘 +𝑔𝑖𝑘,𝑗 −𝑔𝑗𝑘,𝑖. (11) of nonextremal counterparts has as well been analyzed. In this investigation, we demonstrate that the state-space The only nonzero Riemann curvature tensor is correlations of nonextremal black holes modulate relatively more swiftly to an equilibrium statistical basis than those of 𝑁 𝑅 = , the corresponding extremal solutions. 𝑞𝑝𝑞𝑝 𝐷 (12) 4 Advances in High Energy Physics where Ruppenier has in particular noticed for the black holes in general relativity that the scalar curvature 𝑁:=𝑆𝑝𝑝 𝑆𝑞𝑞𝑞𝑆𝑞𝑝𝑝 +𝑆𝑞𝑝𝑆𝑞𝑞𝑝𝑆𝑞𝑝𝑝 𝑑 𝑅 (𝑋) ∼𝜉 , (18) +𝑆 𝑆 𝑆 −𝑆 𝑆 𝑆 𝑞𝑞 𝑞𝑞𝑝 𝑝𝑝𝑝 𝑞𝑝 𝑞𝑞𝑞 𝑝𝑝𝑝 (13) where 𝑑 is the spatial dimension of the statistical system 2 2 and the 𝜉 fixes the physical scale6 [ ]. The limit 𝑅(𝑋) → −𝑆𝑞𝑞𝑆 −𝑆𝑝𝑝 𝑆 , 𝑞𝑝𝑝 𝑞𝑞𝑝 ∞ indicates the existence of certain critical points or phase 2 2 transitions in the underlying statistical system. The fact that 𝐷:=(𝑆 𝑆 −𝑆 ) . (14) 𝑞𝑞 𝑝𝑝 𝑞𝑝 “all the statistical degrees of freedom of a black hole live on the black hole event horizon” signifies that the state- 𝑅 The scalar curvature and the corresponding 𝑖𝑗𝑘𝑙 of an space scalar curvature, as the intrinsic geometric invariant, arbitrary two-dimensional intrinsic state-space manifold indicates an average number of correlated Plank areas on (𝑀 (𝑅), 𝑔) 2 may be given as the event horizon of the black hole [8]. In this concern, [9] 2 offers interesting physical properties of the thermodynamic 𝑅 (𝑞,) 𝑝 = 󵄩 󵄩𝑅 (𝑞,) 𝑝 . scalar curvature and phase transitions in Kerr-Newman black 󵄩𝑔󵄩 𝑞𝑝𝑞𝑝 (15) 󵄩 󵄩 holes. Ruppeiner [6] has further conjectured that the global correlations can be expressed by the following arguments: (a) 3.3. Stability Conditions. For a given set of variables the zero state-space scalar curvature indicates certain bits of 1 2 𝑛 {𝑋 ,𝑋 ,...,𝑋 }, the local stability of the underlying state- information on the event horizon, fluctuating independently space configuration demands the positivity of the heat of each other; (b) the diverging scalar curvature signals a capacities: phase transition indicating highly correlated pixels of the information. 𝑖 {𝑔𝑖𝑖 (𝑋 )>0;∀𝑖=1,2,...,𝑛}. (16) 4. Some Physical Motivations Physically, the principle components of the state-space metric 𝑖 tensor {𝑔𝑖𝑖(𝑋 ) | 𝑖 = 1,2,...,𝑛} signifyasetofdefi- 4.1. Extremal Black Holes. The state-space of the extremal nite heat capacities (or the related compressibilities), whose black hole configuration is a reduced space comprising of positivity apprises that the black hole solution complies an the states which respect the extremality (BPS) condition. The underlying, locally in equilibrium, statistical configuration. state-spaces of the extremal black holes show an intrinsic Notice further that the positivity of principle components geometric description. Our intrinsic geometric analysis offers is not sufficient to insure the global stability of the chosen a possible zero temperature characterization of the limiting configurationandthusonemayonlyachievealocally extremal black brane attractors. From the gauge/gravity stable equilibriumstatistical configuration. In fact, the global correspondence, the existence of state-space geometry could stability condition constraint over the allowed domain of the be relevant to the boundary gauge theories, which have parameters of black hole configurations requires that all the finitely many countable sets of conformal field theory states. principle components and all the principle minors of the metric tensor must be strictly positive definite [6]. The above 4.2. Nonextremal Black Holes. We will analyze the state- stability conditions require that the following set of equations space geometry of nonextremal black holes by the addition must be simultaneously satisfied: of antibrane charge(s) to the entropy of the corresponding extremal black holes. To interrogate the stability of a chosen 𝑝0 := 1, black hole system, we will investigate the question that the 𝑔 (𝑋 )=−𝜕𝜕 𝑆(𝑋 ,𝑋 ,...,𝑋 ) 𝑝 := 𝑔 >0, underlying metric 𝑖𝑗 𝑖 𝑖 𝑗 1 2 𝑛 should 1 11 provide a nondegenerate state-space manifold. The exact 󵄨 󵄨 dependence varies case to case. In the next section, we will 󵄨𝑔11 𝑔12󵄨 𝑝2 := 󵄨 󵄨 >0, proceedinouranalysiswithanincreasingnumberofthe 󵄨𝑔12 𝑔22󵄨 brane charges and antibrane charges. 󵄨 󵄨 󵄨𝑔11 𝑔12 𝑔13󵄨 (17) 󵄨 󵄨 𝑝3 := 󵄨𝑔12 𝑔22 𝑔23󵄨 >0, 4.3. Chemical Geometry. The thermodynamic configurations 󵄨 󵄨 󵄨𝑔13 𝑔23 𝑔33󵄨 of nonextremal black holes in string theory with small statistical fluctuations in a “canonical” ensemble are stable if . . the following inequality holds: 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩𝜕𝑖𝜕𝑗𝑆(𝑋1,𝑋2,...,𝑋𝑛)󵄩 <0. (19) 𝑝𝑛 := 󵄩𝑔󵄩 >0. The thermal fluctuations of nonextremal black holes, when 3.4. Long Range Correlations. The thermodynamic scalar considered in the canonical ensemble, give a closer approxi- curvature of the state-space manifold is proportional to the mation to the microcanonical entropy: correlation volume [6]. Physically, the scalar curvature sig- 1 𝑆=𝑆 − (𝐶𝑇2)+⋅⋅⋅. nifies the interaction(s) of the underlying statistical system. 0 2 ln (20) Advances in High Energy Physics 5

In (20), the 𝑆0 is the entropy in the “canonical” ensemble theory configuration, the choice of compactification [30] and 𝐶 is the specific heat of the black hole statistical chosen is the factorization of the type M(3,1) ×𝑀6,where configuration. At low temperature, the quantum effects 𝑀6 is a compact internal manifold. From the perspective of dominate and the above expansion does not hold anymore. statistical ensemble theory, we will express the entropy of a The stability condition of the canonical ensemble is just nonextremal black hole as the function of the numbers of 𝐶>0. In other words, the Hessian function of the inter- branes and antibranes. Namely, for the charged black holes, nal energy with respect to the chemical variables, namely, the electric and magnetic charges (𝑞𝑖,𝑝𝑖) form a coordinate {𝑥1,𝑥2,...,𝑥𝑛}, remains positive definite. Hence, the energy chart on the state-space manifold. In this case, for a given as the function of the {𝑥1,𝑥2,...,𝑥𝑛} satisfies the following ensemble of 𝐷-branes, the coordinate 𝑞𝑖 is defined as the condition: number of the electric branes and 𝑝𝑖 as the number of the 󵄩 󵄩 magnetic branes. Towards the end of this paper, we will 󵄩𝜕 𝜕 𝐸(𝑥 ,𝑥 ,...,𝑥 )󵄩 >0. 󵄩 𝑖 𝑗 1 2 𝑛 󵄩 (21) offer further motivation for the consideration of the state- 𝑖 space geometry of large charged nonspherical horizon black The state-space coordinates {𝑋 } and intensive chemical holes in spacetime dimensions 𝐷≥5.Inthisconcern, variables {𝑥𝑖} are conjugate to each other. In particular, the [68]playsacentralroletowardstheformationofthe 𝑖 {𝑋 } are defined as the Legendre transform of {𝑥𝑖},andthus lower dimensional black hole configuration. Namely, for the we have torus compatifications, the exotic branes play an important 𝜕𝑆 (𝑥) role concerning the physical properties of supertubes, the 𝑋𝑖 := . 𝐷 𝐹 𝜕𝑥 (22) 0- 1 system and associated counting of the black hole 𝑖 microstates. In what follows, we consider the four-dimensional string 4.4. Physics of Correlation. Geometrically, the positivity of theory black holes in a given duality basis of the charges 𝐶>0 the heat capacity turns out to be the positivity condition (𝑞𝑖,𝑝𝑖). From the perspective of string theory, the exotic 𝑔 >0 𝑖 of 𝑖𝑗 ,foragiven . In many cases, the state-space stability branes and nongeometric configurations offer interesting restriction on the parameters of the black hole corresponds fronts for the black holes in three spacetime dimensions. In to the situation away from the extremality condition; namely, general, such configurations could carry a dipole or a higher 𝑟 =𝑟 + −. Far from the extremality condition, even at the pole charge, and they leave the four-dimension black hole zero antibrane charge or angular momentum, we find that configuration asymptotically flat. In fact, for the spacetime thereisafinitevalueofthethermodynamicscalarcurvature, dimensions 𝐷≥4,[68]showsthatachargeparticle unlike the nonrotating or only brane-charged configurations. corresponds to an underlying gauge field, modulo 𝑈-duality It turns out that the state-space geometry of the two charge transformations. From the perspective of nonextremal black extremal configurations is flat. Thus, the Einstein-Hilbert holes, by taking appropriate boundary condition, namely, contributions lead to a noninteracting statistical system. the unit asymptotic limit of the harmonic function which Atthetreelevel,someblackholeconfigurationsturnout defines the spacetime metric, one can choose the spacetime to be ill-defined, as well. However, we anticipate that the regions such that the supertube effects arising from nonexotic corresponding state-space configuration would become well- branes can effectively be put off in an asymptotically flat defined when a sufficient number of higher derivative cor- space [68].ThisallowsonetocomputetheArnowitt-Deser- rections [64–67]aretakenintoaccountwithrespecttothe Misner (ADM) mass of the asymptotic black hole. From the 𝛼󸀠 𝑙 -corrections and the string loop 𝑠 corrections. For the BTZ viewpoint of the statistical investigation, the dependence of black holes [13], we notice that the large entropy limit turns the mass to the entropy of a nonextremal black hole comes out to be the stability bound, beyond which the underlying from the contribution of the antibranes to the counting quantum effects dominate. degeneracy of the states. Fortheblackholeinstringtheory,theRicciscalarof the state-space geometry is anticipated to be positive definite with finitely many higher order corrections. For nonextremal 5. Extremal Black Holes in String Theory black brane configurations, which are far from the extremal- 5.1. Two Charge Configurations. The state-space geometry of ity condition, such effects have been seen from the nature 𝑅(𝑆(𝑋 ,𝑋 ,...,𝑋 )) the two charge extremal configurations is analyzed in terms of of the state-space scalar curvature 1 2 𝑛 . the winding modes and the momentum modes of an excited Indeed, [12, 14] indicate that the limiting state-space scalar 𝑛 𝑛 𝑅(𝑆(𝑋 ,𝑋 ,...,𝑋 ))| =0̸ string carrying 1 winding modes and 𝑝 momentum modes. curvature 1 2 𝑛 no anticharge gives a set of In the large charge limit, the microscopic entropy obtained stability bounds on the statistical parameters. Thus, our by the degeneracy of the underlying conformal field theory consideration yields a classification of the domain of the states reduces to the following expression: parameters and global correlation of a nonextremal black hole. 𝑆 =2√2𝑛 𝑛 . micro 1 𝑝 (23)

4.5. String Theory Perspective. In this subsection, we recall The microscopic counting can be accomplished by consid- a brief notion of entropy of a general string theory black ering an ensemble of weakly interacting D-branes [54]. The brane configuration from the viewpoint of the counting of counting entropy and the macroscopic attractor entropy of the black hole microstates [53, 53–59, 68]. Given a string the two charge black holes in string theory which have a 𝑛4 6 Advances in High Energy Physics

number of 𝐷4 branes and a 𝑛0 number of 𝐷0 branes match 5.2. Three Charge Configurations. From the consideration of and thus we have the two derivative Einstein-Hilbert action, [53] shows that the leading order entropy of the three charge 𝐷1-𝐷5-𝑃 extremal 𝑆 =2𝜋√𝑛 𝑛 =𝑆 . micro 0 4 macro (24) black holes is 𝑆 =2𝜋√𝑛 𝑛 𝑛 =𝑆 . In this case, the components of underlying state-space metric micro 1 5 𝑝 macro (30) tensor are The concerned components of state-space metric tensor are 𝜋 𝑛 𝜋 1 given in Appendix A. Hereby, it follows further that the local 𝑔 = √ 4 ,𝑔=− , 𝑛0𝑛0 𝑛0𝑛4 state-space metric constraints are satisfied as 2𝑛0 𝑛0 2 √𝑛0𝑛4 (25) 𝑔𝑛 𝑛 >0 ∀𝑖∈{1, 5, 𝑝} |𝑛𝑖 >0. (31) 𝜋 𝑛 𝑖 𝑖 𝑔 = √ 0 . 𝑛4𝑛4 𝑖, 𝑗 ∈ {1, 5} 𝑝 2𝑛4 𝑛4 For distinct and , the list of relative correlation functions is depicted in Appendix A. Further, we see that the local stabilities pertaining to the lines and two-dimensional The diagonal pair correlation functions remain positive surfacesofthestate-spacemanifoldaremeasuredas definite: 𝑛 𝑛 2 𝑔 >0 ∀𝑖∈{0, 4} |𝑛 >0, 𝑔 >0, ∀(𝑛 ,𝑛 ) . 𝜋 5 𝑝 𝜋 2 3 𝑛𝑖𝑛𝑖 𝑖 𝑛4𝑛4 0 4 𝑝 = √ ,𝑝=− (𝑛 𝑛 +𝑛 ). (32) 1 2𝑛 𝑛 2 4𝑛 𝑛2𝑛 𝑝 1 5 (26) 1 1 1 5 𝑝 The stability of the entire equilibrium phase-space configura- For distinct 𝑖, 𝑗 ∈ {0,, 4} the state-space pair correlation tions of the 𝐷1-𝐷5-𝑃 extremal black holes is determined by functions admit the 𝑝3 := 𝑔 determinant of the state-space metric tensor: 2 𝑔 𝑛𝑗 𝑔𝑖𝑗 𝑛 𝑖𝑖 = ( ) , =− 𝑖 . 󵄩 󵄩 1 3 −1/2 (27) 󵄩𝑔󵄩 =− 𝜋 (𝑛1𝑛5𝑛𝑝) . (33) 𝑔𝑗𝑗 𝑛𝑖 𝑔𝑖𝑖 𝑛𝑗 󵄩 󵄩 2

The global properties of fluctuating two charge 𝐷0-𝐷4 The universal nature of statistical interactions and the other extremal configurations are determined by possible principle properties concerning Maldacena, Strominger, and Witten minors.Thefirstminorconstraint𝑝1 >0directly follows (MSW) rotating black branes [55] are elucidated by the state- from the positivity of the first component of metric tensor: space scalar curvature: 3 𝑅(𝑛 ,𝑛 ,𝑛 )= . 𝜋 𝑛4 1 5 𝑝 4𝜋 𝑛 𝑛 𝑛 (34) 𝑝1 = √ . (28) √ 1 5 𝑝 2𝑛0 𝑛0 The constant entropy (or scalar curvature) curve defining the The determinant of the metric tensor 𝑝2 := 𝑔(𝑛0,𝑛4) vanishes state-space manifold is the higher dimensional hyperbola: identically for all allowed values of the parameters. Thus, 2 the leading order large charge extremal black branes having 𝑛1𝑛5𝑛𝑝 =𝑐, (35) (i) a 𝑛0 number of 𝐷0-branes and a 𝑛4 number of 𝐷4 or 𝑐 (𝑐 ,𝑐 )=(𝑆/2𝜋, 3/4𝜋𝑅 ) (ii) excited strings with a 𝑛1 number of windings and a where takesrespectivevaluesof 𝑆 𝑅 0 0 . 𝑛𝑝 number of momenta, where either set of charges forms In [12, 14, 17, 18], we have shown that similar results hold local coordinates on the state-space manifold, find degenerate for the state-space configuration of the four charge extremal intrinsic state-space configurations. For a given configuration black holes. entropy 𝑆0 := 2𝜋𝑐, the constant entropy curve can be depicted as the rectangular hyperbola 6. Nonextremal Black Holes in String Theory

2 𝑛0𝑛4 =𝑐. (29) 6.1. Four Charge Configurations. The state-space configuration of the nonextremal 𝐷1–𝐷5 black holes is considered with nonzero momenta along the clockwise and anticlockwise The intrinsic state-space configuration depends on the 1 attractor values of the scalar fields which arise from the directions of the Kaluza-Klein compactification circle 𝑆 . chosen string compactification. Thus, the possible state- Following [56], the microscopic entropy and the macroscopic space Ruppenier geometry may become well-defined against entropy match for given total mass and brane charges : 󸀠 further higher derivative 𝛼 -corrections. In particular, the 𝑆 =2𝜋√𝑛 𝑛 ( 𝑛 + √𝑛 )=𝑆 . determinant of the state-space metric tensor may take micro 1 5 √ 𝑝 𝑝 macro (36) positive/negative definite values over the domain of brane charges. We will illustrate this point in a bit more detail in the The state-space covariant metric tensor is defined as anega- subsequent consideration with a higher number of charges tive Hessian matrix of the entropy with respect to the number and anticharges. of 𝐷1, 𝐷5 branes {𝑛𝑖 |𝑖=1,5}and clockwise-anticlockwise Advances in High Energy Physics 7

Kaluza-Klein momentum charges {𝑛𝑝, 𝑛𝑝}.Herewith,wefind on the state-space manifold (𝑀4,𝑔). This implies an almost that the components of the metric tensor take elegant forms. everywhere weakly interacting statistical basis. In this case, The corresponding expressions are given in Appendix B.As theconstantentropyhypersurfaceisdefinedbythecurve in the case of the extremal configurations, the state-space metric satisfies the following constraints: 𝑐2 2 =(√𝑛 + √𝑛 ) . 𝑛 𝑛 𝑝 𝑝 (44) 𝑔 > 0, ∀𝑖 = 1, 5; 𝑔 >0, ∀𝑎=𝑝,𝑝. 1 5 𝑛𝑖𝑛𝑖 𝑛𝑎𝑛𝑎 (37) As in the case of two charge 𝐷0-𝐷4 extremal black holes Furthermore, the scaling relations for distinct 𝑖, 𝑗 ∈ and 𝐷1-𝐷5-𝑃 extremal black holes, the constant 𝑐 takes the {1, 5} and 𝑝, concerning the list of relative correlation 2 2 same value of 𝑐:=𝑆0/4𝜋 . For a given state-space scalar functions, are offered in Appendix B.Inthiscase,we curvature 𝑘, the constant state-space curvature curves take find that the stability criteria of the possible surfaces and the following form: hypersurfaces of the underlying state-space configuration are determined by the positivity of the following principle 6 minors: 𝑓 (𝑛𝑝, 𝑛𝑝) =𝑘√𝑛1𝑛5(√𝑛𝑝 + √𝑛𝑝) . (45) 𝜋 𝑛 𝑝 =1, 𝑝 = 5 ( 𝑛 + √𝑛 ), 0 1 √ 3 √ 𝑝 𝑝 6.2. Six Charge Configurations. We now extrapolate the state- 2 𝑛1 spacegeometryoffourchargenonextremal𝐷1–𝐷5 solutions (38) 1 𝜋3 for nonlarge charges, where we are no longer close to an 𝑝2 =0, 𝑝3 =− (√𝑛𝑝 + √𝑛𝑝). ensemble of supersymmetric states. In [57], the computation 2𝑛𝑝 √𝑛1𝑛5 of the entropy of all such special extremal and near-extremal black hole configurations has been considered. The leading 𝐷 𝐷 The complete local stability of the full nonextremal 1– 5 order entropy as a function of charges {𝑛𝑖} and anticharges black brane state-space configuration is ascertained by the {𝑚𝑖} is positivity of the determinant of the metric tensor: 𝑆(𝑛1,𝑚1,𝑛2,𝑚2,𝑛3,𝑚3) 1 𝜋4 2 𝑔(𝑛 ,𝑛 ,𝑛 , 𝑛 )=− ( 𝑛 + √𝑛 ) . 1 5 𝑝 𝑝 3/2 √ 𝑝 𝑝 4 (39) := 2𝜋 (√𝑛1 + √𝑚1)(√𝑛2 + √𝑚2)(√𝑛3 + √𝑚3). (𝑛𝑝𝑛𝑝) (46) The global state-space properties concerning the four charge 𝐷 𝐷 For given charges 𝑖, 𝑗 ∈𝐴1 := {𝑛1,𝑚1}; 𝑘, 𝑙 ∈𝐴2 := {𝑛2,𝑚2}; nonextremal 1– 5 black holes are determined by the 𝑚,𝑛 ∈𝐴 := {𝑛 ,𝑚 } regularity of the invariant scalar curvature: and 3 3 3 , the intrinsic state-space pair correlations are in precise accordance with the underlying 9 −6 macroscopic attractor configurations which are being dis- 𝑅(𝑛 ,𝑛 ,𝑛 , 𝑛 )= (√𝑛 + √𝑛 ) 𝑓(𝑛 , 𝑛 ), 1 5 𝑝 𝑝 4𝜋√𝑛 𝑛 𝑝 𝑝 𝑝 𝑝 closed in the special leading order limit of the nonextremal 1 5 𝐷 𝐷 (40) 1– 5 solutions. The components of the covariant state- space metric tensor over generic nonlarge charge domains are where the function 𝑓(𝑛𝑝, 𝑛𝑝) of two momenta (𝑛𝑝, 𝑛𝑝) run- not difficult to compute, and, indeed, we have offered their 1 corresponding expressions in Appendix C. ning in opposite directions of the Kaluza-Klein circle 𝑆 has For all finite (𝑛𝑖,𝑚𝑖), 𝑖 = 1, 2,, 3 the components involving been defined as 𝑔 brane-brane state-space correlations 𝑛𝑖𝑛𝑖 and antibrane- 5/2 3/2 1/2 2 antibrane state-space correlations 𝑔𝑚 𝑚 satisfy the following 𝑓(𝑛𝑝, 𝑛𝑝):=𝑛𝑝 + 10𝑛𝑝 𝑛𝑝 +5𝑛𝑝 𝑛𝑝 𝑖 𝑖 (41) positivity conditions: +5𝑛2 𝑛 1/2 + 10𝑛 𝑛 3/2 + 𝑛 5/2. 𝑝 𝑝 𝑝 𝑝 𝑝 𝑔 >0, 𝑔 >0. 𝑛𝑖𝑛𝑖 𝑚𝑖𝑚𝑖 (47) BynoticingthePascalcoefficientstructurein(41), we see that The distinct {𝑛𝑖,𝑚𝑖 | 𝑖 ∈ {1, 2, 3}} describing six charge string the above function 𝑓(𝑛𝑝, 𝑛𝑝) canbefactorizedas theory black holes have three types of relative pair correlation 5 functions. The corresponding expressions of the relative 𝑓(𝑛 , 𝑛 )=(𝑛 + 𝑛 ) . 𝑝 𝑝 𝑝 𝑝 (42) statistical correlation functions are given in Appendix C.

Thus, (40) leads to the following state-space scalar curvature: Notice hereby that the scaling relations remain similar to those obtained in the previous case, except that (i) the num- 9 1 ber of relative correlation functions has been increased, and 𝑅(𝑛 ,𝑛 ,𝑛 , 𝑛 )= ×( ). 1 5 𝑝 𝑝 (43) (ii) the set of cross ratios, namely, {𝑔𝑖𝑗 /𝑔𝑘𝑙,𝑔𝑘𝑙/𝑔𝑚𝑛,𝑔𝑖𝑗 /𝑔𝑚𝑛} 4𝜋√𝑛1𝑛5 𝑛 + 𝑛 √ 𝑝 √ 𝑝 being zero in the previous case, becomes ill-defined for the six charge state-space configurations. Inspecting the specific pair In the large charge limit, the nonextremal 𝐷1–𝐷5 black of distinct charge sets 𝐴𝑖 and 𝐴𝑗,therearenow24typesof branes have a nonvanishing small scalar curvature function nontrivial relative correlation functions. The set of principle 8 Advances in High Energy Physics

components denominator ratios computed from the above among the given brane-antibrane pairs {(𝑛1,𝑚1), (𝑛2, 𝑚 ), (𝑛 ,𝑚 )} 𝑅 state-space metric tensor reduces to 2 3 3 .Thecomponent 𝑛1𝑛2𝑚3𝑚4 diverges at the roots 𝑔 of the two variables polynomials defined as the functions of 𝑖𝑗 brane and antibrane charges: = 0, ∀𝑖, 𝑗, 1𝑘 ∈{𝑛 ,𝑚1,𝑛2,𝑚2,𝑛3,𝑚3}. (48) 𝑔𝑘𝑘 4 3 7/2 3 4 𝑓1 (𝑛2,𝑚2)=𝑛2𝑚2 +2(𝑛2𝑚2) +𝑛2𝑚2, For given 𝑖, 𝑗 ∈𝐴1 := {𝑛1,𝑚1}; 𝑘, 𝑙 ∈𝐴2 := {𝑛2,𝑚2}; 𝑚,𝑛 ∈ (54) 𝐴 := {𝑛 ,𝑚 } 𝑔 =0 𝑓 (𝑛 ,𝑚 )=𝑚9/2𝑛4 +𝑛4𝑚9/2. 3 3 3 ,and 𝑛𝑖𝑚𝑖 , there are the total 15 types of 2 3 3 3 3 3 3 trivial relative correlation functions. There are five such trivial {𝐴 | 𝑖 = 1, 2, 3} 𝑅 ratios in each family 𝑖 .Thelocalstabilityofthe However, the component 𝑛3,𝑚3,𝑛3,𝑚3 with an equal number higher charged string theory nonextremal black holes is given of brane and antibrane charges diverges at a root of a single by higher degree polynomial: 𝜋 𝑝 = (√𝑛 + √𝑚 )(√𝑛 + √𝑚 ), 𝑓(𝑛1,𝑚1,𝑛2,𝑚2,𝑛3,𝑚3) 1 3/2 2 2 3 3 2𝑛1 := 𝑛4𝑚3𝑛9/2𝑚4 +𝑛4𝑚3𝑛4𝑚9/2 2 2 2 3 3 2 2 3 3 1 𝜋 2 2 (55) 𝑝 = (√𝑛 + √𝑚 ) (√𝑛 + √𝑚 ) , 7/2 7/2 9/2 4 7/2 7/2 4 9/2 2 3/2 2 2 3 3 4 +2𝑛2 𝑚2 𝑛3 𝑚3 +2𝑛2 𝑚2 𝑛3𝑚3 (𝑛1𝑚1) 3 4 9/2 4 3 4 4 9/2 3 (49) +𝑛 𝑚 𝑛 𝑚 +𝑛 𝑚 𝑛 𝑚 . 1 𝜋 3 2 2 3 3 2 2 3 3 𝑝3 = √𝑚2(√𝑛3 + √𝑚3) 8 (𝑛 𝑚 𝑛 )3/2 1 1 2 Herewith, from the perspective of state-space global invari- ×(√𝑛 + √𝑚 )(√𝑛 + √𝑚 ), ants,wefocusonthelimitingnatureoftheunderlyingensem- 2 2 1 1 ble. Thus, we may choose the equal charge and anticharge 𝑚 := 𝑘 𝑛 := 𝑘 𝑝4 =0. limit by defining 𝑖 and 𝑖 for the calculation of the Ricci scalar. In this case, we find the following small negative The principle minor 𝑝5 remains nonvanishing for all values of curvature scalar: charges on the constituent brane and antibranes. In general, 15 1 𝑅 (𝑘) =− . by an explicit calculation, we find that the hyper-surface 16 𝜋𝑘3/2 (56) minor 𝑝5 takes the following nontrivial value: Further, the physical meaning of taking an equal value of 1 𝜋5 𝑝 =− (√𝑛 + √𝑚 )3 the charges and anticharges lies in the ensemble theory, 5 8 3/2 1 1 namely, in the thermodynamic limit, all the statistical fluctu- (𝑛1𝑚1𝑛2𝑚2) 𝑛3 (50) ations of the charges and anticharges approach to a limiting 3 3 ×(√𝑛2 + √𝑚2) (√𝑛3 + √𝑚3) . Gaussian fluctuations. In this sense, we can take the average over the concerned individual Gaussian fluctuations. This Specifically, for an identical value of the brane and antibrane shows that the limiting statistical ensemble of nonextremal 𝐷 𝐷 charges, the minor 𝑝5 reduces to nonlarge charge 1– 5 solutions yields an attractive state- space configuration. Finally, such a limiting procedure is 𝜋5 indeed defined by considering the standard deviations of the 𝑝 (𝑘) =−64 . (51) 5 𝑘5/2 equal integer charges and anticharges, and thus our interest in calculating the limiting Ricci scalar in order to know The global stability on the full state-space configuration is the nature of the long range interactions underlying in the carried forward by computing the determinant of the metric system. tensor: For a given entropy 𝑆0, the constant entropy hypersurface is again some nonstandard curve: 6 󵄩 󵄩 1 𝜋 4 󵄩𝑔󵄩 =− (√𝑛1 + √𝑚1) (√𝑛 + √𝑚 )(√𝑛 + √𝑚 )(√𝑛 + √𝑚 ) =𝑐, 16 (𝑛 𝑚 𝑛 𝑚 𝑛 𝑚 )3/2 1 1 2 2 3 3 (57) 1 1 2 2 3 3 (52) 4 4 where the real constant 𝑐 takes the precise value of 𝑆0/2𝜋. ×(√𝑛2 + √𝑚2) (√𝑛3 + √𝑚3) .

The underlying state-space configuration remains nonde- 6.3. Eight Charge Configurations. From the perspective of the generate for the domain of given nonzero brane antibrane higher charged and anticharged black hole configurations in charges, except for extreme values of the brane and antibrane string theory, let us systematically analyze the underlying sta- charges {𝑛𝑖,𝑚𝑖}, when they belong to the set tistical structures. In this case, the state-space configuration of the nonextremal black hole involves finitely many non-

𝐵:={(𝑛1,𝑛2,𝑛3,𝑚1,𝑚2,𝑚3)| triviallycircularlyfiberedKaluza-Kleinmonopoles.Inthis (53) process, we enlist the complete set of nontrivial relative state- (𝑛𝑖,𝑚𝑖)=(0, 0) , (∞, ∞) , some 𝑖} , space correlation functions of the eight charged anticharged Advances in High Energy Physics 9 configurations, with respect to the lower parameter configu- pair correlation functions have distinct types of relative rations, as considered in [12, 14]. There have been calculations correlation functions. Apart from the zeros, infinities, and of the entropy of the extremal, near-extremal, and general similar factorizations, we see that the nontrivial relative nonextremal solutions in string theory; see, for instance, correlation functions satisfy the following scaling relations: [58, 59]. Inductively, the most general charge anticharge 3/2 nonextremal black hole has the following entropy: 𝑔𝑥 𝑥 𝑥𝑗 √𝑛𝑗 + √𝑚𝑗 𝑖 𝑖 =( ) , 𝑔 𝑥 √𝑛 + √𝑚 4 𝑥𝑗𝑥𝑗 𝑖 𝑖 𝑖 𝑆(𝑛1,𝑚1,𝑛2,𝑚2,𝑛3,𝑚3,𝑛4,𝑚4)=2𝜋∏ (√𝑛𝑖 + √𝑚𝑖). −1/2 𝑖=1 𝑔𝑥 𝑥 𝑥 𝑥 ∏ (√𝑛𝑝 + √𝑚𝑝) 𝑖 𝑗 =( 𝑖 𝑗 ) 𝑖 =𝑝̸=𝑗̸ , (58) 𝑔 𝑥 𝑥 ∏ ( 𝑛 + 𝑚 ) (61) 𝑥𝑘𝑥𝑙 𝑘 𝑙 𝑘 =𝑞̸=𝑙̸√ 𝑞 √ 𝑞 For the distinct 𝑖, 𝑗, 𝑘 ∈ {1, 2, 3,4}, we find that the compo- 𝑔𝑥 𝑥 𝑥 ∏𝑝 =𝑖̸(√𝑛𝑝 + √𝑚𝑝) nents of the metric tensor are 𝑖 𝑖 =−√( 𝑘 ) . 2 𝜋 𝑔𝑥 𝑥 𝑥𝑖 ∏ (√𝑛 + √𝑚 ) 𝑔 = ∏ (√𝑛 + √𝑚 ), 𝑖 𝑘 𝑖 =𝑞̸=𝑘̸ 𝑞 𝑞 𝑛𝑖𝑛𝑖 3/2 𝑗 𝑗 2𝑛𝑖 𝑗 =𝑖̸ 𝜋 As noticed in [12, 14], it is not difficult to analyze the statistical 𝑔 =− ∏ (√𝑛 + √𝑚 ), stability properties of the eight charged anticharged nonex- 𝑛𝑖𝑛𝑗 1/2 𝑘 𝑘 2(𝑛𝑖𝑛𝑗) 𝑖 =𝑘̸=𝑗̸ tremal black holes; namely, we can compute the principle minors associated with the state-space metric tensor and 𝑔 =0, thereby argue that all the principle minors must be positive 𝑛𝑖𝑚𝑖 𝜋 definite, in order to have a globally stable configuration. In 𝑔 =− ∏ (√𝑛 + √𝑚 ), (59) the present case, it turns out that the above black hole is 𝑛𝑖𝑚𝑗 1/2 𝑘 𝑘 2(𝑛𝑖𝑚𝑗) 𝑖 =𝑘̸=𝑗̸ stable only when some of the charges and/or anticharges areheldfixedortakespecificvaluessuchthat𝑝𝑖 >0for 𝜋 𝑔 = ∏ (√𝑛 + √𝑚 ), all the dimensions of the state-space manifold. From the 𝑚𝑖𝑚𝑖 3/2 𝑗 𝑗 2𝑚𝑖 𝑗 =𝑖̸ definition of the Hessian matrix of the associated entropy concerning the most general nonextremal nonlarge charged 𝜋 𝑔 =− ∏ (√𝑛 + √𝑚 ). black holes, we observe that some of the principle minors 𝑝𝑖 𝑚𝑖𝑚𝑗 1/2 𝑘 𝑘 2(𝑚𝑖𝑚𝑗) 𝑖 =𝑘̸=𝑗̸ are indeed nonpositive. In fact, we discover a uniform local stability criteria on the three-dimensional hypersurfaces, From the above depiction, it is evident that the principle two-dimensional surface, and the one-dimensional line of {𝑔 ,𝑔 | the underlying state-space manifold. In order to simplify the components of the state-space metric tensor 𝑛𝑖𝑛𝑖 𝑚𝑖𝑚𝑖 𝑖 = 1, 2, 3, 4} essentially signify a set of definite heat factors of the higher principle, we may hereby collect the ( 𝑛 + 𝑚 ) capacities (or the related compressibilities) whose positivity powers of each factor √ 𝑖 √ 𝑖 appearing in the expression in turn apprises that the black brane solutions comply with of the entropy. With this notation, Appendix D provides an underlying equilibrium statistical configuration. For an the corresponding principle minors for the most general arbitrary number of the branes {𝑛𝑖} and antibranes {𝑚𝑖},we nonextremal nonlarge charged anticharged black hole in find that the associated state-space metric constraints as the string theory involving finitely many nontrivially circularly diagonal pair correlation functions remain positive definite. fibered Kaluza-Klein monopoles. In particular, ∀𝑖 ∈ {1, 2, 3, 4}; it is clear that we have the Notice that the heat capacities, as the diagonal com- 𝑔 𝑝 𝑝 𝑝 following positivity conditions: ponents 𝑖𝑖,surfaceminor 2,hypersurfaceminors 3, 5, 𝑝6,and𝑝7, and the determinant of the state-space metric 𝑔 >0|𝑛,𝑚 >0, 𝑔 >0|𝑛,𝑚 >0. 𝑝 𝑛𝑖𝑛𝑖 𝑖 𝑖 𝑚𝑖𝑚𝑖 𝑖 𝑖 (60) tensor, as the highest principle minor 8 are examined as the functions of the number of branes 𝑛 and antibranes 𝑚.Thus, As observed in [12, 14], we find that the ratios of diagonal they describe the nature of the statistical fluctuations in the components vary inversely with a multiple of a well-defined vacuum configuration. The corresponding scalar curvature factor in the underlying parameters, namely, the charges and is offered for an equal number of branes and antibranes anticharges, which changes under the Gaussian fluctuations, (𝑛 = ,𝑚) which describes the nature of the long range whereas the ratios involving off diagonal components in effect statistical fluctuations. As per the above evaluation, we have uniquely inversely vary in the parameters of the chosen set 𝐴𝑖 obtained the exact expressions for the components of the of equilibrium black brane configurations. This suggests that metric tensor, principle minors, determinant of the metric thediagonalcomponentsweakeninarelativelycontrolled tensor, and the underlying scalar curvature of the fluctuating fashion into an equilibrium, in contrast with the off diagonal statistical configuration of the eight parameter black holes components, which vary over the domain of associated in string theory. Qualitatively, the local and the global parameters defining the 𝐷1-𝐷5-𝑃-𝐾𝐾 nonextremal nonlarge correlation properties of the limiting vacuum configuration charge configurations. In short, we can easily substantiate, canberealizedunderthestatisticalfluctuations.Thefirst for the distinct 𝑥𝑖 := (𝑛𝑖,𝑚𝑖) | 𝑖 ∈ {1,2,3,4} describing seven principle minors describe the local stability properties, eight (anti)charge string theory black holes, that the relative and the last minor describes the global ensemble stability. 10 Advances in High Energy Physics

The scalar curvature describes the corresponding phase the state-space geometric acquisitions with an appropriate space stability of the eight parameter black hole configura- comprehension of the required parameters, for example, the tion. In general, there exists an akin higher degree polynomial charges and anticharges {𝑛𝑖,𝑚𝑖}, which define the coordinate equation on which the Ricci scalar curvature becomes null, charts. From the consideration of the state-space geometry, and exactly on these points the state-space configuration of we have analyzed state-space pair correlation functions and the underlying nonlarge charge nonextremal eight charge the notion of stability of the most general nonextremal black black hole system corresponds to a noninteracting statistical hole in string theory. From the perspective of the intrinsic system. In this case, the corresponding state-space manifold Riemannian geometry, we find that the stability of these black (𝑀8,𝑔) becomes free from the statistical interaction with a branes has been divulged from the positivity of principle vanishing state-space scalar curvature. As in case of the six minors of the space-state metric tensor. charge configuration, we find interestingly that there exists Herewith, we have explicitly extended the state-space an attractive configuration for the equal number of branes analysis for the four charge and four anticharge nonextremal 𝑛:=𝑘and antibranes 𝑚:= 𝑘 .Inthelimitofalarge𝑘,the black branes in string theory. The present consideration of corresponding system possesses a small negative value of the the eight parameter black brane configurations, where the state-space scalar curvature: underlying leading order statistical entropy is written as a function of the charges {𝑛𝑖} and anticharges {𝑚𝑖}, describes 21 1 𝑅 (𝑘) =− . the stability properties under the Gaussian fluctuations. The 2 (62) 32 𝜋𝑘 present consideration includes all the special cases of the Interestingly, it turns out that the system becomes noninter- extremal and near-extremal configurations with a fewer acting in the limit of 𝑘→∞.Forthecaseofthe𝑛=𝑘=𝑚, number of charges and anticharges. In this case, we obtain we observe that the corresponding principle minors reduce the standard pattern of the underlying state-space geometry to the following constant values: and constant entropy curve as that of the lower parameter nonextremal black holes. The local coordinate of the state- 8 2 3 5 6 space manifold involves four charges and four anticharges {𝑝𝑖}𝑖=1 = {4𝜋, 16𝜋 , 32𝜋 , 0, −2048𝜋 , −16384𝜋 , (63) of the underlying nonextremal black holes. In fact, the −163840𝜋7, −1048576𝜋8}. conclusion to be drawn remains the same, as the underlying state-space geometry remains well-defined as an intrinsic 𝑁:=𝑀\ 𝐵̃ 𝐵̃ In this case, we find that the limiting underlying statistical Riemannian manifold 8 ,where is the set of system remains stable when at most three of the parameters, roots of the determinant of the metric tensor. In particular, namely, {𝑛𝑖 =𝑘=𝑚𝑖},areallowedtofluctuate.Herewith, the state-space configuration of eight parameter black brane wefindforthecaseof𝑛:=𝑘and 𝑚:= 𝑘 that the state- solutions remains nondegenerate for various domains of space manifold of the eight parameter brane and antibrane nonzero brane antibrane charges, except for the values, when {𝑛 } {𝑚 } configuration is free from critical phenomena, except for the the brane charges 𝑖 and antibrane charges 𝑖 belong to roots of the determinant. Thus, the regular state-space scalar the set curvature is comprehensively universal for the nonlarge 𝐵:={(𝑛̃ ,𝑛 ,𝑛 ,𝑛 ,𝑚 ,𝑚 ,𝑚 ,𝑚 )| charge nonextremal black brane configurations in string 1 2 3 4 1 2 3 4 (65) theory. In fact, the above perception turns out to be justified (𝑛𝑖,𝑚𝑖)=(0, 0) , (∞, ∞)}, from the typical state-space geometry, namely, the definition of the metric tensor as the negative Hessian matrix of the for a given brane-antibrane pair, 𝑖 ∈ {1,2,3,4}.Ouranalysis duality invariant expression of the black brane entropy. In this indicates that the leading order statistical behavior of the case, we may nevertheless easily observe, for a given entropy 𝑆 black brane configurations in string theory remains intact 0, that the constant entropy hypersurface is given by the under the inclusion of the Kaluza-Klein monopoles. In short, following curve: we have considered the eight charged anticharged string theory black brane configuration and analyzed the state-space (√𝑛1 + √𝑚1)(√𝑛2 + √𝑚2)(√𝑛3 + √𝑚3)(√𝑛4 + √𝑚4) =𝑐, (64) pair correlation functions, relative scaling relations, stability conditions, and the corresponding global properties. Given where 𝑐 is a real constant taking the precise value of a general nonextremal black brane configuration, we have 𝑆0/2𝜋. Under the vacuum fluctuations, the present analysis exposed (i) for what conditions the considered black hole indicates that the entropy of the eight parameter black configuration is stable, (ii) how its state-space correlations brane solution defines a nondegenerate embedding in the scaleintermsofthenumbersofbranesandantibranes. viewpoints of intrinsic state-space geometry. The above state- space computations determine an intricate set of statistical 7. Conclusion and Outlook properties, namely, pair correlation functions and correlation volume, which reveal the possible nature of the associated The Ruppenier geometry of two charge leading order parameters prescribing an ensemble of microstates of the extremal black holes remains flat or ill-defined. Thus, the dual conformal field theory living on the boundary of the statistical systems are, respectively, noninteracting or require black brane solution. For any black brane configuration, higher derivative corrections. However, an addition of the the above computation hereby shows that we can exhibit third brane charge and other brane and antibrane charges Advances in High Energy Physics 11 indicates an interacting statistical system. The statistical In general, the black brane configurations in string the- fluctuations in the canonical ensemble lead to an interacting ory can be categorized as per their state-space invariants. statistical system, as the scalar curvature of the state-space The underlying subconfigurations turn out to be well- takes a nonzero value. We have explored the state-space defined over possible domains, whenever there exists a geometric description of the charged extremal and associated respective set of nonzero state-space principle minors. The charged anticharged nonextremal black holes in string the- underlying full configuration turns out to be everywhere ory. well-defined, whenever there exists a nonzero state-space Our analysis illustrates that the stability properties of the determinant. The underlying configuration corresponds to specific state-space hypersurface may exactly be exploited an interacting statistical system, whenever there exists a in general. The definite behavior of the state-space prop- nonzero state-space scalar curvature. The intrinsic state- erties, as accounted in the specific cases, suggests that the space manifold of extremal/nonextremal and supersym- underlying hypersurfaces of the state-space configuration metric/nonsupersymmetric string theory black holes may include the intriguing mathematical feature. Namely, we find intrinsically be described by an embedding: well-defined stability properties for the generic extremal and ̃ nonextremal black brane configurations, except for some (𝑀(𝑛),𝑔) 󳨅→ (𝑀(𝑛+1), 𝑔) . (67) specific values of the charges and anticharges. With and withoutthelargechargelimit,wehaveprovidedexplicit The extremal state-space configuration may be examined as forms of the higher principle minors of the state-space a restriction to the full counting entropy with an intrinsic 𝑔 󳨃→ 𝑔|̃ metric tensor for various charged, anticharged, extremal and state-space metric tensor 𝑟+=𝑟− .Furthermore,the nonextremal black holes in string theory. In this concern, the state-space configurations of the supersymmetric black holes state-space configurations of the string theory black holes are may be examined as the BPS restriction of the full space generically well-defined and indicate an interacting statistical of the counting entropy with an understanding that the 𝑔:=𝑔|̃ basis. Interestingly, we discover the state-space geometric intrinsic state-space metric tensor is defined as 𝑀=𝑀0 . 𝑟 = nature of all possible general black brane configurations. From the perspective of string theory, the restrictions + 𝑟 𝑀=𝑀(𝑃 ,𝑄) From the very definition of the intrinsic metric tensor, the − and 0 𝑖 𝑖 should be understood as the present analysis offers a definite stability character of string fact that it has been applied to an assigned entropy of the theory vacua. nonextremal/nonsupersymmetric (or nearly extremal/nearly Significantly, we notice that the related principle minors supersymmetric) black brane configuration. This allows one and the invariant state-space scalar curvature classify the to compute the fluctuations in ADM mass of the black hole. underlying statistical fluctuations. The scalar curvature of In the viewpoint of the present research on the state-space a class of extremal black holes and the corresponding geometry, it is worth mentioning that the dependence of nonextremal black branes is everywhere regular with and the mass to the entropy of a nonextremal black hole comes 󸀠 without the stringy 𝛼 -corrections. A nonzero value of from the contribution of the antibranes, see, for instance, the state-space scalar curvature indicates an interacting Section 4.5, and so we may examine the corresponding underlying statistical system. We find that the antibrane Weinhold chemical geometry, as mentioned in Section 4.3. corrections modify the state-space curvature, but do not induce phase transitions. In the limit of an extremal Future Directions and Open Issues. The state-space instabili- black hole, we construct the intrinsic geometric realiza- ties and their relation to the dual microscopic conformal field tion of a possible thermodynamic description at the zero theoriescouldopenupanumberofnewrealizations.The temperature. state-space perspective includes the following issues. Importantly, the notion of the state-space of the con- (i) Multicenter Gibbons-Hawking solutions [69, 70]with sidered black hole follows from the corresponding Wald generalized base space manifolds having a mixing and Cardy entropies. The microscopic and macroscopic of positive and negative residues, see [71, 72]for entropies match in the large charge limit. From the a perspective development of state-space geometry perspective of statistical fluctuations, we anticipate the by invoking the role of foaming of black holes and intrinsic geometric realization of two point local corre- plumbing the Abyss for the microstates counting of lation functions and the corresponding global correlation black rings. length of the underlying conformal field theory configurations. In relation to the gauge-gravity correspondence (ii) Dual conformal field theories and string duality and extremal black holes, our analysis describes state-space symmetries, see [61]foraquantummechanicalper- geometric properties of the corresponding boundary gauge spective of superconformal black holes and [73, 74] theory. for the origin of gravitational thermodynamics and theroleofgiantgravitonsinconformalfieldtheory. General Remarks.Fordistinct{𝑖, 𝑗}, the state-space pair (iii) Stabilization against local and/or global perturba- correlations of an extremal configurations scale as tions, see [75–80] for black brane dynamics, stability, and critical phenomena. Thus, the consid- 𝑋 2 𝑔 eration of state-space geometry is well-suited for 𝑔𝑖𝑖 𝑗 𝑖𝑗 𝑋𝑖 =( ) , =− . (66) examining the domain of instability. This includes 𝑔𝑗𝑗 𝑋𝑖 𝑔𝑖𝑖 𝑋𝑗 Gregory-Laflamme (GL) modes, chemical potential 12 Advances in High Energy Physics

fluctuations, electric-magnetic charges and dipole A. Correlations for Three charges, rotational fluctuations, and the thermody- Charge Configurations namic temperature fluctuations for the near-extremal and nonextremal black brane solutions. We leave this Following the notion of the fluctuations, we see from the perspective of the state-space geometry open for a Hessian of the entropy equation (30) that the components of future research. state-space metric tensor are In general, various 𝐷 dimensional black brane configurations, 𝐷>5 𝜋 𝑛 𝑛 𝜋 𝑛 see, for instance, [75–80] for black rings in spacetime 𝑔 = √ 5 𝑝 ,𝑔=− √ 𝑝 , 1 𝐷−3 𝑛1𝑛1 𝑛1𝑛5 dimensions with 𝑆 ×𝑆 horizon topology, and the higher 2𝑛1 𝑛1 2 𝑛1𝑛5 1 1 2 3 3 horizon topologies, for example, 𝑆 ×𝑆 ×𝑆 , 𝑆 ×𝑆 ,andso forth offer a platform to extend the consideration of the state- 𝜋 𝑛 𝜋 𝑛1𝑛𝑝 𝑔 =− √ 5 ,𝑔= √ , 𝑛1𝑛𝑝 𝑛5𝑛5 (A.1) space geometry. 2 𝑛1𝑛𝑝 2𝑛5 𝑛5 On the other hand, the bubbling black brane solutions, namely, Lin, Lunin, and Maldacena (LLM) geometries [81], 𝜋 𝑛1 𝜋 𝑛1𝑛5 are interesting from the perspective of Mathur’s Fuzzball 𝑔𝑛 𝑛 =− √ ,𝑔𝑛 𝑛 = √ . 5 𝑝 2 𝑛 𝑛 𝑝 𝑝 2𝑛 𝑛 conjecture(s). From the perspective of the generalized hyper- 5 𝑝 𝑝 𝑝 Kahler¨ manifolds, Mathur’s conjecture [82–85]reducesto classifying and counting asymptotically flat four-dimensional For distinct 𝑖, 𝑗 ∈ {1, 5} and 𝑝,thelistofrelative hyper Kahler¨ manifolds [71] which have moduli regions of correlation functions follows the scaling relations: uniform signature (+,+,+,+)and (−,−,−,−). Finally, the new physics at the length of the Planck scale 𝑔 𝑛 2 𝑔 𝑛 2 𝑔 𝑛 anticipates an analysis of the state-space configurations. In 𝑖𝑖 =( 𝑗 ) , 𝑖𝑖 =( 𝑝 ) , 𝑖𝑖 =−( 𝑗 ), particular, it materializes that the state-space geometry may 𝑔𝑗𝑗 𝑛𝑖 𝑔𝑝𝑝 𝑛𝑖 𝑔𝑖𝑗 𝑛𝑖 be explored with the parameters of the foam geometries 𝑔 𝑛 𝑔 𝑛 𝑔 𝑛 𝑛 [71], and the corresponding empty space virtual black holes, 𝑖𝑖 =−( 𝑝 ), 𝑖𝑝 =( 𝑗 ), 𝑖𝑖 =−( 𝑗 𝑝 ), 2 see [81] for the notion of bubbling AdS space and 1/2 BPS 𝑔𝑖𝑝 𝑛𝑖 𝑔𝑗𝑝 𝑛𝑖 𝑔𝑗𝑝 𝑛𝑖 geometries. In such cases, the local and global statistical correlations, among the parameters of the microstates of the 𝑔 𝑛 𝑔 𝑛 𝑔 𝑛2 𝑖𝑝 =−( 𝑝 ), 𝑖𝑗 =( 𝑝 ), 𝑖𝑗 =−( 𝑝 ). black hole conformal field theory60 [ , 61], would involve the 𝑔 𝑛 𝑔 𝑛 𝑔 𝑛 𝑛 foams of two spheres. From the perspective of the string 𝑝𝑝 𝑖 𝑖𝑝 𝑗 𝑝𝑝 𝑖 𝑗 theory, the present exploration thus opens up an avenue (A.2) for learning new insights into the promising structures of the black brane space-time configurations at very small B. Correlations for Four scales. Charge Configurations Appendices For the given entropy as in (36), we find that the components of the metric tensor are In these appendices, we provide explicit forms of the state- space correlation arising from the metric tensor of the 𝜋 𝑛5 charged (non)extremal (non)large black holes in string the- 𝑔𝑛 𝑛 = √ (√𝑛𝑝 + √𝑛𝑝), 1 1 2 𝑛3 ory. In fact, our analysis illustrates that the stability prop- 1 erties of the specific state-space hypersurface may exactly 𝜋 be exploited in general. The definite behavior of state-space 𝑔𝑛 𝑛 =− (√𝑛𝑝 + √𝑛𝑝), 1 5 2√𝑛 𝑛 properties, as accounted in the concerned main sections, 1 5 suggests that the various intriguing hypersurfaces of the state- 𝜋 𝑛5 𝜋 𝑛5 space configuration include the nice feature that they do have 𝑔𝑛 𝑛 =− √ ,𝑔𝑛 𝑛 =− √ , 1 𝑝 2 𝑛 𝑛 1 𝑝 2 𝑛 𝑛 definite stability properties, except for some specific values of 1 𝑝 1 𝑝 thechargesandanticharges. 𝜋 𝑛 𝜋 𝑛 As mentioned in the main sections, these configura- 𝑔 = √ 1 (√𝑛 + √𝑛 ), 𝑔 =− √ 1 , 𝑛5𝑛5 3 𝑝 𝑝 𝑛5𝑛𝑝 tions are generically well-defined and indicate an interact- 2 𝑛5 2 𝑛5𝑛𝑝 ing statistical basis. Herewith, we discover that the state- space geometry of the general black brane configurations 𝜋 𝑛 𝜋 𝑛 𝑛 𝑔 =− √ 1 ,𝑔= √ 1 5 , in string theory indicates the possible nature of the under- 𝑛5𝑛𝑝 𝑛𝑝𝑛𝑝 3 2 𝑛5𝑛𝑝 2 𝑛𝑝 lying statistical fluctuations. Significantly, we notice from the very definition of the intrinsic metric tensor that the 𝜋 𝑛1𝑛5 related statistical pair correlation functions and relative 𝑔𝑛 𝑛 =0, 𝑔𝑛 𝑛 = √ . 𝑝 𝑝 𝑝 𝑝 2 𝑛 3 statistical correlation functions take the following exact 𝑝 expressions. (B.1) Advances in High Energy Physics 13

𝜋 For distinct 𝑖, 𝑗 ∈ {1, 5},and𝑘, 𝑙 ∈ {𝑝, 𝑝} describing four 𝑔𝑚 𝑚 = (√𝑛2 + √𝑚2)(√𝑛3 + √𝑚3), 1 1 2𝑚3/2 charge nonextremal 𝐷1–𝐷5-𝑃-𝑃 black holes, the statistical 1 pair correlations consist of the following scaling relations: 𝜋 𝑔 =− (√𝑛 + √𝑚 ), 𝑚1𝑛2 3 3 2√𝑚1𝑛2 𝜋 𝑛 2 𝑔 =− (√𝑛 + √𝑚 ), 𝑔𝑖𝑖 𝑗 𝑔𝑖𝑖 𝑛𝑘 𝑚1𝑚2 3 3 =( ) , = √𝑛 (√𝑛 + √𝑛 ), 2√𝑚1𝑚2 𝑔 𝑛 𝑔 𝑛2 𝑘 𝑝 𝑝 𝑗𝑗 𝑖 𝑘𝑘 𝑖 𝜋 𝑔𝑚 𝑛 =− (√𝑛2 + √𝑚2), 𝑔 𝑛𝑗 1 3 2 𝑚 𝑛 𝑖𝑖 =− , √ 1 3 𝑔𝑖𝑗 𝑛𝑖 𝜋 𝑔 =− (√𝑛 + √𝑚 ), 𝑚1𝑚3 2 2 𝑔 √𝑛 2√𝑚1𝑚3 𝑖𝑖 =− 𝑘 (√𝑛 + √𝑛 ), 𝑔 𝑛 𝑝 𝑝 𝜋 𝑖𝑘 𝑖 𝑔 = (√𝑛 + √𝑚 )(√𝑛 + √𝑚 ), (B.2) 𝑛2𝑛2 3/2 1 1 3 3 𝑔 𝑛 𝑔 𝑛 2𝑛 𝑖𝑘 = 𝑗 , 𝑖𝑖 =− 𝑗 𝑛 ( 𝑛 + √𝑛 ), 2 2 √ 𝑘 √ 𝑝 𝑝 𝑔𝑗𝑘 𝑛𝑖 𝑔𝑗𝑘 𝑛𝑖 𝜋 𝑔𝑛 𝑚 =0, 𝑔𝑛 𝑛 =− (√𝑛1 + √𝑚1) 2 2 2 3 2 𝑛 𝑛 𝑔 √ 2 3 𝑔𝑖𝑘 𝑛𝑘 𝑖𝑗 √𝑛𝑘 =− , = (√𝑛𝑝 + √𝑛𝑝), 𝜋 𝑔𝑘𝑘 𝑛𝑖 𝑔𝑖𝑘 𝑛𝑗 𝑔 =− (√𝑛 + √𝑚 ), 𝑛2𝑚3 1 1 2√𝑛2𝑚3 𝑔𝑖𝑗 𝑛 =− 𝑘 √𝑛 (√𝑛 + √𝑛 ). 𝜋 𝑔 𝑛 𝑛 𝑘 𝑝 𝑝 𝑔 = (√𝑛 + √𝑚 )(√𝑛 + √𝑚 ), 𝑘𝑘 𝑖 𝑗 𝑚2𝑚2 3/2 1 1 3 3 2𝑚2 𝜋 𝑔𝑚 𝑛 =− (√𝑛1 + √𝑚1), 2 3 2 𝑚 𝑛 Notice that the list of other mixed relative correlation func- √ 2 3 tions concerning the nonextremal 𝐷1–𝐷5-𝑃-𝑃 black holes 𝜋 𝑔 =− (√𝑛 + √𝑚 ), read as 𝑚2𝑚3 1 1 2√𝑚2𝑚3 𝜋 𝑔𝑛 𝑛 = (√𝑛1 + √𝑚1)(√𝑛2 + √𝑚2), 𝑔𝑛 𝑚 =0, 3 3 2𝑛3/2 3 3 𝑔 𝑛 𝑔 𝑛𝑗 𝑛 𝑔 3 𝑖𝑘 = √ 𝑙 , 𝑖𝑘 = √ 𝑙 , 𝑘𝑙 =0, 𝑔𝑖𝑙 𝑛𝑘 𝑔𝑗𝑙 𝑛𝑖 𝑛𝑘 𝑔𝑖𝑗 𝜋 𝑔𝑚 𝑚 = (√𝑛1 + √𝑚1)(√𝑛2 + √𝑚2). (B.3) 3 3 2𝑚3/2 𝑔 𝑔 𝑛 3/2 𝑔 3 𝑘𝑙 =0, 𝑘𝑘 =( 𝑙 ) , 𝑘𝑙 =0. (C.1) 𝑔𝑖𝑖 𝑔𝑙𝑙 𝑛𝑘 𝑔𝑘𝑘

In this case, from the definition of the relative statistical C. Correlations for Six Charge Configurations correlation functions, for 𝑖, 𝑗 1∈ {𝑛 ,𝑚1},and𝑘, 𝑙 2∈ {𝑛 ,𝑚2}, the relative correlation functions satisfy the following scaling Over generic nonlarge charge domains, we find from the relations: entropy equation (46) that the components of the covariant state-space metric tensor are given by the following expressions: 𝑔 𝑗 3/2 𝑔 𝑘 3/2 √𝑛 + √𝑚 𝑖𝑖 =( ) , 𝑖𝑖 =( ) ( 2 2 ), 𝑔𝑗𝑗 𝑖 𝑔𝑘𝑘 𝑖 √𝑛3 + √𝑚3 𝜋 𝑔 𝑔 = (√𝑛 + √𝑚 )(√𝑛 + √𝑚 ), 𝑔 =0, 𝑖𝑗 =0, 𝑛1𝑛1 3/2 2 2 3 3 𝑛1𝑚1 2𝑛1 𝑔𝑖𝑖 𝜋 𝑔 =− (√𝑛 + √𝑚 ), 𝑔 √𝑘 𝑔 𝑗 𝑛1𝑛2 3 3 𝑖𝑖 𝑖𝑘 √ 2√𝑛1𝑛2 =− (√𝑛2 + √𝑚2), = , 𝑔𝑖𝑘 𝑖 𝑔𝑗𝑘 𝑖 𝜋 𝑔𝑛 𝑚 =− (√𝑛3 + √𝑚3), 1 2 2√𝑛 𝑚 𝑔 √𝑗𝑘 1 2 𝑖𝑖 =− (√𝑛 + √𝑚 ), 3/2 2 2 𝜋 𝑔𝑗𝑘 𝑖 𝑔𝑛 𝑛 =− (√𝑛2 + √𝑚2), 1 3 2√𝑛 𝑛 1 3 𝑔 √𝑖 𝑔𝑖𝑗 𝑔𝑖𝑗 𝑘𝑘 =− (√𝑛 + √𝑚 ), =0, =0. 𝜋 𝑔 𝑘 2 2 𝑔 𝑔 𝑔 =− (√𝑛 + √𝑚 ), 𝑖𝑘 𝑖𝑘 𝑘𝑘 𝑛1𝑚3 2 2 2√𝑛1𝑚3 (C.2) 14 Advances in High Energy Physics

The other concerned relative correlation functions are involving finitely many nontrivially circularly fibered Kaluza- 𝑔 𝑙 𝑔 𝑗𝑙 𝑔 Klein monopoles, the principle minors take the following 𝑖𝑘 = √ , 𝑖𝑘 = √ , 𝑖𝑗 = . ., expressions: 𝑔 𝑘 𝑔 𝑖𝑘 𝑔 n d 𝑖𝑙 𝑗𝑙 𝑘𝑙 𝜋 (C.3) 𝑝 = (√𝑛 + √𝑚 )(√𝑛 + √𝑚 )(√𝑛 + √𝑚 ), 𝑔 𝑔 𝑙 3/2 𝑔 1 2𝑛3/2 2 2 3 3 4 4 𝑘𝑙 =0, 𝑘𝑘 =( ) , 𝑘𝑙 =0. 1 𝑔 𝑔 𝑘 𝑔 𝑖𝑖 𝑙𝑙 𝑘𝑘 𝜋2 𝑝 = ( 𝑛 + 𝑚 )2( 𝑛 + 𝑚 )2 For 𝑘, 𝑙 2∈ {𝑛 ,𝑚2},and𝑚,𝑛∈{𝑛3,𝑚3},wehave 2 √ 2 √ 2 √ 3 √ 3 4(𝑛 𝑚 )3/2 3/2 1 1 𝑔𝑘𝑘 𝑚 √𝑛3 + √𝑚3 =( ) ( ), 2 𝑔𝑚𝑚 𝑘 √𝑛2 + √𝑚2 ×(√𝑛4 + √𝑚4) ,

𝑔 𝑔 √𝑚 3 𝑘𝑙 =0, 𝑘𝑘 =− (√𝑛 + √𝑚 ), 𝜋 3 3 3 3 𝑝3 = (√𝑛3 + √𝑚3) (√𝑛4 + √𝑚4) 𝑔𝑘𝑘 𝑔𝑘𝑚 𝑘 3/2 8(𝑛1𝑚1𝑛2) 𝑔 𝑙 𝑔 √𝑙𝑚 𝑘𝑚 = √ , 𝑘𝑘 =− (√𝑛 + √𝑚 ), (C.4) ×(√𝑛 + √𝑚 ) √𝑚 (√𝑛 + √𝑚 ), 3/2 3 3 2 2 2 1 1 𝑔𝑙𝑚 𝑘 𝑔𝑙𝑚 𝑘 𝑝 =0, √ 4 𝑔𝑚𝑚 𝑘 =− (√𝑛2 + √𝑚2), 5 𝑔𝑘𝑚 𝑚 𝜋 3 𝑝5 =− (√𝑛2 + √𝑚2) 𝑔 𝑔 8(𝑛 𝑛 𝑚 𝑚 )3/2𝑛 𝑘𝑙 =0, 𝑘𝑙 =0. 1 2 2 1 3 𝑔𝑘𝑚 𝑔𝑚𝑚 ×(√𝑛 + √𝑚 )3(√𝑛 + √𝑚 )5(√𝑛 + √𝑚 )3, The other concerned relative correlation functions are 3 3 4 4 1 1 𝑔 𝑛 𝑔 𝑙𝑛 𝑔 𝜋6 𝑘𝑚 = √ , 𝑘𝑚 = √ , 𝑘𝑙 = . ., 𝑝 =− (√𝑛 + √𝑚 )4 n d 6 3/2 2 2 𝑔𝑘𝑛 𝑚 𝑔𝑙𝑛 𝑘𝑚 𝑔𝑚𝑛 16(𝑛 𝑛 𝑚 𝑚 𝑛 𝑚 ) (C.5) 1 2 1 2 3 3 𝑔 𝑔 𝑛 3/2 𝑔 𝑚𝑛 =0, 𝑚𝑚 =( ) , 𝑚𝑛 =0. 4 6 4 ×(√𝑛3 + √𝑚3) (√𝑛4 + √𝑚4) (√𝑛1 + √𝑚1) , 𝑔𝑘𝑘 𝑔𝑛𝑛 𝑚 𝑔𝑚𝑚 7 However, for 𝑖, 𝑗 1∈ {𝑛 ,𝑚1},and𝑚,𝑛∈{𝑛3,𝑚3},wehave 𝜋 𝑝 =− (√𝑛 + √𝑚 )5 3/2 𝑔 7 3/2 2 2 𝑔 𝑚 √𝑛3 + √𝑚3 𝑖𝑗 32(𝑛 𝑚 𝑛 𝑚 𝑛 𝑚 𝑛 ) 𝑖𝑖 =( ) ( ), =0, 1 1 2 2 3 3 4 𝑔𝑚𝑚 𝑖 √𝑛1 + √𝑚1 𝑔𝑖𝑖 5 5 ×(√𝑛3 + √𝑚3) (√𝑛4 + √𝑚4) (4√𝑛4 + √𝑚4) 𝑔𝑖𝑖 √𝑚 =− (√𝑛3 + √𝑚3), 𝑔𝑖𝑚 𝑖 5 ×(√𝑛1 + √𝑚1) , 𝑔 𝑗 𝑔 √𝑗𝑚 𝑖𝑚 √ 𝑖𝑖 𝜋8 = , =− (√𝑛3 + √𝑚3), 6 𝑔 𝑖 𝑔 𝑖3/2 𝑝8 =− (√𝑛2 + √𝑚2) 𝑗𝑚 𝑗𝑚 4 3/2 16(∏𝑖=1𝑛𝑖𝑚𝑖) 𝑔 √𝑖 𝑚𝑚 6 6 6 =− (√𝑛1 + √𝑚1), (C.6) ×(√𝑛 + √𝑚 ) (√𝑛 + √𝑚 ) (√𝑛 + √𝑚 ) . 𝑔𝑖𝑚 𝑚 3 3 4 4 1 1 (D.1) 𝑔 𝑔 𝑔 𝑛 𝑖𝑗 =0, 𝑖𝑗 =0, 𝑖𝑚 = √ , 𝑔 𝑔 𝑔 𝑚 𝑖𝑚 𝑚𝑚 𝑖𝑛 Conflict of Interests 𝑔 𝑔𝑖𝑚 𝑗𝑛 𝑖𝑗 The author declares that there is no conflict of interests = √ , = n.d., 𝑔𝑗𝑛 𝑖𝑚 𝑔𝑚𝑛 regarding the publication of this paper. 𝑔 𝑔 𝑚𝑛 =0, 𝑚𝑛 =0. Acknowledgments 𝑔𝑖𝑖 𝑔𝑚𝑚 This work has been supported in part by the European D. Principle Minors for Eight Research Council Grant no. 226455, “SUPERSYMMETRY, Charge Configurations QUANTUM GRAVITY AND GAUGE FIELDS (SUPER- FIELDS).” Bhupendra Nath Tiwari would like to thank Prof. For the entropy equation (58) of the most general nonex- V. Ravishankar for his support and encouragements towards tremal nonlarge charged anticharged black hole in string the research in string theory. This work was conducted during Advances in High Energy Physics 15

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Research Article Quantum Tunnelling for Hawking Radiation from Both Static and Dynamic Black Holes

Subenoy Chakraborty and Subhajit Saha

Department of Mathematics, Jadavpur University, Kolkata, West Bengal 700032, India

Correspondence should be addressed to Subenoy Chakraborty; [email protected]

Received 31 December 2013; Revised 12 March 2014; Accepted 25 March 2014; Published 23 April 2014

Academic Editor: Christian Corda

Copyright © 2014 S. Chakraborty and S. Saha. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP3.

The paper deals with Hawking radiation from both a general static black hole and a nonstatic spherically symmetric black hole. In case of static black hole, tunnelling of nonzero mass particles is considered and due to complicated calculations, quantum corrections are calculated only up to the first order. The results are compared with those for massless particles near the horizon. On the other hand, for dynamical black hole, quantum corrections are incorporated using the Hamilton-Jacobi method beyond semiclassical approximation. It is found that different order correction terms satisfy identical differential equation and are solved by a typical technique. Finally, using the law of black hole mechanics, a general modified form of the black hole entropy is obtained considering modified Hawking temperature.

1. Introduction where 𝐸 is the energy associated with the tunnelling particle and 𝑇𝐻 is the usual Hawking temperature. Hawking radiation is one of the most important effects in The alternative way of looking into this aspect is known as black hole (BH) physics. Classically, nothing can escape from complexpathsmethoddevelopedbySrinivasanetal.[10, 11]. the BH across its event horizon. But in 1974, there was a In this approach, the differential equation of the action 𝑆(𝑟, 𝑡) dramatic change in view when Hawking and Hartle [1, 2] of a classical scalar particle can be obtained by plugging the showed that BHs are not totally black; they radiate analogous scalar field wave function 𝜙(𝑟, 𝑡) = exp{−(𝑖/ℏ)𝑆(𝑟, 𝑡)} into the to thermal black body radiation. Since then, there has been Klein-Gordon (KG) equation in a gravitational background. lots of attraction to this issue and various approaches have Then, the Hamilton-Jacobi (HJ) method is employed to solve been developed to derive Hawking radiation and its corre- the differential equation for 𝑆.Finally,Hawkingtemperature sponding temperature [3–7]. However, in the last decade, two is obtained using the “principle of detailed balance” [10–12] distinct semiclassical methods have been developed which (time-reversal invariant). It should be noted that the first enhanced the study of Hawking radiation to a great extent. method is limited to massless particles only.Also, this method The first approach developed by Parikh and Wilczek [8, 9] is applicable to such coordinate system only in which there is based on the heuristic pictures of visualisation of the is no singularity across the horizon. On the other hand, in source of radiation as tunnelling and is known as radial null complex paths method, the emitted particles are considered geodesic method. The essence of this method is to calculate without self-gravitation and the action is assumed to satisfy the imaginary part of the action for the s-wave emission the relativistic HJ equation. Here tunnelling of both massless (across the horizon) using the radial null geodesic equation and massive particles is possible and it is applicable to any and is then related to the Boltzmann factor to obtain Hawking coordinate system to describe the BH. radiation by the relation: Most of the studies [13–18] dealing with the Hawking 2 𝐸 radiation are connected to semiclassical analysis. Recently, Γ∝exp {− (Im 𝑆out − Im 𝑆in)} = exp {− }, (1) ℎ 𝑇𝐻 Banerjee and Majhi [19]andCordaetal.[20, 21]initiated 2 Advances in High Energy Physics

2 the calculation of Hawking temperature beyond the semiclas- The radial null geodesic (characterized by 𝑑𝑠 =0= 2 sical limit. Mostly, both groups have considered tunnelling 𝑑Ω2 ) has the differential equation (using3 ( )): of massless particle and evaluated the modified Hawking temperature with quantum corrections. 𝑑𝑟 𝐴 = √ [±1 − √1−𝐵(𝑟,) 𝑡 ], (5) In the present work, at first we consider a general 𝑑𝑡 𝐵 nonstatic metric for dynamical BH. HJ method is extended beyond semiclassical approximation to consider all the terms where outgoing or ingoing geodesic is identified by the + or in the expansion of the one particle action. It is found − sign within the square bracket in (4). In the present case, that the higher order terms (quantum corrections) satisfy we deal with the absorption of particles through the horizon identical differential equations as the semiclassical action (i.e., + sign only) and according to Parikh and Wilczek [8], and the complicated terms are eliminated considering BH the imaginary part of the action is obtained as horizon as one way barrier. We derive the modified Hawking 𝑟 𝑟 𝑝 temperature using both the above approaches which are out out 𝑟 󸀠 Im 𝑆=Im ∫ 𝑝𝑟 𝑑𝑟 = Im ∫ ∫ 𝑑𝑝𝑟 𝑑𝑟 𝑟 𝑟 0 found to be identical at the semiclassical level. Also, modified in in form of the BH entropy with quantum correction has been 𝑟 𝐻 󸀠 (6) out 𝑑𝐻 evaluated. = Im ∫ {∫ }𝑑𝑟. 𝑟 0 𝑑𝑟/𝑑𝑡 Subsequently, in the next section, we consider tun- in nelling of particles having nonzero mass beyond semiclassical approximation. Due to nonzero mass, the imaginary part Note that in the last step of the above derivation we have of the action cannot be evaluated using first approach; only used the Hamilton’s equation 𝑟=(𝑑𝐻/𝑑𝑝̇ 𝑟)|𝑟,where(𝑟,𝑝𝑟) HJ method will be applicable. Further, the complicated form are canonical pair. Further, it is to be mentioned that in of the equations involved restricted us to only first order quantum mechanics, the action of a tunnelled particle in a quantum correction. potential barrier having energy larger than the energy of the particle will be imaginary as 𝑝𝑟 = √2𝑚(𝐸 .Forthe− 𝑉) present nonstatic BH, the mass of the BH is not constant 2. Method of Radial Null Geodesic: 󸀠 and hence the 𝑑𝐻 integration extends over all the values of A Survey of Earlier Works energy of outgoing particle, from zero to 𝐸(𝑡) [22](say).Aswe Thissectiondealswithabriefsurveyofthemethodofradial are dealing with tunnelling across the BH horizon, so using 𝑟 null geodesics method [8] considering the picture of Hawking Taylor series expansion about the horizon ℎ we write radiation as quantum tunnelling. In a word, the method 󵄨 𝜕𝐴(𝑟, 𝑡)󵄨 2󵄨 correlates the imaginary part of the action for the classically 𝐴(𝑟, 𝑡)| = 󵄨 (𝑟 − 𝑟 )+𝑂(𝑟−𝑟 ) 󵄨 , 𝑡 𝜕𝑟 󵄨 ℎ ℎ 󵄨𝑡 forbidden process of s-wave emission across the horizon 󵄨𝑡 󵄨 (7) with the Boltzmann factor for the black body radiation at 𝜕𝐵(𝑟, 𝑡)󵄨 2󵄨 𝐵(𝑟, 𝑡)| = 󵄨 (𝑟 − 𝑟 ) + 𝑂(𝑟 −𝑟 ) 󵄨 . the Hawking temperature. We start with a general class of 𝑡 󵄨 ℎ ℎ 󵄨𝑡 𝜕𝑟 󵄨𝑡 nonstatic spherically symmetric BH metric of the form

2 So, in the neighbourhood of the horizon, the geodesic 2 2 𝑑𝑟 2 2 𝑑𝑠 =−𝐴(𝑟,) 𝑡 𝑑𝑡 + +𝑟 𝑑Ω , (2) equation (4) can be approximated as 𝐵 (𝑟,) 𝑡 2 𝑑𝑟 1 𝑟 𝐴(𝑟 ,𝑡) = 0 = 𝐵(𝑟 ,𝑡) ≈ √𝐴󸀠 (𝑟 ,𝑡)𝐵󸀠 (𝑟 ,𝑡)(𝑟−𝑟 ) . (8) where the horizon ℎ is located at ℎ ℎ 𝑑𝑡 2 ℎ ℎ ℎ and the metric has a coordinate singularity at the horizon. To remove the coordinate singularity, we make the following Substituting this value of 𝑑𝑟/𝑑𝑡 in the last step of (5)wehave Painleve-type transformation of coordinates: 2𝜋𝐸 (𝑡) 𝑆= , 1−𝐵 Im (9) 𝑑𝑡 󳨀→ 𝑑𝑡− √ 𝑑𝑟 (3) √𝐴󸀠 (𝑟 ,𝑡)𝐵󸀠 (𝑟 ,𝑡) 𝐴𝐵 ℎ ℎ 𝑟 and as a result metric (2)transformsto where the choice of contour for -integration is on the upper half complex plane to avoid the coordinate singularity at 𝑟ℎ. 1 Thus, the tunnelling probability is given by 𝑑𝑠2 =−𝐴𝑑𝑡2 +2√𝐴( −1)𝑑𝑡𝑑𝑟 +𝑑𝑟2 +𝑟2𝑑Ω 2. (4) 𝐵 2 2 4𝜋𝐸 (𝑡) Γ∼exp {− Im 𝑆} = exp {− }, (10) This metric (i.e., the choice of coordinates) has some distinct ℏ ℏ√𝐴󸀠𝐵󸀠 features over the former one, as follows. which in turn equates with the Boltzmann factor exp{𝐸(𝑡)/𝑇}; (i) The metric is singularity free across the horizon. the expression for the Hawking temperature is (ii) At any fixed time, we have a flat spatial geometry. √ 󸀠 󸀠 (iii) Both the metric will have the same boundary geome- ℏ 𝐴 (𝑟ℎ,𝑡)𝐵 (𝑟ℎ,𝑡) 𝑇 = . (11) try at any fixed radius. 𝐻 4𝜋 Advances in High Energy Physics 3

From the above expression, it is to be noted that 𝑇𝐻 is time Using the standard ansatz for the semiclassical wave function, dependent. namely, Recently, a drawback of the above approach 𝑖 has been noted [23–25]. It has been𝑟 shown that 𝜓 (𝑟,) 𝑡 = exp {− 𝑆 (𝑟,) 𝑡 }, (15) Γ∼ {−(2/ℏ) 𝑆} = {−(2/ℏ) ∫ out 𝑝 𝑑𝑟} ℏ exp Im exp Im 𝑟 𝑟 is not in canonically invariant and hence is not a proper observable; it the differential equation for the action 𝑆 is should be modified as exp{− Im ∮𝑝𝑟𝑑𝑟/ℏ}.Theclosedpath 𝜕𝑆 2 𝜕𝑆 2 goes across the horizon and back. For tunnelling across the ( ) −𝐴𝐵( ) ordinary barrier, it is immaterial whether the particle goes 𝜕𝑡 𝜕𝑟 from the left to the right or the reverse path. So in that case 𝜕2𝑆 1 𝜕 (𝐴𝐵) 𝜕𝑆 1 𝜕 (𝐴𝐵) 𝜕𝑆 𝜕2𝑆 𝑟 +𝑖ℏ[ − − −𝐴𝐵 ]. out 𝜕𝑡2 2𝐴𝐵 𝜕𝑡 𝜕𝑡 2 𝜕𝑟 𝜕𝑟 𝜕𝑟2 ∮ 𝑝𝑟 𝑑𝑟 =2 ∫ 𝑝𝑟 𝑑𝑟 (12) 𝑟 in (16) and there is no problem of canonical invariance. But difficulty To solve this partial differential equation we expand the arises for BH horizon which behaves as a barrier for particles 𝑆 ℏ going from inside of the BH to outside but it does not act action in powers of Planck’s constant as as a barrier for particles going from outside to the inside. 𝑘 𝑆 (𝑟,) 𝑡 =𝑆 (𝑟,) 𝑡 +Σℏ 𝑆 (𝑟,) 𝑡 , (17) So relation (12) is no longer valid. Also, using tunnelling the 0 𝑘 probability is Γ∼exp{− Im ∮𝑝𝑟𝑑𝑟/ℏ},sotherewillbea with 𝑘 being a positive integer. Note that, in the above problem of factor two in Hawking temperature [24, 26, 27]. expansion, terms of the order of Planck’s constant and its Further, the above analysis of tunnelling approach higher powers are considered as quantum corrections over remains incomplete unless effects of self-gravitation and back the semiclassical action 𝑆0. Now substituting ansatz (17)for𝑆 reaction are taken into account. But unfortunately, no general into (16)andequatingdifferentpowersofℏ on both sides, we approaches to account for the above effects are there in the obtain the following set of partial differential equations: literature; only few results are available for some known BH 2 2 solutions [26–32]. 0 𝜕𝑆 𝜕𝑆 ℏ :( ) − 𝐴𝐵( ) =0, (18) Finally, it is worth mentioning that so far the above 𝜕𝑡 𝜕𝑟 tunnelling approach is purely semiclassical in nature and 𝜕𝑆 𝜕𝑆 𝜕𝑆 𝜕𝑆 quantum corrections are not included. Also, this method is ℏ1 : 0 1 −𝐴𝐵 0 1 applicable for Painleve-type coordinates only; one cannot use 𝜕𝑡 𝜕𝑡 𝜕𝑟 𝜕𝑟 the original metric coordinates to avoid horizon singularity. 𝑖 𝜕2𝑆 1 𝜕 (𝐴𝐵) 𝜕𝑆 Lastly, the tunnelling approach is not applicable for massive + [ 0 − 0 2 (19) particles [19]. 2 𝜕𝑡 2𝐴𝐵 𝜕𝑡 𝜕𝑡 1 𝜕 (𝐴𝐵) 𝜕𝑆 𝜕2𝑆 − 0 −𝐴𝐵 0 ]=0, 3. Hamilton-Jacobi Method: 2 𝜕𝑟 𝜕𝑟 𝜕𝑟2 Quantum Corrections 𝜕𝑆 2 𝜕𝑆 𝜕𝑆 𝜕𝑆 2 𝜕𝑆 𝜕𝑆 ℏ2 :( 1 ) +2 0 2 − 𝐴𝐵( 1 ) −2𝐴𝐵 0 2 Wewillnowfollowthealternativeapproachasmentionedin 𝜕𝑡 𝜕𝑡 𝜕𝑡 𝜕𝑟 𝜕𝑟 𝜕𝑟 the introduction, that is, the HJ method to evaluate the imaginary part of the action and hence the Hawking temperature. 𝜕2𝑆 1 𝜕 (𝐴𝐵) 𝜕𝑆 +𝑖[ 1 − 1 We will analyze the beyond semiclassical approximation by 𝜕𝑡2 2𝐴𝐵 𝜕𝑡 𝜕𝑡 incorporating possible quantum corrections. As this method is not affected by the coordinate singularity at the horizon so 1 𝜕 (𝐴𝐵) 𝜕𝑆 𝜕2𝑆 − 1 −𝐴𝐵 1 ]=0, we will use the general BH metric (2)forconvenience. 2 𝜕𝑟 𝜕𝑟 𝜕𝑟2 In the background of the gravitational field described by the metric (2), massless scalar particles obey the Klein- (20) Gordon equation and so on. 2 ℏ 𝜇] Apparently, different order partial differential equations 𝜕[𝑔 √−𝑔𝜕]]𝜓=0. (13) √−𝑔 are very complicated but fortunately there will be lot of simplifications if, in the partial differential equation corre- 𝑘 For spherically symmetric BH, as we are only considering sponding to ℏ , all previous partial differential equations radial trajectories, so we will consider (𝑡, 𝑟)-sector in the are used and finally we obtain identical partial differential spacetime given by (2);thatis,weconcentrateontwo- equation, namely, dimensional BH problems. Using (2), the above Klein- 𝑘 𝜕𝑆 𝜕𝑆 Gordon equation becomes ℏ : 𝑘 =±√𝐴 (𝑟,) 𝑡 𝐵 (𝑟,) 𝑡 𝑘 , (21) 𝜕𝑡 𝜕𝑟 𝜕2𝜓 1 𝜕 (𝐴𝐵) 𝜕𝜓 1 𝜕 (𝐴𝐵) 𝜕𝜓 𝜕2𝜓 − − −𝐴𝐵 =0. (14) 𝜕𝑡2 2𝐴𝐵 𝜕𝑡 𝜕𝑡 2 𝜕𝑟 𝜕𝑟 𝜕𝑟2 for 𝑘=0,1,2,.... 4 Advances in High Energy Physics

Thus, quantum corrections satisfy the same differential Thus, from (15), using solutions (27)and(28)into(17), the equation as the semiclassical action 𝑆0.Hence,thesolutions wave functions for absorption and emission of scalar particle will be very similar. To solve 𝑆0,itistobenotedthatdueto can be expressed as nonstatic BHs the metric coefficients are functions of 𝑟 and 𝑡 𝑖 𝑡 and hence standard HJ method cannot be applied; some 𝜓 (𝑟,) 𝑡 = {− [(∫ 𝜔 (𝑡󸀠)𝑑𝑡󸀠 emm. exp 0 generalization is needed. We start with a general metric [22] ℏ 0 𝑡 𝑘 󸀠 󸀠 𝑡 +Σ𝑘ℏ ∫ 𝜔𝑘 (𝑡 )𝑑𝑡) 󸀠 0 𝑆0 (𝑟,) 𝑡 = ∫ 𝜔0 (𝑡 )𝑑𝑡+𝐷0 (𝑟,) 𝑡 . (22) 0 𝑟 𝑘 𝑑𝑟 −(𝜔0 (𝑡) +Σ𝑘ℏ 𝜔𝑘 (𝑡)) ∫ ]} , 0 √𝐴𝐵 Here 𝜔0(𝑡) behaves as the energy of the emitted particle 𝑖 𝑡 andthejustificationofthechoiceoftheintegralisthatthe 𝜓 (𝑟,) 𝑡 = {− [(∫ 𝜔 (𝑡󸀠)𝑑𝑡󸀠 abs. exp 0 outgoing particle should have time-dependent continuum ℏ 0 energy. 𝑡 𝑆 (𝑟, 𝑡) 𝑘 󸀠 󸀠 Nowsubstitutingtheaboveansatzfor 0 into (18)and +Σ𝑘ℏ ∫ 𝜔𝑘 (𝑡 )𝑑𝑡) using the radial null geodesic in the usual metric from (2), 0 𝑟 namely, 𝑘 𝑑𝑟 +(𝜔0 (𝑡) +Σ𝑘ℏ 𝜔𝑘 (𝑡)) ∫ ]} , 0 √𝐴𝐵 𝑑𝑟 (29) =±√𝐴𝐵, (23) 𝑑𝑡 respectively. Due to tunnelling across the horizon, there will be a change of sign of the metric coefficients in the (𝑟, 𝑡)-part we have of the metric and as a result, function of 𝑡 coordinate has an imaginary part which will contribute to the probabilities. So 𝜕𝐷 𝜕𝐷 𝑑𝑡 𝑑𝑡 we write 0 + 0 =∓𝜔 (𝑡) . 0 (24) 󵄨 󵄨2 𝜕𝑟 𝜕𝑡 𝑑𝑟 𝑑𝑟 𝑃 = 󵄨𝜓 (𝑟,) 𝑡 󵄨 abs. 󵄨 abs. 󵄨 𝑡 𝑡 2 Im 󸀠 󸀠 𝑘 󸀠 󸀠 that is, = exp { [(∫ 𝜔0 (𝑡 )𝑑𝑡 +Σ𝑘ℏ ∫ 𝜔𝑘 (𝑡 )𝑑𝑡) ℏ 0 0 𝑟 𝑑𝐷 𝜔 (𝑡) 𝑘 𝑑𝑟 0 0 +(𝜔0 (𝑡) +Σ𝑘ℏ 𝜔𝑘 (𝑡)) ∫ ]} , =∓ , (25) 0 √𝐴𝐵 𝑑𝑟 √𝐴𝐵 (30) 󵄨 󵄨2 𝑃 = 󵄨𝜓 (𝑟,) 𝑡 󵄨 which gives emm. 󵄨 emm. 󵄨 𝑡 𝑡 2 Im 󸀠 󸀠 𝑘 󸀠 󸀠 𝑟 = exp { [(∫ 𝜔0 (𝑡 )𝑑𝑡 +Σ𝑘ℏ ∫ 𝜔𝑘 (𝑡 )𝑑𝑡) 𝑑𝑟 ℏ 0 0 𝐷0 =∓𝜔0 (𝑡) ∫ . (26) 0 √𝐴𝐵 𝑟 𝑘 𝑑𝑟 −(𝜔0 (𝑡) +Σ𝑘ℏ 𝜔𝑘 (𝑡)) ∫ ]} . 0 √𝐴𝐵 Hence, the complete semiclassical action takes the form (31)

To have some simplification, we will now use the physical fact 𝑡 𝑟 𝑑𝑟 𝑆 (𝑟,) 𝑡 = ∫ 𝜔 (𝑡󸀠)𝑑𝑡󸀠 ∓𝜔 (𝑡) ∫ . that all incoming particles certainly cross the horizon; that is, 0 0 0 (27) 𝑃 =1 0 0 √𝐴𝐵 abs. .Sofrom(30),

𝑡 𝑡 󸀠 󸀠 𝑘 󸀠 󸀠 − Im (∫ 𝜔0 (𝑡 )𝑑𝑡 +Σ𝑘ℏ ∫ 𝜔𝑘 (𝑡 )𝑑𝑡) Here the (or +) sign corresponds to absorption (or emis- 0 0 sion) particle. As solution (27) contains an arbitrary time- (32) 𝑟 𝑑𝑟 dependent function 𝜔0(𝑡), so a general solution for 𝑆𝑘 can be 𝑘 =−Im (𝜔0 (𝑡) +Σ𝑘ℏ 𝜔𝑘 (𝑡)) ∫ written as 0 √𝐴𝐵

and hence 𝑃 . simplifies to 𝑡 𝑟 𝑑𝑟 emm 𝑆 (𝑟,) 𝑡 = ∫ 𝜔 (𝑡󸀠)𝑑𝑡󸀠 ∓𝜔 (𝑡) ∫ , 𝑘=1,2,3,.... 𝑘 𝑘 0 √ 4 𝑟 𝑑𝑟 0 0 𝐴𝐵 𝑃 = {− (𝜔 (𝑡) +Σ ℏ𝑘𝜔 (𝑡)) ∫ } . emm. exp 0 𝑘 𝑘 Im (33) (28) ℏ 0 √𝐴𝐵 Advances in High Energy Physics 5

Then from the principle of “detailed balance” [10–12](which Thisisrelatedtotheoneloopbackreactioneffectsinthe states that transitions between any two states take place with spacetime [6, 33]withtheparameter𝑎 corresponding to trace equal frequency in either direction at equilibrium), we write anomaly. Higher order loop corrections to the surface gravity can be obtained similarly by suitable choice of the functions 𝜔0 (𝑡) 𝜔0 (𝑡) 𝑃 = {− }𝑃 = {− }. 𝜔𝑘(𝑡). For static BHs, Banerjee and Majhi [19]havestudied emm. exp 𝑇 in exp 𝑇 (34) ℎ ℎ these corrections in detail. Lastly, it is worth mentioning that So, comparing (33)and(34), the temperature of the BH is identical result for BH temperature may be obtained if we use given by the Painleve coordinate system as in the previous section. −1 𝑟 −1 ℏ 𝑘 𝜔𝑘(𝑡) 𝑑𝑟 𝑇ℎ = [1+Σ𝑘ℏ ] [Im ∫ ] , (35) 4. Entropy Function and Quantum Correction 4 𝜔0(𝑡) 0 √𝐴𝐵 We will now examine how the semiclassical Bekenstein- where 𝑆 =(𝐴/4ℏ) 𝐴 Hawking area law, namely, BH ( is the area of the ℏ 𝑟 𝑑𝑟 −1 horizon) is modified due to quantum corrections described 𝑇ℎ = [Im ∫ ] (36) 4 0 √𝐴𝐵 in the previous section. The first law of the BH mechanics, which is essentially the energy conservation relation, related is the usual Hawking temperature of the BH. Thus, due to 𝑀 𝑆 the change of BH mass ( )tothechangeofitsentropy( BH), quantum corrections, the temperature of the BH is modified electric charge (𝑄), and angular momentum (𝐽)as from the Hawking temperature and both temperatures are 𝑡 𝑟 𝑑𝑀 =𝑇 𝑑𝑆 +Φ𝑑𝑄+Ω𝑑𝐽. functions of and .Notethat(36) is the standard expression ℎ BH (40) for semiclassical Hawking temperature and it is valid for nonspherical metric also. However, for spherical metric, one Here, Ω is the angular velocity and Φ is the electrostatic can use the Taylor series expansions (7) near the horizon potential. So, for nonrotating uncharged BHs, the entropy has and obtain 𝑇𝐻 as given in (11)byperformingthecontour the simple form integration.Theambiguityoffactoroftwo(asmentioned 𝑑𝑀 earlier) in the Hawking temperature does not arise here. 𝑆 =∫ , BH 𝑇 (41) Further, one may note that solutions (27)or(28)arethe ℎ unique solutions to (18)or(21) except for a premultiplication or using (35)for𝑇ℎ,weget factor. This arbitrary multiplicative factor does not appear in the expression for Hawking temperature; only the particle 𝜔 (𝑡) 𝑑𝑀 𝑆 = ∫ [1+Σ ℏ𝑘 𝑘 ] . energy (𝜔0)or𝜔𝑘 is rescaled. As quantum correction term BH 𝑘 (42) 𝜔0 (𝑡) 𝑇𝐻 contains 𝜔𝑘/𝜔0,soitdoesnotinvolvethearbitrarymultiplica- tive factor and hence it is unique. For choice (38) corresponding to one loop back reaction Tohave some interpretation about the arbitrary functions effects, we have from (42) the quantum corrected BH entropy 𝜔𝑘(𝑡) appearing in the quantum correction terms, we make as 𝑆 ℏ use of dimensional analysis. As 0 has the dimension ,sothe 2 2 −𝑘 𝑎ℏ 𝑎 ℏ 𝑑𝑀 𝜔𝑘(𝑡) ℏ 𝑆 = ∫ [1 + + +⋅⋅⋅] . arbitrary function has the dimension .Instandard BH 2 (43) 2 𝑀 𝑀 𝑇𝐻 choice of units, namely, 𝐺=𝑐=𝐾𝐵 =1, ℏ∼𝑀𝑝 and so −2𝑘 𝜔𝑘 ∼𝑀 ,where𝑀 is the mass of the BH. The first term is the usual semiclassical Bekenstein-Hawking Similar to the Hawking temperature, the surface gravity entropy and the subsequent terms are the quantum cor- of the BH is modified due to quantum corrections. If 𝜅𝑐 is rections of different order. For static BHs, Banerjee and the semiclassical surface gravity corresponding to Hawking Majhi [19] have shown the correction terms of which the temperature, that is, 𝜅𝑐 =2𝜋𝑇𝐻, then the quantum corrected leading one gives the standard logarithmic correction. On the surface gravity 𝜅=2𝜋𝑇𝐻 is related to the semiclassical value other hand, for nonstatic BHs, as the proportionality factors by the relation: are time-dependent and arbitrary (see (42)) so the leading −1 order correction term may not be logarithmic. For future 𝑘 𝜔 (𝑡) 𝜅=𝜅[1+Σ ℏ 𝑘 ] . (37) work, we will attempt to determine physical interpretation of 𝑐 𝑘 𝜔 (𝑡) 0 the arbitrary time-dependent proportionality factors so that Moreover, based on the dimensional analysis, if we choose, quantum corrections may be evaluated. for simplicity, 𝑎𝑘𝜔 (𝑡) 5. Hamilton-Jacobi Method for Massive 𝜔 (𝑡) = 0 , 𝑎 , Particles: Quantum Corrections 𝑘 𝑀2𝑘 “ ” is a dimensionless parameter (38) The KG equation for a scalar field 𝜓 describing a scalar 𝑚 then expression (37) is simplified to particle of mass 0 has the form [10] −1 2 ℏ𝑎 𝑚0 𝜅=𝜅(1− ) . (39) (◻ + )𝜓=0, (44) 0 𝑀2 ℏ2 6 Advances in High Energy Physics where the box operator "◻" is evaluated in the background of (50)andequatingdifferentpowersofℏ on both sides, we a general static BH metric of the form obtain the following set of partial differential equations:

2 2 2 2 2 𝑑𝑟 2 2 0 1 𝜕𝑆 𝜕𝑆 2 𝑑𝑠 =−𝐴(𝑟) 𝑑𝑡 + +𝑟 𝑑Ω . (45) ℏ : ( ) −𝐵( ) −𝐸 (𝑟) =0, (52) 𝐵 (𝑟) 2 𝐴 𝜕𝑡 𝜕𝑟 0 2 𝜕𝑆 𝜕𝑆 𝜕𝑆 𝜕𝑆 ℏ1 : 0 1 −2𝐵 0 1 TheexplicitformoftheKGequationforthemetric(45)is 𝐴 𝜕𝑡 𝜕𝑡 𝜕𝑟 𝜕𝑟 2 2 2 2 1 1 𝜕 𝑆 𝜕 𝑆 1 𝜕 𝜓 𝜕 𝜓 1 𝜕 (𝐴𝐵) 𝜕𝜓 2𝐵 𝜕𝜓 − [ 0 −𝐵2 0 − +𝐵 + + 2 2 (53) 𝐴 𝜕𝑡2 𝜕𝑟2 2𝐴 𝜕𝑟 𝜕𝑟 𝑟 𝜕𝑟 𝑖 𝐴 𝜕𝑡 𝜕𝑟 1 𝜕 𝜕𝜓 1 𝜕 (𝐴𝐵) 2𝐵 𝜕𝑆 + ( 𝜃 ) 0 2 sin (46) −{ + } ]=0, 𝑟 sin 𝜃 𝜕𝜃 𝜕𝜃 2𝐴 𝜕𝑟 𝑟 𝜕𝑟 2 2 1 𝜕 𝜓 𝑚 1 𝜕𝑆 2 2 𝜕𝑆 𝜕𝑆 𝜕𝑆 2 𝜕𝑆 𝜕𝑆 + = 0 𝜓 (𝑡, 𝑟, 𝜃, 𝜙). ℏ2 : ( 1 ) + 0 2 −𝐵( 1 ) −2𝐵 0 2 𝑟2sin2𝜃 𝜕𝜙2 ℏ2 𝐴 𝜕𝑡 𝐴 𝜕𝑡 𝜕𝑡 𝜕𝑟 𝜕𝑟 𝜕𝑟 1 1 𝜕2𝑆 𝜕2𝑆 Due to spherical symmetry, we can decompose 𝜙 in the form − [ 1 −𝐵2 1 𝑖 𝐴 𝜕𝑡2 𝜕𝑟2 𝜓 (𝑡, 𝑟, 𝜃, 𝜙)=Φ (𝑡, 𝑟) 𝑌𝑚 (𝜃, 𝜙) , 𝑙 (47) 1 𝜕 (𝐴𝐵) 2𝐵 𝜕𝑆 −{ + } 1 ]=0, 2𝐴 𝜕𝑟 𝑟 𝜕𝑟 𝜙 where satisfies [10] (54)

1 𝜕2𝜓 𝜕2𝜓 1 𝜕 (𝐴𝐵) 𝜕𝜓 2𝐵 𝜕𝜓 −𝐵 − − and so on. 2 2 𝐴 𝜕𝑡 𝜕𝑟 2𝐴 𝜕𝑟 𝜕𝑟 𝑟 𝜕𝑟 To solve the semiclassical action 𝑆0, we start with the (48) standard separable choice [10] 𝑙 (𝑙+1) 𝑚2 +{ + 0 }Φ(𝑡, 𝑟) =0. 𝑟2 ℏ2 𝑆0 (𝑟,) 𝑡 =𝜔0𝑡+𝐷0 (𝑟) . (55)

If we substitute the standard ansatz for the semiclassical wave Substituting this choice in (52), we obtain function, namely, 𝑟 𝜔2 −𝐴𝐸2 𝑖 𝐷 =±∫ √ 0 0 𝑑𝑟 = ±𝐼 ( ), (56) 𝜙 (𝑡, 𝑟) = {− 𝑆 (𝑟,) 𝑡 }, (49) 0 0 say exp ℏ 0 𝐴𝐵

− then the action 𝑆 will satisfy the following differential equa- where + or sign corresponds to absorption or emission of 𝑆 tion: scalar particle. Now substituting this choice for 0 in (53), we have the differential equation for first order corrections 𝑆1 as 1 𝜕𝑆 2 𝜕𝑆 2 [ ( ) −𝐵( ) −𝐸2 (𝑟)] 𝐴 𝜕𝑡 𝜕𝑟 0 𝜕𝑆 𝐴𝐸2 𝜕𝑆 1 ∓ √𝐴𝐵√1− 0 1 𝜕𝑡 𝜔2 𝜕𝑟 ℏ 1 𝜕2𝑆 𝜕2𝑆 1 𝜕 (𝐴𝐵) 2𝐵 𝜕𝑆 0 − [ −𝐵2 −{ + } ]=0, 𝑖 𝐴 𝜕𝑡2 𝜕𝑟2 2𝐴 𝜕𝑟 𝑟 𝜕𝑟 √𝐴𝐵 [ 1 𝐴𝐸2 (50) ∓ [ − √1− 0 𝑖 𝑟 𝜔2 [ 2 2 2 2 2 2 where 𝐸0 =𝑚0 +(𝐿/𝑟 ) and 𝐿 = 𝑙(𝑙 + 1)ℏ is the angular 2 2 2 2 3 momentum. To incorporate quantum corrections over the (𝜕𝐴/𝜕𝑟) (𝐸0/𝜔 ) − (2𝐴𝐿 /𝜔0𝑟 )] semiclassical action, we expand the actions in powers of + ] =0. 4√1−(𝐴𝐸2/𝜔2) Planck constant ℏ as 0 ] (57) 𝑘 𝑆 (𝑟,) 𝑡 =𝑆0 (𝑟,) 𝑡 +Σ𝑘ℏ 𝑆𝑘 (𝑟,) 𝑡 , (51) As before, 𝑆1 can be written in separable form as where 𝑆0 is the semiclassical action and 𝑘 is a positive integer. Now substituting this ansatz for 𝑆 in the differential equation 𝑆1 =𝜔1𝑡+𝐷1 (𝑟) , (58) Advances in High Energy Physics 7

𝑃 where and as a result emm. simplifies to 𝑟 𝑑𝑟 𝐷 = ∫ 1 [ 4𝜔 0 √𝐴𝐵√1−(𝐴𝐸2/𝜔2) 𝑃 = [ − 0 0 0 emm. exp ℏ [ [ √𝐴𝐵 × [ ±𝜔 − { 2 1 𝑖 { 𝑟 𝑑𝑟 𝐴𝐸 × ∫ (√1− 0 [ Im { 2 0 √𝐴𝐵 𝜔0 { { 1 𝐴𝐸2 × − √1− 0 { 2 } { 𝑟 𝜔 ℏ(𝜔1/𝜔0) }] { + ) }] . √1 − (𝐴𝐸2/𝜔2) } 0 0 }] 2 2 2 2 3 } (𝜕𝐴/𝜕𝑟) (𝐸0/𝜔 ) − (2𝐴𝐿 /𝜔0𝑟 )}] (66) + }] 4√1 − (𝐴𝐸2/𝜔2) } 0 }] Using the principle of “detailed balance” [10, 11, 20, 21], namely, =±𝐼 −𝐼. 1 2 𝐸 𝐸 (59) 𝑃 = {− }𝑃 = {− }, emm. exp in exp (67) 𝑇ℎ 𝑇ℎ Now due to complicated form, if we retain terms up to first order quantum corrections, that is, the temperature of the BH is given by 𝑆=𝑆 +ℏ𝑆 =(𝜔 +ℏ𝜔 )𝑡+{𝐷 +ℏ𝐷 (𝑟)}, { 0 1 0 1 0 1 (60) ℏ𝐸 [ { 𝑟 𝑑𝑟 𝑇 = [ ∫ ℎ Im { then the wave function denoting absorption and emission 4𝜔0 0 √𝐴𝐵 solutions of the KG equation (48)using(49) are of the form [ { −1 𝑖 2 } 𝜙 = {− (𝜔 +ℏ𝜔 𝑡+𝐼 +ℏ𝐼 −ℏ𝐼)} , 𝐴𝐸 ℏ(𝜔 /𝜔 ) }] abs. exp ℏ 0 1 0 1 2 ×(√1− 0 + 1 0 ) ] , 𝜔2 } (61) 0 √1 − (𝐴𝐸2/𝜔2) 𝑖 0 0 }] 𝜙 = {− (𝜔 +ℏ𝜔 𝑡−𝐼 +ℏ𝐼 −ℏ𝐼)} . emm. exp ℏ 0 1 0 1 2 (68) It is to be noted that in course of tunnelling across the hori- where the semiclassical Hawking temperature of the BH has zon, the coordinate nature changes; that is, more precisely the expression (𝑟, 𝑡) the signs of the metric coefficients in the -hyperplane are −1 altered. Thus, we can interpret this as that the time coordinate ℏ𝐸 𝑟 𝑑𝑟 𝐴𝐸2 𝑇 = [ ∫ √1− 0 ] . has an imaginary part in crossing the horizon and accordingly 𝐻 Im 2 (69) 4𝜔 0 √𝐴𝐵 𝜔 the temporal part has contribution to the probabilities [19, 0 [ 0 ] 33]. Thus, absorption and emission probabilities are given by Now, to obtain the modified form of the surface gravity of the 󵄨 󵄨2 2 BH, we start with the usual relation between surface gravity 𝑃 . = 󵄨𝜙 󵄨 = exp { (Im 𝜔0 +ℏ𝜔1𝑡) abs in ℏ and Hawking temperature, namely, (62) 𝜅 =2𝜋𝑇 , + Im 𝐼0 +ℏ𝐼1 − Im ℏ𝐼2}, 𝐻 𝐻 (70) 𝑇 󵄨 󵄨2 𝑖 where 𝐻 is given by (69). 𝑃 = 󵄨𝜙 󵄨 = {− ( 𝜔 +ℏ𝜔 𝑡) emm. 󵄨 out󵄨 exp ℏ Im 0 1 Sothequantumcorrectedsurfacegravityisgivenby (63) 𝜅 =2𝜋𝑇. QC ℎ (71) − Im 𝐼0 +ℏ𝐼1 − Im ℏ𝐼2}. Further, for the present nonrotating, uncharged, static BHs, In the classical limit ℏ→0, there is no reflection, so all using the law of BH thermodynamics 𝑑𝑀ℎ =𝑇 𝑑𝑆,wehave ingoing particles should be absorbed and hence [33] the expression for the entropy of the BH as 𝑃 =1. 4𝜔 ℏ𝜔 𝑟 𝑑𝑟 lim abs. (64) 𝑆 = ∫ 0 (1 + 1 )𝑑𝑀∫ . ℏ→0 BH (72) ℏ𝐸 𝜔0 0 √𝐴𝐵 So, from (62), we must have Finally, it is easy to see from (68) that near the horizon 2 Im 𝜔0𝑡=Im 𝐼0, Im (𝜔1𝑡−𝐼2)=Im 𝐼1 (65) the presence of 𝐸0 term can be neglected as it is multiplied 8 Advances in High Energy Physics by the metric coefficient 𝐴. Therefore, the quantum corrected Also, it will be interesting to calculate temperature of the (up to first order) temperature of the BH (in68 ( )) reduces to BH for tunnelling nonzero mass particle with full quantum correction and examine whether the result agrees with that −1 𝑟 −1 ℏ𝐸 ℏ𝜔1 𝑑𝑟 of Banerjee and Majhi [19] near the horizon. Finally, it will be 𝑇ℎ = (1 + ) [∫ ] (73) 4𝜔0 𝜔0 0 √𝐴𝐵 nice to determine quantum corrected entropy of the BH in a convenient form. and the Hawking temperature (given in (69)) becomes

ℏ𝐸 𝑟 𝑑𝑟 −1 Conflict of Interests 𝑇𝐻 = [∫ ] . (74) 4𝜔0 0 √𝐴𝐵 The authors declare that there is no conflict of interests regarding the publication of this paper. So we have ℏ𝜔 −1 Acknowledgments 𝑇 =(1+ 1 ) 𝑇 . (75) ℎ 𝜔 𝐻 0 The authors are thankful to IUCAA, Pune, India, for their warm hospitality and research facilities as the work has We see that if the energy of the tunnelling particle is chosen as 𝜔 𝐸=𝜔 𝜔 =𝛽/𝑀 been done there during a visit. Also, Subenoy Chakraborty 0 (i.e., 0)and 1 1 (for notations see Banerjee acknowledges the UGC-DRS Programme in the Department and Majhi [19]) then the Hawking temperature given by of Mathematics, Jadavpur University. The author Subhajit (74) is the usual one derived for massless particles and 𝑇 Saha is thankful to UGC-BSR Programme of Jadavpur the quantum corrected temperature ℎ given in (75)agrees University for awarding research fellowship. The authors are with that of Banerjee and Majhi [19] for massless particle. thankful to Ritabrata Biswas and Nairwita Mazumder for Therefore, Hawking temperature near the horizon remains theirhelpattheinitialpartofthework. the same for both massless and nonzero mass tunnelling particles and it agrees with the claim of Srinivasan and Padmanabhan [10] and Banerjee and Majhi [19]. For future References work, it will be interesting to calculate the temperature of the [1] S. W. Hawking, “Particle creation by black holes,” Communica- BH for tunnelling nonzero mass particle with full quantum tions in Mathematical Physics,vol.43,no.3,pp.199–220,1975. corrections and examine whether the result agrees with that [2] J. B. Hartle and S. W. Hawking, “Path-integral derivation of of Banerjee and Majhi [19] near the horizon. Finally, it will be black-hole radiance,” Physical Review D,vol.13,no.8,pp.2188– nice to determine quantum corrected entropy of the BH in a 2203, 1976. convenient form. [3] J. D. Bekenstein, “Generalized second law of thermodynamics in black-hole physics,” Physical Review D,vol.9,p.2188,1974. 6. Summary of the Work [4] T. Damour and R. Ruffini, “Black-hole evaporation in the Klein- Sauter-Heisenberg-Euler formalism,” Physical Review D,vol.14, Thisworkisanattempttostudyquantumcorrectionsto p. 332, 1976. Hawking radiation of massless particle from a dynamical BH [5] S. M. Christensen and S. A. Fulling, “Trace anomalies and the as well as for massive particle from a static BH. At first, Hawking effect,” Physical Review D,vol.15,no.8,pp.2088–2104, radial null geodesic tunnelling approach has been used with 1977. Painleve-type choice of coordinate system to derive semiclas- [6] J. W.York, “Black hole in thermal equilibrium with a scalar field: sical Hawking temperature. Then full quantum mechanical the back-reaction,” Physical Review D,vol.31,p.755,1985. calculations have performed writing action in a power series [7] Z. Zhao and J. Y. Zhu, “Nernst theorem and Planck absolute of the Planck constant ℏ to evaluate the quantum correc- entropy of black holes,” Acta Physica Sinica,vol.48,no.8,pp. tions to the Hawking temperature. Subsequently, quantum 1558–1564, 1999. corrected surface gravity has been calculated and it is found [8] M. K. Parikh and F. Wilczek, “Hawking radiation as tunneling,” that one loop back reaction effects in the spacetime can be Physical Review Letters,vol.85,no.24,pp.5042–5045,2000. obtained by suitable choice of the arbitrary functions and [9] M. K. Parikh, “Energy conservation and Hawking radiation,” parameters. Finally, an expression for the quantum corrected http://arxiv.org/abs/hep-th/0402166. entropy of the BH has been evaluated. It is found that, due [10] K. Srinivasan and T. Padmanabhan, “Particle production and to the presence of the arbitrary functions in the expression complex path analysis,” Physical Review D,vol.60,no.2,Article ID024007,p.20,1999. for entropy, the leading order quantum correction may not [11] S. Shankaranarayanan, K. Srinivasan, and T. Padmanabhan, be logarithmic in nature. On the other hand, in the case “Method of complex paths and general covariance of Hawking of Hawking radiation of massive particle from static BH, it radiation,” Modern Physics Letters A. Particles and Fields, Grav- is found that Hawking temperature near the horizon does itation, Cosmology, Nuclear Physics,vol.16,no.9,pp.571–578, not depend on the mass term as predicted by Srinivasan 2001. and Padmanabhan [10] and Banerjee et al. [16–18]. For [12] R. Banerjee, B. R. Majhi, and S. Samanta, “Noncommutative future work, we will try to find a solution for the partial black hole thermodynamics,” Physical Review D. Particles, differential equation18 ( ) in a more simple form so that more Fields, Gravitation, and Cosmology,vol.77,no.12,ArticleID physical interpretations can be done from the BH parameters. 124035, p. 8, 2008. Advances in High Energy Physics 9

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Research Article Analyzing Black Hole Super-Radiance Emission of Particles/Energy from a Black Hole as a Gedankenexperiment to Get Bounds on the Mass of a Graviton

A. Beckwith

Department of Physics, Chongqing University, Chongqing 400044, China

Correspondence should be addressed to A. Beckwith; [email protected]

Received 18 December 2013; Revised 10 February 2014; Accepted 10 February 2014; Published 3 April 2014

Academic Editor: Christian Corda

Copyright © 2014 A. Beckwith. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP3.

Use of super-radiance in BH physics, so 𝑑𝐸/𝑑𝑡 <0 specifies conditions for a mass of a graviton being less than or equal to 65 10 grams, allows for determing what role additional dimensions may play in removing the datum that massive gravitons lead to 3/4th the bending of light past the planet Mercury. The present document makes a given differentiation between super-radiance in the case of conventional BHs and Braneworld BH super-radiance, which may delineate whether Braneworlds contribute to an admissible massive graviton in terms of removing the usual problem of the 3/4th the bending of light past the planet Mercury whichisnormallyassociatedwithmassivegravitons.Thisleadstoaforkintheroadbetweentwoalternativeswiththepossibility of needing a multiverse containment of BH structure or embracing what Hawkings wrote up recently, namely, a redo of the event horizon hypothesis as we know it.

1. Introduction: Massive Gravity and graviton results in 5 polarizations plus other problems [4, How to Get It to Commensurate with Black 5]; that is, in [4], there is a description of how a massive Hole Physics graviton leads to 3/4th the calculated bending of light pass the mass of Mercury, as seen in the 1919 experiment. Reference We are now attempting to come up with criteria for either [5] has details on the five polarization states, which are massless or massive gravitons. Our preferred way to do it another problem. One cannot go from a massive graviton and distinguishing between the two forms of super-radiance. One eliminate mass from the graviton and then neatly recover the built about Kerr black holes [1], and the other involving easier spin dynamics (2 polarization states) and vastly simpler brane theory [2]; with the brane theory version of super- situation where one has recovered the Schwartzshield metric. radiance, perhaps correcting a problem as to when a massive As [5] discusses, in its page 92, that this easy recovery of graviton would, without brane theory, lead to 3/4th the the Schwartzshield metric, if a graviton mass goes to zero, angular bending of light and be seen experimentally. We is impossible. Also note that note [4]hasadiscussionon briefly allude to both of these cases in the introduction below, how the bending of light is not commensurate with GR for before giving more details to this phenomenon in Sections 2 massive graviton, which is equivalent to a discussion on a and 3. phenomenological ghost state for the trace of ℎ,whichis In general, relativity of the metric 𝑔𝑎𝑏(𝑥, 𝑡) is a set of given by [6] and occurs regardless of whether the mass for numbers associated with each point which gives the distance graviton nearly goes to zero. In [7], Csakietal.,havegiven´ to neighboring points. That is, general relativity is a classical a temporary fix to restore the bending of light for massive theory. As it is designated by GR traditionalists [3], the gravitons and to remove the 3/4th angle deflection from the graviton is usually stated to be massless, with two spin 1919 GR test value, and this is by the use of brane theory. states and with two polarizations. Adding a mass to the What this document will do will be to try to establish massive 2 Advances in High Energy Physics gravitons as super-radiant emission candidates from black thereby making the case, due to the mass dependence of the holes [8]and,indoingso,provideanotherframeworkfor black hole, that super-radiance would almost certainly not be their analysis which would embed them in GR. In doing so, observable but would firmly embed massive gravitons in GR one should keep in mind that this is a thought experiment in spite of the view point offered in [3]. Doing so would mean and that the author is fully aware of how hard it would that [1] has the following formulation; with respect to when be to perform experimental measurements. In coming up 𝑑𝐸/𝑑𝑡 <0, which occurs for super radiance; in (1)below; with criteria as to graviton mass, we are also, by extension, we set c1 as a constant, radiation frequency omega as the considering the Myers-Perry higher dimensional model of frequency of radiation approaching a black hole, the number black holes [9] and commenting upon its applications, some 𝑚 as a quantum number, and a definite given value for the of which are in [10] and all of which start with the implications angular velocity of a black hole. Here, after the Padmanabhan of 𝑑𝐸/𝑑𝑡, <0 leading to “leakage” from a black hole. That derivation of what 𝑑𝐸/𝑑𝑡 <0 means, there will be a separate, is, energy of the black hole “decreases” in time. That is, there brane theory derivation of BH super-radiance [2]whichhas ℎ ℎ(𝑖, 𝑗) 𝑁/2 are ghost states, where is the trace of which is a GW provisionally 0<𝜔<∑𝐽=1 𝑚𝐽 ⋅(Ω𝐻)𝐽 [2], where 𝑁 is perturbation of the flat Euclidian metric, a possibility that the number of dimensions. The 2nd frequency dependence brane theory and higher dimensions may remove the 3/4th for when 𝑁 cangouptoatleast10orsowillberemarked angle of bent light calculated for massive gravitons, and a after we finish the Padmanabhan frequency dependence for suitable thought experiment as given below may allow for super-radiance, as given below for a “classical” Kerr BH. To 𝑑𝐸/𝑑𝑡 allowing us to determine whether higher dimensional initiate our analysis of the physics happening in due to [1], we modelsarejustifiable.Thisisthereasonwhythesuper- formulate super-radiance by (1) given below radiance phenomenology is being investigated, that is, of 𝑑𝐸 bending of angle of light divergence from GR models using =𝑐1 ⋅𝜔⋅[𝜔−𝑚⋅Ω𝐻] . (1) massive Gravitons. Does 𝑑𝐸/𝑑𝑡 <0 imply that there are brane 𝑑𝑡 theory states which may remove the 3/4th bending of light In this case, according to Padmanabhan, 𝑐1 is a constant, divergence from GR by massive gravitons? And can a super- which is defined via writing1 ( )via[1]. Consider radiance model for when 𝑑𝐸/𝑑𝑡 <0 imply conditions for 𝑑𝐸 𝑀⋅𝑟𝐻 2 2 which brane worlds have to be considered [2], as opposed to = ⋅[∫ (𝑆 (𝜃) ⋅ sin 𝜃 ⋅ 𝑑𝜃) ⋅𝑑𝜙] the simpler model proposed by Padmanabhan [1]. 𝑑𝑡 2𝜋 The paper will differentiate between Kerr BH[1]versions ⋅𝜔⋅[𝜔−𝑚⋅Ω ] of super-radiance and brane theory BH super-radiance [2] 𝐻 and, secondly, afterwards, inquire about whether a BH in = flux-of-energy-through-horizon brane theory configuration is satisfied, if the simper Kerr BH super-radiance criteria is not satisfied. After these two 𝑀⋅𝑟𝐻 2 2 ⇐⇒ 𝑐1 ≡ ⋅[∫ (𝑆 (𝜃) ⋅ sin 𝜃⋅𝑑𝜃)⋅𝑑𝜙]=const., versions are distinguished, we will then discuss experimental 2𝜋 criteria which may result in determining whther Kerr BH (2) super-radiance occurs [1] or brane theory super-radiance where, for a massless scalar field, one has the function 𝑆 for a occurs [2]; if only Kerr BH super-radiance occurs, the “surface area” function, defined as follows1 [ ]: likelihood of massive Gravitons is remote. If brane theory BH 2 −1/2 −1/2 𝑎𝑏 super-radiance occurs, then there may [2] be conditions for ∃𝑆 (𝜃) ⇐⇒ ( − 𝑔 ) 𝜕𝑏 [(−𝑔) ⋅𝑔 ⋅𝜕𝑎Φ] = 0 which the 3/4th error in light bending is removed, permitting (3) massive Gravitons. ⇐⇒ Φ ≡𝑒 −𝑖𝜔𝑡𝑒𝑖𝑚⋅𝜙 ⋅𝑅(𝑟) ⋅𝑆(𝜃) .

In this case, mass 𝑀 is for the source, that is, later for the 2. What is Super-Radiance in Black Hole mass 𝑀 of a GW generator, in this case a BH. Also, here, Physics? First: The Padmanabhan 𝑟𝐻 is the horizon radius, as specified. And this will have its Treatment for Kerr BHs application to the issue of gravitons of a small mass spirialing into a BH, with the BH subsequently releasing radiation via We, first of all, consider a simplified version of super- 𝑑𝐸/𝑑𝑡 <0, with the given versions of a BH set of parameters radiance. In simple language, super-radiance involves having [1]foraKERRBH.Thefollowing(4)comesasfarasangular incoming radiation scattered off the horizon of a BH and velocity of the BH, as well as the following sets of parameters. radiated outward, so the net flow of energy is 𝑑𝐸/𝑑𝑡 <0 Here, the phenomenon of super-radiance is impossible, if (6) radiation energy with a frequency bounded by 0<𝜔< below is zero. More on this point about when super-radiance 𝑚⋅Ω𝐻 [1]. In this case 𝑚 is a quantum number, the frequency is impossible will be discussed later in the text. Consider 𝜔 is for radiation infalling to the event horizon of the BH, 𝑎 Ω Ω = , and the term 𝐻 is the angular velocity of a KERR black 𝐻 2𝑀 ⋅𝑟 (4) hole [2]. This paper, first of all, examines Padmanabhan’s BH 𝐻 derivation of super-radiance [2] stating its application to 𝑟 =𝑀 − √𝑀 2 −𝑎2, (5) the graviton, with mass, and making then a referral to the 𝐻 BH BH likelihood of measurement which ties in with the metric 𝑔𝜇] √ 2 2 being perturbed from flat space values by ℎ00, ℎ0𝑖,andℎ𝑖𝑗 [7], 𝑎= 𝑥 +𝑦 . (6) Advances in High Energy Physics 3

Then, 3. Could Super-Radiance Be Observed 𝜔 Experimentally, and What Good Is This 0< <Ω . 𝑚 𝐻 (7) Thought Experiment? Super-radiance is really about the same as particle production Note that there are conditions, based upon (4)above,which from a BH. From Padmanabhan [1]isavitalresultwhichis go to zero, due to the numerator, in a manner which means given in the following quote. when there is no angular velocity for the black hole; that the frequency of the incoming radiation is set equal to zero and If we think of super-radiance as stimulated emis- that there is, effectively no super-radiance. This will obviously sionofradiationbytheblackholeincertain lead to the classical description of BH physics. A problem, modes, owing the presnce [sic] of the incoming though,isthat,recently,Hawkingshasstatedthatnotallis wave, it seems natural to expect spontaneous well in BH event horizons and that scrambled information emission of radiation in various modes by the could possibly leave a BH, in opposition. black hole in quantum field theory. The black hole evaporation (then) can be thought of as 2.1. Examining Super-Radiance When There Is More than 4 spontaneous emission of particles that survives Dimensions as to BH Physics. As said before [2], even in the limit of zero angular momentum of the black hole. 𝑁/2 Furthermore, on the same page, page 623 of [1]statesthe 0<𝜔<∑𝑚𝐽 ⋅(Ω𝐻)𝐽. (8) 𝐽=1 following.

𝑁 It seems natural to assume that this source of In doing so, as given above is a measure of dimensions as energy radiated to infinity is the mass of the totheBH,andthedifferenceinthisfrom(7)inpartisalso collapsing structure. due to 󵄨 Leading to Formula 14.143 of [1]thatthe“entropy”ofa 𝑎 󵄨 Ω = 󵄨 BH is given by, where 𝑀 is the mass of the BH, 𝐿𝑃 is the 𝐻 󵄨 2𝑀 ⋅𝑟𝐻 󵄨 𝐴 BH Kerr-BH Plamck length, and hor is the area of the Event horizon of 󵄨 (9) 𝑎 󵄨 a black hole. This area of a BH event horizon is relevant since 󳨀󳨀󳨀󳨀󳨀󳨀󳨀󳨀󳨀󳨀󳨀󳨀󳨀󳨀󳨀󳨀󳨀󳨀󳨀→ 󵄨 . 2 2 󵄨 it directly connects, as we will mention later, to [2] version of Kerr-BH → Myers-Perry-BH 𝑎 +𝑟 󵄨 𝐻 󵄨Myers-Perry-BH super-radiance. Reference [1] version of entropy would also hold for [2] as well, and we state the entropy as The numerator of the above is still defined by the square root 𝑥2 +𝑦2 1 𝐴 of and could go to zero for certain quantum numbers, 𝑆=4𝜋𝑀2 = ⋅( hor ). 𝑚 2 (13) 𝐽, and we would then paraphrase the right hand side of (9) 4 𝐿𝑃 as functionally being Here, in [2]wehave(27), that its main result is about the 𝑎 Ω = 𝐽 differential of the area of an event horizon which is given as 𝑗 2 2 (10) 𝑎𝐽 +𝑟𝐻 follows, if there is a brane theory connection to the formation of BHs: as frequency of BH arises due to the jth component of BH 8𝜋𝑟 1 𝑁/2 angular Momentum 𝐽𝑗. 𝑑𝐴 = 𝐻 𝑑𝑀 ⋅(1− ∑𝑚 ⋅Ω). hor 𝐵 BH 𝜔 𝑗 𝑗 (14) So, then, one has a rewrite of (8)asgiven,withaslightly 𝑗=1 different angular frequency for BHs as by[2], The positive definite nature of this expression for the differ- 𝑁/2 󵄨 𝑎𝑗 󵄨 ential of the area of an event horizon would then be [2]since 0<𝜔<∑𝑚 󵄨 ⋅ 𝑗 2 2 󵄨 (11) 𝑑𝑀,thenby[ <0 2], so then by (15) below, we recover (11), 𝑗=1 𝑎𝑗 +𝑟𝐻 󵄨 󵄨Myers-Perry-BH by [2]; if 𝑑𝐴 >0,then

This is to be compared with the Padmanabhan version of 1 𝑁/2 𝑑𝑀 ⋅(1− ∑𝑚 ⋅Ω)>0 super-radiance as given by [1]: BH 𝑗 𝑗 𝜔 𝑗=1 󵄨 (15) 𝑎 󵄨 0<𝜔<𝑚⋅ 󵄨 . (12) 1 𝑁/2 2𝑀 ⋅𝑟 󵄨 BH 𝐻 Kerr-BH ⇐⇒ ( 1 − ∑𝑚𝑗 ⋅Ω𝑗)<0. 𝜔 𝑗=1 We will be commenting upon what the experimental signatures of both (11)and(12) could be and why in the next Wemakethefollowing3claimsasfortheanalogytoBH section. physics. 4 Advances in High Energy Physics

Claim 1. Entropy in both Kerr and Myers-Perry BHs has situation for which if there exists higher than 4 dimensional 𝑑𝐴 >0,where𝐴 is the event horizon, and brane theory, one may correct the 3/4th deficiency. But if Claim 1 part (i) is not true, then the solution allowing for [7] (i) for Myers-Perry BHs, the following are true (dimen- 𝑁=10 is likely not to be true. sions up to 10, say, i.e., ): Note that the super-radiance phenomenon as referenced 1 𝐴 in Claim 1 part (i) and part (ii) has its roots in entropy. Note 𝑆=4𝜋𝑀2 = ⋅( hor ) that entropy of a black hole with its surface area is stated to be 4 𝐿2 𝑃 a precondition for initial conditions for super-radiance. And, more than that, one needs a spinning black hole. No black 8𝜋𝑟 1 𝑁/2 𝑑𝐴 = 𝐻 𝑑𝑀 ⋅(1− ∑𝑚 ⋅Ω) hole spin, with a commensurate treatment, could lead to just hor BH 𝑗 𝑗 (16) 𝐵 𝜔 𝑗=1 black hole evaporation, as noted above, but BH evaporation is not the same as the super-radiance phenomenon. 1 𝑁/2 (1 − ∑𝑚 ⋅Ω)<0; 𝑗 𝑗 3.1. Minimum Experimental Bounds Which Can Affect the 𝜔 𝑗=1 ResultsofourInquiry,ProvidedThatClaim 1 Is True (as well as That Claim 3 Holds). That Is, Myers-Perry as a (ii)forKerrBH,onecouldarguablyhavemuchthesame Higher Representation of Black Holes. Presumably Allowing thing; that is, Massive Gravitons. IMO, as stated above, the Meyers-Perry 1 𝐴 condition for BHs is, as a gateway, a probable candidate to 𝑆=4𝜋𝑀2 = ⋅( hor ) experimental observations for BHs. As mentioned earlier, 4 𝐿2 𝑃 forhigherdimensionalBHswhichmayallowformassive 8𝜋𝑟 1 gravitons, here are the perturbations due to GW due to higher 𝑑𝐴 = 𝐻 𝑑𝑀 ⋅(1− 𝑚⋅Ω ) (17) hor 𝐵 BH 𝜔 𝐻 dimensional black holes. We state these as follows. The subsequent values by ℎ00, ℎ0𝑖,andℎ𝑖𝑗 make the case, 1 (1 − 𝑚⋅Ω )<0. due to mass dependence of the black holes in the Myers-Perry 𝜔 𝐻 black holes, has an explicit mass dependence on the mass of the black hole included. [4]has Proof. By (15), (9), (10), and (11), we next consider the 16𝜋𝐺 𝑀 following. ℎ ≈ ⋅ BH , 00 𝑑−3 (𝑑−2) ⋅Ω𝑑−2 𝑟 𝑎 Claim 2. If is zero, then super-radiance as made possible in 16𝜋𝐺 𝑀 Claim 1 part (ii) is impossible for Kerr Black holes. ℎ ≈ ⋅ BH ⋅𝛿 , 𝑖𝑗 𝑑−3 𝑖𝑗 (19) (𝑑−2) ⋅ (𝑑−3) ⋅Ω𝑑−2 𝑟 𝑎 Proof. goes to zero and mean numerator of (9)goestozero. 𝑘 8𝜋𝐺 𝑥 𝑘𝑖 Hence, (1 − (1/𝜔)𝑚𝐻 ⋅Ω )<0does not happen. Hence, for ℎ ≈− ⋅ ⋅𝐽 . 𝑜𝑖 𝑑−1 nonzero frequency of incoming radiation, 0<𝜔<𝑚⋅Ω𝐻 Ω𝑑−2 𝑟 does not hold. Hence, there is no BH super-radiance. The coefficient 𝑑 is for dimensions, 4 or above, and in this 𝑘𝑖 Claim 3. Onecouldhavethefollowing:Claim 1 part (ii) may situation, with angular momentum 𝐽 . Here, the term put in, be false, but Claim 1 part (i) may be true. namely (20) is for angular area, and it has no relationship with the formula for angular velocity of BHs; namely, (20)hasno Proof. For 𝑁≥4or so, the following decomposition may be relations with (4)and(9)above. true: 2𝜋(𝑑−1)/2 𝑁/2 𝑁/2 Ω = , (20) 1 1 1 𝑑−2 Γ ((𝑑−1) /2) (1 − ∑𝑚𝑗 ⋅Ω𝑗)=[1− 𝑚⋅Ω𝐻]− ∑𝑚𝑗 ⋅Ω𝑗 <0. 𝜔 𝑗=1 𝜔 𝜔 𝑗=2 𝑘𝑖 𝑘 𝑖0 𝑑−1 (18) 𝐽 =2⋅∫ 𝑥 ⋅𝑇 ⋅𝑑 𝑥. (21)

If the first term in []in the RHS of the above formula is equal 𝑖0 The 𝑇 above is a stress energy tensor as part of a 𝑑 to 0, Claim 1 part (ii) is false, but one could still have Claim 1 dimensional Einstein equation given in [5]as part (i) as true. That is, one could write the following. 𝑁/2 Consider 0<𝜔<∑ 𝑚𝐽 ⋅(Ω𝐻) .Then,Claim 1 part (i) 1 𝐽=2 𝐽 𝑅 − ⋅𝑔 ⋅𝑅=8𝜋𝐺𝑇𝑗𝑙. will be true, that is, super-radiance for brane theory BHs. 𝑗𝑙 2 𝑗𝑙 (22)

The significance of the three claims is as follows. As Also, the mass of the black hole is, in this situation scaled as given by [4], there is a problem, if a massive graviton exists, follows: if 𝜇 is a rescaled mass term [5], the bending of light, say about Mercury, the Eddinton 1919 experiment is calculated to be 3/4th the value seen in the 1919 (𝑑−2) 𝑀 =𝜇⋅Ω ⋅ . (23) experiment which proved classical GR. By [7], there can be a BH 𝑑−2 16𝜋𝐺 Advances in High Energy Physics 5

Moregenerally,themassoftheblackholeiswrittenas only the giant ones would do. With that, we next then explore the frequency ranges which could lead to certain Graviton 𝑀 ≡ ∫ 𝑇 𝑑𝑑−1𝑥. masses, as could be linked to super-radiance. That is, it would BH 00 (24) mean that a very large SMBH, of about 100 solar masses of a 𝑍∼10 We will next go to the minimum size of a black hole which redshift of the order of at or less than a billion years would survive as up to 13.6 billion years and then say after the creation of the universe, would lead to the values of something about the relative magnitude of the terms in (22) (29) above, which could be conceivably detected, which then and then their survival today. The variance of black hole leads us to the question of what frequencies of the graviton, 15 if presumably massive, would be involved. This would then masses, from super massive BHs to those smaller than 10 allow us to make inquiry as to what the Meyers-Perry values grams will be discussed, in the context of (19), and stress for super-radiance and absorption/subsequent reflection of strength, with commentary as to what we referred to earlier, GW radiation which could conceivably be detected for strain namely, strain for detecting GW, is given by ℎ(𝑡) given below, 𝑖𝑗 values of the order given by (29)above. with 𝐷 as the detector tensor, that is, a constant term, so that, by [4, page 336], we write 3.3. Frequency and Wavelengths for Ultra Low “Massive 𝑖𝑗 ℎ (𝑡) =𝐷 ℎ𝑖𝑗 . (25) Graviton” Masses. To get the appropriate estimates, we turn to [11], by Goldhaber and Nieta, which can be used to give Equation (25) means that the magnitude of strain, ℎ,is a set of frequency and mass equivalences for the “massive” effected by (19), (20), and (21) and its magnitude, seen next. graviton; on the order of having the following equivalent Note that the magnitude of the strain, ℎ,asbeingbroughtup, values as paired together, namely, starting off, with graviton may be affected by the mass of a graviton, due to 𝑇,whichisa mass, graviton wavelength, and resulting graviton frequency, feed into (19) above. Namely, consider that the mass assumed we observe the dual pairing of the following, if one also looks −65 for the graviton is of the order of 10 grams, which is given at Valev’s estimates [12], by [5]; if ℎ does not equal zero, then the stress energy tensor −65 𝑚𝑔 ∼2 × 10 grams𝜆𝑔 of the massive graviton is for nonzero 𝑇𝑢V which corresponds to a nonzero concentration in interstellar space, with [5] 22 ∼2 × 10 meters 2 𝜅 𝑚𝑔 =− 𝑇 −4 (30) 6ℎ ∼10 ⋅ radius-of-universe𝜔𝑔 (26) 𝑇= 𝑇 . 3 trace 𝑢V ∼( )×10−14/ . 2 second We will get explicit upper bounds to (26)andusethemas commentary in the conclusion of this paper. That will affect Obviously, with regard to this, if such an extremely low value the infalling frequency 𝜔 whichwillbepartofthesuper- for resultant frequency is obtained, and then one is obtaining radiance discussion. the value that is inevitable, just in terms of frequency, to have for any spinning Kerr BH, 3.2. Values of the Meyers-Perry ℎ00, ℎ0𝑖,andℎ𝑖𝑗 in Magnitude 3 ℎ 𝐷 𝜔 ∼( )×10−14/ <Ω . Lead to Nominal Values. If below is redshift corrected 𝑔 2 second 𝐻 (31) distance, in a rough sense, it leads to an approximation of ℎ as roughly proportional to ℎ00 with the roughly scaled results If we evaluate further for any reasonable value of 𝑎,wewill of find that, for a SMBH of about 100 solar masses, one will still −14 𝐺𝑀 have, realistically, 𝜔𝑔 ∼ (3/2) × 10 /second ≪Ω𝐻.The ℎ∼ . 𝑐2𝐷 (27) author has found that for supermassive black holes for masses up to a million times the mass of the sun, the freauency the 𝑖𝑗 −14 Note that the tensor 𝐷 is approximately unity, with the becomes, 𝜔𝑔 ∼ (3/2) × 10 /second ≤Ω𝐻 which leads to resultsasgivenby Claim 4. 󵄨 15 −40 𝜔 ∼ (3/2) × 𝑀 󵄨 . ∝10 grams ⇐⇒ ℎ 00,ℎ𝑖𝑖 ∝10 Claim 4. For super-radiance (Kerr style), 𝑔 BH min-life time −14 (28) 10 /second ≪Ω𝐻 (easy super-radiance), for SMBH for BHs;𝑍(redshift) ∼ 10. −14 100timessolarmassand𝜔𝑔 ∼ (3/2) × 10 /second ≤ Ω 6 whereas super massive black holes of about 100 times the 𝐻 (problematic super-radiance), for SMBH 10 times solar massofoursun,atZ(redshift)ofabout10leadtosignificantly mass. larger values of ℎ𝑖𝑖 as seen below, The proof is in the definition of Ω𝐻 = (𝑎/(2𝑀 ⋅𝑟𝐻))| ,withaverysmallnumerator. 󵄨 BH Kerr-BH 𝑀 󵄨 ⇐⇒ ℎ ,ℎ ∝10−20. BH󵄨100 00 𝑖𝑖 (29) -solar-mass Claim 4 means that, before the formation of massive It is easy from inspection to infer from this that the most spiral galaxies, the super-radiance is doable. However, the early formed black holes would not be accessible and that author fails to understand how it is possible on another 6 Advances in High Energy Physics theoretical ground; that is, what does super-radiance mean a new big bang for each of the 𝑁 universes represented by 𝑖≡1 for BHs for which {Ξ𝑖}𝑖≡𝑁. Verification of this mega structure compression and 𝜆 ∼2×1022 expansion of information with nonuniqueness of informa- 𝑔 meters tion placed in each of the 𝑁 universes favors ergodic mixing (32) −4 treatments of initial values for each of 𝑁 universes expanding ∼10 ⋅ radius-of-universe. from a singularity beginning. The 𝑛𝑓 value, will be using (Ng, 𝑆 ∼𝑛 Onthefaceofit,thisisabsurd.Thatis,howcouldwavelengths 2008) entropy 𝑓.[14]. How to tie in this energy expression, 1/10,000 the size of the universe interact with a Kerr Black as in (33), will be to look at the formation of a nontrivial 𝑁 hole? gravitationalmeasureasanewbigbangforeachofthe 𝑛(𝐸 ) We claim that the embedding of black holes in five or universes as by 𝑖 . The density of states at a given energy 𝐸 higher dimensional space time is a way to make a connection 𝑖 for a partition function (Poplawski, 2011) [15]. Consider with a multiverse, as given in the following supposition [13], ∞ 𝑖≡𝑁 𝑖≡𝑁 −𝐸 and that this may be the only way to reconcile what seems 𝑖 {Ξ𝑖}𝑖≡1 ∝{∫ 𝑑𝐸𝑖 ⋅𝑛(𝐸𝑖)⋅𝑒 } . (34) to be an absurd proposition. That is, graviton wavelengths 0 𝑖≡1 1/10,000 the size of the standard 4 dimensional universe is 𝐸 interacting with spinning black holes in 4 dimensional space- Each of 𝑖, identified with (34)above,iswiththeiterationfor 𝑁 time, whereas that moderate 100 times the mass of the sun universes (Ng, 2008) [14]. Then the following claim holds. −14 BHs easily satisfy 𝜔𝑔 ∼ (3/2) × 10 /second ≪Ω𝐻. Claim 5. Consider

𝑁 4. Conclusion 1 󵄨 ⋅ ∑ Ξ𝑗󵄨 𝑁 󵄨𝑗-before-nucleation-regime In one way, it is ridiculously easy to obtain super-radiance 𝑗=1 (35) for massive gravitons, and, in another sense, it is an absurd 󵄨 󳨀󳨀󳨀󳨀󳨀󳨀󳨀󳨀󳨀󳨀󳨀󳨀󳨀󳨀󳨀󳨀󳨀󳨀󳨀→ Ξ𝑖󵄨𝑖 . proposition. Could a multiverse embedding of BHs be a way vacuum-nucleation-tranfer -fixed-after-nucleation-regime outofwhatotherwiseseemsanimpossibleDichotomy?Or For 𝑁 number of universes, with each Ξ𝑗| will we have to embrace Hawkings’ suggestion that the event 𝑗-before-nucleation-regime Horizon has foundationally crippling flaws? for 𝑗=1to 𝑁 being the partition function of each To address this problem, the author looks at two sugges- universe just before the blend into the RHS of (35)above tions. Either that the BH is really embedded in a multiverse for our present universe. Also, each of the independent and has a different geometry in higher dimensions than is universes given by Ξ𝑗| are constructed by 𝑗-before-nucleation-regime supposed, or one goes to the recent Hawkings’ hypothesis the absorption of one to ten million black holes taking in which changes entirely the supposition of the event horizon. energy. That is,(Ng, 2008)14 [ ]. Furthermore, the main point Namely is similar to what was done in [16] in terms of general ergodic We will first of all give a brief introduction to the Penrose mixing. CCChypothesisgeneralizedtoamultiverse. Claim 6. Consider

4.1. Extending Penrose’s Suggestion of Cyclic Universes, Black Max Hole Evaporation, and the Embedding Structure our Universe 󵄨 ̃ 󵄨 Ξ𝑗󵄨 ≈ ∑ Ξ𝑘󵄨 . (36) 󵄨𝑗-before-nucleation-regime 󵄨black-holes-𝑗th-universe Is Contained within, That Is, Using the Implications of (32) for 𝑘=1 a Multiverse. This Multiverse Embeds BHs and May Resolve What Appears to Be an Impossible Dichotomy. There are What is done in Claims 5 and 6 is to come up with a protocol no fewer than 𝑁 universes undergoing Penrose “infinite as to how a multidimensional representation of black hole expansion” (Penrose, 2006) [13] contained in a mega universe physics enables continual mixing of space and time [17] structure. Furthermore, each of the 𝑁 universes has black largely as a way to avoid the Anthropic principle, as to a hole evaporation, with the Hawkings radiation from decaying preferred set of initial conditions. With investigations, this black holes. If each of the 𝑁 universes is defined by a par- complex multiverse may allow bridging what seems to be an 𝑖≡1 tition function, called {Ξ𝑖}𝑖≡𝑁, then there exists information unworkable dichotomy between ultralow graviton frequency, ensemble of mixed minimum information correlated as about 10−65 7 8 corresponding roughly to grams in rest mass, easily 10 -10 bits of information per partition function in the set satisfied by Kerr black holes with rotational frequencies, as {Ξ }𝑖≡1 | 𝑖 𝑖≡𝑁 before, so minimum information is conserved between given in our text as many times greater, combined with a set of partition functions per universe. Consider the absurdity of what (32)is.Howcanagravitonwitha 10−4 𝑖≡1 󵄨 𝑖≡1 󵄨 wavelength the size of the universe interact with a Kere {Ξ𝑖} 󵄨 ≡{Ξ 𝑖} 󵄨 . (33) 𝑖≡𝑁󵄨before 𝑖≡𝑁󵄨after black hole, spatially? Embedding the BH in a multiverse settingmaybetheonlywayout. However, there is nonuniqueness of information put into 𝑖≡1 each partition function {Ξ𝑖}𝑖≡𝑁.Furthermore,Hawkings’ Claim 5 is particularly important. The idea here is to use radiation from the black holes is collated via a strange what is known as CCC cosmology, which can be thought of attractor collection in the mega universe structure to form as follows. Advances in High Energy Physics 7

First. Have a big bang (initial expansion) for the universe. References After redshift 𝑧=10, a billion years ago, SMBH formation starts. Matter-energy is vacuumed up by the SMBHs, which [1] T. Padmanabhan, Gravitation, Foundations and Frontiers,Cam- at a much later date than today (present era) gather up all bridge University Press, New York, NY, USA, 2010. the matter-energy of the universe and recycle it in a cyclic [2] E. Jung, S. H. Kim, and D. K. Park, “Condition for the superra- conformal translation as follows: diance modes in higher-dimensional rotating black holes with multiple angular momentum parameters,” Physics Letters B,vol. 𝐸=8𝜋⋅𝑇+Λ⋅𝑔 (37) 619, no. 3-4, pp. 347–351, 2005. [3] S. Weinberg, Gravitation and Cosmology: Principles and Appli- cations of the General Theory of Relativity, Wiley Scientific, New York, NY, USA, 1972. 𝐸 =sourceforgravitationalfield [4] P. Tinyakov, “Giving mass to the graviton,” in Particle Physics, 𝑇 = mass energy density and Cosmology, the Fabric of Space-Time,F.Bernadeau,C. Grojean, and J. Dalibard, Eds., pp. 471–499, Elsevier, Oxford, 𝑔 = gravitational metric UK, 2007, part of Les Houches, Session 86. Λ = vacuum energy, rescaled as follows [5] M. Maggiore, GravitationalWaves:Volume1.TheoryandPrac- tice, Oxford University press, Oxford, UK, 2008. Λ=𝑐 ⋅ 𝛽, 1 [Temp] (38) [6] P. Creminelli, A. Nicholis, M. Papucci, and E. Trincherini, “Ghosts in massive gravity,” Journal of High Energy Physics,vol. where 𝑐1is a constant. Then 2005, Article ID 0509, 2005. The main methodology in the Penrose proposal has been [7] C. Csaki,´ J. Erlich, and T. J. Hollowood, “Graviton propagators, in (38) evaluating a change in the metric 𝑔𝑎𝑏 by a conformal brane bending and bending of light in theories with quasi- mapping Ω̂ to localized gravity,” Physics Letters B, vol. 481, no. 1, pp. 107–113, 2000. ̂2 𝑔̂𝑎𝑏 = Ω 𝑔𝑎𝑏. (39) [8] M. Meyers, “Meyers—Perry Black holes,” in Black Holes in Higher Dimensions,G.Horowitz,Ed.,pp.101–131,Cambridge Penrose’s suggestion has been to utilize the following [18] University Press, New York, NY, USA, 2012. [9] Black Holes in Higher Dimensions, G. Horowitz, Ed., Cambridge Ω󳨀̂ 󳨀󳨀󳨀→ Ω̂−1. ccc (40) University Press, New York, NY, USA, 2012. [10] M. Porrati, “Fully covariant van Dam-Veltman-Zakharov dis- Infall into cosmic black hopes has been the main mechanism continuity, and absence thereof,” Physics Letters B,vol.534,no. which the author asserts would be useful for the recycling 1–4, pp. 209–215, 2002. apparent in (40) above with the caveat that ℎ is kept constant [11] A. S. Goldhaber and M. M. Nieta, “Photon and graviton mass from cycle to cycle as represented by limits,” Reviews of Modern Physics,vol.82,pp.939–979,2010. ℎ =ℎ . [12] D. Valev, “Phenominological mass relationship for free massive old-cosmology-cycle present-cosmology-cycle (41) stable particles and estimations of neutrino and graviton mass,” http://arxiv.org/pdf/1004.2449v1.pdf. Equation (40) is to be generalized, as given by a weighing [13] R. Penrose, “Before the big bang. An outrageous new perspec- averaging as given by (35) where the averaging is collated over 𝑁 tive and its implications for particle physics,” in Proceedings of perhaps thousands of universes, call that number ,withan European Particle Accelerator Conference (EPAC ’06),pp.2759– ergodic mixing of all these universes, with the ergodic mixing 2763, 2006. represented by (35) to generalize (40)fromcycletocycle. [14] Y. J. Ng, “Spacetime foam: from entropy and holography to infinite statistics and nonlocality,” Entropy,vol.10,no.4,pp. 4.2. Conclusion, Future Prospects. If this does not work, and 441–461, 2008. the multiverse suggestion is unworkable, there, then, has to [15]N.Poplawski,“CosmologicalconstantfromQCDvacuumand be a consideration of the zero option, namely, Hawkings torsion,” Annalen der Physik,vol.523,pp.291–295,2011. throwing out the event horizon as we know it in BH physics. [16] S. W. Hawkings, “Informational preservation and weather See this reference, namely, [16]. forecasting for black holes,” http://arxiv.org/abs/1401.5761. We are in for interesting times. I see turbulence and [17] H. A. Dye, “On the ergodic mixing theorem,” Transactions of the interesting results ahead. American Mathematical Society,vol.118,pp.123–130,1965. [18] R. Penrose, Cycles of Time—An Extrardinary New View of the Conflict of Interests Universe,AlfredA.Knopf,NewYork,NY,USA,2011. The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment ThisworkissupportedinpartbytheNationalNatureScience Foundation of China Grant no. 11375279. Hindawi Publishing Corporation Advances in High Energy Physics Volume 2014, Article ID 697914, 10 pages http://dx.doi.org/10.1155/2014/697914

Research Article Magnetic String with a Nonlinear 𝑈(1) Source

S. H. Hendi1,2 1 Physics Department and Biruni Observatory, College of Sciences, Shiraz University, Shiraz 71454, Iran 2 Research Institute for Astrophysics and Astronomy of Maragha (RIAAM), P.O. Box 55134-441, Maragha, Iran

Correspondence should be addressed to S. H. Hendi; [email protected]

Received 21 January 2014; Revised 28 February 2014; Accepted 6 March 2014; Published 2 April 2014

Academic Editor: Christian Corda

Copyright © 2014 S. H. Hendi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP3.

Considering the Einstein gravity in the presence of Born-Infeld type electromagnetic fields, we introduce a class of 4-dimensional static horizonless solutions which produce longitudinal magnetic fields. Although these solutions do not have any curvature singularity and horizon, there exists a conic singularity. We investigate the effects of nonlinear electromagnetic fields onthe properties of the solutions and find that the asymptotic behavior of the solutions is adS. Next, we generalize the static metric tothe case of rotating solutions and find that the value of the electric charge depends on the rotation parameter. Furthermore, conserved quantities will be calculated through the use of the counterterm method. Finally, we extend four-dimensional magnetic solutions to higher dimensional solutions. We present higher dimensional rotating magnetic branes with maximum rotation parameters and obtain their conserved quantities.

1. Introduction string model to some experimental data on the inclusive pion asymmetries has been studied in [26, 27]. Some arguments One of the interesting topological defects is cosmic string about the magnetic strings in the Yang-Mills plasma have which may be originated during the early universe phase been found in [28]. transitions [1](seeKibblemechanismformoredetails[2]). On gravitational aspect, the horizonless solutions and Furthermore, considering the inflationary models3 [ , 4], it spacetime with conical singularity have been investigated in has been proposed that cosmic strings can form at the end gravitating electromagnetic field background (see [7, 8, 29– of inflation. Moreover, one of the predictions of supersym- 49] and references therein). Interesting properties of the metric hybrid inflation5 [ ] (and also grand unified models magnetic string in branes, M-theory, and string theory have of inflation [6]) is the cosmic string. Interesting properties been investigated [50, 51]. Calculations of the vacuum energy and interaction of the superconducting cosmic string with of two different fields in the background of a magnetic string astrophysical magnetic fields have been found in7 [ –9]. have been analyzed in [52, 53]. Besides, magnetic strings have been studied in Brans-Dicke One of the generalizations of the Einstein-Maxwell field theory as well as dilaton gravity [10–13]. From cosmological equations is gravitating nonlinear electrodynamics (NLED), point of view, one can find the properties of the magnetic whose most popular theory is Born-Infeld [54–59]. In addi- (cosmic) string in various literatures [14–16]. tion to Lorentz and 𝑈(1) gauge invariances, we know that In addition to cosmic strings, other kinds of strings may the Lagrangian of the Maxwell electrodynamics contains only be considered in QCD and also gravity. Properties of the quadratic forms of gauge potential and its first derivative. QCD static strings have been investigated extensively in [17– One can consider both invariances and leave out the third 20] and it has been shown that QCD magnetic string can condition to obtain NLED [60]. From historical point of contribute to hadron dynamics [21]. Applications of magnetic view,NLEDwereintroducedtoeliminateinfinitequantities string in quantum theories have been presented in [22–24]. in theoretical analysis of charged point-like particles [54– Magnetic strings in antiferromagnetic crystals have been 59]. Recently, we have more motivations for considering investigated in [25]. Application of the (chromo)magnetic NLED theories, for example, various limitations of the linear 2 Advances in High Energy Physics electrodynamics [61, 62], clarification of the self-interaction It is notable that these field equations can be obtained from of virtual electron-positron pairs [63–65], explanation of variation of the following action: electrodynamics on D-branes [66–68], and description of 1 4 radiation propagation inside specific materials [69–72]. In 𝐼𝐺 =− ∫ 𝑑 𝑥√−𝑔 [𝑅−2Λ+𝐿(F)] 16𝜋 M addition, from astrophysical viewpoint, we know that the (4) 1 effects of NLED become indeed quite important in super- − ∫ 𝑑3𝑥√−𝛾Θ (𝛾) , strongly magnetized compact objects, such as pulsars, and 8𝜋 𝜕M particular neutron stars (some examples include the so-called magnetars and strange quark magnetars) [73–75]. Moreover, where the bulk action (first term) is supplemented with a NLED modifies in a fundamental basis the concept of Gibbons-Hawking surface term (second term) whose variation will cancel the extra normal derivative term in deriving gravitational redshift and its dependency on any background Θ 𝛾 magnetic field as compared to the well-established method the equation of motion. The quantities and denote the trace of the extrinsic curvature and the induced metric for introduced by standard general relativity. Furthermore, it has 𝜕M beenrecentlyshownthatNLEDobjectscanremovebothof the boundary ,respectively. the big bang and black hole singularities [76–78]. In this work, we take into account the recently proposed Amongst the nonlinear generalization of Maxwell elec- BI type models of NLED [88, 89]. They have been nominated trodynamics, the so-called BI type NLED, whose first non- the Exponential form of Nonlinear Electromagnetic Field linear correction is quadratic function of Maxwell invariant, (ENEF) and the Logarithmic form of Nonlinear Electromag- is completely special. It has been shown that BI type NLED netic Field (LNEF), in which their Lagrangians are maybearisenasalowenergylimitofheteroticstring F {𝛽2 ( (− )−1), , theory [66, 68, 79–83], which led to an increased interest { exp 𝛽2 ENEF 𝐿 (F) = for BI type NLED theories. In addition, BI type theories { F (5) {−8𝛽2 (1 + ), . have some interesting properties; for example, these theories ln 2 LNEF enjoy the birefringence phenomena, free of the shock waves { 8𝛽 [84, 85] and electric-magnetic duality [86]. Furthermore, Here, we want to obtain magnetic solutions. It is well considering the relation between AdS/CFT correspondence known that the electric field comes from the time component and superconductivity phenomenon, it was shown that the BI of the vector potential (𝐴𝑡), while the magnetic field is type theories make a crucial effect on the condensation, the associated with the angular component (𝐴𝜙). Hence one critical temperature, and energy gap of the superconductors expects that a magnetic solution may be written in a metric [87]. gaugeinwhichthecomponents𝑔𝑡𝑡 and 𝑔𝜙𝜙 interchange In this paper, we investigate the horizonless magnetic their roles relatively to that present in the Schwarzschild stringsinthepresenceoftwokindsoftheBItypeNLED gaugeusedtodescribeelectricsolution.Therefore,westart [88, 89]. One of the elemental motivations for analyzing the with a class of the four-dimensional metrics which produces horizonless string solutions is that they may be interpreted as longitudinal magnetic fields along the 𝑧 direction [48]: cosmic strings. 𝜌2 𝑑𝜌2 𝜌2 𝑑𝑠2 =− 𝑑𝑡2 + +𝑙2𝑓 (𝜌) 𝑑𝜑2 + 𝑑𝑧2, 𝑙2 𝑓(𝜌) 𝑙2 (6) 2. Basic Field Equations where 𝑓(𝜌) is an arbitrary function of coordinate 𝜌.Itis Our goal in this work is to construct a class of four- notable that this metric may be obtained from the horizon dimensional solutions to the Einstein equations with negative flat Schwarzschild-like metric: cosmological constant in the presence of nonlinear electro- 𝑑𝜌2 𝜌2 magnetic source, 𝐿(F), which describes a magnetic string. 𝑑𝑠2 =−𝑓(𝜌)𝑑𝑡2 + +𝜌2𝑑𝜑2 + 𝑑𝑧2, 𝑓(𝜌) 𝑙2 (7) The Euler-Lagrange equations of motion for the metric 𝑔𝜇] 𝐴 andthegaugepotential 𝜇 may be written as [90] with the following local transformation: 𝑖𝑡 𝑡󳨀→𝑖𝑙𝜑, 𝜑󳨀→ . 1 𝑙 (8) 𝑅 − 𝑔 (𝑅−2Λ) =𝑇 , 𝜇] 2 𝜇] 𝜇] (1) Since the mentioned transformation is not a global mapping 𝜕 (√−𝑔𝐿 𝐹𝜇])=0, and metric (7)canbelocallymappedtometric(6), one can 𝜇 F (2) find that both (6)and(7) do not describe a unique spacetime. Using the nonlinear Maxwell equation (2)withthemetric(6), 2 𝜇] one can obtain where Λ=−3/𝑙, 𝐿F =𝑑𝐿(F)/𝑑F, F =𝐹𝜇]𝐹 denotes 2𝐹 2 2𝐹 the Maxwell invariant, and the energy-momentum tensor is 𝜙𝜌 󸀠 𝜙𝜌 [1 − ( ) ]𝐹 + =0, ENEF, given by 𝛽𝑙 𝜙𝜌 𝜌 (9) 𝐹 2 𝐹 2 𝐹 𝜙𝜌 󸀠 𝜙𝜌 𝜙𝜌 1 𝜆 [1 − ( ) ]𝐹 +[4+( ) ] =0, 𝑇 =( 𝑔 𝐿 (F) −2𝐿 𝐹 𝐹 ). 𝜙𝜌 LNEF 𝜇] 2 𝜇] F 𝜇𝜆 ] (3) 2𝛽𝑙 𝛽𝑙 2𝜌 Advances in High Energy Physics 3

0.18 2.5 0.16

0.14 2

0.12

1.5 0.10

0.08 1 0.06

0.04 0.5 0.02

0 0 234561 1.5 2 2.5

(a) ENEF (b) LNEF 𝑇 𝜌 𝑙=1𝑞=1 𝛽=1 𝛽 = 1.3 𝛽=3 Figure 1: ̂𝑡̂𝑡 versus for , ,and (solid line), (bold line), and (dashed line). “ENEF branch (a) and LNEF branch (b).” with the following solutions: reduce the concentration volume of the energy density, we

2 should increase the nonlinearity parameter. {2𝑞𝑙 1 Now, we should obtain the metric function 𝑓(𝜌).One { exp (− 𝐿𝑊), ENEF, { 𝜌2 2 can take into account (6)and(10)inthegravitationalfield 𝐹 = 𝜙𝜌 { 2 2 (10) equation (1) to obtain its nonzero components as {𝛽 𝜌 (1−Γ) , LNEF, { 𝑞 1 (Ψ (𝜌)+𝜌𝛽2) (− 𝐿 )−𝜌𝛽2 [1+𝐻2 (𝜌)] = 0, , exp 2 𝑊 ENEF where the prime denotes differentiation with respect 𝜌 𝑞 2 2 to ,theparameter is an integration constant, 2 𝛽𝜌 (1−Γ) 2 2 2 4 Ψ (𝜌) + 8𝜌𝛽 ln [1+( ) ]−𝐽(𝜌) = 0, LNEF, 𝐿𝑊 = Lambert 𝑊(−16𝑙 𝑞 /𝛽 𝜌 ) which satisfies 2𝑞𝑙 Lambert 𝑊(𝑥) exp[Lambert 𝑊(𝑥)] =𝑥 [91, 92], and 2 (13) Γ=√1 − (2𝑙𝑞/𝛽𝜌2) .Itisworthwhiletonotethatinorderto 󸀠 2 have a real electromagnetic field, we should consider 𝜌>𝜌0, where Ψ(𝜌) = 2𝑓 (𝜌) + 𝑔(𝜌) − (6𝜌/𝑙 ) and where 𝜌𝑓 󸀠󸀠 (𝜌) , 𝑡𝑡 (𝑧𝑧) , 1 { component 2𝑙𝑞 {√2 ( ), , { exp ENEF 𝑔(𝜌)= 𝜌0 = √ × 4 (11) { 𝛽 { {2𝑓 (𝜌) 1, LNEF. , 𝜌𝜌 (𝜑𝜑) component, { { 𝜌 Here,weusetheorthonormalcontravariant(hatted)basis {0, 𝑡𝑡 (𝑧𝑧) component, vectors to study the effect of nonlinearity on the energy 𝐻(𝜌)= 4𝑞𝑙 1 density. Considering the mentioned diagonal metric in this { (− 𝐿 ) , 𝜌𝜌 (𝜑𝜑) , ̂̂ 𝛽𝜌2 exp 2 𝑊 component basis, one should apply ê𝑡 = (𝑙/𝜌)(𝜕/𝜕𝑡) and therefore the 𝑡𝑡 { component of the stress-energy tensor is {0, 𝑡𝑡 (𝑧𝑧) component 2 5 4 2 𝛽2 2𝐹 𝐽(𝜌)= 4𝜌 𝛽 (1−Γ) { 𝜙𝜌 { , 𝜌𝜌 (𝜑𝜑) component. { [1 − exp (− )] , ENEF, 2 2 2 2 { 2 𝑙2𝛽2 { 𝑙 𝑞 [1 + (𝛽𝜌 (1−Γ) /2𝑞𝑙) ] 𝑇 = ̂𝑡̂𝑡 { 𝐹2 (12) (14) { 2 𝜙𝜌 {𝛽 ln (1 + ), LNEF. { 4𝑙2𝛽2 After some calculations one can show that these equations have the following solutions: We plot 𝑇̂𝑡̂𝑡 versus 𝜌>𝜌0 in Figure 1 and find that, for a fixed 2 3 2 2 value of 𝜌, as nonlinearity parameter increases, the energy 𝜌 2𝑚𝑙 𝜒𝛽 𝜌 2𝛽𝑞𝑙 𝑓 (𝜌) = + − − Υ (𝜌) , (15) density of the spacetime decreases and therefore, in order to 𝑙2 𝜌 6 𝜌 4 Advances in High Energy Physics

where 𝑚 is the integration constant which is related to mass 𝜌>𝑟0 to 𝜌<𝑟0. In order to get rid of this incorrect extension, parameter, 𝜒 is equal to 1 and −16 for ENEF and LNEF one may introduce a new radial coordinate 𝑟 in the following branches, respectively, and form: 2 2 2 −1/2 𝑟 =𝜌 −𝑟 , {∫ (−𝐿 ) [1 − 𝐿 ]𝑑𝜌, , 0 { 𝑊 𝑊 ENEF { { 𝑟2 (20) {2𝛽 𝛽2 (1−Γ) 𝜌4 𝑑𝜌2 = 𝑑𝑟2. Υ(𝜌)= ∫ [𝜌2 ( ) 𝑟2 +𝑟2 { 𝑞𝑙 ln 2𝑙2𝑞2 (16) 0 { { 2 2 { 4𝑞 𝑙 Considering this suitable coordinate transformation, the { + ]𝑑𝜌, LNEF, { 𝛽2𝜌2 (1−Γ) electromagnetic field can be written as 2𝑞𝑙2 1 where one may calculate these integrations. We should note { (− 𝐿󸀠 ), , { 2 2 exp 𝑊 ENEF that the obtained solutions are the same as asymptotically {𝑟 +𝑟0 2 𝐹𝜙𝑟 = (21) anti-de Sitter magnetic solution of Einstein-Maxwell gravity {𝛽2 (𝑟2 +𝑟2) { 0 󸀠 [49], asymptotically (large values of radial coordinate). In { (1 − Γ ), LNEF, addition, one may expect to recover the solution of [49]for { 𝑞 𝛽→∞. 󸀠 2 2 2 2 2 2 󸀠 Taking into account the metric (6), it is clearly desirable where 𝐿𝑊 = Lambert 𝑊(−16𝑙 𝑞 /𝛽 (𝑟 +𝑟0 ) ) and Γ = to have an examination on the geometric structure of the 2 2 2 √1 − (2𝑙𝑞/𝛽(𝑟 +𝑟 )) .Moreover,themetric(6)inthenew solutions. The first step is investigation of the spacetime 0 coordinate is curvature. It is easy to show that the Kretschmann scalar is 𝑟2 +𝑟2 𝑟2 𝑑𝑟2 𝑑2𝑓(𝜌) 2 𝑑𝑓 (𝜌) 2 𝑓(𝜌) 2 𝑑𝑠2 =− 0 𝑑𝑡2 +𝑙2𝑓 (𝑟) 𝑑𝜙2 + 𝜇]𝜆𝜅 1 𝑙2 (𝑟2 +𝑟2) 𝑓 (𝑟) 𝑅𝜇]𝜆𝜅𝑅 =( ) +4( ) +4( ) . 0 𝑑𝜌2 𝜌 𝑑𝜌 𝜌2 (22) 𝑟2 +𝑟2 (17) + 0 𝑑𝑧2, 𝑙2 Numerical calculations show that the Kretschmann is finite for nonzero 𝜌. Furthermore, we can show that with 0≤𝑟<∞,and𝑓(𝑟) is now given as

2 6 2 2 2 𝜇]𝜆𝜅 48𝑀 𝑙 𝐴(𝑀,𝑞,𝛽,𝑙) 1 2 2 3 𝜒𝛽 (𝑟 +𝑟 ) 𝑅 𝑅 = + +𝑂( ) , 𝑟 +𝑟0 2𝑚𝑙 0 lim 𝜇]𝜆𝜅 6 5 4 𝑓 (𝑟) = + − 𝜌→0 𝜌 𝜌 𝜌 2 𝑙 2 2 6 √𝑟 +𝑟0 (23) (18) 2𝛽𝑞𝑙 − Υ (𝑟) , 8Λ2 48𝑀2𝑙6 384𝑀𝑙5𝑞2 √𝑟2 +𝑟2 𝑅 𝑅𝜇]𝜆𝜅 = + − 0 𝜌→∞lim 𝜇]𝜆𝜅 3 𝜌6 𝜌7 (19) where 𝐵(𝑀,𝑞,𝛽,𝑙) 1 + +𝑂( ), Υ (𝑟) 𝜌8 𝜌11 { [1 − 𝐿󸀠 ]𝑟 where 𝐴 and 𝐵 are different functions of metric parameters { 𝑊 {∫ 𝑑𝑟, ENEF, 𝑀, 𝑞, 𝑙,and𝛽.Equation(19) confirms that the asymptotic { √−(𝑟2 +𝑟2)𝐿󸀠 { 0 𝑊 behavior of the solutions is adS. In addition, one may take { 2 { ((𝛽2 (1 − Γ󸀠)(𝑟2 +𝑟2) )/2𝑙2𝑞2) into account (18) to think about the existence of a curvature {2𝛽 ln 0 𝜌=0 = ∫ ( singularity located at and therefore conclude that there { 𝑞𝑙 (𝑟2 +𝑟2)−1/2 𝜌>𝜌 { 0 are magnetically charged black hole solutions. Since 0, { 𝛽 { one concludes that, for charged solutions with finite ,the { 2 2 𝜌=0 { 4𝑞 𝑙 spacetime never achieves .Inaddition,weshouldobtain { + )𝑟𝑑𝑟 LNEF. −1 { 2 󸀠 2 2 3/2 the zeroes of the function 𝑓(𝜌) =𝜌𝜌 (𝑔 ) . Considering the 𝛽 (1 − Γ )(𝑟 +𝑟 ) { 0 largest positive real root of 𝑓(𝜌) =0 by 𝑟0 (suppose 𝑟0 >𝜌0; (24) for 𝑟0 <𝜌0, the metric function 𝑓(𝜌) is positive definite which we are not interested in), one can find that the function 𝑓(𝜌) 𝜌<𝑟 𝑔 𝑔 Numerical calculations show that not only Kretschmann is negative for 0.Weshouldnotethat 𝜌𝜌 and 𝜙𝜙 are scalar but also other curvature invariants are finite in the 𝑓(𝜌)−1 =𝑔 =𝑙−2𝑔 related by 𝜌𝜌 𝜙𝜙, and therefore negativity range 0≤𝑟<∞(𝑟0 ≤𝜌<∞) and therefore the of 𝑔𝜌𝜌 (which occurs for 𝜌<𝑟0) leads to negativity of 𝑔𝜙𝜙 mentioned spacetime has no curvature singularity and no and hence the signature of the metric changes from +2 to −2. horizon. It is notable that the above-mentioned magnetic This indicates that we could not extend the spacetime from solutions differ from the electric solutions and the properties Advances in High Energy Physics 5

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(a) ENEF (b) LNEF

Figure 2: The deficit angle versus 𝛽 for 𝑟0 =1, 𝑙=1,and𝑞=1. “ENEF branch (a) and LNEF branch (b).” of electric and magnetic solutions are distinct. For example, Following the Vilenkin procedure [93], one can identify the the electric solutions lead to black objects interpretation, near origin metric (28) with a cosmic string and interpret 𝜇 while the magnetic solutions do not. as the mass per unit length of the string [93]. In spite of the fact that the obtained magnetic solu- Here, we are in a position to investigate the effect of tions have no essential singularity, one can show that nonlinearity parameter on the deficit angle 𝛿𝜙.Atfirst,we 𝛿𝜙 >𝛽 lim𝑟→0(1/𝑟)√𝑔𝜙𝜙/𝑔𝑟𝑟 =1̸and so, when 𝑟 goes to zero, the should note that is a smooth real function for ext,where limitoftheratio“circumference/radius”isnot2𝜋.This 1 indicates that there is a conic singularity located at 𝑟=0. 2𝑙𝑞 {2 exp ( ), ENEF, 𝛽 = × 2 (29) In order to remove the conic singularity, one can identify the ext 𝑟2 { 0 1, LNEF. angular coordinate 𝜙 with the period { −1 Second, it is interesting to note that the minimum and 1 𝑔𝜙𝜙 Period𝜙 =2𝜋(lim √ ) = 2𝜋 (1 − 4𝜇) , (25) maximum values of the deficit angle are 𝑟→0𝑟 𝑔𝑟𝑟 󵄨 𝛿𝜙󵄨 = lim 𝛿𝜙, where the conical singularity has a deficit angle 𝛿𝜙 = 8𝜋𝜇. Min 𝛽→∞ 𝑟→0 Expanding the metric function for ,onecanshowthat 󵄨 (30) 𝛿𝜙󵄨Max = lim 𝛿𝜙, 󵄨 𝛽→𝛽+ 󵄨 𝑑𝑓 (𝑟)󵄨 ext 𝑓 (𝑟)󵄨𝑟=0 = 󵄨 =0, 𝑑𝑟 󵄨𝑟=0 which means that increasing the nonlinearity parameter 𝛽 󵄨 (26) 𝑑2𝑓 (𝑟)󵄨 leads to decreasing the deficit angle (see Figure 2 for more 𝑓󸀠󸀠 (0) = 󵄨 =0,̸ 2 󵄨 clarifications). 𝑑𝑟 󵄨𝑟=0 𝜇 and therefore is given by 3. Spinning Magnetic String 1 2 𝜇= (1− ) . In this section, we apply a local rotation boost to the static 4 𝑙𝑟 𝑓󸀠󸀠 (0) (27) 0 metric (22) to obtain rotating spacetime solutions. In 4- 𝑆𝑂(3) It is easy to show that the near origin metric can be written as dimensional spacetime the rotation group is ,andso one can find that there is only one independent rotation 󵄨 𝑟2 𝑟2𝑙2 𝑑2𝑓 (𝑟)󵄨 parameter. In order to apply rotation, one may use the 𝑑𝑠2 =− 0 𝑑𝑡2 + 󵄨 𝑑𝜙2 2 2 󵄨 following local transformation in the 𝑡−𝜙plane: 𝑙 2 𝑑𝑟 󵄨𝑟=0 (28) 2 2 2 2 2 𝑑𝑟 𝑟0 2 𝑎 𝑎 𝑎 + 󵄨 + 𝑑𝑧 . 𝑡 󳨃󳨀→ √1+ 𝑡−𝑎𝜙, 𝜙󳨃󳨀→ √1+ 𝜙− 𝑡, (31) 2 2 2 󵄨 𝑙2 2 2 2 𝑟0 (𝑑 𝑓 (𝑟) /𝑑𝑟 )󵄨𝑟=0 𝑙 𝑙 𝑙 6 Advances in High Energy Physics where 𝑎 is a rotation parameter. Taking into account the static 4. Magnetic Brane Solutions metric (20)andapplying(31), we can obtain 𝑛+1 2 Here, we start with a class of the ( )-dimensional metrics 𝑟2 +𝑟2 𝑎2 𝑟2𝑑𝑟2 to obtain magnetic brane solutions with the following ansatz: 𝑑𝑠2 =− 0 (√1+ 𝑑𝑡 − 𝑎𝑑𝜙) + 2 2 2 2 𝑙 𝑙 (𝑟 +𝑟0 )𝑓(𝑟) 2 2 2 2 𝜌 2 𝑑𝜌 2 2 𝜌 2 𝑑𝑠 =− 𝑑𝑡 + +𝑙 𝑓(𝜌)𝑑𝜑 + 𝑑𝑋 , (37) 2 𝑙2 𝑓(𝜌) 𝑙2 1 𝑎 𝑎2 𝑟2 +𝑟2 +𝑙2𝑓 (𝑟) ( 𝑑𝑡 − √1+ 𝑑𝜙) + 0 𝑑𝑧2, 𝑙2 𝑙2 𝑙2 2 𝑛−2 𝑖 2 where 𝑑𝑋1 = ∑𝑖=1 (𝑑𝑥 ) is the Euclidean metric on the (𝑛 − (32) 2)-dimensional submanifold. Using the nonlinear Maxwell equation (2)withthemetric(37), we find that the nonzero where the metric function 𝑓(𝑟) isthesameasthatin(23). components of Maxwell field are According to the mentioned transformation, one can find that, in spite of the static case, 𝐹𝑟𝑡 does not vanish for 2𝑞𝑙𝑛−1 1 rotating solutions. Straightforward calculations show that the { (− 𝐿 ), , { 𝜌𝑛−1 exp 2 𝑊 ENEF nonvanishing components of the electromagnetic fields are 𝐹 =−𝐹 = 𝜙𝜌 𝜌𝜙 { 2 𝑛−1 (38) 𝑎 𝑎 {𝛽 𝜌 𝐹𝑟𝑡 =− 𝐹𝑟𝜙 = (1−Γ) , LNEF, 𝑙2√1+𝑎2/𝑙2 √1+𝑎2/𝑙2 { 𝑞 2𝑞 1 { (− 𝐿󸀠 ), , 𝐿 = 𝑊(−16𝑙2(𝑛−2)𝑞2/𝛽2𝜌2(𝑛−1)) Γ= { 2 2 exp 𝑊 ENEF (33) where 𝑊 Lambert and {𝑟 +𝑟0 2 × √1 − (2𝑙𝑞/𝛽𝜌𝑛−1)2 {𝛽2 (𝑟2 +𝑟2) . It is notable that considering the real { 0 󸀠 𝜌>𝜌 (1 − Γ ), LNEF. electromagnetic field leads to 0,where { 𝑞𝑙2 2 2(𝑛−3) Considering (22), (31), and (32), one may think that there 2𝑛−2 2𝑙𝑞 4𝑙 exp (1) , ENEF, 𝜌0 =( ) ×{ (39) is a one-to-one correspondence between static and rotating 𝛽 1, LNEF. spacetimes and so they are the same. But this statement is not correct. It is worthwhile to mention that the coordinate 𝜙 Now, we are in a position to obtain the metric function is periodic and therefore (31) is not a proper coordinate 𝑓(𝜌). Considering (37)with(38), we find that the solution of transformation on the entire manifold. In other words, the the gravitational field equation1 ( )is metrics (22)and(32)canbelocallymappedintoeachother but not globally, and so (31) generates a new metric (for some 𝜌2 2𝑚𝑙3 𝜒𝛽2𝜌2 4𝛽𝑞𝑙𝑛−2 details about this local transformation see, e.g., [94]). 𝑓(𝜌)= + − − Υ(𝜌), 2 𝑛−2 𝑛−2 In order to finalize this section, we should discuss 𝑙 𝜌 𝑛 (𝑛−1) (𝑛−1) 𝜌 the conserved quantities of the magnetic string. Using the (40) counterterm method [95–98]andfollowingtheprocedureof magnetic solutions papers [7, 8, 29–49], one can find that the where massandangularmomentumperunitlengthofthestringcan be written as −1/2 {∫ (−𝐿𝑊) [1 − 𝐿𝑊]𝑑𝜌, ENEF, 2 { 𝜋 3𝑎 { 𝑀= (1 + )𝑚, { 2 2(𝑛−1) 2 (34) { 2𝛽 𝛽 (1−Γ) 𝜌 2 𝑙 ∫ [𝜌𝑛−1 ( ) Υ(𝜌)= { 𝑛−2 ln 2 2 {𝑞𝑙 2𝑙 𝑞 { 3𝜋𝑚𝑎 𝑎2 { 2 2 𝐽= √1+ . (35) { 4𝑞 𝑙 2 + 𝑑𝜌] LNEF. 2 𝑙 { 𝛽2𝜌𝑛−1 (1−Γ) Equation (35) shows that considering 𝑎=0leads to vanishing (41) angular momentum and it confirms that 𝑎 is the rotational parameter of the spacetime. In addition, it is interesting to It is easy to show that the Kretschmann scalar diverges when calculatetheelectricchargeofthesolutions.UsingGauss’slaw 𝜌→0and is finite for 𝜌 =0̸.Followingthesamemethod,we andcalculatingthefluxoftheelectricfieldatinfinity,wefind find that one could not extend the spacetime from 𝜌>𝑟0 to that the electric charge per unit length 𝑄 canbegivenby 𝜌<𝑟0 in which 𝑟0 is the largest positive real root of 𝑓(𝜌). =0 Therefore, we can use radial coordinate transformation (20) 𝑄=𝜋𝑞𝑎. (36) to obtain a real well-defined spacetime for 0≤𝑟<∞.Here, We should note that the electric charge may be originated we leave details for reasons of economy. from the electric field. Since, for rotating solutions, besides Final step is generalization of static magnetic branes to the magnetic field along the 𝜙 coordinate, there is also a spinning ones. We know that the rotation group in 𝑛+ radial electric field 𝐹𝑡𝑟 (see (33)), one may expect to obtain 1 dimensions is 𝑆𝑂(𝑛) andhencethemaximumnumber an electric charge which is related to the rotating parameter. of independent rotation parameters is integer part of 𝑛/2. Advances in High Energy Physics 7

Generalization of static solutions to the spinning case with that the mass, angular momentum, and electric charge per 𝑘≤[𝑛/2]rotation parameters leads to the following metric: unit volume of the magnetic branes may be written as

2 2 2 𝑘 2 𝑟 +𝑟0 𝑖 𝑑𝑠 =− (Ξ𝑑𝑡 − ∑𝑎 𝑑𝜙 ) (2𝜋)𝑘 𝑙2 𝑖 𝑀= [𝑛 (Ξ2 − 1) + 1] 𝑚, 𝑖=1 4 2 Ξ 𝑘 (2𝜋)𝑘 +𝑓(𝑟) (√Ξ2 − 1𝑑𝑡 − ∑𝑎 𝑑𝜙𝑖) 𝐽 = 𝑛Ξ𝑚𝑎 , (46) 2 𝑖 𝑖 𝑖 √Ξ −1𝑖=1 4 𝑘 2 2 2 2 𝑘 (2𝜋) 𝑞𝑙 𝑟 𝑑𝑟 𝑟 +𝑟 2 𝑄= √Ξ2 −1. + + 0 ∑(𝑎 𝑑𝜙 −𝑎𝑑𝜙 ) 2 2 2 2 𝑖 𝑗 𝑗 𝑖 2 (𝑟 +𝑟0 )𝑓(𝑟) 𝑙 (Ξ −1)𝑖<𝑗

𝑟2 +𝑟2 + 0 𝑑𝑋2, We should note that the electric charge is proportional 𝑙2 2 to the rotation parameter and is zero for the case of static (42) magnetic branes. This is due the fact that radial electric field 𝐹𝑡𝑟 vanishes for the static solutions. √ 𝑘 2 2 2 where Ξ= 1+∑𝑖 𝑎𝑖 /𝑙 ; 𝑑𝑋2 is the Euclidean metric on the (𝑛−𝑘−1) 𝑉 -dimensional submanifold with volume 𝑛−𝑘−1.The 5. Conclusions nonvanishing components of electromagnetic field tensor and the metric function are, respectively, At the first step, we introduced a class of static magnetic string solutions in Einstein gravity in the presence of negative (Ξ2 −1) (Ξ2 −1) cosmological constant with two types of NLED. In order to 𝐹𝑟𝑡 =− 𝐹𝑟𝜙𝑖 = Ξ𝑎𝑖 Ξ𝑎𝑖 have real solutions, we obtained a lower limit for the radial coordinate, 𝜌. Furthermore, we nominated the largest real 2𝑞𝑙𝑛−1 { 1 󸀠 root of the metric function as 𝑟0 and, in order to get rid of { exp (− 𝐿𝑊), ENEF, (43) { 2 2 (𝑛−1)/2 2 signature changing, we introduced a new radial coordinate 𝑟. (𝑟 +𝑟0 ) × { (𝑛−1)/2 Calculations of geometric quantities showed that {𝛽2(𝑟2 +𝑟2) { 0 󸀠 although these solutions do not have curvature singularity, { (1 − Γ ), LNEF, { 𝑞 there is a conical singularity at 𝑟=0with a deficit angle 𝛿𝜙 = 8𝜋𝜇, where one can interpret 𝜇 as the mass per unit 2 2 3 2 2 2 𝑟 +𝑟 2𝑚𝑙 𝜒𝛽 (𝑟 +𝑟0 ) length of the string. Moreover, we found that, unlike the 𝑓 (𝑟) = 0 + − 𝑙2 2 2 (𝑛−2)/2 𝑛 (𝑛−1) power Maxwell invariant solutions [45–47], the nonlinearity (𝑟 +𝑟0 ) (44) does not have any effect on the asymptotic behavior of 4𝛽𝑞𝑙𝑛−2Υ (𝑟) the solutions and, in other words, obtained solutions are − , asymptotically adS. (𝑛−1) (𝑟2 +𝑟2)(𝑛−2)/2 0 In addition, we investigated the effects of nonlinearity parameter on the energy density and deficit angle, separately, 𝐿󸀠 = 𝑊(−16𝑙2(𝑛−2)𝑞2/𝛽2(𝑟2 +𝑟2)𝑛−1) Γ󸀠 = where 𝑊 Lambert 0 , and found that when one increases the nonlinearity param- √ 2 2 2 2 2 𝑛−1 1−4𝑙 𝑞 /𝛽 (𝑟 +𝑟0 ) ,andthefunctionΥ(𝑟) is eter, the concentration volume of the energy density and the deficit angle reduce. Υ (𝑟) Using a suitable local transformation, we added an angular momentum to the spacetime and found that for rotating { [1 − 𝐿󸀠 ]𝑟 solutions there is an electric field in addition to the magnetic { 𝑊 {∫ 𝑑𝑟, ENEF, one. { √−(𝑟2 +𝑟2)𝐿󸀠 { 0 𝑊 Next, we used the counterterm method and Gauss’s law to { 𝑛−1 { (𝛽2 (1 − Γ󸀠)(𝑟2 +𝑟2) /2𝑙2𝑞2) obtain conserved quantities and electric charge, respectively. { 2𝛽 ln 0 = ∫ ( It is interesting to note that these quantities depend on the {𝑞𝑙𝑛−2 (𝑟2 +𝑟2)1−𝑛/2 rotation parameter and the static string has no net electric { 0 { { charge. { 2 2 At the final step, we studied magnetic solutions in { 4𝑞 𝑙 { + )𝑟𝑑𝑟 LNEF. higher dimensions. We generalized static magnetic branes to { 𝛽2 (1 − Γ󸀠)(𝑟2 +𝑟2)𝑛/2 { 0 spinning ones and obtained consistent electromagnetic field (45) as well as metric function. Moreover, we obtained conserved quantities of the magnetic branes and found that the electric Following the known counterterm procedure and Gauss’s charge vanishes for the static magnetic branes. In addition, we law, it is easy to calculate the conserved quantities of the found that, for 𝑛=3, the conserved quantities of the magnetic magnetic brane solutions. Straightforward calculations show branes reduce to those of magnetic string, as we expected. 8 Advances in High Energy Physics

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Research Article Holographic Brownian Motion in Three-Dimensional Gödel Black Hole

J. Sadeghi,1 F. Pourasadollah,1 and H. Vaez2

1 Sciences Faculty, Department of Physics, Mazandaran University, P.O. Box 47416-95447, Babolsar, Iran 2 Young Researchers Club, Islamic Azad University, Ayatollah Amoli Branch, Amol, Iran

Correspondence should be addressed to H. Vaez; [email protected]

Received 7 January 2014; Accepted 21 February 2014; Published 2 April 2014

Academic Editor: Deyou Chen

Copyright © 2014 J. Sadeghi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP3.

By using the AdS/CFT correspondence and Godel¨ black hole background, we study the dynamics of heavy quark under a rotating plasma. In that case we follow Atmaja (2013) about Brownian motion in BTZ black hole. In this paper we receive some new results 2 2 for the case of 𝛼 𝑙 =1̸. In this case, we must redefine the angular velocity of string fluctuation. We obtain the time evolution of displacement square and angular velocity and show that it behaves as a Brownian particle in non relativistic limit. In this plasma, it seems that relating the Brownian motion to physical observables is rather a difficult work. But our results match with Atmaja work 2 2 in the limit 𝛼 𝑙 →1.

1. Introduction Langevin equation which phenomenologically describes the forceactingonBrownianparticles[10–12]whichisgivenby In the last several years, the holographic AdS/CFT [1–4]has 𝑝̇(𝑡) =−𝛾0𝑝 (𝑡) +𝑅(𝑡) , (1) been exploited to study strongly coupled systems, in particular quark gluon plasmas [5–7]. The quark gluon plasma where 𝑝 is momentum of Brownian particle and 𝛾0 is the (QGP) is produced, when two heavy ions collide with each friction coefficient. These forces originate from losing energy otheratveryhightemperature.Arelativelyheavyparticle, to medium due to friction term (first term) and getting a for example, a heavy quark, immerses in a soup of quarks random kick from the thermal bath (second term). One and gluons with small fluctuations due to its interaction with can learn about the microscopic interaction between the constituent of QGP.The random motion of this particle is well Brownian particle and the fluid constituents, if these forces ⟨𝑚𝑥̇2⟩=𝑇 knownasBrownianmotion[8–10]. The Brownian motion is a be clear. By assuming ,thetimeevolutionof universal phenomenon in finite temperature systems and any displacement square is given as follows [10]: particle immersed in a fluid at finite temperature undergoes 𝑇 1 { 𝑡2,(𝑡≪ ), Brownian motion. The Brownian motion opens a wide view 2 2 {𝑚 𝛾 ⟨𝑠(𝑡) ⟩=⟨[𝑥 (𝑡) −𝑥(0)] ⟩≈{ 0 (2) from microscopic nature. It offers a better understanding of { 1 the microscopic origin of thermodynamics of black holes. 2𝐷𝑡, (𝑡 ≫ ), { 𝛾0 Therefore, it is a natural step to study Brownian motion using the AdS/CFT correspondence. Particularly, the AdS/CFT where 𝐷 = 𝑇/𝛾0𝑚 is diffusion constant, 𝑇 is the temperature, correspondence can be utilized to investigate the Brownian and 𝑚 is the mass of Brownian particle. At early time, 𝑡≪ motion for a quark in the quark gluon plasma. 1/𝛾0 (ballistic regime), the Brownian particle moves with InthefieldtheoryorboundarysideofAdS/CFTstory,a constant velocity 𝑥∼̇ √𝑇/𝑚,whileatthelatetime,𝑡≫1/𝛾0 mathematical description of Brownian motion is given by the (diffusive regime), the particle undergoes a random walk. 2 Advances in High Energy Physics

In the gravity or bulk side of AdS/CFT version for model for the bosonic part of five-dimensional supergravity Brownian motion, we need a gravitational analog of a quark theory, so it can be an advantage in development of string immersed in QGP. This is achieved by introducing a bulk theory. Three-dimensional Godelblackholesareliketheir¨ fundamental string stretching between the boundary at infin- higher dimensional counterparts in special properties. The ity and event horizon of an asymptotically AdS black hole action of Einstein-Maxwell-Chern-Simons theory in three background [13–17]. The dual statement of a quark in QGP on dimensions is given by [21] the boundary corresponds to the black hole environment that 1 2 1 𝛼 𝐼= ∫𝑑3𝑥[√−𝑔 (𝑅 + − 𝐹 𝐹𝜇])− 𝜖𝜇]𝜌𝐴 𝐹 ]. excites the modes of string. In the context of this duality, the 2 𝜇] 𝜇 ]𝜌 endofstringattheboundarycorrespondstothequarkwhich 16𝜋𝐺 𝑙 4 2 shows Brownian motion and its dynamics is formulated by (3) Langevin equation. In the formulation of AdS/CFT corre- A general spherically symmetric static solution to the above spondence, fields of gravitational theory would be related to action in various cases for the 𝛼 parameter can be written by the corresponding boundary theory operators [3, 4]. In this [22] way, instead of using the boundary field theory to obtain the correlation function of quantum operators, we can determine 𝑑𝑟2 𝑑𝑠2 = +𝑝𝑑𝑡2 +2ℎ𝑑𝑡𝑑𝜙+𝑞𝑑𝜙2, these correlators by the thermal physics of black holes and ℎ2 −𝑝𝑞 (4) use them to compute the correlation functions. In [14– 17], the Brownian motion has been studied in holographic where 𝑝, 𝑞,andℎ are functions of 𝑟 as setting and the time evolution of displacement square. If we consider different gravity theories, we know that strings 𝑝 (𝑟) =8𝐺𝜇, live in a black hole background and excitation of the modes −4𝐺𝐽 𝛾2 is done by Hawking radiation of black hole. So, different 𝑞 (𝑟) = +2𝑟−2 𝑟2, 2 (5) theories of gravity can be associated with various plasma 𝛼 𝑙 in the boundary. In this paper we follow different works ℎ (𝑟) =−2𝛼𝑟, to investigate Brownian motion of a particle in rotating plasmas. We try to consider the motion of a particle in two- with dimensional rotating plasma whose gravity dual is described by three-dimensional Godel¨ metric background. In this case, 1−𝛼2𝑙2 𝛾=√ . (6) we will see that, for the 𝛼 parameter in the Godel¨ metric, 8𝐺𝜇 new conditions for Brownian motion will be provided. As we know, the Brownian motion of a particle in two-dimensional The gauge potential is given by rotating plasma with the corresponding gravity of BTZ black 𝐴=𝐴 (𝑟) 𝑑𝑡 +𝐴 (𝑟) 𝑑𝜙, hole has been studied in [16]. The Godel¨ metric background 𝑡 𝜙 (7) in the special case receives to the BTZ black hole [18], so the with comparison of our results with [16] gives us motivation to understanding the Brownian motion in rotating plasmas in 𝛼2𝑙2 −1 −4𝐺𝑄 𝛾 𝐴 (𝑟) = +𝜀, 𝐴 (𝑟) = +2 𝑟. general form of background as Godel¨ black hole. 𝑡 𝛾𝛼𝑙 𝜑 𝛼 𝑙 (8) This paper is arranged as follows. In Section 2,wegive some review of three-dimensional Godel¨ black hole and The parameters 𝜇 and 𝐽 are mass and angular momentum. derive string action from this metric background. Section 3 is The arbitrary constant 𝜀 isapuregauge.Wecanrewritemetric devoted to investigate a holographic realization of Brownian (4)intheADMformasfollows: motion and obtain the solution for equation of motion of 2 2 2 Δ 2 𝑑𝑟 ℎ string in Godel¨ black hole geometry. We study the Hawking 𝑑𝑠 =− 𝑑𝑡 + +𝑞(𝑑𝜙 + 𝑑𝑡) , (9) radiation of the transverse modes near the outer horizon 𝑞 Δ 𝑞 of Godel¨ black hole to describe the random motion of the external quark in Section 4.InSection 5,wemakesome where summery about our results. 2(1+𝛼2𝑙2) Δ=ℎ2 −𝑝𝑞=𝜆(𝑟−𝑟 )(𝑟−𝑟 ), 𝜆= , + − 𝑙2

2. Background and String Action 8𝐺 𝜇𝐽𝜆 𝑟 = [𝜇±√𝜇2 − ] . ± 𝜆 2𝛼 2.1. Godel¨ Black Hole. Three-dimensionalodel G¨ spacetime [ ] is an exact solution of Einstein-Maxwell theory with a (10) negative cosmological constant and a Chern-Simons term [19]. When the electromagnetic field acquires a topological The Hawking temperature that gives the temperature of the mass 𝛼 Maxwell equation will be modified by an additional plasma is [23] term. In that case, we receive to Einstein-Maxwell-Chern- Simons system, and the geometry is the Godel¨ space time 𝜆(𝑟+ −𝑟−) 𝑇𝐻 = . (11) [20]. This theory can be viewed as a lower dimensional toy 8𝜋𝛼𝑟+ Advances in High Energy Physics 3

Here 𝑟− is the inner horizon and 𝑟+ is the outer horizon. In metric background (9) by the Nambu-Goto action (14)inthe 2 2 the sector 𝛼 𝑙 >1, we have real solution only for 𝜇 negative. following form: In this regime, there are Godel¨ particles and theory supports 2 2 1 time-like constants fields. When 𝛼 𝑙 <1, 𝜇 has positive 𝑆 =− values. In this case black hole will be will be constructed and NG 2𝜋𝛼́ 2 2 theory supports space-like constants fields. For 𝛼 𝑙 =1, 2 2 2 𝑟 2 𝑞(𝑟 )𝑟 metric (4) reduces to BTZ metric as can be explicitly seen × ∫ 𝑑𝑥2 ( +Δ(𝑟2)𝜙󸀠 − 2 2 by transforming to the standard frame that is nonrotating at 𝑞(𝑟 ) Δ(𝑟 ) (16) infinity with respect to anti-de Sitter space: 2 1/2 ℎ(𝑟2) 2 ×(𝜙+𝛼+̇ ) ) . 𝑟 2𝐺𝐽 2 𝜙󳨀→𝜙+𝛼𝑡, 𝑟󳨀→ + . (12) 𝑞(𝑟 ) 2 𝛼

We have obtained the above relation in the standard frame. In the standard frame, energy and angular momentum The equation of motion for 𝜙 derived from (16)is become 𝑀=𝜇−𝛼𝐽and 𝐽,insteadof𝜇 and 𝐽 in rotating frame. 𝜕 𝑟2𝑞 ℎ 𝜕 Δ𝜙󸀠 − [ (𝜙̇+𝛼+ )] + [ ] =0. (17) 𝜕𝑡 Δ√−𝑔 𝑞 𝜕𝑟 √−𝑔 2.2. String Action. In the general case for a 𝑑+2-dimensional black hole metric background is 3.1. Trivial Solution. The Nambu-Goto action up to quadratic terms after subsisting the small fluctuation 𝜙→𝐶+𝜙under

2 𝜇 ] 𝐼 𝐽 Godel¨ metric background in the standard frame is given by 𝑑𝑠 =𝑔𝜇] (𝑥) 𝑑𝑥 𝑑𝑥 +𝐺𝐼𝐽 (𝑥) 𝑑𝑥 𝑑𝑥 . (13) 2 1 Δ3/2𝜙󸀠 𝑆(2) =− ∫ 𝑑𝑡 𝑑𝑟 𝜇 NG 4𝜋𝛼́ 2 1/2 Here 𝑥 =𝑟,𝑡stands for the string worldsheet coordinates 𝑟3[Δ/𝑞 − 𝑞(ℎ/𝑞 +𝛼) ] 𝐼 𝐼 and 𝑋 =𝑋(𝑥)(𝐼,𝐽 = 1,...,𝑑 −2) for the spacetime (18) 𝑟 Δ1/2𝜙2̇ coordinates. If we stretch a string along the direction and − . 𝑋𝐼 3/2 consider small fluctuation in the transverse direction ,the 𝑟[Δ/𝑞−𝑞(ℎ/𝑞+𝛼)2] dynamics of this string follows from the Nambu-Goto action [14]: 2 By changing coordinate to 𝑠=𝑥−𝑥+ (where 𝑥=𝑟/2 + −𝑖𝜔𝑡 2𝐺𝐽/𝛼) and defining 𝜙(𝑡, 𝑠) =𝑒 𝑓𝜔(𝑠),onecanwritethe 1 2 𝑆 =− ∫ 𝑑𝑥2√(𝑋𝑋̇ 󸀠) − 𝑋̇2𝑋󸀠2. (14) equation of motion as follows: NG 2𝜋𝛼́ 1 𝑊 (𝑠) 𝑍 (𝑠) 𝜕2𝑓 + [3𝜕𝑍 (𝑠) 𝑊 (𝑠) −𝜕𝑊(𝑠) 𝑍 (𝑠)] 𝜕 𝑓 𝑠 𝜔 2 𝑠 𝜔 𝐼 If the scalars 𝑋 do not fluctuate too far from their equilib- 𝐼 2 rium values (𝑋 =0), we can expand the above action up to +𝜔 𝑓𝜔 =0, 𝐼 quadratic order in 𝑋 : (19)

where 1 𝜕𝑋𝐼 𝜕𝑋𝐽 𝑆 ≈− ∫ 𝑑𝑥2√−𝑔 (𝑥)𝑔𝜇]𝐺 . (15) NG 4𝜋𝛼́ 𝐼𝐽 𝜕𝑥𝜇 𝜕𝑥] Δ (𝑠) ℎ (𝑠) 2 𝑊 (𝑠) = −𝑞(𝑠) ( +𝛼) ,𝑍(𝑠) =Δ(𝑠) . 𝑞 (𝑠) 𝑞 (𝑠) In fact, this quadratic fluctuation Lagrangian can be inter- (20) preted as taking the nonrelativistic limit, so we must use the dual Langevin dynamics on boundary in the nonrelativistic The solution for this equation of motion is the trivial solution case. for the relation (17). In general, solution of this equation is very complicated. However, for the extremal case 𝜇=𝐽(1+ 2 2 2 𝛼 𝑙 )/𝛼𝑙 ,wecanfindananalyticalsolutionas 3. Strings in Gödel Black Hole ± −1±𝐷 𝑓𝜔 =𝑠 𝑌 (𝑠) , As we said in the Introduction, an external quark is dual to an open string that extends from the boundary to the 𝜔2𝑙2 𝜇 (21) horizon of the black hole [24]. We can obtain the dynamics 𝐷=√1+ 𝑀= , where with 2 2 of this string in a threedimensional Godel¨ black hole with the 16𝐺𝑀 1+𝛼 𝑙 4 Advances in High Energy Physics where 𝑌(𝑠) is obtained from the following relation: According to [16] we set dominator to zero, so we have

2 2 2 1 𝑧 𝑑 ℎ𝛼 + 𝑝 𝑠(𝑐𝑠 +4𝑧(𝑠− )) 𝑌 (𝑠) 𝜋2 =Δ=( ) ,ℎ=ℎ(𝑟2 | ). (27) 4 𝛼2 𝑑𝑠2 𝜙 𝛼 NH 𝑤=0 1 𝑧 +(2𝜒(𝑐𝑠2 +4𝑧(𝑠− )) The external force 𝐹 canbeobtainedbyconsideringthe 4 𝛼2 ext (22) rotation and the topological mass of black hole which is given by 𝑧2 𝑑 +2𝑐𝑠2 + 10𝑠𝑧 −3 ) 𝑌 (𝑠) 𝛼2 𝑑𝑠 𝜋𝜙 ℎ𝛼 + 𝑝 𝐹 = = . (28) ext 2𝜋𝛼́2𝜋𝛼𝛼́ +𝜒((𝜒−1)(𝑐𝑠 + 4𝑧) + 2𝑐𝑠 + 10𝑧) 𝑌 (𝑠) =0, After extracting this external force, we can derive the friction 2 where 𝜒=−1±𝐷and 𝑧=2𝑥+/𝑙 .Thecompletesolution coefficient 𝛾0 for nonzero 𝑤, by considering the relation 𝑝𝜙 = of 𝑌(𝑠) is derived in the Appendix. With similar argument 𝑚0𝑤,as as in [16], this solution is not acceptable, because it does 2 not have oscillatory modes in radial coordinates and also, 𝑞 𝑟2 − (2𝛾2/𝑙2)(𝑟2 /2 + 2𝐺𝐽/𝛼) 𝛾 𝑚 = NH = NH NH . (29) for this trivial constant solution, one can investigate that the 0 0 2𝜋𝛼́ 2𝜋𝛼́ square root determinant of the worldsheet metric is not real 2 2 everywhere; therefore, it is not a physical solution. Due to With 𝛼 𝑙 =1this coefficient reduces to the excepted value nonphysical motivation about the mentioned solution, we 𝛾 =𝑟2 /2𝜋𝛼𝑚́ 0 NH 0 for BTZ black hole [16]. have to consider another approach, which is linear solution. The Nambu-Goto action, with the small fluctuation, 𝜙→ 𝑤𝑡 + 𝜂(𝑟), +𝜙 under the Godel¨ background becomes 3.2. Linear Solution. We can take the linear ansatz for the 2 small fluctuation in the transverse direction 𝜙 to achieve a 1 Δ3/2𝜙󸀠 𝑆(2) =− ∫ 𝑑𝑡𝑑𝑟 nontrivialsolution,soweexpanditas NG 1/2 4𝜋𝛼́ 𝑟3[Δ/𝑞 − 𝑞(ℎ/𝑞 + 𝛼 +𝑤)2] 𝜙 (𝑡, 𝑟) =𝑤𝑡+𝜂(𝑟) , (23) (30) Δ1/2𝜙2̇ 𝑤 − . where is a constant angular velocity. By replacing this 2 3/2 relation into (17), the solution for 𝜂 is obtained as follows: 𝑟[Δ/𝑞−𝑞(ℎ/𝑞+𝛼+𝑤) ]

𝜋 (𝑟2/𝑞) (Δ − 𝑞2(ℎ/𝑞 + 𝛼 2+ 𝑤) ) The equation of motion from the above Nambu-Goto action 𝜂󸀠 (𝑟) =− 𝜙 √ , 2 (24) is given by Δ Δ−𝜋𝜙 −𝑟Δ1/2𝜙̈ 𝜋𝜙 where isaconstantwhichhasaconceptasthetotalforce 2 3/2 to keep string moving with linear angular velocity 𝑤 and also [Δ/𝑞 − 𝑞(ℎ/𝑞 + 𝛼 +𝑤) ] is related to momentum conjugate of 𝜙 in 𝑟 direction. At 𝑟= 2 (31) 𝑟 , the numerator becomes zero, so the denominator should 𝜕 Δ3/2𝜙󸀠 NH + =0. also vanish there, because the string solution (23)mustbereal 𝜕𝑟 2 1/2 𝑤=0 𝛼2𝑙2 =1̸ 𝑟 𝑟[Δ/𝑞 − 𝑞(ℎ/𝑞 +𝛼𝑤) ] everywhere along the worldsheet. For and , NH is given by Solving this equation is quite complicated for more values of 2 2 𝑤. However, one can find that there are some values like 2 𝑙 √ 8𝐺𝛾 4𝐺𝐽 𝑟 =− [1± 1+ (2𝜇 − 𝐽𝛼)] − . (25) NH 𝛾2 𝑙2𝛼2 𝛼 𝑥 +(𝑥 −𝑥 )((1−𝛼2𝑙2)/2) [ ] 𝑤= − + − 𝛼𝑥 𝑙2 2 2 + When 𝑤 =0̸ and 𝛼 𝑙 =1,wereceivetheexceptedrelationfor (32) 2 2 2 2 BTZ black hole [16]. For 𝑤 =0̸ and 𝛼 𝑙 =1̸,weobtain 𝑟− +(𝑟+ −𝑟−)((1−𝛼 𝑙 )/2) = , 2 𝑙2 𝛼𝑟+𝑙 𝑟2 = NH 𝛾2 (𝛼+𝑤) where this makes it possible to solve the equation of motion. In derivation of the right-hand side of the above relation we 16𝐺𝛾2 𝐽(𝛼+𝑤)2 𝑟 𝑟 =4𝐺𝐽/𝛼 𝑟 ×[(𝑤−𝛼) ± √(𝑤−𝛼)2 + >(𝜇− )] use − + .Thespecialradius NH approaches the 𝑙2 2𝛼 outer horizon of the Godel¨ black hole for this value of angular [ ] velocity 𝑤,whereΔ(𝑟+)=0(and 𝜋𝜙 =0); then the steady 4𝐺𝐽 state solution is the case that 𝑟 =𝑟+.Fromrelation(32) − . NH 𝛼 fortheangularvelocity,itisevidentthatwecanreceiveto 2 2 (26) 𝑤=𝑟−/𝑟+ for BTZ black hole (𝛼 𝑙 =1).Furthermore,for Advances in High Energy Physics 5

2 2 2 2 (𝛼 𝑙 >1),wecancheckthat𝑤 <𝛼,buttheremustbe 4. Displacement Square 2 2 2 2 some condition on 𝜇 and 𝐽 to have 𝑤 <𝛼 for (𝛼 𝑙 <1).We can write the equation of motion for this terminal angular So far, we have succeeded to drive oscillation modes for a velocity with changing coordinate to 𝑠=𝑥−𝑥+ as string moving in the Godel¨ black hole background. In the following, we follow the same procedure as in [14, 16]to 󸀠󸀠 1 󸀠 ̈ 𝑊 (𝑠) 𝑍 (𝑠) 𝜙𝑠 + [3𝜕𝑍 (𝑠) 𝑊 (𝑠) −𝜕𝑊(𝑠) 𝑍 (𝑠)] 𝜙𝑠 − 𝜙𝑠 =0, compute the displacement square for Brownian motion. In 2 order to achieve this, we write the solutions for bulk equation (33) ± of motion as a linear combination of 𝑓𝜔 : where + − −𝑖𝜔𝑡 𝑝 𝑓𝜔 (𝑠) =𝐴[𝑓𝜔 (𝑠) +𝐵𝑓𝜔 (𝑠)]𝑒 , (41) 𝑊 (𝑠) = [𝑠 (𝑐𝑠 +) 𝜆𝜁 ], 𝑍(𝑠) =𝜆𝑠(𝑠+𝜁) , 2 (2𝛼𝑥+) where 𝐴 and 𝐵 are constants. By exerting the Neumann boundary condition near the boundary, 𝜕𝑠𝑓𝑠(𝜔) = 0 with 𝑐=𝜆−4𝛼2,𝜁=𝑥−𝑥 . + − 𝑠=𝑠𝑐 ≫0, to put the UV-cutoff, we obtain (34) (4𝜆𝜁)2𝑖𝜗(𝑐𝑠 +𝜆𝜁)−2𝑖𝜗𝑠2𝑖𝜗 (1−2𝑖𝜗) 𝜙(𝑡, 𝑠)−𝑖𝜔𝑡 =𝑒 𝑓 (𝑠) 𝐵= 𝑐 𝑐 As before, we take 𝜔 ,so(33)reduceto 4𝑖𝜗 1 [1 + √1+(𝜆−𝑐) 𝑠 /(𝑐𝑠 +𝜆𝜁)] (1+2𝑖𝜗) 𝑊 (𝑠) 𝑍 (𝑠) 𝑓󸀠󸀠 + [3𝑍󸀠 (𝑠) 𝑊 (𝑠) −𝑊󸀠 (𝑠) 𝑍 (𝑠)]𝑓󸀠 +𝜔2𝑓 𝑐 𝑐 𝜔 2 𝜔 𝜔 (42) =0. [1 + 2𝑖𝜗√1+(𝜆−𝑐) 𝑠𝑐/(𝑐𝑠𝑐 +𝜆𝜁)] 𝑖𝜃 × ≡𝑒 𝜔 . (35) [1 − 2𝑖𝜗√1+(𝜆−𝑐) 𝑠𝑐/(𝑐𝑠𝑐 +𝜆𝜁)] Consequently, the independent linear solutions to the above equation are obtained as below: Note that the constant 𝐵 is a pure phase, so by using (40)in ± ±𝑖𝜗 3 (−𝜆 +) 𝑐 𝑠 the near horizon we can write 𝑓𝜔 (𝑠) = (𝜆𝜁𝑠) 2𝐹1 (±𝑖𝜗, ± 𝑖𝜗; 1 ± 2𝑖𝜗, ) 2 𝑐𝑠 + 𝜆𝜁 𝑖𝜔𝑡 −𝑖𝜔(𝑡−𝑠⋆) 𝑖𝜃𝜔 −𝑖𝜔(𝑡+𝑠⋆) Φ (𝑡, 𝑠) =𝑓𝜔 (𝑠) 𝑒 ∼𝑒 +𝑒 𝑒 . (43) × 𝑐𝑠 + 𝜆𝜁 ∓𝑖𝜗, ( ) To regulate the theory, we implement another cutoff near the (36) outer horizon at 𝑠ℎ =𝜖, 𝜖≪1, which is called IR-cutoff; we where obtain 2𝛼𝜔𝑥 2𝑖𝜗 −2𝑖𝜗 ln(1/𝜖) 𝜗= + , 𝐵≈𝜖 =𝑒 . (44) 𝜆𝜁√𝑝 (37) If we take 𝐵 in the terms of 𝜔 by relation (41)only,then 2 2 √ or with 𝜉=2𝜁=𝑟+ −𝑟− and 𝑥+/√𝑝 = 4𝐺𝐽/(𝛼𝑟− 𝜆),wehave 𝐵 has continuous values, since the 𝜔 canhaveanyvalue. 3/2 𝜗 = 16𝐺𝐽𝜔/𝜆 𝜉𝑟−. By considering the following relation Using relation (43)for𝐵 will satisfy our requirements to have for hypergeometric functions, discrete values in 𝜖≪1. In this case, the discreteness is [14, 16] 3 𝜋 2𝐹1 (𝜅, 𝜅 + ,2𝜅+1,𝑧) Δ𝜗 = , (45) 2 ln (1/𝜖) 2𝜅 (38) 2 −2𝜅 1 𝜔 = [1 + (1−𝑧)1/2] [ +𝜅(1−𝑧)−1/2], where, in terms of ,itisgivenby 𝜅+1/2 2 3/2 𝜆 𝜋𝑟−𝜉 (36)reducesas Δ𝜔 = . (46) 16𝐺𝐽 ln (1/𝜖) (4𝜆𝜁)±𝑖𝜗 [1 ± 2𝑖𝜗/√1+(𝜆−𝑐) 𝑠/ (𝑐𝑠 +) 𝜆𝜁 ] 𝑓± (𝑠) = Following the above processes and using IR-cutoff to discrete 𝑤 1±2𝑖𝜗 ±2𝑖𝜗 [1 + √1+(𝜆−𝑐) 𝑠/ (𝑐𝑠 +) 𝜆𝜁 ] (39) the continuous spectrum makes it easy to find normalized bases of modes and to quantize 𝜙(𝑡, 𝑟) by extending in these × (𝑐𝑠 +) 𝜆𝜁 ∓𝑖𝜗𝑠±𝑖𝜗, modes. which gives oscillation modes. We have the following asymp- toticbehaviorfromthesolutionsneartheouterhorizon(𝑠 → 4.1. Brownian Particle Location. In this section we are going 0) and the boundary (𝑠→∞): to use quantized modes of the string near the outer horizon of Godel¨ black hole to describe the Brownian motion of ±𝑖𝜔𝑠 {𝑒 ⋆ (𝑠󳨀→0) an external quark. Therefore we consider the Nambu-Goto { ±𝑖𝜗 𝑓± (𝑠) ∼ (4𝜆𝜁) (1 ± 2𝑖𝜗/√𝜆/𝑐) action for certain amount of terminal angular velocity, near 𝑤 { (𝑠󳨀→∞) , (40) (𝑠→0) { ±2𝑖𝜗 the outer horizon : (1±2𝑖𝜗) (1 + √𝜆/𝑐) { 1 2 𝑆2 ∼ ∫ 𝑑𝑡𝑑𝑠 (Φ̇2 −Φ󸀠 ), 3/2 NG ⋆ (47) with 𝑠⋆ =(𝜗/𝜔)ln(𝑠) = (16𝐺𝐽/𝜆 𝜉𝑟−) ln(𝑠). 2 6 Advances in High Energy Physics

√ where Φ ≡ (8𝐺𝐽/(𝑟− 2𝜋𝜆𝛼))𝜙́. Thus, according to the radiation [25, 26] and incoming modes (𝜔<0) which fall into same procedure for standard scalar fields, we introduce the black hole. The outgoing mode correlators are determined by following mode expansions: the thermal density matrix:

† ∗ −𝛽𝐻 Φ (𝑡, 𝑠) = ∑ [𝑎𝜔𝑢𝜔 (𝑡, 𝑠) +𝑎𝜔𝑢𝜔(𝑡, 𝑠) ], 𝑒 (48) 𝜌 = ,𝐻=∑𝜔𝑎† 𝑎 , 𝜔>0 0 −𝛽𝐻 𝜔 𝜔 (52) 𝑇𝑟 (𝑒 ) 𝜔>0 with and the expectation value of occupation number is given by 𝜆3/2𝜉𝑟 √ − + − −𝑖𝜔𝑡 the Bose-Einstein distribution: 𝑢𝜔 (𝑡, 𝑠) = [𝑓𝜔 (𝑠) +𝐵𝑓𝜔 (𝑠)]𝑒 , 32𝐺𝐽𝜔 ln (1/𝜖) (49) 𝛿𝜔𝜔́ ⟨𝑎† 𝑎 ⟩ = , † 𝜔 𝜔́ 𝛽𝜔 (53) [𝑎𝜔,𝑎𝜔́]=𝛿𝜔𝜔́. 𝑒 −1

Now, by considering the above quantum modes on the probe with 𝛽=1/𝑇. Using the knowledge of relation (52)about string in the bulk, we want to work out the dynamics of outgoing modes correlators in the bulk, we can investigate the endpoint which corresponds to an external quark. We the motion of the endpoint of the string at 𝑆=𝑅≫1.We investigate the wave-functions of the world-sheet fields in can also determine the behavior of the Brownian motion, by the two interesting regions: (i) near the black hole horizon computing displacement square, as came in (2). So we can and (ii) close to the boundary. From (40), near the horizon predict the nature of Brownian motion of external particle (𝑆 ∼ 0), expansion (48)becomes on the boundary. For this purpose, we compute the modes correlators at 𝑆=𝑅≫1as 𝑟3/2√2𝜋𝛼́𝜆5/2 𝜉 𝜙 (𝑡, 𝑆 󳨀→) 0 = − 𝑟3 𝜋𝛼𝜆́5/2𝜉 3/2 ⟨𝜙 (𝑡) 𝜙 (0)⟩=∑ − (16𝐺𝐽) √ln (1/𝜀) 𝑅 𝑅 3 𝜔>0 (16𝐺𝐽) 𝜔 ln (1/𝜖) ∞ 1 −𝑖𝜔(𝑡−𝑆⋆) 𝑖𝜃𝜔 −𝑖𝜔(𝑡+𝑆⋆) 2 × ∑ (𝑒 +𝑒 𝑒 )𝑎𝜔. [1 + 4𝜗(𝜔) ] 𝜔=−∞ √𝜔 × 2 (50) [1 + 4𝜗(𝜔) (𝜆 (𝑅+𝜉) / (𝑐𝑅 +)) 𝜆𝜁 ]

2 2 2 𝜔𝑡 −𝑖𝜔𝑡 We used 𝑆=2𝑠=𝑟 −𝑟+. On the other hand, expansion (48) ×[ cos +𝑒 ]. at 𝑆=𝑅(the location of the regulated boundary) is given by 𝑒𝛽𝜔 −1 (54) 𝜙 (𝑡, 𝑆 =𝑅) By utilizing (46), we can write the above relation in the 𝑟3/2√2𝜋𝛼́𝜆5/2 𝜉 integralform.Weseethattheintegralisdiverging.Sowe = − † † 3/2 regularize it by normally ordering the 𝑎, 𝑎 oscillators: 𝑎𝜔𝑎𝜔 ≡ (16𝐺𝐽) √ln (1/𝜖) † 𝑎𝜔𝑎𝜔;thenwehave 1 × ∑ ⟨: 𝜙 (𝑡) 𝜙 (0) :⟩ 𝜔>0 √𝜔 𝑅 𝑅 2 8𝜆𝑟−𝛼́ 𝑖𝜗 −𝑖𝜗 = × [ (21+2𝑖𝜗(𝜆𝜉𝑅) (𝑐𝑅 + 𝜆𝜉) (1−2𝑖𝜗) (16𝐺𝐽)2 [ ∞ 𝑑𝜔 (16𝐺𝐽)2𝜔2 × ∫ (1 + 4 ) 2𝑖𝜗 3 2 2 (𝜆−𝑐) 𝑅 0 𝜔 𝜆 𝜉 𝑟 (55) ×([1+√1+ ] − 𝑐𝑅 + 𝜆𝜉 −1 (16𝐺𝐽)2𝜔2 𝜆 (𝑅+𝜉) ×(1+4 ( )) 3 2 2 −1 𝜆 𝜉 𝑟− 𝑐𝑅 + 𝜆𝜉 (𝜆−𝑐) 𝑅 ×[1−2𝑖𝜗√1+ ]) ) 2 (𝜔𝑡) 𝑐𝑅 + 𝜆𝜉 ×[ cos ], 𝑒𝛽𝜔 −1

−𝑖𝜔𝑡 ] and the displacement square becomes ×𝑒 𝑎𝜔 +ℎ.𝑐. . ] 2 2 𝑆 (𝑡) ≡⟨:[𝜙𝑅 (𝑡) −𝜙𝑅 (0)] :⟩ (51) reg 16𝜆𝑟2 𝛼́(𝜆−𝑐) 𝑅 𝑐𝑅 + 𝜆𝜉 (56) Onecanseethattherearetwomodesinthesolutions.The = − [ 𝐼 + 𝐼 ], 2 𝜆 (𝑅+𝜉) 1 𝜆 (𝑅+𝜉) 2 outgoing modes (𝜔>0)thatareexcitedbecauseofHawking (16𝐺𝐽) Advances in High Energy Physics 7 with One can check that the displacement square (61)isconsistent 2 2 2 with BTZ black hole in [16]bysetting𝑐=0or 𝛼 𝑙 =1. ∞ 𝑑𝑦 sin (𝑘𝑦/2) 𝐼 =4∫ , In that case, the 𝑤 vanishes for 𝐽=0(or 𝑟− =0), but when 1 2 2 𝑦 2 2 2 2 2 0 𝑦(1+𝑎 𝑦 ) 𝑒 −1 𝛼 𝑙 =1̸,the𝑤 will have zero value only for 𝐽/𝛼𝑙 =(𝛼𝑙 −1)𝜇 (57) (see relation (32)). Then our static solution is achieved by this ∞ 𝑑𝑦 2 (𝑘𝑦/2) 𝐼 =4∫ sin , condition. The diffusion constant from (61)isgivenby 2 𝑦 0 𝑦 𝑒 −1 𝜆𝜋𝛼𝛼́2 𝐷= 𝑇. andwehavedefined 2 (62) 2𝑟+ 𝑡 16𝐺𝐽 2 𝑅+𝜉 𝑦=𝛽𝜔, 𝑘= ,𝑎2 =4( ) ( ). So, the relaxation time of Brownian particle is as follows: 𝛽 𝜆𝜉𝑟−𝛽 𝑐𝑅 + 𝜆𝜉 1 𝜆𝑚 𝜋𝛼𝛼́2 (58) 𝑡 = = 0 . 𝑐 2 (63) 𝛾0 2𝑟 The evaluation of these integrals and their behavior for 𝑅≫1 + 𝑎≫1 and can be found in Appendix B of [14]. From relation The mass of external particle, 𝑚0,canbecomputedbyusing (58), we can see that when 𝑅≫1,wehave𝑎∝1/𝑐.Thusin the total energy and momentum of string [27] under the general case for 𝑎, we use the following relations for integrals metric background (4): (57): 1 0 1 0 1 𝑘 𝑘 𝐸= ∫ 𝑑𝑟𝜋𝑡 ,𝑝𝜙 = ∫ 𝑑𝑟𝜋𝜙, (64) 𝐼 = [𝑒𝑘/𝑎𝐸𝑖 (− )+𝑒−𝑘/𝑎𝐸𝑖 ( )] 2𝜋𝛼́ 2𝜋𝛼́ 1 2 𝑎 𝑎 with 1 1 1 + [𝜓 (1 + )+𝜓(1− )] 𝜙󸀠2 𝑔 2 2𝜋𝑎 2𝜋𝑎 𝜋0 = (𝑔2 −𝑔 𝑔 )− 𝑟𝑟 (𝑔 +𝑔 𝜙)̇ , 𝑡 √−𝑔 𝑡𝜙 𝑡𝑡 𝜙𝜙 √−𝑔 𝑡𝑡 𝑡𝜙 𝜋 1 2𝑎 𝜋𝑘 𝑒−2𝜋|𝑘| (65) − (1 − 𝑒|𝑘|/𝑎) + ( sinh )+ 𝑔 2 cot2𝑎 log 𝑘 2 𝜋0 = 𝑟𝑟 (𝑔 +𝑔 𝜙)̇ . 𝜙 √−𝑔 𝑡𝜙 𝜙𝜙 −2𝜋|𝑘| 2𝐹1 (1, 1 + 1/2𝜋𝑎, 2 + 1/2𝜋𝑎; 𝑒 ) ×[ Thenwehave 1 + 1/2𝜋𝑎 𝛼 𝑐(𝑠+𝑥 )+𝜆𝑥 𝐸= ∫ 𝑑𝑠 + + , 𝐹 (1, 1 − 1/2𝜋𝑎, 2 − 1/2𝜋𝑎;−2𝜋|𝑘| 𝑒 ) 2𝜋𝛼́ + 2 1 ], √𝜆𝑝 (𝑠+𝜁)(𝑐𝑠 +) 𝜆𝜁 1 − 1/2𝜋𝑎 (66) 1 −𝑐 (𝑠 +𝑥 )+𝜆𝑥 𝑝 = ∫ 𝑑𝑠 + − . sinh 𝜋𝑘 𝜙 𝐼 = ( ). 2𝜋𝛼́ √𝜆𝑝 (𝑠+𝜁)(𝑐𝑠 +) 𝜆𝜁 2 log 𝜋𝑘 (59) One can check that, after putting 𝑐=0in the above relation, 2 2 However, for 𝑅≫1and ≪1(𝛼𝑙 →1),then𝑎≫1,one the result of integral is as excepted for BTZ black hole. However, for 𝑐 =0̸ we obtain can utilize the following relation for 𝐼1 and 𝐼2: 2 2 √(𝑐𝑅 + 𝜆𝜉)(𝑅+𝜉) − √𝜆𝜉 { 𝜋𝑘 −2 𝐸=𝛼 { +𝑂(𝑎 ) √𝜆𝑝 𝐼 = 2𝑎 1 { { 𝜋𝑘+𝑂(log 𝑘) , 𝜆+𝑐 𝑝 √𝑐𝑅 + 𝜆𝜉 + √𝑐 (𝑅+𝜉) (60) +2𝛼 √ ln [ ], 𝜆 𝜆𝑐 √𝑐𝜉 + √𝜆𝜉 𝑂(𝑎0), (𝑘≪𝑎) 𝐼2 ={ (67) 𝜋𝑘+𝑂(log 𝑘) , (𝑘≫𝑎) . √(𝑐𝑅 + 𝜆𝜉)(𝑅+𝜉) − √𝜆𝜉2 𝑝 =− 𝜙 √𝜆𝑝 𝑆 (𝑡)2 Therefore, reg has the following form: 𝜆−𝑐 𝑝 √𝑐𝑅 + 𝜆𝜉 + √𝑐 (𝑅+𝜉) 16𝑟3 𝜆𝜋𝛼́𝜉 (𝜆−𝑐)(𝑐𝑅 + 𝜆𝜉)1/2 +2 √ ln [ ], { − [ ]𝑡2 𝜆 𝜆𝑐 √𝑐𝜉 + √𝜆𝜉 { 3 1/2 { (16𝐺𝐽) 𝛽 4(𝑅+𝜉) { { 𝑐𝑅 + 𝜆𝜉 where √𝑝/𝜆 += (𝑟 +𝑟−)/2. Then the mass is defined as ⟨𝑆 (𝑡)2⟩= +𝑂( ), (𝑡≪𝛽), reg { 𝑅+𝜉 { 2 2 2 { 𝑚 =𝐸 −𝑝 . { 16𝑟2 𝜆𝜋𝛼́ 𝑡 0 𝜙 (68) { − 𝑡+𝑂( ), (𝑡≫𝛽). 2 log { (16𝐺𝐽) 𝛽 𝛽 From the above relations, we see that relating the physical (61) mass to displacement square is difficult. 8 Advances in High Energy Physics

5. Summary × 2𝐹1 (𝐸 (𝛼)(𝑇 (𝛼) +𝑈(𝛼)) , In this paper, by using AdS/CFT correspondence, we studied the Brownian motion of an external quark in plasma. It is 𝐸 (𝛼)(𝑇 (𝛼) +𝑈(𝛼)) , corresponded to a string stretched from horizon of AdS to boundary. By using the Nambu-Goto action, we obtained the 𝑝 (𝛼) (1 − ); equation of motion for this string in the Godel¨ background. 𝑞 (𝛼) 𝑟 (𝛼) For an acceptable solution with oscillatory modes, we had to redefine the terminal angular velocity. We found that turning 𝑐𝛼√𝜆𝑠 on a finite density for a conserved 𝑈(1) charge (reflected ) byaCSterminthebulk)andtherotationofblackhole (−𝑧√𝜆+𝛼(𝑐𝑠 + 2𝑧))𝑟(𝛼) influence oscillatory modes. For realization of the Brownian motion, we derived the time evolution of the displacement +𝐶 𝑠−3/2−𝜒√4 𝑠 (𝑐𝑠 + 4𝑧) 𝛼2 −𝑧2 square from the modes correlators. We showed that in general 2 (𝛼2𝑙2 =1̸ case Godel¨ black hole), our results for displacement 𝑠 (1/2)(1+𝑝(𝛼)/𝑟(𝛼)𝑞(𝛼)) square are different in comparison with16 [ ]. However, in ×( ) 2 2 𝛼 𝑙 =1limit (BTZ black hole), we confirmed that our −𝑐𝛼𝑠 − 2𝑧𝛼√ +𝑧 𝜆 results agree with the work of Atmaja [16]. We derived the (1/2)(1+3√−𝑐2/4𝑟(𝛼)𝑞(𝛼)) physical mass, but we found that relating the displacement −𝛼√𝜆𝑠 + 2𝑠𝛼2 −𝑧 ×( ) square to physical observables is a difficult work. This is −𝑧√𝜆+𝛼(𝑐𝑠 + 2𝑧) the problem that we would like to consider in future work. AlsowewouldliketoinvestigatetheBrownianmotion × (−𝑐𝛼𝑠 − 2𝑧𝛼√ +𝑧 𝜆) of external quarks in different environments, in particular plasmas which correspond to metric backgrounds as Lifshitz geometry [28] and metric backgrounds with hyperscaling × 2𝐹1 (𝐸(𝛼)(𝑇 (𝛼) +𝑅(𝛼)) , violation [29, 30].

𝐸 (𝛼)(𝑇 (𝛼) +𝑅(𝛼)) , Appendix 𝑝 (𝛼) (1 + ); The solution to the following differential equation 𝑞 (𝛼) 𝑟 (𝛼)

2 𝑐𝛼√𝜆𝑠 2 1 𝑧 𝑑 ), 𝑠(𝑐𝑠 +4𝑧(𝑠− )) 𝑌 (𝑠) √ 4 𝛼2 𝑑𝑠2 (−𝑧 𝜆+𝛼(𝑐𝑠 + 2𝑧))𝑟(𝛼) 1 𝑧 (A.2) +(2𝜒(𝑐𝑠2 +4𝑧(𝑠− )) 4 𝛼2

𝑧2 𝑑 (A.1) where 𝐸, 𝑅, 𝑇, 𝑈, 𝑟, 𝑞,and𝑝(𝛼) are given by relations (A.2)– +2𝑐𝑠2 + 10𝑠𝑧 −3 ) 𝑌 (𝑠) 𝛼2 𝑑𝑠 (A.6):

+𝜒((𝜒−1)(𝑐𝑠 + 4𝑧) + 2𝑐𝑠 + 10𝑧) 𝑌 (𝑠) 1 =0 𝐸 (𝛼) = 2√−𝑐 − 8𝛼2 +4𝛼√𝜆(𝛼+1/2√𝜆) (2𝛼√𝜆+𝜆) 1 canbeobtainedanalyticallyas = , 4√𝜆𝑟2 (𝛼) 𝑞 (𝛼) (A.3) −3/2−𝜒√4 2 2 𝑌 (𝑠) =𝐶1𝑠 𝑠 (𝑐𝑠 + 4𝑧) 𝛼 −𝑧 𝑟 (𝛼) = √−𝑐 − 8𝛼2 +4𝛼√𝜆, 𝑠 (1/2)(1−𝑝(𝛼)/𝑟(𝛼)𝑞(𝛼)) ×( ) 1 −𝑐𝛼𝑠 − 2𝑧𝛼√ +𝑧 𝜆 𝑞 (𝛼) =(𝛼+ √𝜆) , (A.4) 2 √ 2 2 (1/2)(1+3 −𝑐 /4𝑟(𝛼)𝑞(𝛼)) −𝛼√𝜆𝑠 + 2𝑠𝛼 −𝑧 2 ×( ) 𝑝 (𝛼) = √−(𝜒 + 1) 𝑐2, −𝑧√𝜆+𝛼(𝑐𝑠 + 2𝑧) 1 √ 𝑇 (𝛼) = ((2𝛼2 + 𝑐) √𝜆+𝛼𝜆)√−𝑐 − 8𝛼2 +4𝛼√𝜆, (A.5) ×(−𝑐𝛼𝑠−2𝑧𝛼+𝑧 𝜆) 4 Advances in High Energy Physics 9

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Research Article Holographic Screens in Ultraviolet Self-Complete Quantum Gravity

Piero Nicolini1,2 and Euro Spallucci3

1 Frankfurt Institute for Advanced Studies (FIAS), Ruth-Moufang-Straße 1, 60438 Frankfurt am Main, Germany 2 Institut fur¨ Theoretische Physik, Johann Wolfgang Goethe-Universitat,¨ Max-von-Laue-Straße 1, 60438 Frankfurt am Main, Germany 3 Dipartimento di Fisica, Sezione di Fisica Teorica, UniversitadegliStudidiTriesteeINFN,SezionediTrieste,` Strada Costiera 11, 34151 Trieste, Italy

Correspondence should be addressed to Piero Nicolini; [email protected]

Received 29 January 2014; Accepted 3 March 2014; Published 27 March 2014

Academic Editor: Christian Corda

Copyright © 2014 P. Nicolini and E. Spallucci. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP3.

This paper studies the geometry and the thermodynamics ofa holographic screen in the framework of the ultraviolet self-complete quantum gravity. To achieve this goal we construct a new static, neutral, nonrotating black hole metric, whose outer (event) horizon coincides with the surface of the screen. The spacetime admits an extremal configuration corresponding to the minimal holographic screen and having both mass and radius equalling the Planck units. We identify this object as the spacetime fundamental building block, whose interior is physically unaccessible and cannot be probed even during the Hawking evaporation terminal phase. In agreement with the holographic principle, relevant processes take place on the screen surface. The area quantization leads to a discrete mass spectrum. An analysis of the entropy shows that the minimal holographic screen can store only one byte of information, while in the thermodynamic limit the area law is corrected by a logarithmic term.

1. Introduction theory is to give up the very idea of point-like building blocks of matter and replace them with one-dimensional “Quantum gravity” is the common tag for any attempt vibrating strings. As there does not exist any physical object to reconcile gravity and quantum mechanics. Since the smallerthanastring;therearenophysicalwaystoprobe early proposals by Wheeler [1, 2]andDeWitt[3], up to distances smaller than the length of the string itself. In the recent ultraviolet (UV) self-complete scenario [4], the this regard two properties of fundamental strings are worth diverse formulations of a would-be quantum theory of mentioning: gravity have shown a common feature, that is, a fundamental length/energy scale where the smooth manifold model (i) string excitations correspond to different mass and of spacetime breaks down. Let us refer to this scale as spin “particle” states; 19 the “Planck scale” irrespectively whether it is 10 GeV or − 2 (ii) highly excited strings share various physical proper- 10 10 TeV. The very concept of distance becomes physically ties with black holes. meaningless at the Planck scale and spacetime “evaporates” into something different, a sort of “foamy” structure, a Thus, we infer that string theory provides a bridge between spin network, a fractal dust, and so forth, according to the particle-like objects and black holes (see for instance [6]). chosen model [5].Asamatteroffact,oneofthemost However, it is important to remark that while the Comp- powerful frameworks for describing the Planckian phase ton wavelength of a particle-type excitation decreases by of gravity is definitely (Super) String Theory. The price to increasing the mass, the Schwarzschild radius of a black hole pay to have a perturbatively finite, anomaly-free quantum increases with its mass. Thus, the first tenet of high energy 2 Advances in High Energy Physics particle physics, which is “the higher the energy the shorter 4 the distance,” breaks down when gravity comes into play and turns a “particle” into a black hole. The above remark is the foundation of the UV self-complete quantum gravity scenario, where the Planckian and sub-Planckian length 3 scales are permanently shielded from observation due to the production of black hole excitations at Planck energy scattering [7]. Accordingly the Planck scale assumes the additional meaning of scale at which matter undergoes a transition C 2 between its two admissible “phases,”that is, the particle phase 𝜆 andtheblackholephase[8–10]. From this perspective, trans- Particle Planckian physics is dominated by larger and larger black hole configurations. It follows that only black holes larger than, or at most equal to, Planck size objects can self-consistently fit 1 into this scheme. However, classical black hole solutions do Black hole not fulfill this requirement, that is, the existence of a lower bound for their mass and size (see Figure 1). A first attempt to overcome this limitation is offered 0 bythenoncommutativegeometryinspiredsolutionsofthe 0 1 2 3 4 Einstein equations [11]. The latter are a family of regular black M holes which span all possible combinations of parameters, Figure 1: The dotted and the solid curves represent the particle such as mass [12], charge [13], and angular momentum Compton wavelength 𝜆𝐶 and the Schwarzschild radius as a function [14, 15]. In addition such regular geometries admit a vari- of the energy 𝑀 in Planck units (quantities are rescaled). The ety of complementary gravitational configurations such as squared bullet is the Planck scale. The grey area of the diagram is traversable wormholes [16], dirty black holes [17], dilaton actually excluded, meaning that a particle cannot be compressed gravity black holes [18], and collapsing matter shells [19]. at distances smaller than the Planck length: at trans-Planckian Recently this family of black holes has been recognized as energy only black hole form. The arrow shows the inadequacy of viable solutions of nonlocal gravity [20, 21], that is, a set of the Schwarzschild metric: black holes have no lower mass bounds, theories exhibiting an infinite number of derivative terms of can have size smaller than the Planck length, and can expose the the curvature scalar [22–24] in place of the mere Ricci scalar curvature singularity by decaying through the Hawking process. as in the standard Einstein-Hilbert action. More importantly extensions of noncommutative geometry inspired metrics to the higher dimensional scenario [25, 26] are currently under scrutiny at the LHC for their unconventional phenomenology input from quantum gravity: in the latter case it has been [27]: specifically the terascale black holes described by such shownthatsuchextremalblackholesfitprettywellinthe regular metrics tend to have a slower evaporation rate [28] UV self-complete scenario providing a stable, minimum size and emit only soft particles mainly on the four-dimensional probe at the transition point between particles and black holes brane [29]. A characteristic feature of this type of solutions [36]. is that the minimum size configuration is given by the Inthispaperwewanttotakeastepfurtherinthe extremal black hole configuration which exists even in the realization of this program by avoiding the introduction of neutral nonspinning case [30–32]. This fact automatically an additional principle to justify the presence of a minimal implies a minimum energy for black hole production in length, rather we demand the radius of a Planck size extremal particle collisions [33] without any further need of correcting black hole to provide the natural UV cutoff of a quantum formulas of cross sections with ad hoc threshold functions. spacetime. In this framework gravity is expected to be self- Extremal configurations play a crucial role in the physics regular in the sense that the actual regulator cutting off sub- ofthedecayingdeSitteruniverseviathenucleationof Planckian length scales is given in terms of the gravitational √ microscopic black holes. It has been shown that Planck coupling constant; that is, 𝐺=𝐿𝑃.Thepaperisorganizedas size noncommutative inspired black holes might have been follows. In Section 2 we derive a black hole metric, consistent copiously produced during inflationary epochs [34]. This fact with the above discussion and the concept of holographic has further phenomenological repercussions: being stable, screen. The latter coincides with the outer horizon of the noninteracting objects, extremal black holes turn out to be black hole whose mass spectrum is bounded from below by a reliable candidate for dark matter component. On the the mass of the extremal configuration equalling the Planck theoretical side, extremal configurations in the presence mass. Once trans-Planckian length scales are cut-ff, the of a negative cosmological term can provide a short scale “interior” of the black hole loses its physical meaning in the completion of the Hawking-Page diagram which switches to sense that all the relevant degrees of freedom are necessarily a more realistic Van der Waals phase diagram [35]. located on the horizon itself. In Section 3 we discuss the Extremal configurations can be either descending from thermodynamics of the screen. We find that the area law the introduction of a fundamental length in the line element is modified by logarithmic corrections and that there exists and can alternatively be interpreted as a phenomenological a minimal holographic screen with zero thermodynamic Advances in High Energy Physics 3

−1 entropy. Finally we propose a “holographic quantization” The condition for the metric coefficients 𝑔00 =−𝑔11 scheme where the area of the extremal configuration provides determines the equation of state, namely, the relation between the quantum of surface. In Section 4 we offer the reader a brief the energy density and the radial pressure, 𝑝𝑟 =−𝜌.The summary of the main results of this work. angular pressure is specified by the conservation of the stress tensor and reads 𝑝⊥ =𝑝𝑟 + (𝑟/2)𝜕𝑟𝑝𝑟. 2. Self-Regular Holographic Screen By plugging the tensor (5) in Einstein equations, one finds that the metric reads (𝐺 = 1) A simple but intriguing model of singularity-free black hole 2𝑚 (𝑟) has been “guessed” in [37], in the sense that the metric was 𝑑𝑠2 =−(1− )𝑑𝑡2 𝑟 assigned as an input for the Einstein equations. Sometimes (6) this inverted procedure is called “engineering” because the 2𝑚 (𝑟) −1 +(1− ) 𝑑𝑟2 +𝑟2𝑑Ω2, actual source term of field equations is not known a priori. 𝑟 The distinctive feature of the solution is the presence in the line element of a free parameter with dimension of a with length, acting as a short distance regulator for the spacetime 󸀠 󸀠 2 󸀠 curvature, allowing a safe investigation of back-reaction 𝑚 (𝑟) =4𝜋∫ 𝑑𝑟 (𝑟 ) 𝜌(𝑟 ). (7) effects of the Hawking radiation. In38 [ ] a higher dimensional extension of this model has been proposed; it was also shown At large distances 𝑟≫𝐿𝑃, the above energy density has that, by a numerical rescaling of the short-distance regulator, to quickly vanish; that is, 𝜌(𝑟) →0 in order to match the it is possible to identify this fundamental length scale with “vacuum” Schwarzschild metric. Conversely, at shorter scales the radius of the extremal configuration. With hindsight, 𝑟≳𝐿𝑃, the density 𝜌(𝑟) (and accordingly ℎ(𝑟)) has to depart we are going to take a step forward to improve this inverse from the point-particle profile in order to fulfill the following procedure. Specifically, we want to follow thedirect “ way” requirements: by building up a consistent source for Einstein equations: we introduce a physically motivated energy momentum tensor (i)nocurvaturesingularityintheorigin; which allows for transitions between particle-like objects and (ii) self-implementation of a characteristic scale 𝑙0 in the black holes as consistently required by UV self-complete spacetimegeometrybymeansoftheradiusofthe quantum gravity. extremal configuration 𝑟0;thatis,𝑟0 =𝑙0. We start from the energy density for a point-particle in spherical coordinates as The latter condition is crucial. For instance noncommutative 𝑀 geometry inspired black holes [11] are derived by the direct 𝜌 (𝑟) = 𝛿 (𝑟) , way; they enjoy (i) but fail to fulfill the condition (ii). This 𝑝 4𝜋𝑟2 (1) means that the characteristic length scale of the system where 𝛿(𝑟) is the Dirac delta. The energy distribution (1) 𝑙0 and the extremal configuration radius 𝑟0 are indepen- implies a black hole for any value of mass 𝑀 even for sub- dent quantities. Indeed noncommutative geometry is the Planckian values where one expects just particles. Before underlying theory which provides the scale 𝑙0 in terms proceeding,wewouldliketorecallthataDiracdeltafunction of an “external” parameter, namely, the noncommutative can be represented as the derivative of a Heaviside step- parameter 𝜃.Inotherwordsoneneedstoinvokeaprinciple, function Θ: like a modification of commutators in quantum mechanics, 𝑑 or the emergence of a quantum gravity induced fundamental 𝛿 (𝑟) = Θ (𝑟) . (2) 𝑑𝑟 length to achieve the regularity of the geometry at short scales.Againstthisbackground,wewantjusttouse𝑟0 as Against this background, we want to accommodate both fundamental scale, getting rid of any 𝑙0 as emerging from any particles and black holes by a suitable modification of the theory or principle not included in Einstein field equations. energy distribution in order to overcome the ambiguities This is a step forward since it opens the possibility for of the Schwarzschild metric in the sub-/trans-Planckian Einstein gravity to be self-protected in the ultraviolet regime. regimes (see also Figure 1). This can be done by considering ℎ(𝑟) Toemphasizethispoint,weintroducedtheword“self - a“smooth”function in place of the Heaviside step: implementation” in (ii). Since there exists actually only one Θ (𝑟) 󳨀→ ℎ (𝑟) . (3) additional scale beyond 𝑟0, that is, the Planck length 𝐿𝑃 = √ √ 𝐺, or the Planck mass 𝑀𝑃 =1/ 𝐺, we can implement the The new profile 𝜌(𝑟) of the energy density is defined through condition (ii) in the most natural way by setting 𝑟0 =𝐿𝑃 and ℎ(𝑟) by the relation accordingly 𝑀0 =𝑀𝑃,where𝑀0 ≡𝑀(𝑟0) is the extremal 𝑀 𝑑 0 black hole mass. 𝜌 (𝑟) = ℎ (𝑟) ≡𝑇. (4) 4𝜋𝑟2 𝑑𝑟 0 Despite the virtues of the above line of reasoning, we feel 𝜇] that the set of conditions (i) and (ii) can be relaxed and a By means of the conservation equation ∇𝜇𝑇 =0one can further simplification is possible. Having in mind that for determine the remaining components of the stress tensor, extremal black hole configurations the Hawking emission which turns to be out of the form ] stops we just need to find a metric for which only the 𝑇𝜇 = diag (−𝜌,𝑟 𝑝 ,𝑝⊥,𝑝⊥). (5) condition (ii) holds. This would be enough for completing 4 Advances in High Energy Physics the program of the UV self-complete quantum gravity by (ii) The line element (10)admitsapairofhorizons protecting the short distance behavior of gravity during the provided 𝑀≥𝑀𝑃.Theradii𝑟± of the horizons are final stages of the evaporation process. In this regard, the given by resulting extremal black hole is just the smallest object one 2 √ 2 2 can use to probe short-distance physics. In other words, in 𝑟± =𝐿𝑃 (𝑀 ± 𝑀 −𝑀𝑃). (12) the framework of UV self-complete quantum gravity, it is notphysicallymeaningfultoaskaboutcurvaturesingularity For 𝑀=𝑀𝑃 thetwohorizonsmergeintoasingle inside the horizon as the very concept of spacetime is no 𝑟 =𝑟 =𝐿 𝑀≫ longer defined below this length scale. (degenerate) null surface at ± 0 𝑃.For 𝑀𝑃 the outer horizon approaches the conventional According with such a line of reasoning, we can deter- 𝑟 ≃ ℎ(𝑟) value of the Schwarzschild geometry; that is, + mine the function by dropping the condition (i) and 2𝑀𝐿2 keeping just the condition (ii). Inside the class of all admis- 𝑃. sible profiles for ℎ(𝑟), the most natural and algebraically (iii) By inserting (11)into(12)onefinds𝑟+ =𝑟ℎ, 𝑟− = 2 compact choice is given by 𝐿𝑃/𝑟ℎ. We see that the holographic screen surface coincides with the (outer) black hole horizon 𝑟+, 𝐿2 ℎ (𝑟) =1− 𝑃 . while the inner Cauchy horizon has a radius which 2 2 (8) 𝑟 +𝐿𝑃 is always smaller or equal to the Planck length. This fact lets us circumvent the issue of potential blue shift Asimilarprocedurehasbeenalreadyusedin[33]and instabilities [43, 44] (see, i.e., recent analyses for non- accounts for the fact that in the presence of 𝐿𝑃 the step cannot commutative inspired [45, 46]andotherquantum 𝑟 be any longer sharp. Thus, the smeared energy density 𝜌(𝑟) gravity corrected metrics [47, 48]) because − simply turns out to be loses its physical meaning being not accessible to any sort of measurement process. In what follows we can 2 𝑀 𝐿𝑃 identify the holographic screen with the black hole 𝜌 (𝑟) = . (9) 2𝜋𝑟 (𝑟2 +𝐿2 )2 outer horizon without distinguishing between the two 𝑃 surfaces any longer. 𝑀<𝑀 As a result we find the following metric which is derived from (iv) “Light” objects, with 𝑃,are“particles” a stress tensor modeling a particle-black hole system (5): rather than holographic screens. By particles we mean localized lumps of energy of linear size given by the 2 2𝑀𝐿 𝑟 Compton wavelength 𝜆𝐶 =1/𝑀that can never 𝑑𝑠2 =−(1− 𝑃 )𝑑𝑡2 𝑟2 +𝐿2 collapse into a black hole. Rather they give rise to 𝑃 horizonless metrics (see Figure 2)andcannotprobe −1 (10) 𝜆 2𝑀𝐿2 𝑟 distances smaller than 𝐶. The “transition” particle +(1− 𝑃 ) 𝑑𝑟2 +𝑟2𝑑Ω2, → black holes is discussed below in terms of critical 𝑟2 +𝐿2 𝑃 surface density. where the arbitrary constant 𝑀 is defined as follows: As a further analysis of this result, it is interesting to consider the surface energy density of the holographic screen which is 1 𝑀≡ (𝑟2 +𝐿2 ). defined as 2𝐿2 𝑟 ℎ 𝑃 (11) 𝑃 ℎ 𝑀 1 𝑟2 +𝐿2 𝜎 ≡ = + 𝑃 . ℎ 4𝜋𝑟2 8𝜋𝐿2 𝑟3 (13) We give 𝑀 the physical meaning of mass for a spherical, ℎ 𝑃 + holographic screen with radius 𝑟ℎ. The basic idea is that gravi- From the above relation we see that 𝜎ℎ is a monotonically tational phenomena taking place in three-dimensional space decreasing function of the screen radius. We notice that there can be projected on a two-dimensional “viewing screen” with exists a minimal screen encoding the physically maximum no loss of information [39]. The idea of holographic screen attainable energy density, that is, the Planck (surface) density: hasbeenproposedin[40]andithasmathematicallybeenfor- mulated in [41]: the holographic screen plays the role of “basic 1 𝑀 𝜎 (𝑟 =𝐿 )= = 𝑃 . constituent of space where the Newton potential is constant.” ℎ + 𝑃 3 2 (14) 4𝜋𝐿𝑃 4𝜋𝐿𝑃 Along this line of reasoning, the idea of holographic screen has been used also in the context of noncommutative inspired We stress that there is no physically meaningful “interior” for metric to derive compelling deviations to Newton’s law [42]. the minimal screen; that is, the “volume” of such an object For what concerns the current discussion, however, we just is not even defined, in the sense that it can never be probed. need to recall that a special case of holographic screen is given Thus, we can only consider energy per unit area, rather than by an event horizon where the entropy is maximized. per unit volume. If we, formally, define a surface energy for a Several remarks are in order. particle as 𝑀≥𝑀 𝑀 1 (i)Itiseasytoshowthat 𝑃 and equals the Planck 𝜎 ≡ = 𝑟 =𝐿 𝑝 2 3 (15) mass only for ℎ 𝑃. 4𝜋𝜆𝐶 4𝜋𝜆𝐶 Advances in High Energy Physics 5

we see that the two curves (13)and(15)crossat𝜆𝐶 = 4 𝐿𝑃 =𝑟+. This result offers an additional interpretation for Sub-Planckian Trans-Planckian the Planck length which consistently turns to be the minimal regime regime size for a particle as well for a black hole (see Figure 2). Accordingly, the Planck density (14)isthecritical density for a 3 particle to collapse into a black hole. This argument is usually formulated in terms of volume energy density having in mind the picture of macroscopic body gravitationally collapsing under their own weight. From our holographic vantage point, C 2 where “surfaces” are the basic dynamical objects, it is natural 𝜆 Black holes to reformulate this reasoning in terms of areal densities Particle [39]. In addition holography offers a way to circumvent potential conflicts between the mechanism of spontaneous dimensional reduction [49, 50]andtheUVself-complete 1 paradigm. If we perform the limit for 𝑟→0the metric (10) wouldapparentlyreduceintoaneffectivetwo-dimensional spacetime: 0 𝑟2 1 2 3 4 𝑑𝑠2 󳨀→ − (1 − 2𝑀𝑟) 𝑑𝑡2 + (1 − 2𝑀𝑟)−1𝑑𝑟2 + O ( ). 2 M 𝐿𝑃 (16) Figure 2: The plot shows a length/energy relation consistent with the self-complete quantum gravity arguments in Planck units. As explained in [51], this mechanism would lead the for- Particles (dotted line) and black holes (solid line) cannot probe mation of lower dimensional black holes for length scales length shorter than the Planck length. The grey area is permanently below the Planck length, in contrast with the predicted inaccessible and accordingly represents the minimal spacetime time semiclassical regime of trans-Planckian black holes in four region or fundamental constituent, that is, the “atom” the spacetime dimensions. However, contrary to the Schwarzschild metric is supposed to be made of. that eventually reduces into dilaton gravity black holes when 𝑟≃𝐿𝑃 (for reviews of the mechanism see [52, 53]), the presence of the holographic screen forbids the access to 𝑟<𝐿 length scales 𝑃 and safely protects the arguments at the One can check that for large distances, that is, 𝑟+ ≫𝐿𝑃,both basis of the UV self-complete quantum gravity. (17)and(18) coincide with the conventional results of the 2 Schwarzschild metric; that is, 𝑇𝐻 ≈1/4𝜋𝑟+ and 𝐶≈−2𝜋𝑟+ 3. Thermodynamics, Area Quantization, and (see Figures 3 and 4). On the other hand at Planckian scales, Mass Spectrum contrary to the standard result for which a Planckian black hole has a temperature 𝑇𝐻 =𝑀𝑃/8𝜋,wehavethat𝑇𝐻 →0 𝑟 →𝑟 =𝐿 In this section we would like to investigate the thermody- as + 0 𝑃 as expected for any extremal configurations. namics of the black hole described by (10) and determine This discrepancy with the classical picture is consistent with the relation between entropy and area of the event horizon. the genuine quantum gravitational character of the black It is customary to consider the area law for granted in any holeandisreminiscentofthemodifiedthermodynamicsof case, but this assumption leads to an inconsistency with the noncommutative inspired black holes [54, 55]. third law of thermodynamics: extremal black holes have zero The Hawking emission is a semiclassical decay where temperature but nonvanishing area. Here, we stick to the gravity is considered just in terms of a classical spacetime textbook definition of thermodynamical entropy and not to background. Such a semiclassical approximation convention- more exotic quantity like Renyi,´ or entanglement entropy. To ally breaks down as the Planck scale is approached. On the √ √ cure this flaw, we will derive the relation between entropy and other hand for our metric, at 𝑟+ =𝑟𝑀 = 2+ 5𝐿𝑃 ≃ areafromthefirstlaw,ratherthanassumingit.TheHawking 2.058𝐿𝑃 the temperature admits a maximum corresponding temperature associated to the metric (10)canbecalculatedby to a pole in the heat capacity. In the final stage of the 𝜅 evaluating the surface gravity : evaporation, that is, 𝐿𝑃 <𝑟+ <𝑟𝑀, the heat capacity is positive; the Hawking emission slows down and switches off 2 𝜅 1 𝑑𝑔00 1 2𝐿𝑃 at 𝑟+ =𝐿𝑃.Fromanumericalestimateofthemaximum 𝑇𝐻 = = ( ) = (1 − ), (17) 2𝜋 4𝜋 𝑑𝑟 4𝜋𝑟 𝑟2 +𝐿2 temperature one finds 𝑇𝐻(𝑟𝑀) = 0.0239𝑀𝑃. This implies that 𝑟=𝑟+ + + 𝑃 the ratio temperature/mass is 𝑇𝐻/𝑀 <𝐻 𝑇 (𝑟𝑀)/𝑀0 ≃ 0.0239. As a consequence, no relevant back reaction occurs during while the heat capacity 𝐶≡𝜕𝑈/𝜕𝑇𝐻 is all the evaporation processes and the metric can consistently 2 2 2 describe the system “black hole + radiation” for all 𝑟+ ≥𝐿𝑃. 𝜕𝑀 𝑟2 −𝐿2 (𝑟 +𝐿 ) 𝐶≡ =−2𝜋𝑟 ( + 𝑃 ) + 𝑃 . (18) We can summarize the process with the following + 2 4 2 2 4 𝜕𝑇𝐻 𝐿𝑃 𝑟+ −4𝐿𝑃𝑟+ −𝐿𝑃 scheme: 6 Advances in High Energy Physics

0.04 the major approaches to quantum gravity, which universally foresee a logarithmic term as a correction to the classical area law. For brevity we recall that this is the case for string theory [57, 58], loop quantum gravity [59–61], and other results 0.03 based on generic arguments [62, 63], on Cardy’s formula [64], conformal properties of spacetimes [65], and other mechanisms for counting microstates [66–68]. We can check that this is the case for the metric (10) by performing the limit 𝑟+ ≫𝐿𝑃 for (20)toobtain H 0.02 T A A 𝑆(A )≈ + +𝜋 ( + ). + 2 ln 2 (21) 4𝐿𝑃 4𝜋𝐿𝑃 0.01 Conversely for 𝑟+ →𝐿𝑃 the entropy vanishes; that is,

4𝜋 2 𝑆 (A+) ≈ (𝑟+ −𝐿𝑃) +𝑂((𝑟+ −𝐿𝑃) ) . (22) 𝐿𝑃 0 012345This result is consistent both with the third law of thermodynamics and the entropy statistical meaning. The Planck r+ size, zero temperature, black hole configuration is the unique Figure 3: The solid curve represents the Hawking temperature 𝑇𝐻 ground state for holographic screens. Thus, it is a zero entropy and as a function of the horizon radius 𝑟+ in Planck units. The dotted state as there is only one way to realize this configuration. To curve represents the corresponding classical result in terms of the see this we promote the extremal configuration area to the Schwarzschild metric. fundamental quantum of area: 2 A+ ≡ A𝑛−1 =𝑛A0 =4𝜋𝑛𝐿𝑃, (23)

(i) “large”,far-from-extremality, black holes are semiclas- 2 sical objects which radiate thermally; where 𝐿𝑃 represents the basic information pixel and 𝑛= 1, 2, 3 . . . isthenumberofbytes(weborrowherethenames (ii) “small”, quasi-extremal, black holes are quantum of some units of digital information. In the present context, objects; 2 each byte consists of 4𝜋 bits. Each bit, represented by 𝐿 𝑟=𝑟 𝑃 (iii) 𝑀 is “critical point” where the heat capacity is the basic capacity of information of the holographic 𝐶>0 𝑟 <𝑟 <𝑟 diverges (see Figure 4). Since for 0 + 𝑀 screen. In the analogy with the theory of information for 𝐶<0 𝑟 <𝑟 and for 𝑀 +,weconcludethataphase which a byte represents the minimum amount of bits for transition takes place from large thermodynamically encoding a single character of text, here the byte represents unstable black holes to small stable black holes. 2 the minimum number of basic pixel 𝐿𝑃 for encoding the As a matter of fact, the black hole emission preceding the smallest holographic screen). From the above condition one evaporation switching off (often called “SCRAM phase” [11]) obtains might not be thermal. It has been argued that such a quantum 1/2 𝑟𝑛−1 ≡𝑛 𝐿𝑃, regime might be characterized by discrete jumps towards the 1 (24) ground state [7, 56]. To clarify the nature of this mechanism 𝑀 ≡ (𝑛1/2 +𝑛−1/2)𝑀 . we proceed by studying the black hole entropy profile and the 𝑛−1 2 𝑃 related area quantization. By integrating the first law, taking 𝑟 =𝐿 intoaccountthatnoblackholecanhavearadiussmallerthan Consistently the ground state of the system is 0 𝑃 and 𝑀 =𝑀 𝑛≫1 𝑟0 =𝐿𝑃,thatis, 0 𝑃, while for one finds a continuous spectrum ofvalues.Thiscanbecheckedthroughthefollowingrelation: 𝑟 + 𝑑𝑀 𝜋 𝑟 𝑆 (𝑟 ) = ∫ = (𝑟2 −𝐿2 ) +2𝜋 ( + ) , 1 + 2 + 𝑃 ln (19) −1/2 𝑟 𝑇 𝐿 𝐿 Δ𝑀 ≡𝑀 −𝑀 ∼ 𝑛 𝑀 . (25) 0 𝐻 𝑃 𝑃 𝑛 𝑛 𝑛−1 4 𝑃 we can cast the entropy in terms of the area of the event 𝑛≤4 2 We notice that for weareintheregimeofpositive horizon A+ ≡4𝜋𝑟+ as heat capacity 𝐶>0and discrete mass spectrum, while for 𝑛>4we approach the semiclassical limit characterized by A 𝜋 + negative heat capacity 𝐶<0and continuous mass spectrum; 𝑆(A+)= (A+ − A0)+𝜋ln ( ), (20) A0 A0 that is, Δ𝑀𝑛/𝑀𝑛 ≤1/12.Thisconfirmsthatat𝑟+ =𝑟𝑀, the system undergoes a phase transition from a semiclassical 2 where A0 =4𝜋𝐿𝑃 is the area of the extremal event horizon. regime to a genuine quantum gravity regime. As a conclusion We remark that the modifications to the Schwarzschild we have that large black holes decay thermally, while small metric, encoded in our model, are in agreement with all objects decay quantum mechanically, by emitting quanta of Advances in High Energy Physics 7

1000 energy scales one has just a quantum particle able to probe at the most distances of the order of its Compton wavelength. By increasing the degree of compression of the particle, onetraversesthePlanckscalewhereacollapseintoablack 500 hole occurs, before probing a semiclassical regime at trans- Planckian energies. The virtual curvature singularity of the geometry in 𝑟=0is therefore wiped out since in such a context sub-Planckian lengths have no physical meaning. C 0 From this vantage point spacetime stops to exist beyond the 1234567 Planck scale as there is no physical way to access this regime.

r+ Thus, the curvature singularity problem is ultimately resolved by giving up the very concept of spacetime at sub-Planckian length scales. −500 The study of the associated thermodynamic quantities confirmed that at trans-Planckian energies black holes radiate thermally before undergoing a phase transition to smaller, quantum black holes. The latter decay by emitting a discrete −1000 spectrum of quanta of energy and reach the ground state of the evaporation corresponding to the minimal holographic Figure 4: The solid curve represents the black hole heat capacity 𝐶 𝑟 screen. We came to this conclusion by quantizing the black as a function of the horizon radius + in Planck units. The dotted hole horizon area in terms of the minimal holographic screen curve represents the corresponding classical results in terms of the which actually plays the role of a basic information byte. We Schwarzschild metric. showed that in the thermodynamic limit, the area law for the black hole entropy acquires a logarithmic correction in agreement with all the major quantum gravity formulations. energy (for a recent phenomenological analysis of such kind In conclusion, we stress that the line element (10)notonly of decay see [69]). The end point of the decay is a Planck mass, captures the basic features of more “sophisticated” models of holographic screen. quantum gravity improved black holes (e.g., noncommutative The quantization of the area of the holographic screen lets geometry inspired black holes [11], loop quantum gravity us disclose further features of the informational content of the black holes [70, 71], asymptotically safe gravity black holes holographic screen. We have that the surface density can be [72, 73],andotherstudiesaboutcollapsesinquantumgravity written as [74, 75]) but overcomes some of their current weak points: 1 1 1 𝑀 𝜎 (𝑛) = ( + ) 𝑃 , specifically there is no longer any concern for potential ℎ 1/2 3/2 2 (26) 2 𝑛 𝑛 4𝜋𝐿𝑃 Cauchy instabilities or for conflicts between the gravity self- completeness and the Planck scale spontaneous dimensional while the entropy reads 𝑆(𝑛) = 𝜋(𝑛 + ln(𝑛) − 1).From reductionmechanism,aswellasthescenariooftheterminal thisrelationwelearnthat,whiletheentropyincreaseswith phase of the evaporation for static, nonrotating, neutral black the number 𝑛 of bytes, the surface density decreases. This holes. In addition, for its compact form the new metric allows confirms that the extremal configuration is nothing but straightforward analytic calculations and opens the route to a single byte, zero entropy, Planckian density holographic testable predictions. screen. Conflict of Interests 4. Discussion and Conclusions The authors declare that there is no conflict of interests In this paper we have presented a neutral nonspinning regarding the publication of this paper. black hole geometry admitting an extremal configuration whose mass and radius coincide with the Planck units. Wehavereachedthisgoalbysuitablymodellingastress References tensor able to accommodate both the particle and black hole [1] J. A. Wheeler, “On the nature of quantum geometrodynamics,” configurations, undergoing a transition at the Planck scale. Annals of Physics,vol.2,no.6,pp.604–614,1957. We showed that the horizon of the degenerate black hole represents the minimal holographic screen, within which [2] J. A. Wheeler, “Physics at the planck length,” International we cannot access any information about the matter-energy Journal of Modern Physics A,vol.8,pp.4013–4018,1993. contentofspacetime. [3] B. S. DeWitt, “Quantum theory of gravity. I. The canonical We showed that a generic holographic screen is described theory,” Physical Review, vol. 160, p. 1113, 1967. in terms of the outer horizon of the metric (10), while the [4] G. Dvali and C. Gomez, “Self-completeness of Einsteingravity,” inner horizon lies within the prohibited region, that is, inside http://arxiv.org/abs/1005.3497. the minimal holographic screen. The whole scheme fits into [5] L. J. Garay, “Quantum gravity and minimum length,” Interna- the gravity self-completeness scenario. For sub-Planckian tional Journal of Modern Physics A,vol.10,no.2,p.145,1995. 8 Advances in High Energy Physics

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Research Article Black Holes and Quantum Mechanics

B. G. Sidharth1,2 1 International Institute for Applicable Mathematics & Information Sciences, Udine, Italy 2 BMBirlaScienceCentre,AdarshNagar,Hyderabad500063,India

Correspondence should be addressed to B. G. Sidharth; [email protected]

Received 27 January 2014; Accepted 23 February 2014; Published 20 March 2014

Academic Editor: Christian Corda

Copyright © 2014 B. G. Sidharth. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP3.

We look at black holes from different, novel perspectives.

1. Introduction 2. Black Holes We will first show that black holes, generally, thought to be Our starting point is the observation that a Planck mass, −5 −33 a general relativistic phenomena could also be understood 10 gms at the Planck length 10 cms, satisfies2 ( )and without invoking general relativity at all. (Indeed, Laplace had as such a Schwarzschild black hole is. Rosen has used anticipated these objects.) nonrelativistic quantum theory to show that such a particle We start by defining a black hole as an object at the surface is a mini universe [2]. of which the escape velocity equals the maximum possible We next come to stellar scales. It is well known that for an velocity in the universe, namely, the velocity of light. We next electron gas in a highly dense mass we have [3, 4] use the well-known equation of Keplerian orbits [1] 4/3 2/3 2 𝑀 𝑀 𝑀 𝐾( − )=𝐾󸀠 , 1 𝐺𝑀 4 2 4 (3) = (1+𝑒 𝜃) , 𝑅 𝑅 𝑅 𝑟 𝐿2 cos (1) where 𝐿 𝑅𝑐 where ,theso-calledimpactparameter,isgivenby ,where 𝐾 27𝜋 ℏ𝑐 𝑅 is the point of closest approach, in our case a point on the ( )= ( )( )≈1040, 𝐾󸀠 64𝛼 𝛾𝑚2 (4) surface of the object, and 𝑐 is the velocity of approach, in our 𝑃 case the velocity of light. 9𝜋 𝑀 𝑅 Choosing 𝜃=0and 𝑒≈1,wecandeducefrom(1) 𝑀= , 𝑅= , (5) 8 𝑚𝑃 (ℏ/𝑚𝑒𝑐) 2𝐺𝑀 𝑀 𝑅 𝑚 𝑚 𝑅= . (2) is the mass, the radius of the body, 𝑃 and 𝑒 are 𝑐2 the proton and electron masses, and ℏ is the reduced Planck constant. From (3)and(4), it is easy to see that for 𝑀< 60 Equation (2) gives the Schwarzschild radius for a black hole 10 , there are highly condensed planet sized stars. (In fact these considerations lead to the Chandrasekhar limit in stellar and can be deduced from the full general relativity theory as 60 well. theory.) We can also verify that for 𝑀 approaching 10 36 We will now use (2) to exhibit black holes at three different corresponding to a mass ∼10 gms, or roughly a hundred to scales, the micro-, the macro-, and the cosmic scales. a thousand times the solar mass, the radius 𝑅 gets smaller and 2 Advances in High Energy Physics

8 smaller and would be ∼10 cms, so as to satisfy (2)andgivea Once such a minimum spacetime scale is invoked, then black hole in broad agreement with theory and observation. we have a noncommutative geometry as shown by Snyder [9, Finallyfortheuniverseasawhole,usingonlythetheory 10] of Newtonian gravitation, we had deduced [5] 𝚤𝑎2 𝚤𝑎2 [𝑥,𝑦]=( )𝐿𝑧, [𝑡, 𝑥] =( )𝑀𝑥, etc, 2𝐺𝑀 ℏ ℏ𝑐 𝑅∼ ; 2 (6) (9) 𝑐 2 𝑎 2 [𝑥,𝑥 𝑝 ]=𝚤ℏ[1+( ) 𝑝𝑥]. 28 ℏ that is, (2)wherethistime𝑅∼10 cms is the radius of 𝑀∼1055 the universe and gms is the mass of the universe. The relations (9) are compatible with special relativity. (6) can be deduced alternatively from general relativistic Indeed, such minimum spacetime models were studied considerations also as noted. for several decades, precisely to overcome the divergences Equation (6)isthesameas(2) and suggests that the encountered in quantum field theory4 [ , 10–13]. universe itself is a black hole. (This will still be true if there Allthisissymptomaticofthefactthatwecannotmeasure is dark matter.) arbitrary small intervals of spacetime in quantum theory, as It is remarkable that if we consider the universe to be a indeed argued by Dirac himself [14]. Indeed subsequently Schwarzschild black hole as suggested by (6),thetimetaken Salecker and Wigner argued that time within the Compton byarayoflighttotraversetheuniverse,thatis,fromthe −5 scale has no physical meaning [15] (and for a detailed horizon to the singularity, namely, 10 (𝑀/𝑀0),equalsthe 17 discussion cf. [16]). Indeed this quantum mechanical feature age of the universe ∼10 secs as shown elsewhere [5]. 𝑀0 is explainswhatMisneretal.termedthegreatestcrisisof the mass of the sum. We will deduce this result alternatively physics [8], namely, the singularity of the black hole. All this alittlelater. has been the matter of detailed study (cf. [16]).

3. Micro Black Holes 4. Black Hole Thermodynamics Attempts have been made to express elementary particles The author has approached this problem from the point of as tiny black holes by several authors, notably, Markov and view of oscillations at the Planck scale [16]. Briefly, if there Recami [6, 7]. These black holes do not reproduce charge or are 𝑁 such oscillators with an amplitude Δ𝑥,thenwehave spin which are so essential. Letus,instead,observethatifwetreatanelectronasa 𝑅=√𝑁Δ𝑥2. (10) Kerr-Newman black hole, then we get the correct quantum mechanical 𝑔=2factor, but the horizon of the black hole This leads to becomes complex [4, 8]. Consider 𝑚 𝑅=√𝑁𝑙 ,𝑀=𝑃 , 𝑃 √𝑁 (11) 2 2 2 1/2 𝐺𝑀 𝐺 𝑀 𝐺𝑄 2 𝑟 = +𝚤𝑏, 𝑏≡( − −𝑎 ) (7) 𝑀 𝑅 𝑙 𝑚 + 𝑐2 𝑐4 𝑐4 where is the arbitrary mass, the extent, and 𝑃 and 𝑃 are the Planck length and Planck mass, respectively. We now use the fact that 𝑙𝑃 is the Schwarzschild radius of the Planck mass 𝐺 𝑀 with being the gravitational constant, being the mass, as was shown by Rosen [2]. Substitution in the above gives us 𝑎 ≡ 𝐿/𝑀𝑐 𝐿 and , being the angular momentum. While the Schwarzschild radius; that is (4) (7) exhibits a naked singularity and as such has no physical meaning, we note that from the realm of quantum mechanics 2𝐺𝑀 𝑅= . (12) the position coordinate for a Dirac particle is given by 𝑐2

𝚤 Itcanbeimmediatelyseenfrom(11)that 𝑥=(𝑐2𝑝 𝐻−1𝑡) + 𝑐ℏ (𝛼 −𝑐𝑝 𝐻−1)𝐻−1 1 2 1 1 (8) 𝑅𝑀𝑃 =𝑙 𝑚𝑃. (13) an expression that is very similar to (7). In the above, the It must be mentioned that the above is completely consistent various symbols have their usual meaning. In fact as was with the mass and radius of an arbitrary black hole, including argued in detail [4], the imaginary parts of both (7)and(8) theuniverseitself. are the same, being of the order of the Compton wavelength. From the theory of black hole thermodynamics we have It is at this stage that a proper physical interpretation as it is well known [17] begins to emerge. Dirac himself observed that to interpret (8)meaningfullyitmustberememberedthatquantum ℏ𝑐3 𝑇= , (14) mechanical measurements (unlike classical ones) are really 8𝜋𝑘𝑚𝐺 averaged over the Compton scale. Within the scale there are theunphysicalZitterbewegungeffects:forapointelectronthe namely, the Beckenstein temperature. Interestingly, (14)can velocity equals that of light. be deduced alternatively from our above theory of oscillations Advances in High Energy Physics 3 at the Planck scale. For this we use the following relations for the author deduced in 1995 [20, 21], this behaviour in two a Schwarzschild black hole [17] dimensions is given by

→ → 𝑘𝑐 𝜎 𝑑𝑀 = 𝑇𝑑𝑆, 𝑆= 𝐴, (15) ]𝐹 ⋅ ∇ 𝜓 (𝑟) =𝐸𝜓(𝑟) , (20) 4ℏ𝐺 ] ∼106 / 𝑐 𝑇 𝑆 𝐴 where 𝐹 m sistheFermivelocityreplacing ,the where is the Bekenstein temperature, the entropy, and 𝜓(𝑟) √ velocity of light, and is a two-component wave function, is the area of the black hole. In our case, the mass 𝑀= 𝑁𝑚𝑃 → 𝐴=𝑁𝑙2 𝑁 𝜎 and 𝐸, denoting the Pauli matrices and energy. and 𝑃,where is arbitrary for an arbitrary black hole. ] This follows from11 ( ). Whence, Though this resembles the neutrino equation, 𝐹 is some three hundred times less than the velocity of light. However 𝑑𝑀 4ℏ𝐺 𝑑𝑀 the author has argued that for a sufficiently large sheet of 𝑇= = . 2 (16) graphene, this would approximate the neutrino equation 𝑑𝑆 𝑘𝑙𝑃𝑐 𝑑𝑁 itself, that is, the usual Minkowski spacetime. From this point If we use the fact that 𝑙𝑃 is the Schwarzschild radius for the ofview,ablackholecanbesimulatedbya“grapheneball.” Planck mass 𝑚𝑃 andusetheexpressionfor𝑀,theabove ItmaybementionedthatveryrecentlyHawkinghas reduces to (14), the Bekenstein formula. proposed rather shockingly that black holes may not have Equation (14) gives also the thermodynamic temperature event horizons [22]. of a Planck mass black hole. Further, in this theory as it is known [17], Conflict of Interests 𝑑𝑀 𝛽 =− , (17) The author declares that there is no conflict of interests 𝑑𝑡 𝑀2 regarding the publication of this paper. with 𝑀 being the mass. Before proceeding, we observe that we have deduced a string of 𝑁 Planck oscillators, 𝑁 arbitrary, References √𝑁𝑚 =𝑀 form a Schwarzschild black hole of mass 𝑃 .We [1] H. Goldstein, Classical Mechanics, Addison-Wesley Press, Read- can now deduce that ing, Mass, USA, 1951. 𝑑𝑀 𝑚 [2] N. Rosen, “Quantum mechanics of a miniuniverse,” Interna- = 𝑃 , 𝑑𝑡 𝑡 tional Journal of Theoretical Physics,vol.32,no.8,pp.1435–1440, 𝑃 1993. (18) 𝑚𝑃 [3] K. Huang, Statistical Mechanics, John Wiley & Sons, New York, 𝑀= ( )⋅𝑡, NY, USA, 2nd edition, 1987. 𝑡𝑃 [4]B.G.Sidharth,Chaotic Universe: From the Planck to the Hubble where 𝑡 is the “Hawking-Bekenstein decay time.” For the Scale, Nova Science, New York, NY, USA, 2001. [5] B. G. Sidharth, “‘Fluctuational cosmology’ in quantum mechan- Planck mass, 𝑀=𝑚𝑃, the decay time is the Planck time ics and general relativity,” in Proceeding of the 8th Marcell 𝑡=𝑡𝑃. For the universe, the above gives the life time 𝑡 as ∼ 17 Grossmann Meeting on General Relativity,T.Piran,Ed.,World 10 sec,theageoftheuniverseagain. Scientific, Singapore, 1999. Further, we have also seen the emergence of the quantum [6] M. A. Markov, Soviet Physics, JETP,vol.24,no.3,p.584,1967. of area [18]asitisevidentfromthe𝑁 elementary Planck areas 𝑙2 [7]M.A.Markov,Commemoration Issue for the 30th Anniversary 𝑃 fortheblackhole(cf.also[18]). of the Meson Theory by Dr. H. Yukawa,Suppl.ofProgressofTh. It has also been argued that not only does the universe Phys, 1965. mimic a black hole but also the black hole is a two dimen- [8] C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation,W. sional object [16, 19]. Indeed, the interior of a black hole is in H. Freeman, San Francisco, Calif, USA, 1973. any case inaccessible and the two dimensions follow from the [9] H. S. Snyder, “The electromagnetic field in quantized space- area of the black hole which plays a central role in black hole time,” Physical Review,vol.72,pp.68–71,1947. thermodynamics. We have already seen that the area of the [10] H. S. Snyder, “Quantized space-time,” Physical Review,vol.71, black hole is given by pp. 38–41, 1947. [11] D. R. Finkelstein, Quantum Relativity A Synthesis of the Ideas 𝐴=𝑁𝑙2 . 𝑝 (19) of Einstein and Heisenberg, Texts and Monographs in Physics, Springer, Berlin, Germany, 1996. For these quantum gravity considerations, we have to deal [12] C. Wolf, HadronicJournal,vol.13,pp.22–29,1990. with the quantum of area [16, 18]. In other words, we have [13] T. D. Lee, “Can time be a discrete dynamical variable? ” Physics to consider the black hole to be made up of 𝑁 quanta of area. Letters, vol. 122B, no. 3-4, pp. 217–220, 1983. It is remarkable that we can get an opportunity to test these [14] P.A. M. Dirac, The Principles of Quantum Mechanics,Clarendon quantum gravity features in two-dimensional surfaces such Press, Oxford, UK, 3rd edition, 1947. as graphene. [15] H. Salecker and E. P. Wigner, “Quantum limitations of the That is, we could model a black hole as a “graphene” ball. measurement of space-time distances,” Physical Review,vol.109, Indeed, in the case of graphene as it is well known, and as pp. 571–577, 1958. 4 Advances in High Energy Physics

[16] B. G. Sidharth, The Thermodynamic Universe, World Scientific, Singapore, 2008. [17] R. Runi and L. Z. Zang, BasicConceptsinRelativisticAstro- Physics, World Scientific, Singapore, 1983. [18] J. Baez, “Quantum gravity: the quantum of area?” Nature,vol. 421, no. 6924, pp. 702–703, 2003. [19] B. G. Sidharth, “Black-hole thermodynamics and electromag- netism,” Foundations of Physics Letters,vol.19,no.1,pp.87–94, 2006. [20] B. G. Sidharth, A Note on Two Dimensional Fermions in BSC- CAMCS-TR-95-04-01, 1995. [21] B. G. Sidharth, “Anomalous fermions,” Journal of Statistical Physics,vol.95,no.3-4,pp.775–784,1999. [22] S. W. Hawking, “Information preservation and weather forecasting for black holes,” http://arxiv.org/abs/1401.5761 . Hindawi Publishing Corporation Advances in High Energy Physics Volume 2014, Article ID 204016, 7 pages http://dx.doi.org/10.1155/2014/204016

Research Article Particle Collisions in the Lower Dimensional Rotating Black Hole Space-Time with the Cosmological Constant

Jie Yang,1 Yun-Liang Li,1 Yang Li,2 Shao-Wen Wei,1 and Yu-Xiao Liu1

1 Institute of Theoretical Physics, Lanzhou University, Lanzhou 730000, China 2 Department of Physics, Beijing Normal University, Beijing 100875, China

Correspondence should be addressed to Jie Yang; [email protected]

Received 3 January 2014; Accepted 14 February 2014; Published 17 March 2014

Academic Editor: Xiaoxiong Zeng

Copyright © 2014 Jie Yang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP3.

We study the effect of ultrahigh energy collisions of two particles with different energies near the horizon ofa 2+1dimensional BTZ black hole (BSW effect). We find that the particle with the critical angular momentum could exist inside the outer horizon of the BTZ black hole regardless of the particle energy. Therefore, for the nonextremal BTZ black hole, the BSW process is possible on the inner horizon with the fine tuning of parameters which are characterized by the motion of particle, while, for the extremal BTZ black hole, the particle with the critical angular momentum could only exist on the degenerated horizon, and the BSW process could also happen there.

1. Introduction However, all of the works mentioned above have been focused on the black holes embedded in the asymptotically In the recent paper [1], Banados,˜ Silk, and West proposed a flat space-time without cosmological constant. In our pre- mechanism (BSW process) that two particles may collide on vious work [30], we had considered the BSW process in the horizon of an extremal Kerr black hole with ultrahigh the background of the Kerr- (anti-) de Sitter black hole center-of-mass (CM) energy, although it was pointed out with nonzero cosmological constant and had found that the in [2, 3]thatthecollisioninfacttakesaninfiniteproper cosmological constant has an important effect on the result. time. Moreover, there are astrophysical limitations preventing For the case of the general Kerr-de Sitter black hole (with a Kerr black hole from being an extreme one, and the gravitational radiation and backreaction effects should also positive cosmological constant), the collision of two particles be included in this process. Due to the potential interest can take place on the outer horizon of the nonextremal in exploring ultrahigh energy physics, the BSW process has blackholeandtheCMenergyofcollisioncanblowup been studied extensively in other kinds of black holes or arbitrarily if one of the colliding particles has the critical naked singularities [4–29]. To achieve ultrahigh CM energy angular momentum. In the present paper, we extend the under the astrophysical limitation of maximal spin, the investigation of the BSW process to the background of a 2 multiple scattering was taken into account in the nonextremal + 1 dimensional BTZ black hole [31], and our motivation is to Kerr black hole [7, 15]. Another more direct application is to examine whether the BSW effect remains valid in the lower consider different extreme rotating black holes, such as the dimensional case. Actually, in [5, 6], Lake had pointed out Kerr-Newman black holes and the Sen black hole [8, 11]. On the divergence of the CM energy of particle collision on the the other hand, a general explanation of this BSW process inner horizon of the BTZ black hole, but the process was not was tried to give for a rotating black hole [19]andforother discussed in detail. In this paper, we study this process in the black holes [20, 21].Someeffortshadalsobeenmadetodraw BTZ black hole with circumstances. some implications concerning the effects of gravity generated This paper is organized as follows. In Section 2,wegivea by colliding particles in [23]. brief review of the BTZ black hole. In Section 3,westudythe 2 Advances in High Energy Physics

CM energy of the particle collision on the horizon and derive 𝑃𝑡 and 𝑃𝜙 are constants of motion. In fact, they correspond the critical angular momentum to blow up the CM energy. to the test particle’s energy per unit mass 𝐸 and the angular In Section 4,weinvestigatetheradialmotionofcolliding momentum parallel to the symmetry axis per unit mass 𝐿, particles with the critical angular momentum in detail. The respectively. And in the following discussion we will just extremal and nonextremal cases are examined, respectively. regard these two constants of motion as −𝐸 ≡𝑡 𝑃 and 𝐿≡𝑃𝜙 The conclusion is given in the last section. [24]. The affine parameter 𝜆 can be related to the proper time by 𝜏=𝜇𝜆,where𝜏 is given by the normalization condition 2. The 2 + 1 Dimensional BTZ Black Hole 2 𝜇 ] 2 −𝜇 =𝑔𝜇]𝑥̇𝑥̇with 𝜇 =1for time-like geodesics and 2 In this section we would like to study the horizon structure of 𝜇 =0for null geodesics. For a time-like geodesic, the affine the 2 + 1 dimensional BTZ black hole. The metric of the BTZ parameter can be identified with the proper time, and thus, blackholeisusuallywrittenas[31](withunits𝑐=𝐺=1) from (7), we can solve the 2 + 1 dimensional “4-velocity” ̇ ̇ 2 components 𝑡 and 𝜙 (where the dot denotes a derivative with 𝑑𝑠2 =−𝑁2𝑑𝑡2 +𝑁−2𝑑𝑟2 +𝑟2(𝑁 𝑑𝑡 +) 𝑑𝜙 𝑟 𝑟 𝜙 (1) respect to the proper time now) as

2 with 𝑑𝑡 2𝐸 − (𝐽𝐿/𝑟 ) = , 𝑟2 𝐽2 𝑑𝜏 2𝑁2 𝑁2 (𝑟) =−𝑀+ + , 𝑟 𝑟 𝑙2 4𝑟2 (8) (2) 𝑑𝜙 𝐽(−𝐽𝐿+2𝐸𝑟2)+4𝐿𝑟2𝑁2 𝐽 = 𝑟 . 𝑁𝜙 (𝑟) =− , 4 2 2𝑟2 𝑑𝜏 4𝑟 𝑁𝑟 where 𝑀 and 𝐽 are the mass and spin angular momentum For the remaining component 𝑟̇ = 𝑑𝑟/𝑑𝜏 of the radial motion, 2 of the black hole, respectively, and 𝑙 is related to the wecanobtainitfromtheHamilton-Jacobiequationofthe −2 cosmological constant Λ by 𝑙 =−Λ. time-like geodesic: Thehorizonscanbesolvedfrom𝑁𝑟|𝑟=𝑟 =0,andtheyare ℎ 𝜕𝑆 1 𝜇] 𝜕𝑆 𝜕𝑆 given by =− 𝑔 (9) 𝜕𝜏 2 𝜕𝑥𝜇 𝜕𝑥] 𝑙 𝑟 = √ (𝑙𝑀 ± √𝑙2𝑀2 −𝐽2). (3) with the ansatz ± 2 1 𝑆= 𝜏−𝐸𝑡+𝐿𝜙+𝑆𝑟 (𝑟) , (10) Here, 𝑟+ is the outer horizon and 𝑟− is the inner horizon. The 2 existence of the horizon requires where 𝑆𝑟(𝑟) is a function of 𝑟. Inserting the ansatz into (9), |𝐽| ≤𝑀𝑙. (4) with the help of the metric (1), we get 2 2 2 2 2 2 2 2 2 The horizon of the extremal black hole (corresponding to 𝑑𝑆 (𝑟) 2 𝐽 𝐿 −4𝐸𝐽𝐿𝑟 −4𝑟 [𝐿 𝑁 +(−𝐸 +𝑁 )𝑟 ] ( 𝑟 ) = 𝑟 𝑟 . |𝐽| = 𝑀𝑙)isreadas 4 4 𝑑𝑟 4𝑁𝑟 𝑟 𝑀 (11) 𝑟 = √ 𝑙. (5) 𝑒 2 On the other hand, we have 𝑑𝑆 (𝑟) 𝑟̇ 3. The Center-of-Mass Energy for 𝑟 =𝑃 =𝑔 𝑟=̇ . 𝑑𝑟 𝑟 𝑟𝑟 𝑁2 (12) the On-Horizon Collision 𝑟 To investigate the CM energy of the collision on the horizon Thus we get the square of the 4-velocity radial component: of the BTZ black hole, we have to derive the 2 + 1 dimensional 2 2 2 2 2 2 “4-velocity” component of the colliding particle in the back- 𝑑𝑟 𝐾 −4𝑟 𝑁𝑟 (𝐿 +𝑟 ) ( ) = , (13) ground of the 2 + 1 dimensional BTZ black hole. 𝑑𝜏 4𝑟4 The generalized momentum 𝑃𝜇 is where 𝑃 =𝑔 𝑥̇], 𝜇 𝜇] (6) 2 𝐾=𝐽𝐿−2𝐸𝑟. (14) where the dot denotes the derivative with respect to the affine parameter 𝜆 and 𝜇, ] =𝑡,𝑟,𝜙.Thus,thecomponents𝑃𝑡 and Here we have obtained all nonzero 2 + 1 dimensional “4- 𝑃𝜙 of the momentum are turned out to be velocity” components for the geodesic equation. Next we wouldliketostudytheCMenergyofthetwo-particle ̇ ̇ 𝑃𝑡 =𝑔𝑡𝑡𝑡+𝑔𝑡𝜙𝜙, collision in the background of the BTZ black hole. Here (7) we consider a more general case that the two colliding ̇ ̇ 𝑃𝜙 =𝑔𝜙𝜙𝜙+𝑔𝑡𝜙𝑡. particles have different energies 𝐸1 and 𝐸2 and different Advances in High Energy Physics 3

angularmomentaperunitmass𝐿1 and 𝐿2. For simplicity, the where 𝑚 particles under consideration have the same rest mass 0.We 𝐴=𝐽2 [(𝐿 −𝐿 )2 −(𝐸 −𝐸 )2𝑙2 𝐸 1 2 1 2 can compute the CM energy CM of this two-particle collision by using −2(𝐿1 −𝐿2)(𝐿𝐶1 −𝐿𝐶2)] (21) 𝐸 = √2𝑚 √1−𝑔 𝑢𝜇𝑢], +2𝑙[(𝐸 𝐿 −𝐸 𝐿 )2 +(𝐸 −𝐸 )2𝑙2𝑀] CM 0 𝜇] 1 2 (15) 2 1 1 2 1 2

𝜇 ] ×(𝑙𝑀+√𝑙2𝑀2 −𝐽2). where 𝑢1 and 𝑢2 are the “4-velocity” vectors of the two 𝑢=(𝑡,̇𝑟,̇𝜙)̇ particles ( ). With the help of (8)and(13), we 𝐾 =0 obtain the CM energy: Soitcanbeseenthatwhen 𝑖 , the CM energy on the horizon will blow up. Solving 𝐾𝑖 =0,wegetthecritical angular momentum: 𝐸2 1 CM = [(𝐽𝐿 −2𝐸 𝑟2)(𝐽𝐿 −2𝐸 𝑟2) 2𝑚2 4𝑟4𝑁2 1 1 2 2 2𝑟2𝐸 𝐸 𝑙(𝑙𝑀+√𝑙2𝑀2 −𝐽2) 0 𝑟 (16) 𝐿 = ℎ 𝑖 = 𝑖 ,𝑖=1,2. 𝐶𝑖 𝐽 𝐽 +4𝑟2𝑁2 (−𝐿 𝐿 +𝑟2)−𝐻𝐻 ], 𝑟 1 2 1 2 (22) 𝐾 =0 𝐾 =0 where It is easy to prove that when 1 and 2 ,theCM energy is finite. So in order to obtain an arbitrarily high CM energy,oneandonlyoneofthecollidingparticlesshouldhave 𝐻 = √(𝐽𝐿 −2𝐸𝑟2)2 −4𝑟2𝑁2 (𝐿2 +𝑟2) 𝑖 𝑖 𝑖 𝑟 𝑖 the critical angular momentum. For the extremal BTZ black (17) 2 𝐽=𝑙𝑀 𝐸 (𝑖=1,2) . hole ,the CM on the extremal horizon is 2 𝐸 (𝑟 󳨀→ 𝑟 ) 2 CM 𝑒 𝐸 ≡ For simplicity, we can rescale the CM energy as CM 2 2 2 (1/2𝑚2)𝐸2 𝐸 𝑀[(𝐿1 −𝐸1𝑙) − (𝐿2 −𝐸2𝑙)] +2(𝐸2𝐿1 −𝐸1𝐿2) 0 CM. We would like to study CM for the case that =2+ . the particles collide on the black hole’s horizon, which means 2𝑀 (𝐿1 −𝐸1𝑙) (𝐿2 −𝐸2𝑙) 𝑁𝑟 =0. The denominator of the expression on the right hand (23) of (16)iszero,andthenumeratorofitis Obviously,whenoneparticlehasthecriticalangularmomen- 𝐿 =𝐸𝑙 𝐿 =𝐸𝑙 tum C1 1 (or C2 2 ) and the other does not, the CM √ 2√ 2 𝐾1𝐾2 − 𝐾1 𝐾2 , energy on the extremal horizon could be infinite. (18) From the above derivation, it seems that the CM energy 𝐾 =𝐾| ,𝑖=1,2. 𝑖 𝐸=𝐸𝑖,𝐿=𝐿𝑖 couldblowuponthehorizon.However,inordertoget arbitrarily high CM energy on the horizon of the BTZ When 𝐾1𝐾2 ≥0, the numerator will be zero and the value of black hole, the colliding particle with the critical angular 2 𝐸 𝐾 𝐾 < momentum must be able to reach the horizon of the black CM on the horizon will be undetermined, but when 1 2 2 hole. We will investigate this part in the next section. 0 𝐸 , the numerator will be a negative finite value and CM on the horizon will be negative infinity. So it should have 𝐾1𝐾2 ≥0, and, for the CM energy on the horizon, we have to compute 4. The Radial Motion of the the limiting value of (16)as𝑟→𝑟ℎ,where𝑟ℎ is the horizon Particle with the Critical Angular of the black hole. Momentum near the Horizon After some calculations, we get the limiting value of(16): In this section, we will study the radial motion of the particle 2 𝐸 (𝑟 󳨀→ 𝑟 ) with the critical angular momentum and find the region CM ℎ where it can exist. In order for a particle to reach the (𝐿 −𝐿 )2 −𝑙2(𝐸 −𝐸 )2 −2(𝐿 −𝐿 )(𝐿 −𝐿 ) horizon of the black hole, the square of the radial component =2+ 1 2 1 2 1 2 𝐶1 𝐶2 (𝑑𝑟/𝑑𝜏)2 2(𝐿 −𝐿 )(𝐿 −𝐿 ) of the “4-velocity” in (13)hastobepositivein 1 𝐶1 2 𝐶2 the neighborhood of the black hole’s horizon. Obviously, 2 2 2 𝑅(𝑟)| =0 𝑙[(𝐸 𝐿 −𝐸 𝐿 ) +𝑀𝑙 (𝐸 −𝐸 ) ] (𝑙𝑀 + √𝑙2𝑀2 −𝐽2) 𝐿=𝐿𝐶𝑖 on the horizon of the BTZ black hole. For a + 2 1 1 2 1 2 , particle with arbitrary energy 𝐸 and angular momentum 𝐿, 𝐽2 (𝐿 −𝐿 )(𝐿 −𝐿 ) 2 1 𝐶1 2 𝐶2 the explicit form of (𝑑𝑟/𝑑𝜏) , which is denoted by 𝑅(𝑟),reads (19) 𝑑𝑟 2 𝐿2 𝑅 (𝑟) ≡( ) =𝐸2 − +𝑀 which can also be rewritten as 𝑑𝜏 𝑙2 2 2 (24) 2 𝐴 1 𝐽 𝑟 𝐸 (𝑟 󳨀→ 𝑟 )=2+ , + (𝐿2𝑀−𝐸𝐽𝐿− )− . CM ℎ (20) 2 2 2𝐾1𝐾2 𝑟 4 𝑙 4 Advances in High Energy Physics

1 600

400 50 100 150 r −1 200 R(r)

R(r) −2 10 20 30 40 50 −3 −200 r

−4 −400

(a) (b)

2 2 2 2 Figure 1: Behaviors of 𝑅(𝑟) for (a) 𝐿 𝑀−𝐸𝐽𝐿−(𝐽 /4) > 0 and (b) 𝐿 𝑀−𝐸𝐽𝐿−(𝐽 /4) < 0.

2 We draw 𝑅(𝑟) in Figure 1.Itcanbeseenthat,when𝐿 𝑀− When 𝐸>𝐸0, 𝑅(𝑟) =0 has one root 2 𝐸𝐽𝐿−𝐽 /4 > 0, 𝑅(𝑟 → 0) → +∞ and 𝑅(𝑟 → +∞), →−∞ so there is only one positive root for 𝑅(𝑟) =0 and the particle 2 1 can exist in the region inside of the root. When 𝐿 𝑀−𝐸𝐽𝐿− 𝑟 = {𝑙2 [𝐽2𝑀+2𝐸2 2 0 𝐽 /4 < 0, 𝑅(𝑟 → 0) → −∞ and 𝑅(𝑟 → +∞),and →−∞ √2𝐽 there are two positive roots and the particle can exist in the 2 region between the two roots. The bigger root of 𝑅(𝑟) =0 is ×(𝐽 −𝑙𝑀(𝑙𝑀+√𝑙2𝑀2 −𝐽2))]

2 2 2 𝑟2 =(𝑙 (𝐸 +𝑀)−𝐿 +𝑙[(𝑙2𝑀2 −𝐽2)

1/2 2 2 +√[𝑙2𝑀−𝐽𝑙+(𝐿−𝐸𝑙) ][𝑙2𝑀+𝐽𝑙+(𝐿+𝐸𝑙) ]) ×[𝐽4 +8𝐸4𝑙3𝑀(𝑙𝑀+√𝑙2𝑀2 −𝐽2)

√ −1 ×( 2) . +4𝐸2𝐽2𝑙(𝑙𝑀−𝐸2𝑙 (25) 1/2 1/2 We find that it increases with 𝐸 and 𝐿,whichmeansthatthe +√𝑙2𝑀2 −𝐽2)]] } particle can move arbitrarily far from black hole’s horizon with its energy and angular momentum’s increase. (29) Next, we will study the radial motion of the particle with the critical angular momentum: and the particle with the critical angular momentum can exist 2 𝑊 𝑟 2 inside of it. When 𝐸<𝐸0, 𝑅(𝑟) =0 has two roots 𝑅 (𝑟) |𝐿=𝐿 = − +2𝐸 +𝑀 c 𝑟2 𝑙2 (26) 2𝐸2𝑙2𝑀2 +2𝐸2𝑙𝑀√𝑙2𝑀2 −𝐽2 1 − , 𝑟 = {𝑙2 [𝐽2𝑀+2𝐸2 𝐽2 0+ √2𝐽 where ×(𝐽2 −𝑙𝑀(𝑙𝑀+√+𝑙2𝑀2 −𝐽2))] 𝐸2𝑙[2𝑙𝑀(𝑙2𝑀2 −𝐽2) + (2𝑙2𝑀2 −𝐽2) √𝑙2𝑀2 −𝐽2] 𝑊= 𝐽2 +𝑙[(𝑙2𝑀2 −𝐽2) 𝐽2 − . 4 4 3 √ 2 2 2 4 ×(𝐽 +8𝐸 𝑙 𝑀(𝑙𝑀+ 𝑙 𝑀 −𝐽 ) (27) +4𝐸2𝐽2𝑙(𝑙𝑀−𝐸2𝑙 By solving 𝑊=0we get the critical energy 𝐸0: 𝐽2 1/2 1/2 2 2 2 𝐸0 = . +√𝑙 𝑀 −𝐽 ))] } , 2√2𝑙2𝑀(𝑙2𝑀2 −𝐽2)+(2𝑙3𝑀2 −𝐽2𝑙) √𝑙2𝑀2 −𝐽2 (28) (30) Advances in High Energy Physics 5

2.0 0.5 1.5

1.0 2 4 6 8 10 12 r −0.5 R(r)

0.5 R(r) −1.0 2 4 6 8 10 12 −0.5 r −1.5

(a) (b)

Figure 2: The variation of 𝑅(𝑟) versus radius 𝑟 for the case of the nonextremal BTZ black hole (𝑙=10, 𝑀=1,and𝐽=1), 𝐸>𝐸0 ((a) 𝐸 = 0.003)and𝐸<𝐸0 ((b) 𝐸 = 0.002). The vertical lines denote the locations of the outer and inner horizons.

1 2 2 2 𝑟0− = {𝑙 [𝐽 𝑀+2𝐸 0.5 1.0 1.5 2.0 √2𝐽 −1 r

−2 ×(𝐽2 −𝑙𝑀(𝑙𝑀+√+𝑙2𝑀2 −𝐽2))] −3

−𝑙[(𝑙2𝑀2 −𝐽2) R(r) −4 −5 ×(𝐽4 +8𝐸4𝑙3𝑀(𝑙𝑀+√𝑙2𝑀2 −𝐽2) −6

+4𝐸2𝐽2𝑙(𝑙𝑀−𝐸2𝑙 Figure 3: The variation of 𝑅(𝑟) versus radius 𝑟 for the case of the extremal BTZ black hole (𝑙=1, 𝑀=1,and𝐽=1). The vertical line denotes the locations of the degenerated horizon. 1/2 1/2 +√𝑙2𝑀2 −𝐽2))] } ,

(31) 4.2. Extremal BTZ Black Hole. For the extremal BTZ black hole case, 𝑅(𝑟) for particle with the critical angular momen- and the particle with the critical angular momentum can tum becomes very simple: exist between them. The above discussion only concerns the square of the ”4-velocity” radial component. To find whether 𝑟2 𝐽2 𝑅 (𝑟) =𝑀− − . (33) the particle with the critical angular momentum can reach 𝑙2 4𝑟2 the horizon of the BTZ black hole, we should investigate 𝑅(𝑟) =0 the roots of and the horizons of the black hole. We solve 𝑅(𝑟) =0 and get The nonextremal and extremal cases will be considered in the following. 𝑀 𝑟 = √ 𝑙. (34) 0 2 4.1. Nonextremal BTZ Black Hole. For the nonextremal BTZ black hole case, we can prove that the solution (for 𝐸>𝐸0 𝐸<𝐸 𝑅(𝑟) =0 It is just the degenerated horizon of the extremal black hole. case) or the bigger solution (for 0 case) of is 𝑅(𝑟) just the outer horizon of black hole: The behaviors of for the particle with the critical angular momentum are plotted in Figure 2 for the nonextremal black hole and Figure 3 for the extremal black hole. 𝑙 For the nonextremal black hole, we find that the particle with 𝑟 =𝑟 = √ (𝑙𝑀 + √𝑙2𝑀2 −𝐽2). (32) 0 + 2 the critical angular momentum can exist inside the outer horizon. So particle collision on the inner horizon could produce unlimited CM energy. For the extremal black hole, That means that the particle with the critical angular momen- theparticlewiththecriticalangularmomentumcouldonly tum can exist inside the outer horizon of the nonextremal exist on the degenerated horizon, so if such particle exists, BTZ black hole. thenunlimitedCMenergywillbeapproached. 6 Advances in High Energy Physics

5. Conclusion [11] S. W.Wei, Y. X. Liu, H. T. Li, and F. W.Chen, “Particle collisions on stringy black hole background,” Journal of High Energy Inthiswork,wehaveanalyzedthepossibilitythatthe2+1 Physics,vol.12,article066,2010. dimensional BTZ black holes can serve as particle acceler- [12] P. J. Mao, L. Y. Jia, J. R. Ren, and R. Li, “Accelera- ator. We first calculate the CM energy for the two-particle tion of particles in Einstein-Maxwell-Dilaton black hole,” collision. In order to obtain unlimited CM energy, one of http://arxiv.org/abs/1008.2660 . the particles should have the critical angular momentum. [13] T. Harada and M. Kimura, “Collision of an innermost stable Next,westudytheradialmotionfortheparticlewiththe circular orbit particle around a Kerr black hole,” Physical Review critical angular momentum. For the extremal BTZ black hole, D,vol.83,no.2,ArticleID024002,2011. particles with critical angular momentum can only exist on [14] T. Harada and M. Kimura, “Collision of two general geodesic the outer horizon of the BTZ black hole. So if such particle particles around a Kerr black hole,” Physical Review D,vol.83, exists, then unlimited CM energy will be approached. For no. 8, Article ID 084041, 2011. the nonextremal BTZ black hole, particles can collide on the [15]A.A.GribandY.V.Pavlov,“Onparticlecollisionsnearrotating inner horizon with arbitrarily high CM energy. black holes,” Gravitation and Cosmology,vol.17,no.1,pp.42–46, 2011. Conflict of Interests [16] M. Banados,˜ B. Hassanain, J. Silk, and S. M. West, “Emergent flux from particle collisions near a Kerr black hole,” Physical The authors declare that there is no conflict of interests Review D,vol.83,no.2,ArticleID023004,2011. regarding the publication of this paper. [17] M. Patil and P. S. Joshi, “Naked singularities as particle accelerators,” Physical Review D,vol.82,ArticleID104049,2010. Acknowledgments [18] M. Patil, P. S. Joshi, and D. 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Research Article Entropy Spectrum of a KS Black Hole in IR Modified Holava-Lifshitz Gravity

Shiwei Zhou,1,2 Ge-Rui Chen,1 and Yong-Chang Huang1

1 Institute of Theoretical Physics, Beijing University of Technology, Beijing 100124, China 2 Department of Foundation, Academy of Armored Forces Engineering, Beijing 100072, China

Correspondence should be addressed to Shiwei Zhou; [email protected]

Received 6 January 2014; Accepted 10 February 2014; Published 13 March 2014

Academic Editor: Xiaoxiong Zeng

Copyright © 2014 Shiwei Zhou et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP3.

As a renormalizable theory of gravity, Hoˇrava-Lifshitz gravity, might be an ultraviolet completion of general relativity and reduces to Einstein gravity with a nonvanishing cosmological constant in infrared. Kehagias and Sfetsos obtained a static spherically symmetric black hole solution called KS black hole in the IR modified Hoˇrava-Lifshitz theory. In this paper, the entropy spectrum and area spectrum of a KS black hole are investigated based on the proposal of adiabatic invariant quantity. By calculating the action of producing a pair of particles near the horizon, it is obtained that the action of the system is exactly equivalent to the change of black hole entropy, which is an adiabatic invariant quantity. With the help of Bohr-Sommerfeld quantization rule, it is concluded that the entropy spectrum is discrete and equidistant spaced and the area spectrum is not equidistant spaced, which depends on the parameter of gravity theory. Some summary and discussion will be given in the last.

1. Introduction Maggiore gave a new interpretation of black hole quasinormal modes in connection to the quantization of black hole In the 1970’s, with the discovery of Hawking radiation horizon area. An important statement in Maggiore’s work is and Bekenstein’s proposal of black hole entropy, black hole that the periodicity of a black hole in Euclidean time may thermodynamics has been built up successfully, which has be the origin of area quantization [12]. It is well known that opened a new field to study quantum theory and gravity for any background spacetime with a horizon in Kruskal theory [1–4]. Nowadays there has been some trouble on the coordinates, the period with respect to Euclidean time takes statistic origin and quantization of black hole entropy for the form of 𝑇=2𝜋/𝜅,where𝜅 is the surface gravity physicists on black hole thermodynamics. Since Bekenstein of the horizon. Vagenas exclusively used the fact that the proposed that the horizon area of a nonextremal black hole black hole horizon area is an adiabatic invariant quantum is an adiabatic invariant classically and the horizon area and derived an equally spaced entropy spectrum of a black 2 of black hole is quantized in units of 𝑙𝑝 [5–8], there has holewithitsvaluetobeequaltothatofBekenstein[13]. been much attention paid to the quantization of black hole Zeng et al. considered that the action 𝐼,actionvariable𝐼V, entropy spectrum and area spectrum. Hod proposed that if and Hamiltonian 𝐻 of any single periodic system satisfy the one employs Bohr’s corresponding principle, the real part of relation 𝐼=𝐼V −∫𝐻𝑑𝑡. They proposed that the action the quasinormal mode frequency is responsible for the area variable can be quantized with the equally spaced form spectrum of black hole [9, 10]. Combining the proposal by 𝐼V =2𝜋𝑛ℎ. Once the action and Hamiltonian are given, the Bekenstein for the adiabaticity of black hole horizon area quantization action variable can be obtained. With the help andHod’sproposal,Kunstatterprovedthatentropyspectrum of Bohr-Sommerfeld quantization rule, they proved that the of a 𝑑-dimensional black hole is quantized and the result is quantized action variable is nothing but the entropy of black in agreement with that of Hod and Bekenstein [11]. Later, hole; thus the entropy and the horizon area of a black hole 2 Advances in High Energy Physics can be quantized. They emphasized that the action variable which is the adiabatic invariant quantity. With the help of should be adiabatic invariant [14–16]. Recently, Jiang and Han Bohr-Sommerfeld quantization rule, we obtain the quantized argued that the adiabatic invariant quantity ∫𝑝𝑖𝑑𝑞𝑖 is not entropy and area spectrum. Some summary and discussion canonically invariant, and the adiabatic invariant quantum will be given in the latter. shouldbeofthecovarianceform∮𝑝𝑖𝑑𝑞𝑖 =𝑛ℎ.Theyalso obtained the equally spaced entropy spectrum with the form 2. Review of a KS Black Hole in IR Modified Δ𝑆 = 2𝜋 of [17]. Some more works on the quantization of Holava-Lifshitz Gravity entropy spectrum of a black hole can be seen in the paper [18–23]. Using the ADM decomposition of the metric Hoˇrava gravity is a nonrelativistic renormalizable theory 2 2 2 𝑖 𝑖 𝑗 𝑗 of gravitation [24], which is inspired by the anisotropic 𝑑𝑠 =−𝑁𝑑𝑡 +𝑔𝑖𝑗 (𝑑𝑥 +𝑁𝑑𝑡) (𝑑𝑥 +𝑁𝑑𝑡) , (1) scaling between time and space in condensed matter systems in particular in the theory of quantum critical phenomena, 𝑖 where 𝑁 and 𝑁 are the “lapse” and “shift” variables, respec- where the degree of anisotropy between space and time tively, and 𝑔𝑖𝑗 is the spatial metric. On the limit of Λ 𝜔 →0, is characterized by the “dynamical critical exponent” 𝑧.It the action of IR modified Hoˇrava-Lifshitz gravity theory can is well known that relativistic systems automatically satisfy be given by 𝑧=1as a consequence of Lorentz invariance. In Hoˇrava-

Lifshitz theory, systems’ scaling at a short distance exhibits a 3 strong anisotropy between space and time with 𝑧>1.This 𝑆=∫ 𝑑𝑡𝑑 𝑥√𝑔𝑁 will improve the short-distance behavior of the theory. The anisotropy at short distance can be lost for long distance while 2 ×[ (𝐾 𝐾𝑖𝑗 −𝜆𝐾2) the Lorentz symmetry will appear as an emergent symmetry. 𝜅2 𝑖𝑗 The black hole solution of originalˇ Horava-Lifshitz gravity 2 2 does not recover the usual Schwarzschild-anti-de Sitter black 𝜅 𝑖𝑗 𝜅 𝜇 𝑖𝑗𝑘 (3) (3)𝑙 − 𝐶 𝐶 + 𝜖 𝑅 ∇ 𝑅 (2) hole with the detailed-balance condition. A relevant operator 2𝜔4 𝑖𝑗 2𝜔2 𝑖𝑙 𝑗 𝑘 proportional to the 3𝐷 geometry Ricci scalar of the original 𝜅2𝜇2 Hoˇrava-Lifshitz theory action was introduced [25]andit − 𝑅(3)𝑅(3)𝑖𝑗 deviated from detailed balance. This does not modify the 8 𝑖𝑗 ultraviolet properties of the theory. However, it modifies the 2 2 𝜅 𝜇 1−4𝜆 2 infrared Hoˇrava-Lifshitz gravity theory. So a Schwarzschild- + (𝑅(3)) +𝜇4𝑅(3)], anti-de Sitter solution can be realized in infrared modified 8 (1−3𝜆) 4 Hoˇrava-Lifshitz gravity theory and the Minkowski vacuum is also allowed. On the limit of vanishing Λ 𝜔,aspherically which is obtained by introducing a term proportional to the 4 (3) symmetric black hole solution has been obtained by Kehagias Ricci scalar of the three-geometry 𝜇 𝑅 to the original action and Sfetsos [25], which is the analogy of Schwarzschild black of Hoˇrava-Lifshitz gravity [26, 27]. Here 𝐾𝑖𝑗 is the extrinsic hole in general relativity and is exactly asymptotically flat. curvature, defined by After Hoˇrava-Lifshitz gravity was proposed, much attention hasbeenpaidtoit[26–31]. 1 𝐾 = (𝑔̇−∇𝑁 −∇𝑁 ), (3) In this paper, based on Vagenas’ proposal of adiabatic 𝑖𝑗 2𝑁 𝑖𝑗 𝑖 𝑗 𝑗 𝑖 invariant quantity, we investigate entropy spectrum of a KS 𝑖𝑗 blackholeinIRmodifiedHoˇrava-Lifshitz gravity. Vagenas where the dot denotes a derivative with respect to 𝑡, 𝐶 is the pointed that the general coordinates 𝑞𝑖 contain 𝑞0 =𝜏and Cotton tensor defined by 𝑞 =𝑟 1 , and the contribution of the two parts to the adiabatic 1 invariant quantity are equivalent with each other by defining 𝐶𝑖𝑗 =𝜖𝑖𝑘𝑙∇ (𝑅(3)𝑗 − 𝑅(3)𝛿𝑗), 𝑘 𝑙 4 𝑙 (4) 𝑟̇ = 𝑑𝑟/𝑑𝜏. Thus the integration result of adiabatic invariant quantity was directly given by the 𝜏-integration, where the (3) and 𝑅 is the 3-dimensional curvature scalar for 𝑔𝑖𝑗 ; period 𝑇 of gravity system satisfies 𝑇=2𝜋/𝜅.Wefindthat 𝜅, 𝜆, 𝜔,𝜇 are all coupling constant parameters. the periodicity of gravity system can conveniently be used to Comparing the action for the case of 𝜆=1with the calculatetheentropyspectrum.However,thephysicalpicture standard Einstein-Hilbert action, we find that the Lagrangian of periodicity is not clear. We give our explanation about it by will become the usual Einstein-Hilbert Lagrangian when the considering a process that a pair of particles create outside the speed of light 𝑐,Newton’sconstant𝐺, and the cosmological horizon. One period of the system corresponds to the process constant Λ are given by with the outgoing positive energy particle crossing outwards the horizon while the negative energy particle tunnels into 𝜅2𝜇4 𝜅2 3 𝑐2 = ,𝐺= ,Λ=Λ . (5) the black hole. The movement of the two particles can be 2 32𝜋𝑐 2 𝑊 described as a tunneling process proposed by Parikh and Wilczek’s Hawking radiation [32]. After calculating, we find The spherically symmetric asymptotically flat black hole that the action of the system is exactly the black hole entropy solution has been obtained by Cai et al. [27], which is the Advances in High Energy Physics 3 analogy of Schwarzschild black hole in general relativity. The When considering 𝑟=𝑑𝑟/𝑑𝜏=𝑓(𝑟)̇ ,weget metric can be written as

2 2 1 2 2 2 2 2 𝑑𝑠 =−𝑓(𝑟) 𝑑𝑡 + 𝑑𝑟 +𝑟 (𝑑𝜃 + 𝜃𝑑𝜑 ), 𝑚 𝑟 sin (6) out 𝑑𝑟 𝑓 (𝑟) 𝐼 = ∫ ∫ 𝑑𝑚󸀠. 1 2 2 3 󸀠 (11) 0 𝑟 1+𝜔𝑟 −𝑟[𝜔 𝑟 +4𝜔(𝑀−𝑚)] in with

𝑓 (𝑟) =1+𝜔𝑟2 − √𝑟(𝜔2𝑟3 +4𝜔𝑀), (7) 󸀠 󸀠 It is easily found that there is a pole at 𝑟ℎ =(𝑀−𝑚)+ 2 where 𝑀 is an integration constant corresponding to the mass √(𝑀 −󸀠 𝑚 ) −1/2𝜔. We do the integration as follows: of black hole, and 𝜔 is a coupling constant parameter. The condition 𝑓(𝑟±)=0gives the outer and inner horizons at

𝑚 𝑟 1 out 𝑟 =𝑀(1±√1− ) . 𝐼 = ∫ ∫ (𝑑 𝑟 ( ( 𝑟 −𝑟 󸀠 ) ± 2 (8) 1 ℎ 2𝜔𝑀 0 𝑟 in

2 To avoid naked singularity, we should have 𝜔𝑀 ≥1/2.In 2 the regime of traditional general relativity, we have 𝜔𝑀 ≫ [ × [2𝜔𝑟 1, so the outer horizon approaches the usual Schwarzschild horizon 𝑟+ ≃2𝑀, whereas the inner one approaches the [ 𝑟− ≃0 singularity . −1 2𝜔2𝑟3+2𝜔(𝑀−𝑚󸀠) ] 󸀠 3. Entropy Quantization via Adiabatic − ]) )𝑑𝑚 √𝑟(𝜔2𝑟3 +4𝜔(𝑀−𝑚󸀠)) Invariant Action ]

We consider a process that a pair of particles create near the 𝑚 horizon. While the outgoing positive energy particle crossing 󸀠 =2𝜋∫ (1 (2𝜔𝑟ℎ outwards the horizon, the negative energy particle ingoing 0 towards the black hole along the radial direction. We describe the movement of the two particles as a tunneling process −1 proposed by Parikh and Wilczek’s proposal [32]. 2𝜔2𝑟󸀠3 +2𝜔(𝑀−𝑚󸀠) ℎ 󸀠 The action of the system is − ) )𝑑𝑚 √𝑟󸀠 [𝜔2𝑟󸀠3 +4𝜔(𝑀−𝑚󸀠)] 𝑟 𝑟 ℎ ℎ out in 𝐼=∫ 𝑝𝑟𝑑𝑟 = ∫ 𝑝𝑟𝑑𝑟 + ∫ 𝑝𝑟𝑑𝑟 𝑟 𝑟 𝑚 in out 1 2 =2𝜋∫ (( +2𝜔(𝑀−𝑚󸀠) 𝑟 𝑝 𝑟 𝑝 (9) out 𝑟 󸀠 in 𝑟 󸀠 0 2 = ∫ ∫ 𝑑𝑝𝑟𝑑𝑟 + ∫ ∫ 𝑑𝑝𝑟𝑑𝑟. 𝑟 0 𝑟 0 in out 1 +2𝜔(𝑀−𝑚󸀠) √(𝑀 −󸀠 𝑚 )2 − ) The first term is corresponding to the particles with positive 2𝜔 energy, and the second term is the negative energy one. When energy conservation is considered, the black hole mass will −1 2 1 󸀠 decrease with the outgoing particle emitting. The Hamilton ×(2𝜔√(𝑀 −󸀠 𝑚 ) − ) )𝑑𝑚 󸀠 2𝜔 𝐻, ADM energy 𝑀, and the particle’s energy 𝑚 satisfy the 󸀠 󸀠 relation 𝐻=𝑀−𝑚;thatis,𝑑𝐻 = −𝑑𝑚 .Thenbyuse of Hamilton’s equation 𝑟=𝑑𝐻/𝑑𝑝̇ 𝑟,thefirsttermof(9) 𝑚 [ 1 󸀠 becomes =2𝜋[∫ 𝑑𝑚 0 4𝜔√(𝑀 −󸀠 𝑚 )2 −1/2𝜔 𝑟 𝑝 𝑟 𝑀−𝑚 [ out 𝑟 󸀠 out 𝑑𝐻 𝐼1 ≡ ∫ ∫ 𝑑𝑝𝑟𝑑𝑟 = ∫ ∫ 𝑑𝑟 𝑟 0 𝑟 𝑀 𝑟̇ 󸀠 2 in in 𝑚 (𝑀 − 𝑚 ) 󸀠 𝑚 𝑟 (10) + ∫ 𝑑𝑚 out 𝑑𝑟 = ∫ ∫ (−𝑑𝑚󸀠), 0 √(𝑀 −󸀠 𝑚 )2 −1/2𝜔 0 𝑟 𝑟̇ in

󸀠 𝑚 where 𝑟 =𝑟ℎ(𝑀) −, 𝜖 𝑟 =𝑟ℎ(𝑀 − 𝑚),forthereason +𝜖 ] in out + ∫ (𝑀 −󸀠 𝑚 )𝑑𝑚󸀠] of that the black hole horizon will decrease with the particle 0 emitting out. ] 4 Advances in High Energy Physics

󸀠 𝑀 2 1 𝑀−𝑚 󸀠 2 1 coordinates 𝑢=𝑡−𝑟∗, V =𝑡+𝑟∗,wecangetthecoordinates =2𝜋[ √𝑀 − − √(𝑀 − 𝑚 ) − −𝜅 𝑢 𝜅 V 2 2𝜔 2 2𝜔 𝑈=−𝑒 + and 𝑉=𝑒 + [33, 34]. Then define

1 1 1 + [ 𝑀+√𝑀2 − − (𝑀−𝑚) 𝑇= (𝑉+𝑈) 2𝜔 ln 2𝜔 ln 2

1/2 𝜅+/2𝜅− 2 𝜅 𝑟 𝑟−𝑟 𝑟−𝑟 1 𝑚 =𝑒 + ( + ) ( − ) 𝜅 𝑡, +√(𝑀−𝑚)2 − ]+𝑀𝑚− ]. 𝑟 𝑟 sinh + 2𝜔 2 + − 1/2 𝜅 /2𝜅 1 𝑟−𝑟 𝑟−𝑟 + − (12) 𝜅+𝑟 + − 𝑅= (𝑉−𝑈) =𝑒 ( ) ( ) cosh 𝜅+𝑡, 2 𝑟+ 𝑟− The second term of (9) corresponds to ingoing particles (19) with negative energy. After similar calculation as the first term, we find that the contribution is equivalent to that of 𝐼1. where 𝑇, 𝑅 are the Kruskal-like coordinates. That is, Different from18 ( ), that is, 𝑑𝑟 =𝑓(𝑟)−1𝑑𝑟; 𝐼=2𝐼1. (13) ∗ (20)

On the other hand, the Hawking temperature of the outer the two-dimensional KS metric becomes event horizon has been obtained as [31] 2 2 2 𝑑𝑠 =−𝑑𝑡 +𝑑𝑟∗ 2 2𝜔𝑟 −1 𝜅+/𝜅− + −2 −2𝜅 𝑟 𝑟−𝑟 𝑟 𝑇 = =𝜅 𝑒 + ( − )( − ) 𝐻 8𝜋 (𝜔𝑟3 +𝑟 ) + (21) + + 𝑟+ 𝑟−𝑟− (14) √𝑀2 −1/2𝜔 ×[−𝑑𝑇2 +𝑑𝑅2]. = , 2𝜋 (2𝑀2 +2𝑀√𝑀2 − 1/2𝜔 + 1/2𝜔) Transformingthetimecoordinateas𝑡→−𝑖𝜏,wehave 𝜅= which is proportional to the surface gravity 1/2 𝜅+/2𝜅− 𝜅 𝑟 𝑟−𝑟+ 𝑟−𝑟− (1/2)(𝜕𝑓(𝑟)/𝜕𝑟)|𝑟=𝑟 on the event horizon of black hole. 𝑖𝑇 =𝑒 + ( ) ( ) 𝜅 𝜏, + 𝑟 𝑟 sin + Considering the first law of thermodynamics and substituting + − (22) the expression of temperature of black hole, we can obtain 1/2 𝜅 /2𝜅 𝑟−𝑟 𝑟−𝑟 + − 𝜅+𝑟 + − the entropy 𝑅=𝑒 ( ) ( ) cos 𝜅+𝜏. 𝑟+ 𝑟− 𝐴 𝜋 𝐴 𝑆 = + , 𝑇 𝑅 BH 4 𝜔 ln 4 (15) It is easily found that both , are periodic functions withrespecttotheEuclideantime𝜏 with the period of 2 2𝜋/𝜅+. Since we only consider the case of outer horizon, for where 𝐴=𝜋𝑟+ is the area of event horizon. simplicity we write 𝜅+ for 𝜅 from now on. After calculation, we find that the change of black hole When utilizing Vagenas’s adiabatic invariant quantity entropy Δ𝑆, when a particle with energy of 𝑚 emits out of the with the following form: black hole, is exactly equivalent to the action 𝐼;thatis, 𝑝 𝐻 󸀠 𝑖 𝑑𝐻 2 2 𝐼=∫ 𝑝 𝑑𝑞 = ∫∫ 𝑑𝑝󸀠𝑑𝑞 = ∫∫ 𝑑𝑞 Δ𝑆 = 𝜋+ [𝑟 (𝑀) −𝑟+ (𝑀−𝑚)]=𝐼. (16) 𝑖 𝑖 𝑖 𝑖 𝑖 0 0 𝑞𝑖̇ (23) 𝐻 𝐻 𝑑𝐻󸀠 Now, using the periodicity of the black hole, we calculate = ∫∫ 𝑑𝐻󸀠𝑑𝜏 + ∫∫ 𝑑𝑟, the adiabatic invariant quantity. According to the dimen- 0 0 𝑟̇ sional reduction technique, the two-dimensional spacetime ofaKSblackholecanbegivenby where 𝑝𝑖 are the conjugate momentum of the general coordinate 𝑞𝑖 with 𝑖=0,1for 𝑞0 =𝜏and 𝑞1 =𝑟. Considering 2 2 −1 2 𝑟̇ = 𝑑𝑟/𝑑𝜏 𝑑𝑠 =−𝑓(𝑟) 𝑑𝑡 +𝑓(𝑟) 𝑑𝑟 . (17) ,wehave 𝐻 󸀠 When defining the tortoise coordinate as 𝐼=2∫∫ 𝑑𝐻 𝑑𝜏. (24) 0

1 𝑟−𝑟+ 1 𝑟−𝑟− 𝜏 𝑇=2𝜋/ℎ𝜅 𝑟∗ =𝑟+ ln + ln , (18) Because of the periodicity of with , the adiabatic 2𝜅+ 𝑟+ 2𝜅− 𝑟− invariant quantity can be calculated as

󸀠 2 2 𝐻 󸀠 𝐻 󸀠 𝜅± =(𝑓(𝑟)/2)|𝑟=𝑟 =(2𝜔𝑟 −1)/(4𝑟±(𝜔𝑟 +1)) 𝑑𝐻 𝑑𝐻 in which ± ± ± is the 𝐼=2𝜋∫ =ℎ∫ =ℎ𝑆 . 𝜅 𝑇 BH (25) surface gravity on the outer (inner) horizon. Using the null 0 0 BH Advances in High Energy Physics 5

Implementing the Bohr-Sommerfeld quantization condi- Acknowledgments tion ThisworkissupportedbyBeijingPostdoctoralResearch Foundation no. 71006015201201 and National Natural Science ∮ 𝑝𝑑𝑞 = 2𝜋𝑛ℎ, (26) Foundation of China (no. 11275017 and no. 11173028). theblackholeentropyspectrumcanbegivenas References 𝑆 =2𝜋𝑛, 𝑛=1,2,3,..., BH (27) [1] S. W. Hawking, “Particle creation by black holes,” Communica- tions in Mathematical Physics,vol.43,no.3,pp.199–220,1975. and the entropy spectrum is discrete and equidistant spaced [2]S.W.Hawking,“Blackholeexplosions?”Nature,vol.248,no. with 5443, pp. 30–31, 1974. [3] S. W. Hawking, “Information loss in black holes,” Physical Δ𝑆 =2𝜋. BH (28) Review D,vol.72,no.8,ArticleID084013,4pages,2005. [4] J. M. Bardeen, B. Carter, and S. W. Hawking, “The four laws To get the area spectrum, differentiate (15), of black hole mechanics,” Communications in Mathematical Physics,vol.31,pp.161–170,1973. 1 𝜋 4 [5] J. D. Bebenstein, “Black holes and entropy,” Physical Review D, Δ𝑆 = Δ𝐴 + Δ𝐴, (29) BH 4 𝜔 𝐴 vol. 7, no. 8, pp. 2333–2346, 1973. [6] J. D. Bebenstein, “Black holes and the second law,” Lettere Al we have Nuovo Cimento,vol.4,pp.737–740,1972. [7] J. D. Bebenstein, “The quantum mass spectrum of the Kerr black Δ𝑆 4 Δ𝐴 = BH ≃8𝜋(1− ). hole,” Lettere Al Nuovo Cimento,vol.11,pp.467–470,1974. 2 (30) 1/4 + 4𝜋/𝐴𝜔 𝜔𝑟+ [8] J. D. Bebenstein, “Extraction of energy and charge from a black hole,” Physical Review D,vol.7,pp.949–953,1973. Wefindthatareaspectrumisnotequidistantspaced. [9] S. Hod, “Bohr’scorrespondence principle and the area spectrum of quantum black holes,” Physical Review Letters,vol.81,no.20, pp. 4293–4296, 1998. 4. Summary and Conclusion [10] S. Hod, “Gravitation, the quantum and Bohr’s correspondence principle,” General Relativity and Gravitation, vol. 31, no. 11, pp. In this paper, based on the idea of adiabatic invariant 1639–1644, 1999. quantity, we have investigated entropy spectrum of a KS [11] G. Kunstatter, “𝑑-dimensional black hole entropy spectrum blackholeinIRmodifiedHoˇrava-Lifshitz gravity. As a from quasinormal modes,” Physical Review Letters,vol.90,no. modified gravity theory, the entropy of a KS black hole 16, Article ID 161301, 4 pages, 2003. does not satisfy Bekenstein’s entropy-area relation. It consists [12] M. Maggiore, “Physical interpretation of the spectrum of black of two terms: one is the Bekenstein-Hawking entropy, the hole quasinormal modes,” Physical Review Letters,vol.100,no. other is a logarithmic term. The discrepancy between the 14, Article ID 141301, 4 pages, 2008. entropy and the Bekenstein-Hawking entropy is the reflection [13]B.R.MajhiandE.C.Vagenas,“Blackholespectroscopyvia of differences between this modified gravity theory and adiabatic invariance,” Physics Letters B,vol.701,pp.623–625, general relativity. After calculating, we find that the black 2011. hole entropy is an adiabatic invariant quantity. With the [14]X.X.Zeng,X.M.Liu,andW.B.Liu,“Periodicityandarea help of Bohr-Sommerfeld quantization rule, we obtain the spectrum of black holes,” The European Physical Journal C,vol. quantized entropy and area spectrum. It is concluded that 72,p.1967,2012. 𝑆 =2𝜋𝑛 the entropy spectrum can be given as BH with [15] X. X. Zeng and W. B. Liu, “Spectroscopy of a Reissner- 𝑛 = 1, 2, 3, . ., which is discrete and equidistant spaced Nordstrom¨ black hole via an action variable,” The European Δ𝑆 =2𝜋 Physical Journal C,vol.72,p.1987,2012. with BH ;andtheareaspectrumisnotequidistant spaced, which depends on the parameter of gravity theory. [16] X.-M. Liu, X.-X. Zeng, and S.-W. Zhou, “Area spectra of BTZ In addition, by calculating the action of a production of a black holes via periodicity,” Science China Physics, Mechanics pair of particles near the horizon, we find that the action and Astronomy,vol.56,no.9,pp.1627–1631,2013. of the system is exactly equivalent to the change to black [17]Q.-Q.JiangandY.Han,“Onblackholespectroscopyvia hole entropy, which is an adiabatic invariant quantity. The adiabatic invariance,” http://arxiv.org/abs/1210.4002. procession of the particle producing, with positive energy [18] H.-L. Li, R. Lin, and L.-Y. Chen, “Entropy quantization of outgoing towards the horizon while the one with negative Reissner-Nordstrom¨ de Sitter black hole via adiabatic covariant energy is ingoing the horizon, can give a clear explanation to action,” General Relativity and Gravitation,vol.45,pp.865–875, 2013. the periodicity of gravity system. [19] D. Chen and H. Yang, “Entropy spectrum of a Kerr anti-de Sitter black hole,” The European Physical Journal C,vol.72,p.2027, Conflict of Interests 2012. [20] R. Tharanath and V. C. Kuriakose, “Thermodynamics and The authors declare that there is no conflict of interests spectroscopy of charged dilaton black holes,” General Relativity regarding the publication of this paper. and Gravitation,vol.45,no.9,pp.1761–1770,2013. 6 Advances in High Energy Physics

[21] Q. Q. Jiang, “Revisit emission spectrum and entropy quantum of the Reissner-Nordstrom¨ black hole,” The European Physical Journal C,vol.72,p.2086,2012. [22] C.-Z. Liu, “Black hole spectroscopy via adiabatic invariant in a quantum corrected spacetime,” The European Physical Journal C,vol.72,p.2009,2012. [23] T. Tanaka and T. Tamaki, “Area spectrum of horizon and black hole entropy,” The European Physical Journal C,vol.73,p.2314, 2013. [24] P.Hoˇrava, “Quantum gravity at a Lifshitz point,” Physical Review D,vol.79,no.8,ArticleID084008,15pages,2009. [25] A. Kehagias and K. Sfetsos, “The black hole and FRW geometries of non-relativistic gravity,” Physics Letters B,vol.678,no.1, pp. 123–126, 2009. [26] H. Lu,¨ J. Mei, and C. N. Pope, “Solutions to hoˇrava gravity,” Physical Review Letters,vol.103,no.9,ArticleID091301,2009. [27] R.-G. Cai, L.-M. Cao, and N. Ohta, “Topological black holes in Hoˇrava-Lifshitz gravity,” Physical Review D,vol.80,no.2, ArticleID024003,7pages,2009. [28] S. Mukohyama, “Scale-invariant cosmological perturbations from Hoˇrava-Lifshitz gravity without inflation,” Journal of Cosmology and Astroparticle Physics,vol.2009,article001,2009. [29] S. Mukohyama, K. Nakayama, F. Takahashi, and S. Yokoyama, “Phenomenological aspects of Horava-Lifshitz cosmology,” Physics Letters B,vol.679,no.1,pp.6–9,2009. [30] S.-W. Zhou and W.-B. Liu, “Three classical tests of Hoˇrava- Lifshitz gravity theory,” Astrophysics and Space Science,vol.337, no. 2, pp. 779–784, 2012. [31] S.-W. Zhou and W.-B. Liu, “Black hole thermodynamics of Hoˇrava-Lifshitz and IR modified Hoˇrava-Lifshitz gravity theory,” International Journal of Theoretical Physics,vol.50,no.6, pp.1776–1784,2011. [32] M. K. Parikh and F. Wilczek, “Hawking radiation as tunneling,” Physical Review Letters,vol.85,no.24,pp.5042–5045,2000. [33] C. Liang and B. Zhou, Differential Geometry and General Relativity, Science Press, Beijing, China, 2006. [34] Z. Zhao, The Thermal Nature of Black Holes and the Singularity of the Space-Time, Beijing Normal University Press, Beijing, China, 1999. Hindawi Publishing Corporation Advances in High Energy Physics Volume 2014, Article ID 124854, 6 pages http://dx.doi.org/10.1155/2014/124854

Research Article The Critical Phenomena and Thermodynamics of the Reissner-Nordstrom-de Sitter Black Hole

Ren Zhao,1,2 Mengsen Ma,1,2 Huihua Zhao,1,2 and Lichun Zhang1,2

1 Institute of Theoretical Physics, Shanxi Datong University, Datong 037009, China 2 Department of Physics, Shanxi Datong University, Datong 037009, China

Correspondence should be addressed to Ren Zhao; [email protected] and Mengsen Ma; [email protected]

Received 25 October 2013; Revised 26 January 2014; Accepted 5 February 2014; Published 10 March 2014

Academic Editor: Deyou Chen

Copyright © 2014 Ren Zhao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP3.

It is wellknown that there are two horizons for the Reissner-Nordstrom-de Sitter spacetime, namely, the black hole horizon and the cosmological one. Both horizons can usually seem to be two independent thermodynamic systems; however, the thermodynamic quantities on both horizons satisfy the laws of black hole thermodynamics and are not independent. In this paper by considering the relations between the two horizons we give the effective thermodynamic quantities in Reissner-Nordstrom-de Sitter spacetime. The thermodynamic properties of these effective quantities are analyzed; moreover, the critical temperature, critical pressure, and critical volume are obtained. We also discussed the thermodynamic stability of Reissner-Nordstrom-de Sitter spacetime.

1. Introduction Recently, by considering the cosmological constant correspond to pressure in general thermodynamic system, namely, Black hole physics, especially the black hole thermodynamics, 1 3 1 refer directly to the theories of gravity, statistical physics, 𝑃=− Λ= , 2 (1) particle physics, field theory, and so forth. This makes the 8𝜋 8𝜋 𝑙 field concerned by many physicists [1–6]. Although the com- the thermodynamic volumes in AdS and dS spacetime are plete statistical description of black hole thermodynamics obtained [18–24]. The studies on phase transition of black is still unclear, the research on the properties of black hole holes have aroused great interest [25–31]. Connecting the thermodynamics is prevalent, such as Hawking-Page phase thermodynamic quantities of AdS black holes to (𝑃 ∼ 𝑉) in transition [7], and critical phenomena. More interestingly, the the ordinary thermodynamic system, the critical behaviors of research on the charged and nonrotating RN-AdS black hole blackholescanbeanalyzedandthephasediagramlikevan showsthatthereexistsasimilarphasetransitiontothevan der Waals vapor-liquid system can be obtained. This helps der Waals-Maxwell vapor-liquid phase transition [8, 9]. to further understand the black hole entropy, temperature, Motivated by the AdS/CFT correspondence [10], where heat capacities, and so forth. It also has a very important the transitions have been related with the holographic super- significance in completing the geometric theory of black hole conductivity [11, 12], the subject of the phase transitions of thermodynamics. black holes in asymptotically anti-de Sitter (AdS) spacetime Asiswellknown,thereareblackholehorizonandcosmo- has received considerable attention [13–17]. The underlying logical horizon in the appropriate range of parameters for de microscopic statistical interaction of the black holes is also Sitter spacetime. Both horizons have thermal radiation, but expected to be understood via the study of the gauge theory with different temperatures. The thermodynamic quantities living on the boundary in the gauge/gravity duality. on both horizons satisfy the first law of thermodynamics, 2 Advances in High Energy Physics

and the corresponding entropy fulfills the area formula23 [ , horizon (BEH) located at 𝑟=𝑟+,andthecosmologicalevent 32, 33]. In recent years, the research on the thermodynamic horizon (CEH) located at 𝑟=𝑟𝑐,where𝑟𝑐 >𝑟+ >𝑟−,theonly properties of de Sitter spacetime has drawn a lot of attention real, positive zeroes of 𝑓(𝑟). =0 [23, 32–36]. In the inflation epoch of early universe, the The equations 𝑓(𝑟+)=0and 𝑓(𝑟𝑐)=0are rearranged to universe is a quasi-de Sitter spacetime. The cosmological 𝑟2 +𝑟𝑟 +𝑟2 constant introduced in de Sitter space may come from 𝑄2 =𝑟 𝑟 (1 − 𝑐 𝑐 + + Λ) , the vacuum energy, which is also a kind of energy. If the + 𝑐 3 cosmological constant is the dark energy, the universe will (4) 𝑟2 +𝑟2 evolve to a new de Sitter phase. To depict the whole history of 2𝑀 = (𝑟 +𝑟 )(1− 𝑐 + Λ) . evolutionoftheuniverse,weshouldhavesomeknowledge 𝑐 + 3 on the classical and quantum properties of de Sitter space [23, 33, 37, 38]. The surface gravity on the black hole horizon and the Firstly, we expect the thermodynamic entropy to satisfy cosmological horizon is, respectively, the Nernst theorem [34, 35, 39]. At present a satisfactory 󵄨 1 𝑑𝑓 (𝑟)󵄨 explanation to the problem in which the thermodynamic 𝜅 = 󵄨 + 2 𝑑𝑟 󵄨 entropy of the horizon of the extreme de Sitter spacetime does 󵄨𝑟=𝑟+ notfulfilltheNernsttheoremisstilllacking.Secondly,when 𝑟2 −𝑄2 −𝑟4 Λ considering the correlation between the black hole horizon = + + 3 and the cosmological horizon whether the thermodynamic 2𝑟+ quantities in de Sitter spacetime still have the phase transition 2 2 2 andcriticalbehaviorlikeinAdSblackholes.Thusitis (𝑟 −𝑟) (𝑟 +2𝑟+𝑟𝑐 +3𝑟 )(𝑟𝑐𝑟+ −𝑄 ) = + 𝑐 (1 − 𝑐 + ), worthy of our deep investigation and reflection to establish 2 2 2 2𝑟+ 𝑟𝑐𝑟+ (𝑟 +𝑟+𝑟𝑐 +𝑟 ) aconsistentthermodynamicsindeSitterspacetime. 𝑐 + 󵄨 (5) Becausethethermodynamicquantitiesontheblackhole 1 𝑑𝑓 (𝑟)󵄨 𝜅 = 󵄨 horizon and the cosmological one in de Sitter spacetime are 𝑐 󵄨 2 𝑑𝑟 󵄨𝑟=𝑟 the functions of mass 𝑀,electriccharge𝑄, and cosmological 𝑐 constant Λ. The quantities are not independent of each 𝑟2 −𝑄2 −𝑟4Λ = 𝑐 𝑐 other. Considering the relation between the thermodynamic 2𝑟3 quantities on the two horizons is very important for studying 𝑐 the thermodynamic properties of de Sitter spacetime. Based 2 2 2 (𝑟 −𝑟 ) (𝑟 +2𝑟+𝑟𝑐 +3𝑟 )(𝑟𝑐𝑟+ −𝑄 ) ontherelationwegivetheeffectivetemperatureandpressure = 𝑐 + (1 − + 𝑐 ). 2𝑟2 𝑟 𝑟 (𝑟2 +𝑟 𝑟 +𝑟2 ) of Reissner-Nordstrom-de Sitter(R-NdS) spacetime and ana- 𝑐 𝑐 + 𝑐 + 𝑐 + lyze the critical behavior of the equivalent thermodynamic quantities. It is shown that when considering the relation Thethermodynamicquantitiesonthetwohorizonssatisfy between the two horizons in RN-dS spacetime there is the the first law of thermodynamics23 [ , 33, 40, 41]: similar phase transition like the ones in van der Waals liquid- 𝜅 𝛿𝑀 = + 𝛿𝑆 +Φ 𝛿𝑄 +𝑉 𝛿𝑃, gas system and charged AdS black holes. 2𝜋 + + + The paper is arranged as follows. In Section 2 we intro- (6) 𝜅𝑐 duce the Reissner-Nordstrom-de Sitter(R-NdS) spacetime 𝛿𝑀 = 𝛿𝑆𝑐 +Φ𝑐𝛿𝑄𝑐 +𝑉 𝛿𝑃, and give the two horizons and corresponding thermody- 2𝜋 namic quantities. In Section 3 by considering the relations 2 2 where 𝑆+ =𝜋𝑟, 𝑆𝑐 =𝜋𝑟, Φ+ =𝑄/𝑟+, Φ𝑐 =−(𝑄/𝑟𝑐), 𝑉+ = between the two horizons we obtain the effective temperature + 𝑐 (4𝜋/3)𝑟3 𝑉 =(4𝜋/3)𝑟3 𝑃 = −(Λ/8𝜋) and the equivalent pressure. In Section 4 the critical phe- +, 𝑐 𝑐 , . nomena of effective thermodynamic quantities are discussed. Finally we discuss and summarize our results in Section 5 (we 3. Thermodynamic Quantity of 𝐺 =ℎ=𝑘 =𝑐=1 use the units 𝑛+1 𝐵 ). RN-dS Spacetime

2. RN-dS Spacetime In Section 2, we have obtained thermodynamic quantities without considering the relationship between the black The line element of the R-N SdS black holes is given by[32] hole horizon and the cosmological horizon. Because there are three variables 𝑀, 𝑄,andΛ in the spacetime, the 2 2 −1 2 2 2 𝑑𝑠 =−𝑓(𝑟) 𝑑𝑡 +𝑓 𝑑𝑟 +𝑟 𝑑Ω , (2) thermodynamic quantities corresponding to the black hole horizon and the cosmological horizon are functional with where respect to 𝑀, 𝑄,andΛ. The thermodynamic quantities 2𝑀 𝑄2 Λ corresponding to the black hole horizon are related to the 𝑓 (𝑟) =1− + − 𝑟2. (3) 𝑟 𝑟2 3 ones corresponding to the cosmological horizon. When the thermodynamic property of charged de Sitter spacetime is The above geometry possesses three horizons: the black hole studied, we must consider the relationship with the two Cauchy horizon located at 𝑟=𝑟−,theblackholeevent horizons. Recently, by studying Hawking radiation of de Sitter Advances in High Energy Physics 3 spacetime, [42, 43] obtained that the outgoing rate of the The effective pressure is chargeddeSitterspacetimewhichradiatesparticleswith 2 3 4 𝜔 (1−𝑥) (1+3𝑥+3𝑥 +3𝑥 +𝑥 ) energy is 𝑃 = eff 2 2 2 Δ𝑆 +Δ𝑆 8𝜋𝑟 𝑥 (1+𝑥) (1+𝑥+𝑥 ) Γ=𝑒 + 𝑐 , 𝑐 (7) (16) 2 2 5 6 7 Δ𝑆 Δ𝑆 𝑄 (1−𝑥) (1 + 2𝑥 + 3𝑥 −3𝑥 −2𝑥 −𝑥 ) where + and 𝑐 are Bekenstein-Hawking entropy dif- − , 4 3 3 2 ference corresponding to the black hole horizon and the 8𝜋𝑟𝑐 𝑥 (1+𝑥) (1 − 𝑥 )(1+𝑥+𝑥 ) cosmological horizon after the charged de Sitter spacetime 𝑥:=𝑟 /𝑟 0<𝑥<1 radiates particles with energy 𝜔. Therefore, the thermody- where + 𝑐 and . 𝑄=0 namic entropy of the charged de Sitter spacetime is the sum of From (15)and(16), when , the effective temperature the black hole horizon entropy and the cosmological horizon and pressure are both greater than zero. This fulfills the stable entropy: condition of thermodynamic equilibrium. If considering the black hole horizon and the cosmological one as independent 𝑆=𝑆+ +𝑆𝑐. (8) of each other, because of the different radiant temperatures on the two horizons, the spacetime is instable. Substituting (6)into(8), one can obtain Another problem of considering the black hole horizon 1 1 and the cosmological one as independent of each other is that 𝑑𝑆 = 2𝜋 ( + )𝑑𝑀 when the two horizons coincide, namely, 𝜅+ 𝜅𝑐 (9) 1±√1−4𝑄2Λ 𝜑 𝜑 𝜋 𝑟3 𝑟3 𝑟2 = =𝑟2, (17) −2𝜋( + + 𝑐 )𝑑𝑄+ ( + + 𝑐 ) 𝑑Λ. +/𝑐 2Λ 0 𝜅+ 𝜅𝑐 3 𝜅+ 𝜅𝑐 from (4), the surface gravity 𝜅+/𝑐 =0;thusthetemperature For simplicity, we consider the case with constant. In this case on the black hole horizon and the temperature on the the equation above turns into cosmological horizon are both zero. However, both horizons have nonzero area, which means that the entropy for the two 1 1 𝜋 𝑟3 𝑟3 𝑑𝑆 = 2𝜋 ( + )𝑑𝑀+ ( + + 𝑐 ) 𝑑Λ. horizons should not be zero. This conclusion is inconsistent (10) 𝑟2 =𝑟2 𝜅+ 𝜅𝑐 3 𝜅+ 𝜅𝑐 with Nernst theorem. In the extreme case +/𝑐 0 ,the effective temperature from (15)is From (4)and(5), we derive 1 5𝑄2 3 3 𝑇 = (1 − ). (18) 𝑑𝑀 (𝑟𝑐 /3) 𝑑𝑀 (𝑟+/3) eff 12𝜋𝑟 2𝑟2 𝑑𝑟 = + 𝑑Λ, 𝑑𝑟 = + 𝑑Λ. 0 0 𝑐 𝑟 𝜅 2𝑟 𝜅 + 𝑟 𝜅 2𝑟 𝜅 𝑐 𝑐 𝑐 𝑐 + + + + The effective pressure is (11) 𝑃 =0. (19) Recently, the thermodynamic volume of RN-dS spacetime eff [23, 34]isgivenas In this case the volume-thermodynamic system becomes 4𝜋 area-thermodynamic one. According to (19) the pressure of 𝑉= (𝑟3 −𝑟3 ). thermodynamic membrane is zero. However from (18), the 3 𝑐 + (12) temperature of thermodynamic membrane is nonzero. This Substituting (11)into(12), one gets can partly solve the problem that extreme de Sitter black holes do not satisfy the Nernst theorem, when 𝑄=0,(15)and(16) 4 4 𝑟𝑐 𝑟+ 2𝜋 𝑟𝑐 𝑟+ return to the known result [34]. 𝑑𝑉=4𝜋( − )𝑑𝑀+ ( − ) 𝑑Λ. (13) 𝜅𝑐 𝜅+ 3 𝜅𝑐 𝜅+ 4. Critical Behaviour Substituting (13)into(10), one can derive the thermodynamic equation of thermodynamic quantities in de Sitter spacetime 𝑃 → To compare with the van der Waals equation, we set eff [34]: 𝑃, V → V and discuss the phase transition and the critical phenomena when 𝑟𝑐 is invariant. The van der Waals equation 𝑑𝑀 =𝑇 𝑑𝑆 −𝑃 𝑑𝑉, (14) eff eff is 𝑎 where the effective temperature is (𝑃 + )(V − ̃𝑏) = 𝑘𝑇. V2 (20) (𝑥4 +𝑥3 −2𝑥2 +𝑥+1) 𝑇 = Here, V =𝑉/𝑁is the specific volume of the fluid, 𝑃 is eff 4𝜋𝑟 𝑥 (𝑥+1) (𝑥2 +𝑥+1) 𝑐 its pressure, 𝑇 is its temperature, and 𝑘 is the Boltzmann 2 constant. 𝑄 (15) − Substituting (15)into(16), we obtain 4𝜋𝑟3𝑥3 (𝑥+1) (𝑥2 +𝑥+1) 𝑐 𝐵 𝐵 𝐵 −𝐵 𝐵 𝑃 =𝑇 4 + 2 3 1 4 , ×(1+𝑥+𝑥2 −2𝑥3 +𝑥4 +𝑥5 +𝑥6). eff eff 2 (21) 2𝑟𝑐𝐵2 8𝜋𝑟𝑐 𝑥 (1+𝑥) 𝐵2 4 Advances in High Energy Physics

2 2 3 4 where +𝑄 (30 + 165𝑥 + 456𝑥 + 837𝑥 + 1050𝑥 1+𝑥−2𝑥2 +𝑥3 +𝑥4 + 903𝑥5 + 255𝑥6 − 720𝑥7 − 1707𝑥8 𝐵 = , 1 1+𝑥+𝑥2 − 2113𝑥9 − 1743𝑥10 − 969𝑥11 − 314𝑥12 1+𝑥+𝑥2 −2𝑥3 +𝑥4 +𝑥5 +𝑥6 𝐵 = , 13 14 15 16 17 2 1+𝑥+𝑥2 −9𝑥 + 63𝑥 + 44𝑥 + 21𝑥 +6𝑥 1+2𝑥−2𝑥4 −𝑥5 (22) 18 𝐵 = , +𝑥 )] = 0. 3 2 (1+𝑥+𝑥2) (25)

2 5 6 7 1+2𝑥+3𝑥 −3𝑥 −2𝑥 −𝑥 When 𝑟𝑐 =1,from(15), (16), and (25)onecanobtainthe 𝐵4 = . (1+𝑥+𝑥2)2 position of critical point 𝑐 𝑥 = 0.871992. (26) According to (21), combing dimensional analysis [22] with (12), we conclude that we should identify the specific The critical electric charge, specific volume, temperature, and volume V with the critical pressure are, respectively, 𝑄𝑐 = 0.60395, V𝑐 = 0.128008, V =𝑟 (1−𝑥) . 𝑐 (23) (27) 𝑇𝑐 = 0.00436559, 𝑝𝑐 = 0.0000145468. From the two equations 𝑃 V In Figure 1 we give the figure of eff with the change of 𝜕𝑃 𝜕2𝑃 at the constant effective temperature near the critical point. eff =0, eff =0, (24) 𝑟 𝑟 = 𝜕V 𝜕V2 To reflect the influence of 𝑐 onthespacetime,weset 𝑐 5, from which we derived the position of the critical point we first calculate the position of critical points. Then, one can 𝑐 𝑥 = 0.871992. (28) derive 𝜕𝑃 The critical electric charge, specific volume, temperature, and ( eff ) the critical pressure are, respectively, 𝜕V 𝑇 eff 𝑄𝑐 = 3.01975, V𝑐 = 0.640038, 3 =−1×(8𝜋𝑟𝑐5 (1+𝑥) (1+𝑥+𝑥2) (29) 𝑇𝑐 = 0.000873119, 𝑝𝑐 = 5.8187 × 10−7. −1 2 3 4 5 6 𝑃 V ×(1+𝑥+𝑥 −2𝑥 +𝑥 +𝑥 +𝑥 )) Thediagramoftheeffective eff with the change of is depictedinFigure2. ×[𝑟𝑐2𝑥2 (−6 − 21𝑥 − 38𝑥2 − 45𝑥3 − 30𝑥4 − 10𝑥5 For the van der Waals fluid and the RN-AdS black hole, the relation 𝑃𝑐V𝑐/𝑇𝑐 is a universal number and is independent +6𝑥6 +7𝑥7 +4𝑥8 +𝑥9) of the charge 𝑄. For the RN-dS black hole, according to the effective thermodynamic quantities, numerical calculation 2 2 3 4 5 ×𝑄 𝑥 (15 + 54𝑥 + 98𝑥 + 104𝑥 + 59𝑥 + 18𝑥 shows that 𝑃𝑐V𝑐/𝑇𝑐 still has a 𝑄-independent universal value. Certainly,theuniversalvalueisnomorethan3/8,but −9𝑥6 − 12𝑥7 − 10𝑥8 −4𝑥9 −𝑥10)] = 0 ∼0.0004265.

𝜕2𝑃 ( eff ) 5. Discussion and Conclusions 𝜕V2 𝑇 eff Fromabovewecanfindoutthatthevalueof𝑟𝑐 does 4 𝑥 =1×(4𝜋𝑟𝑐6 (1+𝑥) (1+𝑥+𝑥2) not influence the critical point ; namely, for the RN-dS spacetime, the position of critical point is irrelevant to the −1 value of the cosmological horizon. This indicates that 𝑥= 2 3 4 5 6 2 ×(1+𝑥+𝑥 −2𝑥 +𝑥 +𝑥 +𝑥 ) ) 𝑟𝑐/𝑟+ isfixed;howeverthecriticaleffectivetemperature, critical volume, and critical pressure are dependent on the 2 2 3 4 ×[−𝑟𝑐 𝑥 (6 + 36𝑥 + 115𝑥 + 286𝑥 + 486𝑥 value of 𝑟𝑐. The critical effective temperature and pressure for the RN-dS system will decrease as the values of 𝑟𝑐 increase, + 530𝑥5 + 203𝑥6 − 357𝑥7 − 783𝑥8 while the critical electric charge and the critical volume will increase as the values of 𝑟𝑐 increase. − 854𝑥9 − 666𝑥10 − 375𝑥11 − 127𝑥12 𝑃 − V By Figures 1 and 2,the eff curve at constant temperature of RN-dS spacetime is different from the ones 13 14 15 16 17 +3𝑥 + 32𝑥 + 18𝑥 +6𝑥 +𝑥 ) of van der Waals equation and charged AdS black holes. Advances in High Energy Physics 5

𝑃 − V 𝑇 >𝑇𝑐 0.005 (3) On the eff curves with eff the system lies 𝑃 − V 𝑇 <𝑇𝑐 in a phase; on the eff curves with eff the 0.004 same pressure will correspond to two different values of 𝑥, namely, two-phase coexistence region. Using the equal area criterion we can find out the proportion of 0.003 the two phases for the RN-dS system. From the above discussion, in RN-dS spacetime when 0.002 considering the relation between the two horizons there is eff

P the similar phase transition like the one in van der Waals 0.001 equation and charged AdS black holes. The reason is still unclear. This deserves further study. If the cosmological constant is just the dark energy, the universe will evolve into 0.000 0.2 0.4 0.6 0.8 1.0 a new de Sitter phase. According to Figures 1 and 2 the RN- dSsystemcanevolveintodeSitterphasealongisothermal −0.001 curve. However, along the different isothermal curves, which processes will be the evolution of the universe to a new de −0.002 Sitter phase needs further consideration according to the observations. Figure 1: 𝑃 − V diagram of RN-dS black holes. The temperature eff 𝑐 of isotherms decreases from top to bottom and corresponds to 𝑇 + 𝑐 𝑐 𝑐 0.004, 𝑇 + 0.002, 𝑇 , 𝑇 − 0.002. Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper. ×10−6 3 Acknowledgments 2.5 This work is supported by NSFC under Grant nos. 11175109 and 11075098 and by the Doctoral Sustentation Foundation 2 of Shanxi Datong University (2011-B-03).

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Research Article Researching on Hawking Effect in a Kerr Space Time via Open Quantum System Approach

Xian-Ming Liu1 andWen-BiaoLiu2

1 Department of Physics, Hubei University for Nationalities, Enshi Hube 445000, China 2 Department of Physics, Institute of Theoretical Physics, Beijing Normal University, Beijing 100875, China

Correspondence should be addressed to Xian-Ming Liu; [email protected]

Received 21 January 2014; Accepted 30 January 2014; Published 9 March 2014

Academic Editor: Xiaoxiong Zeng

Copyright © 2014 X.-M. Liu and W.-B. Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP3.

It has been proposed that Hawking radiation from a Schwarzschild or a de Sitter spacetime can be understood as the manifestation of thermalization phenomena in the framework of an open quantum system. Through examining the time evolution of a detector interacting with vacuum massless scalar fields, it is found that the detector would spontaneously excite with a probability the same as the thermal radiation at Hawking temperature. Following the proposals, the Hawking effect in a Kerr space time is investigated in the framework of an open quantum systems. It is shown that Hawking effect of the Kerr space time can also be understood as the the manifestation of thermalization phenomena via open quantum system approach. Furthermore, it is found that near horizon local conformal symmetry plays the key role in the quantum effect of the Kerr space time.

1. Introduction information science, modern quantum optics, atomic and many-body systems, soft condensed matter physics, and Hawking radiation arising from the quantization of matter biophysics. Recently, in the paradigm of open quantum field in a curved background space-time with the event system, based on [15], Yu and Zhang proposed a new insight horizon is a prominent quantum effect. The research to to understand Hawking radiation in a Schwarzschild space understand Hawking radiation, which is related to gen- time [16]. Through examining the time evolution of a detector eral relativity, quantum theory, and thermodynamics, has interacting with vacuum massless scalar field, they got a attracted widespread interest in the physics community. Since conclusion that the detector in both Unruh and Hartle- Hawking’s original derivation of black hole thermal radiation Hawking vacua would spontaneously excite with a nonva- [1–3], several alternative methods have been proposed, such nishing probability the same as Hawking thermal radiation as Damour-Ruffini method [4, 5], the tunneling method [6– from the black hole. This new approach has been extended 11], and gravitational anomaly method [12, 13]. to understand the Gibbons-Hawking effect of de Sitter space- However, from a physical viewpoint, the black hole time [17]. However, there remain some challenges to study thermodynamics system should be more like a nonequilib- Hawking radiation from a generic Kerr space time under rium system rather than an equilibrium system. Hawking the manifestation of thermalization phenomena in an open effectshouldbeinvestigatedintheframeofnonequilibrium quantum system. statistics physics. In quantum mechanics and nonequilibrium Our motivation comes from the fact that the near-horizon statistics physics, the open quantum theory system has gotten geometry plays the key role to the character of a black hole a lot of successful development [14]. The quantum dynamics space time [18–29]. In 1998, Strominger [18]discussedthe of an open quantum system characterized by the effects of near-horizon asymptotic symmetry in a Kerr black hole and decoherence and dissipation cannot be represented in terms found that there was a holographic duality between extremal of a unitary time evolution. It has been applied to quantum and near-extremal Kerr black hole and a 2-dimensional 2 Advances in High Energy Physics

conformal field theory. In [23, 24], it was shown that around simplicity, the Hamiltonian of the detector 𝐻𝑆 may be taken the horizon of a Kerr space time, the scalar field theory can as bereducedtoa2-dimensionaleffectivefieldtheory.Now,the 𝜔 𝐻 = 0 𝜎 , thermal radiation of scalar particles from a Kerr black hole 𝑆 2 3 (1) can be derived on the basis of a conformal symmetry arising from the near-horizon geometry [29]. where 𝜎3 is the Pauli matrix and 𝜔0 is the energy level spacing. Using the near-horizon geometry and open quantum The standard Hamiltonian of massless, free scalar field ina system approach, Hawking effect in a Kerr space time will Kerrspacetimecanbedenotedas𝐻𝐵, which will be discussed be investigated. We will examine the time evolution of a in detail in Section 3. The interaction Hamiltonian of the static detector (modeled by a two-level atom) outside a Kerr detector with the scalar field can be denoted as space time immersed in a vacuum massless scalar field. 󸀠 The dynamics of the detector can be obtained from the 𝐻𝐼 =𝜎3𝜙 (𝑥) . (2) complete time evolution describing the total system (detector 𝑆+𝐵 plus external field) by integrating over the field degrees of Therefore, the Hilbert space of the total system is given H = H ⊗ H freedom. Our results show that the detector would sponta- by the tensor product space 𝑆 𝐵.Thetotal 𝐻(𝑡) neously excite in the Unruh vacuum state with a probability Hamiltonian can be taken as thesameasthethermalradiationatHawkingtemperature, 𝐻=𝐻 ⊗𝐼 +𝐼 ⊗𝐻 +𝜆𝐻󸀠, indicating that Hawking radiation from a Kerr space time 𝑆 𝐵 𝑆 𝐵 𝐼 (3) can be understood as the manifestation of thermalization where 𝜆 is the coupling constant and, 𝐼𝑆 and 𝐼𝐵 denote the phenomena in the framework of open quantum systems. The identity operators in H𝑆 and H𝐵,respectively. conformal invariance of the wave equation near the horizon is Now in order to get the reduced dynamics of the subsys- atthekeypointofHawkingquantumeffectintheKerrspace tem 𝑆, we assume that the interaction between the detector time. and the scalar field is weak as 𝜆 is small and the finite time The organization of our paper is as follows. In Section 2, evolution describing the dynamics of the detector takes the we will review the basic formulae, including the master equa- form of a one-parameter semigroup of completely positive tion describing the system of the detector plus external vac- map. uumscalarfieldintheweak-couplinglimitandthereduced Initially, the complete system is described by the total dynamical equation for the finite time evolution of the 𝜌 =𝜌(0)⊗|0⟩⟨0| 𝜌(0) density matrix tot ,where is the initial detector. In Section 3,thedimensionalreductiontechnique reduced density matrix of the detector and |0⟩ is the Kerr is used to investigate the massless scalar field in a Kerr space space-time vacuum state of field 𝜙(𝑥). In the frame of the time, and the Wightman function is obtained. In Section 4, atom, the evolution in the proper time 𝜏 of the total density applying the method and results of the preceding sections to 𝜌 of the complete system satisfies calculate the probability of a spontaneous transition of the tot detector from the ground state to the excited state outside a 𝜕𝜌 (𝜏) tot =−𝑖𝐿 [𝜌 (𝜏)], (4) Kerr space time. Finally, some discussions and conclusions 𝜕𝜏 𝐻 tot will be given in Section 5. which is often referred to the von Neumann or Liouville- von Neumann equation, where 𝐿𝐻 represents the Liouville operator associated with 𝐻 as follows: 2. Review of the Open Quantum System Approach 𝐿𝐻 [𝑆] ≡ [𝐻,] 𝑆 . (5)

In this section, we will review the open quantum system The dynamics of the detector can be obtained by summing approach to get the master equation describing the combined over the degrees of freedom of the field 𝜙; that is, by applying system 𝐵+𝑆, where a static detector (two-level atom) as an 𝜌 (𝜏) 𝑃 to tot with the trace projection operator as follows: open system 𝑆 which is coupled to another quantum system 𝐵 𝜌 (𝜏) =𝑃[𝜌 (𝜏)]≡ [𝜌 (𝜏)]. ofavacuummasslessscalarfieldinaKerrspacetime.Our tot Tr𝜙 tot (6) derivation mostly follows the works in [15–17]. Here, we will consider the evolution of the static detector in the proper In the limit of weak coupling, we can find that the reduced time and assume the combined system (𝐵+𝑆)tobeinitially density obeys an equation in the Kossakowski-Lindblad form prepared in a factorized state, with the detector keeping static [30–32] as follows: in the exterior region of the Kerr black hole and the field 𝜕𝜌 (𝜏) keeping in vacuum state. The static detector is a two-level =−𝑖[𝐻 ,𝜌(𝜏)]+L [𝜌 (𝜏)], eff (7) simplest quantum system whose Hilbert space is spanned 𝜕𝜏 over just two states, an excited state |+⟩,andagroundstate where |−⟩. The Hilbert space of such a system is equivalent to that of a spin-(1/2) system. So the states of the detector can be 1 3 represented by a 2×2density matrix, which is Hermitian L [𝜌] = ∑ 𝑎𝑖𝑗 [2𝜎𝑗𝜌𝜎𝑖 −𝜎𝑖𝜎𝑗𝜌−𝜌𝜎𝑖𝜎𝑗]. (8) † 2 𝜌 =𝜌, and normalized Tr(𝜌) = 1 with det(𝜌) ⩾.For 0 𝑖,𝑗=1 Advances in High Energy Physics 3

𝑎 𝐻 The matrix 𝑖𝑗 and the effective Hamiltonian eff are deter- Equation (15) can be solved exactly and its solution is mined by the Fourier transform G(𝜆) and Hilbert transform 󵄨 −2H𝜏 󵄨 −2H𝜏 󵄨 K(𝜆) of the vacuum field correlation function (the Wight- 󵄨𝜌 (𝜏)⟩ =𝑒 󵄨𝜌 (0)⟩ + (1−𝑒 ) 󵄨𝜌∞⟩, (17) man function) as follows: where + 𝐺 (𝑥−𝑦) = ⟨0| 𝜙 (𝑥) 𝜙 (𝑦) |0⟩, (9) 0 󵄨 1 −1 󵄨 𝐵 󵄨𝜌∞⟩= H 󵄨𝜂⟩ = − (0), (18) andtheyaredefinedas 2 𝐴 1

−2H𝜏 G (𝜆) = ∫ 𝑑𝜏 𝑒𝑖𝜆𝜏𝐺+ (𝑥 (𝜏)) , the matrix 𝑒 is defined by series expansion as usual. How- ever, H obeys a cubic eigenvalue equation, so powers of H (10) 𝑃 G (𝜔) higher than 2 can always be written in terms of combinations K (𝜆) = ∫ 𝑑𝜔 . 2 𝜋𝑖 𝜔−𝜆 of H , H,and𝐼. Actually, three eigenvalues of H are 𝜆1 = 2𝐴, 𝜆± =(2𝐴+𝐶)±𝑖Ω/2.Wecanwrite 𝑎 The coefficients of the Kossakowski matrix 𝑖𝑗 can be written 4 as 𝑒−2H𝜏 = {𝑒−2𝐴𝜏Λ +2𝑒−2(2𝐴+𝐶)𝜏 Ω2 +4𝐶2 1 𝑎𝑖𝑗 =𝐴𝛿𝑖𝑗 − 𝑖𝐵𝜖𝑖𝑗𝑘 𝛿𝑘3 +𝐶𝛿𝑖3𝛿𝑗3, (11) (Ω𝜏) ×[Λ (Ω𝜏) +Λ sin ]} , 2 cos 3 Ω with (19) 1 𝐴= [G (𝜔 )+G (−𝜔 )], where 2 0 0 (Ω)2 1 Λ =[(2𝐴 +) 𝐶 2 + ]𝐼−2(2𝐴 +) 𝐶 H + H2, 𝐵= [G (𝜔 )−G (−𝜔 )], (12) 1 4 2 0 0 1 𝐶=G (0) −𝐴. Λ =−2𝐴(𝐴+𝐶) 𝐼+(2𝐴 +) 𝐶 H − H2, 2 2 𝐻 (20) The effective Hamiltonian eff contains a correction term, Ω2 the so-called Lamb shift, and one can find that it canbe Λ =2𝐴[ −𝐶(2𝐴 +) 𝐶 ]𝐼 3 4 obtained by replacing 𝜔0 in 𝐻𝑠 with a renormalized energy Ω level spacing as follows: Ω2 +[𝐶(4𝐴 +) 𝐶 − ] H −𝐶H2. Ω 4 𝐻 = 𝜎 =𝜔 +𝑖[K (−𝜔 ) − K (𝜔 )] 𝜎 , (13) eff 2 3 0 0 0 3 Equation (19)revealsthatafreelyfallingatominaKerrspace where a suitable subtraction is assumed in the definition of time is subjected to the effects of decoherence and dissipation K(−𝜔0)−K(𝜔0) to remove the logarithmic divergence which by the exponentially decaying factors including the real parts would otherwise be presented. of the eigenvalues of H and oscillating terms associated with To facilitate the discussion of the properties of solutions the imaginary part. These nonunitary effects can be analyzed for (7)and(8), let us express the density matrix in terms of by examining the evolution behavior in time of suitable atom the Pauli matrices as follows: observable. For any observable of the atom represented by a Hermitian operator O, the behavior of its mean value is 1 3 determined by 𝜌 (𝜏) = (1 + ∑𝜌 (𝜏) 𝜎 ). 2 𝑖 𝑖 (14) 𝑖=1 ⟨O⟩ = Tr [O𝜌 (𝜏)]. (21)

Substituting (14)into(8), the Bloch vector |𝜌(𝜏)⟩ of compo- Let the observable O be an admissible atom state 𝜌𝑓,the nents 𝜌1(𝜏),2 𝜌 (𝜏),3 𝜌 (𝜏) satisfies probability P𝑖→𝑓, that the atom evolves to the expected state represented by density matrix 𝜌𝑓(𝜏) from an initial one 𝜌𝑖 ≡ 𝜕 󵄨 󵄨 󵄨 𝜌(0) 󵄨𝜌 (𝜏)⟩ =−2H 󵄨𝜌 (𝜏)⟩ + 󵄨𝜂⟩ , (15) ,shouldbe 𝜕𝜏 󵄨 󵄨 󵄨 P𝑖→𝑓 (𝜏) = Tr [𝜌𝑓𝜌 (𝜏)]. (22) where |𝜂⟩ denotes a constant vector {0, 0, −4𝐵}. The exact form of the matrix H reads If initially the atom is in the ground state, its Bloch vector |𝜌(0)⟩ {0, 0, −1} 𝜌 Ω is , and the final state 𝑓 is the excited state 2𝐴 + 𝐶 0 givenbytheBlochvector|𝜌𝑓⟩ = {0,0,1}, according to (17)– 2 2 (22), we have H =( − 2𝐴 + 𝐶 0). (16) Ω 1 𝐵 P = (1 − 𝑒−4𝐴𝜏)(1− ). 00𝐴 𝑖→𝑓 2 𝐴 (23) 4 Advances in High Energy Physics

1 𝑎2 The probability per unit time of the transition from the +( − )𝜕2 +𝜕Δ𝜕 2 𝜑 𝑟 𝑟 ground state to the excited state, in the limit of infinitely slow sin 𝜃 Δ switching on and off the atom-field interaction, that is, the spontaneous excitation rate, can be calculated by taking the 1 P (𝜏) 𝜏=0 + 𝜕 𝜃𝜕 ] 𝜙+𝑆 . time derivative of 𝑖→𝑓 at as 𝜃 𝜃 sin 𝜃 int sin ] 󵄨 (27) 𝜕 󵄨 Γ𝑖→𝑓 = P𝑖→𝑓 (𝜏)󵄨 =2𝐴−2𝐵=2G (−𝜔0). (24) 𝜕𝜏 󵄨𝜏=0 Now, we transform the radial coordinate 𝑟 into the tortoise coordinate 𝑟∗ defined by 3. Scalar Wave Equation Near the Event 𝑑𝑟 1 𝑟2 +𝑎2 Horizon in a Kerr Space-Time ∗ = ≡ . (28) 𝑑𝑟 𝐹 (𝑟) Δ 3.1. Dimensional Reduction Near the Horizon. In order to find out how the reduced density evolves with proper time from After the transformation, the action (27)canbewrittenas (7), we will investigate the scalar wave equation of the Kerr 1 space time. In Boyer-Lindquist coordinates, the stationary 𝑆=− ∫ 𝑑𝑟 𝑑𝑡 𝑑𝜃 𝑑𝜑 𝜃𝜙 Kerr space time can be written as 2 ∗ sin

Δ 2 ×[−((𝑟2 +𝑎2)−𝐹(𝑟) 𝑎2 2 𝜃) 𝜕2 𝑑𝑠2 =− (𝑑𝑡 −𝑎 2 𝜃𝑑𝜑) sin 𝑡 𝜌2 sin 2 2 𝐹 (𝑟) 𝑎 2 sin 𝜃 2 2 2 −2𝑎(1−𝐹(𝑟)) 𝜕 𝜕 +( − )𝜕 + [(𝑟 +𝑎 )𝑑𝜑−𝑎𝑑𝑡] 𝑡 𝜑 2 𝑟2 +𝑎2 𝜑 𝜌2 (25) sin 𝜃

2 2 2 𝐹 (𝑟) 𝜌 2 2 2 +𝜕 (𝑟 +𝑎 )𝜕 + 𝜕 𝜃𝜕 ]𝜙+𝑆 . + 𝑑𝑟 +𝜌 𝑑𝜃 , 𝑟∗ 𝑟∗ 𝜃 sin 𝜃 int Δ sin 𝜃 (29) Δ=(𝑟−𝑟)(𝑟 − 𝑟 ) 𝜌2 =𝑟2 +𝑎2 2 𝜃 𝑟 = where + − , cos and ± Now we consider this action in the region near the horizon. 2 2 1/2 𝑀±(𝑀 −𝑎 ) . The parameters 𝑀 and 𝑎 represent the Since 𝐹(𝑟+)=0at 𝑟→𝑟+,weonlyretaindominanttermsin mass and the angular momentum per unit mass of the black (29). We have hole, respectively. The event horizon of the Kerr black hole is 1 located at 𝑟=𝑟+.Thelineelementin(25)isstationaryand 𝜇 𝜇 𝑆=− ∫ 𝑑𝑟∗ 𝑑𝑡 𝑑𝜃 𝑑𝜑 sin 𝜃𝜙 axisymmetric, with 𝜕𝑡 and 𝜕𝜑 as the corresponding Killing 2 vector fields. ×[−(𝑟2 +𝑎2)𝜕2 −2𝑎𝜕𝜕 And then, we will show that the scalar field theory in 𝑡 𝑡 𝜑 (30) the background (25)canbereducedtoa2-dimensional field theory in the near-horizon region with the dimen- 2 𝑎 2 2 2 sional reduction technique. This technique firstly has been − 𝜕 +𝜕𝑟 (𝑟 +𝑎 )𝜕𝑟 ]𝜙, 𝑟2 +𝑎2 𝜑 ∗ ∗ employed for the Kerr black hole by Murata and Soda [24] and developed with a more general technique by Iso et al. [23]. 𝑆 𝐹(𝑟 )=0 𝑟→ The action for the scalar field in a Kerr space timeis wherewehaveignored int by using + at 𝑟+. Because the theory becomes high-energy case near the horizon and the kinetic term dominates, we can ignore all the 1 4 𝜇] 𝑆 𝑆= ∫ 𝑑𝑥 √−𝑔𝑔 𝜕𝜇𝜙𝜕]𝜙+𝑆int, (26) terms in int. After this analysis, we return to the expression 2 written in terms of 𝑟.So,wehave 1 where the first term is the kinetic term and the second term 𝑆=− ∫ 𝑑𝑡 𝑑𝑟 𝑑𝜃𝑑𝜑 𝜃(𝑟2 +𝑎2)𝜙 2 sin 𝑆int represents the mass, potential, and interaction terms. By substituting (25)into(26), we obtain 1 2𝑎 𝑎2 ×[− 𝜕2 − 𝜕 𝜕 − 𝜕2 𝐹 (𝑟) 𝑡 Δ 𝑡 𝜑 Δ(𝑟2 +𝑎2) 𝜑 (31) 2 (𝑟2 +𝑎2) 1 [ 2 2 2 𝑆=− ∫ 𝑑𝑟 𝑑𝑡 𝑑𝜃𝑑𝜑 sin 𝜃𝜙 −( −𝑎 sin 𝜃)𝑡 𝜕 2 Δ +𝜕𝑟𝐹 (𝑟) 𝜕𝑟]𝜙. [

2 2 2𝑎 (𝑟 +𝑎 −Δ) FollowingMurataandSoda’smethod[24], we transform − 𝜕 𝜕 Δ 𝑡 𝜑 the coordinates to the corotating coordinate system. They Advances in High Energy Physics 5 employed a locally corotating coordinate system, and we will Its standard ingoing and outgoing orthonormal mode use a globally corotating coordinate system as solutions are

𝑎 −𝑖𝜎(𝜉+𝑟 ) −𝑖𝜎(𝜉−𝑟 ) −𝑖𝜎V −𝑖𝜎𝑢 𝜓=𝜑− 𝑡, 𝜙(𝜉,𝑟 )∼(𝑒 ∗ ,𝑒 ∗ )∼(𝑒 ,𝑒 ), 𝑟2 +𝑎2 ∗ (40) (32) 𝜉=𝑡. where V =𝜉+𝑟∗,𝑢=𝜉−𝑟∗ are null coordinates. These modes are positive frequency modes with respect to the killing vector Under these new coordinates, we can rewrite the action (31) field 𝜕/𝜕𝜉 for 𝜎>0,andtheysatisfy as 1 𝐿 𝜙=−𝑖𝜎𝜙. 𝑆[𝜙]=− ∫ 𝑑𝜉 𝑑𝑟 𝑑𝜓2 𝑑𝜃(𝑟 +𝑎2) 𝜕/𝜕𝜉 (41) 2 (33) It is obvious that the wave equation (39)ismanifestly 1 2 invariant under the infinite-dimensional group of conformal × sin 𝜃𝜙(− 𝜕𝜉 +𝜕𝑟𝐹 (𝑟) 𝜕𝑟)𝜙, 󸀠 󸀠 𝐹 (𝑟) transformation in two dimensions 𝑢→𝑢(𝑢), V → V (V). so the angular terms disappear completely. Using the spher- In the following, we will show how this conformal symmetry does play the key role in the quantum effect of a Kerr space ical harmonics expansion 𝜙=Σ𝑙,𝑚𝜙𝑙𝑚(𝜉, 𝑟)𝑌𝑙𝑚(𝜃, 𝜓),we obtain the effective 2-dimensional action time. 1 Near the event horizon, we only consider the outgoing 𝑆[𝜙]=∑ ∫ (𝑟2 +𝑎2)𝑑𝜉𝑑𝑟𝜙 modes 2 𝑙𝑚 𝑙,𝑚 1 −𝑖𝜎(𝜉−𝑟 ) 1 −𝑖𝜎𝑢 (34) 𝜙out (𝜉, 𝑟 )= 𝑒 ∗ = 𝑒 , 1 ∗ √4𝜋𝜔 √4𝜋𝜔 (42) ×(− 𝜕2 +𝜕𝐹 (𝑟) 𝜕 )𝜙 , 𝐹 (𝑟) 𝜉 𝑟 𝑟 𝑙𝑚 along the rays 𝑢=constant. Quantizing the field 𝜙out in the where we have used the orthonormal condition for the exterior of the black hole, we can expand it as follows spherical harmonics as follows: 𝜙̂𝐼 = ∑ [𝑎𝐼 𝜙out (𝜉, 𝑟 )+𝑎𝐼†𝜙out† (𝜉, 𝑟 )], ∗ 𝜎 𝜎 ∗ 𝜎 𝜎 ∗ (43) ∫ 𝑑𝜓 𝑑𝜃 sin 𝜃𝑌𝑙󸀠𝑚󸀠 𝑌𝑙𝑚 =𝛿𝑙󸀠,𝑙𝛿𝑚󸀠,𝑚. (35) 𝜎

𝐼 𝐼† From the action (34), it is obvious to find that 𝜙 can be where 𝑎𝜎 and 𝑎𝜎 are the annihilation and creation operators considered as a (1 + 1)-dimensional massless scalar field in acting on the 𝐼 vacuum state, which corresponds to the the backgrounds of the dilaton Φ.Theeffective2-dimensional Boulwarevacuum.TheFockvacuumstatecanbedefinedas 𝐼 metric and the dilaton Φ canbewrittenas 𝑎𝜎|0⟩ = 0. So, with the proper 𝑖𝜖 prescription, the Wightman 𝐼 2 2 1 2 function of statecanbewrittenas 𝑑𝑠 =−𝐹(𝑟) 𝑑𝜉 + 𝑑𝑟 , (36) 𝐹 (𝑟) 1 1 1 1 𝐺𝐼+ (𝑥,󸀠 𝑥 )=− =− , 2 2 Kerr 4𝜋2 0 2 1 2 4𝜋2 (Δ𝜉 − 𝑖𝜖)2 Φ=𝑟 +𝑎 . (37) (𝑥 −𝑖𝜖) −(𝑥 ) (44) So far, we have reduced the 4-dimensional field theory to a 2-dimensional case. This is consistent with [23]. This 0 1 where 𝑥 =𝜉, 𝑥 =𝑟∗. 2-dimensional metric tells us that, near the horizon, the geometry of a Kerr space time can be regarded as a Rindler 3.3. The Unruh Vacuum. In order to define the Unruh space time when 𝑟+ >𝑟−. In the extremal case 𝑟+ =𝑟−,the vacuum state, one can write the Kerr line element in the near- near horizon geometry reduces to 𝐴𝑑𝑆2 which is consistent with [19, 23]. The same as the Schwarzschild space time [33], horizon region in terms of Kruskal-like coordinates defined we will define two vacuum states by using the two natural as notions of time translation of this effective 2-dimensional 𝑈=𝑇−𝑅=−𝜅−1𝑒−𝜅𝑢, metric, namely, the Killing time and the proper time as (45) measured by a congruence of freely falling observers. 𝑉=𝑇+𝑅=𝜅−1𝑒𝜅V,

−1 𝜅𝑟∗ −1 𝜅𝑟∗ 3.2. The Boulware Vacuum. Using the tortoise coordinate in where 𝑇=𝜅 𝑒 sinh 𝜅𝜉, 𝑅=𝜅 𝑒 cosh 𝜅𝜉 and 𝜅=(𝑟+ − 2 2 (28), the effective 2-dimensional metric36 ( ) can be changed 𝑟−)/2(𝑟+ +𝑎 ) is the surface gravity of the event horizon. The into effective 2-dimensional space time (36)becomes 2 2 2 𝑑𝑠𝐼 =−𝐹(𝑟) (𝑑𝜉 +𝑑𝑟∗). (38) 2 2 2 𝑑𝑠𝐼𝐼 =𝐶(𝑟) [−𝑑𝑇 +𝑑𝑅 ], (46) We can see that the (𝜉,∗ 𝑟 ) part of the metric has the form −2𝜅𝑟 of Minkowski metric. Now in this 2-dimensional space time, where 𝐶(𝑟) =𝑒 ∗ 𝐹(𝑟), which is a finite constant near the 𝜙(𝜉, 𝑟 ) thewaveequationof ∗ canbewrittenas event horizon 𝑟=𝑟+.The(𝑇, 𝑅) part of the metric also has 2 2 the form of Minkowski metric. The interval of time Δ𝑇 then [𝜕𝜉 −𝜕𝑟 ]𝜙(𝜉,𝑟∗)≡𝜕𝑢𝜕V𝜙 (𝑢, V) =0. (39) ∗ corresponds to the interval of proper time of a radial freely 6 Advances in High Energy Physics falling observer crossing the horizon. The 2-dimensional According to (12), we have space time in the (𝑇, 𝑅) coordinates is well-behaved near the 𝐴=𝐵=0. event horizon. One can obtain the wave equation of 𝜙(𝑇, 𝑅) (53) as So the spontaneous excitation rate can be obtained as [𝜕2 −𝜕2 ]𝜙(𝑇, 𝑅) ≡𝜕 𝜕 𝜙 (𝑈, 𝑉) =0. 𝑇 𝑅 𝑈 𝑉 (47) Γ𝑖→𝑓 =2(𝐴−𝐵) =0. (54) Similar to previous proceeding, we can obtain the out- Therefore, no spontaneous excitation would ever occur in −𝑖𝜔(𝑇−𝑅) −𝑖𝜔𝑈 going wave solution as 𝜙out(𝑇, 𝑅) ∼𝑒 =𝑒 .These the 𝐼 state-Boulware vacuum. In fact, the Boulware vacuum modes are positive frequency modes with respect to the freely corresponds to our familiar notion of a vacuum state. This falling observer for 𝜔>0,satisfying result is consistent with the conclusion in [16].

𝐿𝜕/𝜕 𝜙 = −𝑖𝜔𝜙. 𝑇 (48) 4.2. The 𝐼𝐼 State-Unruh Vacuum. Using the Wightman function (50) and the relation between the proper time and As pointed in previous subsection, there are conformal coordinate time (51), the Fourier transform can be given as transformations from 𝜙(𝑈, 𝑉) to 𝜙(𝑢, V),as(45). 𝜙out(𝑇, 𝑅) 2𝜋𝜅−1𝜆 Subsequently, quantizing the field in the exte- +∞ 𝜆 𝑒 𝑟 G (𝜆) = ∫ 𝑑𝜏𝑒𝑖𝜆𝜏𝐺𝐼𝐼+ (𝑥, 𝑥󸀠)= , rior of the black hole, we can expand it as follows: Kerr Kerr 2𝜋𝜅−1𝜆 −∞ 2𝜋 𝑒 𝑟 −1 𝜙̂𝐼𝐼 = ∑ [𝑎𝐼𝐼𝜙out (𝑇, 𝑅) +𝑎𝐼𝐼†𝜙out† (𝑇, 𝑅)], (55) 𝜔 𝜔 𝜔 𝜔 (49) 𝜔 2 2 where 𝜅𝑟 =𝜅/√𝐹(𝑟)√ =𝜅 (𝑟 +𝑎 )/Δ. 𝐼𝐼 𝐼𝐼† where 𝑎𝜔 and 𝑎𝜔 are the annihilation and creation operators According to (12), we have acting on the 𝐼𝐼 vacuum state, which can be defined as 𝐼𝐼 1 𝜔0 coth (𝜋𝜔0/𝜅𝑟) 𝑎 |0⟩ = 0. This vacuum state is just the so-called Unruh 𝐴= [G (𝜔 )+G (−𝜔 )]= , 𝜔 2 Kerr 0 Kerr 0 4𝜋 vacuum defined in the maximally extended geometry. So, 𝑖𝜖 1 𝜔 with the proper prescription, the Wightman function of the 𝐵= [G (𝜔 )−G (−𝜔 )]= 0 , 𝐼𝐼 statecanbewrittenas 2 Kerr 0 Kerr 0 4𝜋 1 1 𝐺𝐼𝐼+ (𝑥,󸀠 𝑥 )=− 𝜔0 coth (𝜋𝜔0/𝜅𝑟) 2 2 2 𝐶=G (0) − . Kerr 4𝜋 (𝑥0 −𝑖𝜖) −(𝑥1) Kerr 4𝜋 (50) (56) 1 =− , 2 16𝜋2𝜅−2sinh [(𝜉 − 𝜉󸀠)𝜅/2 − 𝑖𝜖] Using (23)and(24), we have

1 −(𝜔 coth(𝜋𝜔 /𝜅 )/𝜋)𝑡 𝑥0 =𝑇 𝑥1 =𝑅 P = (1 − 𝑒 0 0 𝑟 ) where , . 𝑖→𝑓 2 (57) 1 4. Probability of Spontaneous Transition of ×(1− ), (𝜋𝜔 /𝜅 ) the Detector in a Kerr Space-Time coth 0 𝑟 In what follows, we will calculate the spontaneous excitation and the spontaneous excitation rate is 󵄨 rate in the two vacuum states with the open quantum system 𝜕 󵄨 𝜔 Γ = P (𝑡)󵄨 = 0 , approach. 𝑖→𝑓 𝑖→𝑓 󵄨 2𝜋𝜔 /𝜅 (58) 𝜕𝑡 󵄨𝑡=0 𝜋(𝑒 0 𝑟 −1)

4.1. The 𝐼 State-Boulware Vacuum. Firstly, let us turn to which reveals that, the ground state detector in the 𝐼𝐼 vacuum the 𝐼 state-Boulware vacuum case. Thinking of the relation would spontaneously excite with an excitation rate that one between the proper time and the coordinate time, would expect in the case of a flux of thermal radiation at the temperature √ 𝑑𝜏 = 𝐹 (𝑟)𝑑𝜉, (51) 𝜅 𝑇= 𝑟 . 2𝜋 (59) the Fourier transform of the Wightman function (44)with respect to the proper time can be expressed as It is obvious that there is a near horizon conformal symmetry 𝜙(𝑈, 𝑉) 𝜙(𝑢, V) +∞ from to as (45), which is just the reason of G (𝜆) = ∫ 𝑑𝜏 𝑒𝑖𝜆𝜏𝐺𝐼+ (𝑥, 𝑥󸀠) the nonvanishing spontaneously excitation. This proposal is Kerr Kerr −∞ similar to [29]. In fact, the effective temperature 𝑇 in (59) 𝑇=𝜅/2𝜋=(𝑟 − +∞ 1 1 approaches to Hawking temperature + =−∫ 𝑑𝜉√𝐹 (𝑟)𝑒𝑖𝜆√𝐹(𝑟)𝜉 [ ]=0. 𝑟 )/4𝜋(𝑟2 +𝑎2) 𝑟→∞ 2 2 − + as . This suggests that the thermal −∞ 4𝜋 (Δ𝜉 − 𝑖𝜖) radiation emanating from the horizon of a Kerr black hole is (52) just Hawking radiation, which is in agreement with [16, 17]. Advances in High Energy Physics 7

5. Conclusions and Discussions [7] M. Parikh, “A secret tunnel through the horizon,” International Journal of Modern Physics D, vol. 13, no. 10, pp. 2351–2354, 2004. In brief, we have investigated the Hawking radiation from [8] M. K. Parikh, “Energy conservation andHawking radiation,” a Kerr space time through examining the evolution of a http://arxiv.org/abs/hep-th/0402166. detector (modeled by a two-level atom) interacting with [9] K. Xiao and W. B. Liu, “From Schwarzschild to Kerr under thevacuummasslessscalarfieldintheframeworkofopen de Sitter background due to black hole tunnelling,” Canadian quantum systems. Journal of Physics,vol.85,no.8,pp.863–868,2007. First of all, using the dimensional reduction technique, [10] Q. Dai and W. B. 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On the basis of these, we have calculated the time [13]K.Xiao,W.Liu,andH.B.Zhang,“AnomaliesoftheAchucarro- evolution of the detector in the two vacuum states. It is 𝐼𝐼 Ortiz black hole,” Physics Letters B,vol.647,no.5-6,pp.482–485, found that the detector in the state-Unruh vacuum would 2007. spontaneously excite with a nonvanishing probability the [14] H.-P. Breuer and F. Petruccione, The Theory of Open Quantum same as thermal radiation at Hawking temperature from Systems, Oxford University Press, Oxford, UK, 2002. a Kerr black hole. Hawking-Unruh effect of a Kerr space [15] F. Benatti and R. Floreanini, “Entanglement generation in time can be understood as a manifestation of thermalization uniformly accelerating atoms: reexamination of the Unruh phenomenainanopenquantumsystem.Meanwhile,itisalso effect,” Physical Review A,vol.70,no.1,ArticleID012112,12 foundthattheprobabilityofspontaneoustransitionofthe pages, 2004. detector would be vanishing in the 𝐼 state-Boulware vacuum. [16] H. Yu and J. Zhang, “Understanding Hawking radiation in the It suggests that near horizon conformal symmetry plays the framework of open quantum systems,” Physical Review D,vol. key role in the full quantum phenomena in the Kerr space 77,no.2,ArticleID024031,7pages,2008. time. [17] H. Yu, “Open quantum system approach to the Gibbons- Hawking effect of de Sitter space-time,” Physical Review Letters, Conflict of Interests vol.106,no.6,ArticleID061101,4pages,2011. [18] A. Strominger, “Black hole entropy from near-horizon The authors declare that there is no conflict of interests microstates,” Journal of High Energy Physics, vol. 1998, article regarding the publication of this paper. 009, 1998. [19] J. M. Bardeen and G. T. Horowitz, “Extreme Kerr throat 2 geometry: a vacuum analog of AdS2 × S ,” Physical Review D, Acknowledgments vol.60,no.10,ArticleID104030,10pages,1999. The authors would like to thank Dr. Kui Xiao and Dr. [20] S. 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[28] T. Hartman, W. Song, and A. Strominger, “Holographic derivation of Kerr-Newman scattering amplitudes for general charge and spin,” Journal of High Energy Physics,vol.2010,no.3,article 118, 2010. [29] I. Agullo, J. Navarro-Salas, G. J. Olmo, and L. Parker, “Hawking radiation by Kerr black holes and conformal symmetry,” Phys- ical Review Letters,vol.105,no.21,ArticleID211305,4pages, 2010. [30] F. Benatti, R. Floreanini, and M. Piani, “Environment induced entanglement in Markovian dissipative dynamics,” Physical Review Letters,vol.91,no.7,ArticleID070402,4pages,2003. [31]V.Gorini,A.Kossakowski,andE.C.G.Surdarshan,“Com- pletely positive dynamical semigroups of N level systems,” Journal of Mathematical Physics,vol.17,p.821,1976. [32] G. Lindblad, “On the generators of quantum dynamical semigroups,” Communications in Mathematical Physics,vol.48,no. 2,pp.119–130,1976. [33] N. D. Birrell and P. C. W. Davies, Quantum Fields in Curved Space, vol. 7,Cambridge University Press, Cambridge, UK, 1982. Hindawi Publishing Corporation Advances in High Energy Physics Volume 2014, Article ID 907518, 14 pages http://dx.doi.org/10.1155/2014/907518

Review Article The Geometry of Black Hole Singularities

Ovidiu Cristinel Stoica

Horia Hulubei National Institute for Physics and Nuclear Engineering, 077125 Bucharest, Romania

Correspondence should be addressed to Ovidiu Cristinel Stoica; [email protected]

Received 3 January 2014; Accepted 17 January 2014; Published 3 March 2014

Academic Editor: Christian Corda

Copyright © 2014 Ovidiu Cristinel Stoica. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP3.

Recent results show that important singularities in General Relativity can be naturally described in terms of finite and invariant canonical geometric objects. Consequently, one can write field equations which are equivalent to Einstein’s at nonsingular points but, in addition remain well-defined and smooth at singularities. The black hole singularities appear to be less undesirable thanit was thought, especially after we remove the part of the singularity due to the coordinate system. Black hole singularities are then compatible with global hyperbolicity and do not make the evolution equations break down, when these are expressed in terms of the appropriate variables. The charged black holes turn out to have smooth potential and electromagnetic fields in the new atlas. Classical charged particles can be modeled, in General Relativity, as charged black hole solutions. Since black hole singularities are accompanied by dimensional reduction, this should affect Feynman’s path integrals. Therefore, it is expected that singularities induce dimensional reduction effects in Quantum Gravity. These dimensional reduction effects are very similar to those postulated in some approaches to make Quantum Gravity perturbatively renormalizable. This may provide a way to test indirectly the effects of singularities, otherwise inaccessible.

1. Introduction The first reaction to the singularities was to somehow minimize their importance, on the grounds that they are For millennia, space was considered the fixed background exceptionsduetotheperfectsymmetryofthesolutions.This where physical phenomena took place. Special Relativity hope was ruined by the theorems of Penrose [11, 12]and changed this, by proposing spacetime as the new arena. Then, Hawking [13–16], showing that the singularities are predicted while trying to extend the success of Special Relativity to to occur in GR under very general conditions and are not noninertial frames and gravity, Einstein realized that one caused by the perfect symmetry. should let go the idea of an immutable background, and Singularities, hidden by the event horizon or naked, are General Relativity (GR) was born. There is a very deep inter- very well researched in the literature (e.g., [12, 17–25]and dependence between matter and the geometry of spacetime, references therein). encoded in Einstein’s equation. Its predictions were tested Interesting results concerning singularities were obtained with high accuracy and confirmed. in some modified gravity theories, for example, 𝑓(𝑅) gravity However, the task of decoding the way our universe ([26–30] and references therein). Another way to avoid works from something as abstract as Einstein’s equation is singularities was proposed in nonlinear electrodynamics [31]. not easy, and we are far from grasping all of its consequences. In addition to the singularities, infinities occur in GR For instance, even from the beginning, when Schwarzschild when we try to quantize gravity, because gravity is perturba- proposed his model for the exterior of a spherically sym- tively nonrenormalizable [32, 33]. It is expected by many that metric object, Einstein’s equations led to infinities [1, 2]. The a solution to the problem of quantization will also remove the Schwarzschild metric tensor becomes infinite at 𝑟=0and on singularities. For example, Loop quantum cosmology obtained the event horizon, where 𝑟=2𝑚. The big bang also exhibited significant positive results in showing that quantum effects a singularity [3–10]. may prevent the occurrence of singularities [34–37]. 2 Advances in High Energy Physics

There is another possibility: the problem of singularities Table 1: Singular objects and their nonsingular equivalents. maybeinfactnotduetoGRbuttoourlimitedunderstanding 𝑔 of GR. Therefore, it would be useful to better understand Singular Nonsingular When is Γ𝑐 Γ singularities, even in the eventuality that a better theory will 𝑎𝑏 (2nd) 𝑎𝑏𝑐 (1st) Smooth 𝑑 replace GR. In the following we review some recent results 𝑅𝑎𝑏𝑐 𝑅𝑎𝑏𝑐𝑑 Semiregular showing that by confronting singularities, we realize that they 󵄨 󵄨𝑊 𝑅 𝑅 √󵄨 𝑔󵄨 ,𝑊≤2 are not that undesirable [38]. Moreover, new possibilities 𝑎𝑏 𝑎𝑏 󵄨det 󵄨 Semiregular 𝑊 󵄨 󵄨 open also for the Quantum Gravity problem. 𝑅𝑅√󵄨det 𝑔󵄨 ,𝑊≤2 Semiregular Ric Ric ∘𝑔 Quasi-regular 2. The Problem of Singularities in 𝑅𝑅𝑔∘𝑔Quasi-regular General Relativity

2.1. Two Types of Singularities. Not all singularities are born 𝑅 =𝑅𝑠 𝑅=𝑅𝑠 equal. We can roughly classify the singularities in two types: where 𝑎𝑏 𝑎𝑠𝑏 is the Ricci tensor and 𝑠 is the scalar curvature. (1) Malign singularities: some of the components of the In the case of malign singularities, since some of metric’s metric are divergent: 𝑔𝑎𝑏 →∞. components are singular, the geometric objects like the Levi- Civita connection and the Riemann curvature tensor are (2) Benign singularities: 𝑔𝑎𝑏 are smooth and finite but singular too. Therefore, it seems that the situation of malign det 𝑔→0. singularities is hopeless. Benign singularities turn out to be, in many cases, Even in the case of benign singularities, when the metric manageable [39–41]. The infinities simply disappear, if we is smooth, but its determinant det 𝑔→0,theusual use different geometric objects to write the equations and Riemannian objects are singular. For example, the covariant derivative cannot be defined, because the inverse of the describe the phenomena. At points where the metric is 𝑎𝑏 𝑎𝑏 nondegenerate, the proposed description is equivalent to the metric, 𝑔 ,becomessingular(𝑔 →∞when det 𝑔→0). standard one. But, in addition, it works also at the points This makes Christoffel’s symbols of the second kind (1)and where the metric becomes degenerate. the Riemann curvature (2)singular. Malign singularities appear in the black hole solutions. It is therefore understandable why singularities were They appear to be malign because the coordinates in which considered unsolvable problems for so many years. they are represented are singular. In nosingular coordinates, they become benign [42–44]. This is somewhat similar to the 2.3. From Singular to Nonsingular: A Dictionary. The main case of the apparent singularity on the event horizon, which variables which appear in the equations are indeed singular. turned out to be a coordinate singularity and not a genuine Butwecanreplacethemwithnewvariables,whichare one [45, 46]. equivalent to the original ones on the domain where both are defined. Sometimes, we can choose the new variables so that 2.2. What Is Wrong with Singularities? The geometry of theequationsremainvalidatpointswheretheoriginalones spacetime is encoded in the metric tensor. To write down field were singular. equations, we have to use partial derivatives. In curved spaces, The geometric objects of interest that become singular partial derivatives are replaced by covariant derivatives.They when the metric is degenerate are the Levi-Civita connection are defined with the help of the Levi-Civita connection,which (1), the Riemann curvature (2), and the Ricci and the scalar takes into account the parallel translations, to compare fields curvatures. If the metric is nondegenerate, the Christoffel at infinitesimally closed points. The covariant derivative is symbols of the first kind are equivalent to those of the second written using the Christoffel symbol of the second kind, kind,inthesensethatbyknowingoneofthem,wecan 𝑎 obtained from the metric tensor by obtain the other one. Similarly, the Riemann curvature 𝑅𝑏𝑐𝑑 is equivalent to 𝑅𝑎𝑏𝑐𝑑,andtheRicciandscalarcurvaturesare 1 𝑐 𝑐𝑠 equivalent to their densitized versions and to their Kulkarni- Γ𝑎𝑏 = 𝑔 (𝜕𝑎𝑔𝑏𝑠 +𝜕𝑏𝑔𝑠𝑎 −𝜕𝑠𝑔𝑎𝑏) . (1) 2 Nomizu products (see (30)) with the metric. In some important cases, these equivalent objects remain nonsingular even It can be used to define the Riemann curvature tensor: when the metric is degenerate [39, 41]. We summarize these 𝑑 𝑑 𝑑 𝑑 𝑠 𝑑 𝑠 cases in Table 1. 𝑅𝑎𝑏𝑐 =Γ𝑎𝑐,𝑏 −Γ𝑎𝑏,𝑐 +Γ𝑏𝑠Γ𝑎𝑐 −Γ𝑐𝑠Γ𝑎𝑏. (2)

It plays a major part in the Einstein equation: 3. The Mathematical Methods: Singular Semi-Riemannian Geometry 𝐺𝑎𝑏 +Λ𝑔𝑎𝑏 =𝜅𝑇𝑎𝑏, (3) 3.1. Singular Semi-Riemannian Geometry. We review the since main mathematical tool on which the results presented 1 here are based, named Singular Semi-Riemannian Geometry 𝐺 =𝑅 − 𝑅𝑔 , 𝑎𝑏 𝑎𝑏 2 𝑎𝑏 (4) [39, 40]. Singular Semi-Riemannian Geometry is mainly Advances in High Energy Physics 3 concerned with the study of singular semi-Riemannian man- Its components in local coordinates are just Christoffel’s ifolds. symbols of the first kind:

Definition 1 (see [39, 47]). A singular semi-Riemannian man- 1 K𝑎𝑏𝑐 = K (𝜕𝑎,𝜕𝑏,𝜕𝑐)= (𝜕𝑎𝑔𝑏𝑐 +𝜕𝑏𝑔𝑐𝑎 −𝜕𝑐𝑔𝑎𝑏)=Γ𝑎𝑏𝑐. ifold (𝑀, 𝑔) consists in a differentiable manifold 𝑀 and a 2 symmetric bilinear form 𝑔 on 𝑀,namedmetric tensor or (6) metric. If the metric is nondegenerate, one defines the Levi-Civita If 𝑔 is nondegenerate, then (𝑀, 𝑔) is just a semi- connection uniquely, by raising an index of the Koszul object: Riemannian manifold. If in addition 𝑔 is positive definite, ∇ 𝑌=K(𝑋, 𝑌, )♯. (𝑀, 𝑔) is named Riemannian manifold. In General Relativity 𝑋 − (7) semi-Riemannian manifolds are normally used, but when we But if the metric is degenerate, one cannot raise the are dealing with singularities, it is natural to use the Singular index, and we will have to avoid the usage of the Levi-Civita Semi-Riemannian Geometry, which is more general. connection. Luckily, we can do what we do with the Levi- Civita connection and more, just by using the Koszul object instead. 3.2. Properties of the Degenerate Inner Product. Let (𝑉,𝑔) be ∗ an inner product vector space. Let ♭:𝑉 → 𝑉 be the 𝑢 󳨃→𝑢∙ := ♭(𝑢) =♭ 𝑢 = 𝑔(𝑢, −) 3.3.2. The Covariant Derivatives. We define the lower covari- morphism defined by .We 𝑌 define the radical of 𝑉 as the set of isotropic vectors in 𝑉: ant derivative of a vector field in the direction of a vector ⊥ field 𝑋 by 𝑉∘ := ker ♭=𝑉. We define the radical annihilator space ∙ ∗ of 𝑉 as the image of ♭, 𝑉 := im ♭⊂𝑉.Theinnerproduct ♭ ∙ ♭ ♭ (∇𝑋𝑌) (𝑍) := K (𝑋, 𝑌,) 𝑍 . (8) 𝑔 induces on 𝑉 an inner product, defined by 𝑔∙(𝑢1,𝑢1):= 𝑔(𝑢1,𝑢2).Thisoneistheinverseof𝑔 if and only if det 𝑔 =0̸. 𝑉 := 𝑉/𝑉 Thisisnotquiteatruecovariantderivative,becauseitdoes The coannihilator is the quotient space ∙ ∘,givenby not map vector fields to vector fields but to 1-forms. However, the equivalence classes of the form 𝑢+𝑉∘.Onthecoannihilator ∙ wecanuseittoreplacethecovariantderivativeofvector 𝑉∙,themetric𝑔 induces an inner product 𝑔 (𝑢1+𝑉∘,𝑢2+𝑉∘):= 𝑔(𝑢 ,𝑢 ) fields, and it is equivalent to it if the metric is nondegenerate. 1 2 . If the Koszul object satisfies the condition that Let 𝑝∈𝑀. In the following, we will denote by 𝑇∘𝑝𝑀≤ K(𝑋, 𝑌, 𝑊) =0for any 𝑊∈Γ(𝑇∘𝑀), then the singular semi- 𝑇 𝑀 𝑝 𝑇∙ 𝑀≤𝑇∗𝑀 𝑝 the radical of the tangent space at ,by 𝑝 𝑝 Riemannian manifold (𝑀, 𝑔) is named radical stationary. the radical annihilator and by 𝑇∙𝑝𝑀 the coannihilator. In this case, it makes sense to contract in the third slot of We have seen that one important problem which appears the Koszul object and define by this covariant derivatives when the metric becomes degenerate is that it does not of differential forms. The covariant derivative of differential 𝑎𝑏 admit an inverse 𝑔 , and fundamental tensor operations like forms is defined by raising indices and contractions between covariant indices (∇ 𝜔) (𝑌) := 𝑋 (𝜔 (𝑌)) −𝑔 (∇♭ 𝑌,𝜔), are no longer defined. But we can use the reciprocal metric 𝑋 ∙ 𝑋 (9) 𝑔∙ to define metric contraction between covariant indices, 𝜔∈A∙(𝑀) := Γ(𝑇∙𝑀) for tensors that live in tensor products between 𝑇𝑝𝑀 and if . More general, ∙ the subspace 𝑇𝑝𝑀. This turned out to be enough for some ∇𝑋 (𝜔1 ⊗⋅⋅⋅⊗𝜔𝑠) important singularities in General Relativity. (10) := ∇𝑋 (𝜔1)⊗⋅⋅⋅⊗𝜔𝑠 +⋅⋅⋅+𝜔1 ⊗⋅⋅⋅⊗∇𝑋 (𝜔𝑠).

𝑘 ∙ 3.3. Covariant Derivative. Because at points where the metric The covariant derivative of a tensor 𝑇∈Γ(⊗𝑀𝑇 𝑀) is is degenerate there is no inverse metric, the Levi-Civita defined as connection is not defined. Then, how can we derivate? We (∇ 𝑇) (𝑌 ,...,𝑌 )=𝑋(𝑇(𝑌,...,𝑌 )) willseethatinsomecases,whichturnouttobeenoughfor 𝑋 1 𝑘 1 𝑘 our purposes, we still can derivate. 𝑘 − ∑K (𝑋,𝑖 𝑌 ,∙)𝑇(𝑌1,,...,∙,...,𝑌𝑘). 𝑖=1 3.3.1. The Koszul Object. Let 𝑋, 𝑌, 𝑍 be vector fields on 𝑀.We (11) define the Koszul object as 3.4. Riemann Curvature Tensor: Semi-Regular Manifolds. Let (𝑀, 𝑔) be a radical stationary manifold. Then, the Riemann 1 K (𝑋, 𝑌,) 𝑍 := {𝑋 ⟨𝑌,𝑍⟩ +𝑌⟨𝑍, 𝑋⟩ −𝑍⟨𝑋, 𝑌⟩ curvature tensor is defined as 2 𝑅 (𝑋,𝑌,𝑍,𝑇) − ⟨𝑋 [𝑌,𝑍]⟩ + ⟨𝑌 [𝑍, 𝑋]⟩ + ⟨𝑍 [𝑋, 𝑌]⟩} . ♭ ♭ ♭ (12) (5) =(∇𝑋∇𝑌𝑍) (𝑇) −(∇𝑌∇𝑋𝑍) (𝑇) −(∇[𝑋,𝑌]𝑍) (𝑇) . 4 Advances in High Energy Physics

The components of the Riemann curvature tensor in local 4.1. Einstein’s Equation on Semi-regular Spacetimes coordinates are 4.1.1. The Densitized Einstein Equation. Consider the follow- 𝑅 =𝜕K −𝜕K +(K K − K K ). ing densitized version of the Einstein equation: 𝑎𝑏𝑐𝑑 𝑎 𝑏𝑐𝑑 𝑏 𝑎𝑐𝑑 𝑎𝑐∙ 𝑏𝑑∙ 𝑏𝑐∙ 𝑎𝑑∙ (13) 𝐺 det 𝑔+Λ𝑔det 𝑔=𝜅𝑇det 𝑔, (18) The Riemann curvature tensor has the same symmetry properties as in Riemannian geometry and is radical anni- or, in coordinates or local frames: hilator in each of its slots. A singular semi-Riemannian manifold is called semireg- 𝐺𝑎𝑏 det 𝑔+Λ𝑔𝑎𝑏 det 𝑔=𝜅𝑇𝑎𝑏 det 𝑔. (19) ular [39]if

♭ ∙ If the metric is nondegenerate, this equation is equivalent ∇𝑋∇𝑌𝑍∈A (𝑀) . (14) to the Einstein equation, the only difference is the factor det 𝑔 =0̸. But what happens if the metric becomes degener- An equivalent condition is ate? In this case, it is not allowed to divide by det 𝑔,because this is 0. K (𝑋, 𝑌,∙) K (𝑍, 𝑇,∙) ∈ F (𝑀) . (15) On four-dimensional semi-regular spacetimes Einstein tensor density 𝐺 det 𝑔 is smooth [39]. Hence, the pro- ItiseasytoseethattheRiemanncurvatureofsemiregular posed densitized Einstein equation (18)issmooth,andnon- manifolds is smooth. singular. If the metric is regular, this equation is equivalent to the Einstein equation. 3.5. Examples of Semiregular Semi-Riemannian Manifolds. We present some examples of semi-Riemannian manifolds 4.1.2. FLRW Spacetimes. To better understand black hole [39, 40]. singularities,whichwillbediscussedlater,westartbytakinga look at the Friedmann-Lemaˆıtre-Robertson-Walker (FLRW) 3.5.1. Isotropic Singularities. Isotropic singularities have the singularities, which are benign. Black hole singularities are form malign but can be made benign by removing the coordinate singularity (see Sections 5, 6,and7). 2 𝑔=Ω𝑔,̃ (16) FLRW spacetimes are examples of degenerate warped products, with the metric defined by 𝑔̃ 𝑀 where is a nondegenerate bilinear form on . 2 2 2 2 Such singularities were studied in connection to some d𝑠 =−d𝑡 +𝑎 (𝑡) dΣ , (20) cosmological models [48–56]. where 2 3.5.2. Degenerate Warped Products. Warped products are 2 d𝑟 2 2 2 2 dΣ = +𝑟 (d𝜃 + sin 𝜃d𝜙 ) , (21) products of two semi-Riemannian manifolds (𝐵,𝐵 𝑔 ) and 1−𝑘𝑟2 (𝐹,𝐹 𝑔 ), so that the metric on the manifold 𝐹 is scaled by a 3 3 3 scalar function 𝑓 defined on the manifold 𝐵 [57]. The warped where 𝑘=1for 𝑆 , 𝑘=0for R ,and𝑘=−1for 𝐻 .Itfollows product has the form that they are semiregular. Since the FLRW singularities are warped products, they 2 2 2 2 d𝑠 = d𝑠𝐵 +𝑓 (𝑝) d𝑠𝐹. (17) are semiregular. Therefore, we can expect that the densitized Einstein equation holds. In fact, in [58]moreisshownthan Normally, the warping function 𝑓 is taken to be strictly that, as we will see now. positive at all points of 𝐵. However, it may happen to vanish The FLRW stress-energy tensor is at some points, and in this case the result is a singular semi- 𝑎𝑏 𝑎 𝑏 𝑎𝑏 Riemannian manifold. The resulting manifold is semiregular 𝑇 = (𝜌+𝑝) 𝑢 𝑢 +𝑝𝑔 , (22) [40]. Moreover, if the manifolds 𝐵 and 𝐹 are radical stationary ∙ 𝑎 and if d𝑓∈A (𝑀), their warped product is radical where 𝑢 is the time-like vector field 𝜕𝑡,normalized.The ∙ stationary. If 𝐵 and 𝐹 are semiregular, d𝑓∈A (𝑀),and scalar 𝜌 represents the mass density and 𝑝 the pressure ∙ ∇𝑋d𝑓∈A (𝑀) for any vector field 𝑋,andthen𝐵×𝑓 𝐹 is density. From the stress-energy tensor (22), in the case of a semiregular [40]. homogeneous and isotopic universe, follow the Friedmann equation: 4. Einstein Equations at Singularities 3 𝑎2̇+𝑘 𝜌= , 2 (23) We discuss now two equations which are equivalent to 𝜅 𝑎 Einstein’s when the metric is nondegenerate but remains and the acceleration equation: smooth and finite also at some singularities. The first equation remains smooth at semiregular singularities, while the second 6 𝑎̈ 𝜌+3𝑝=− . at quasi-regular singularities. 𝜅 𝑎 (24) Advances in High Energy Physics 5

Equations (23)and(24) show that the scalars 𝜌 and 4.2.2. The Expanded Einstein Equation. For dimension 𝑛=4, 𝑝 are singular for 𝑎=0.But𝜌 and 𝑝 represent the in [41]weintroducedtheexpanded Einstein equation: mass and pressure densities the orthonormal frame obtained (𝜕 ,𝜕 ,𝜕 ,𝜕 ) by normalizing the comoving frame 𝑡 𝑥 𝑦 𝑧 ,where (𝐺 ∘ 𝑔)𝑎𝑏𝑐𝑑 +Λ(𝑔∘𝑔)𝑎𝑏𝑐𝑑 = 𝜅(𝑇 ∘ 𝑔)𝑎𝑏𝑐𝑑 (31) (𝑥,𝑦,𝑧) arecoordinatesonthespacemanifold𝑆.Themass and pressure density can be identified with the scalars 𝜌 and 𝑝 or, equivalently, only in an orthogonal frame. But at the singularity 𝑎=0there is no orthonormal frame, so we should not normalize the 2𝐸𝑎𝑏𝑐𝑑 −6𝑆𝑎𝑏𝑐𝑑 +Λ(𝑔∘𝑔)𝑎𝑏𝑐𝑑 =𝜅(𝑇∘𝑔)𝑎𝑏𝑐𝑑. (32) comoving frame. In general, nonnormalized case, the actual −𝑔(=3 𝑎 𝑔 ) densities contain in fact the factor √ √ Σ : It is equivalent to Einstein’s equation if the metric is nondegenerate but in addition extends smoothly at quasi- 3 𝜌=𝜌̃ √−𝑔=𝜌𝑎√𝑔Σ, regular singularities. (25) 𝑝=𝑝̃ √−𝑔=𝑝𝑎3√𝑔 . Σ 4.2.3. Examples of Quasi-Regular Singularities. As shown in [41], the following are examples of quasi-regular singularities: The Friedmann and the acceleration equations become (i) isotropic singularities, 3 2 𝜌=̃ 𝑎(𝑎̇+𝑘)√𝑔Σ, 𝜅 (ii) degenerate warped products 𝐵×𝑓 𝐹 with dim 𝐵=1 (26) 𝐹=3 6 and dim , 𝜌+3̃ 𝑝=−̃ 𝑎2𝑎√̈𝑔 . 𝜅 Σ (iii) FLRW singularities, as a particular case of degenerate warped products [62], We see that 𝜌̃ and 𝑝̃ are smooth and so is the densitized (iv) Schwarzschild singularities (after removing the coor- stress-energy tensor: dinates singularity, see Section 5). The question whether the Reissner-Nordstrom¨ and Kerr-Newman 𝑇𝑎𝑏√−𝑔=(𝜌+̃ 𝑝)̃ 𝑎 𝑢 𝑢𝑏 + 𝜌𝑔̃ 𝑎𝑏. (27) singularities are quasi-regular, or at least semi-regular, is still open. We obtain a densitized Einstein equation, from which (18) follows by multiplying with √−𝑔. Hence, the FLRW solution is described by smooth den- 4.2.4. The Weyl Curvature Hypothesis and Quasi-Regular sities even at the big bang singularity. Moreover, the solution Singularities. To explain the low entropy at the big bang and extends beyond the singularity. the high homogeneity of the universe, Penrose emitted the Weyl curvature hypothesis, stating that the Weyl curvature tensor vanishes at the big bang singularity [18]. 4.2.Einstein’sEquationonQuasi-RegularSpacetimes From (28), the Weyl curvature tensor is (𝑀, 𝑔) 𝑛 4.2.1. The Ricci Decomposition. Let be an - 𝐶 =𝑅 −𝑆 −𝐸 . dimensional semi-Riemannian manifold. The Riemann 𝑎𝑏𝑐𝑑 𝑎𝑏𝑐𝑑 𝑎𝑏𝑐𝑑 𝑎𝑏𝑐𝑑 (33) curvature decomposes algebraically [59–61]as In [63]itwasshownthatwhenapproachingaquasi- regular singularity, 𝐶𝑎𝑏𝑐𝑑 →0smoothly. Because of this, 𝑅𝑎𝑏𝑐𝑑 =𝑆𝑎𝑏𝑐𝑑 +𝐸𝑎𝑏𝑐𝑑 +𝐶𝑎𝑏𝑐𝑑, (28) any quasi-regular big bang satisfies the Weyl curvature where hypothesis. In [63]ithasalsobeenshownthataverylarge class of big bang singularities, which are not homogeneous or 1 𝑆 = 𝑅(𝑔 ∘ 𝑔) , isotropic, are quasi-regular. 𝑎𝑏𝑐𝑑 𝑛 (𝑛−1) 𝑎𝑏𝑐𝑑 1 4.3. Taming a Malign Singularity. We have seen that when 𝐸 = (𝑆 ∘ 𝑔) , (29) 𝑎𝑏𝑐𝑑 𝑛−2 𝑎𝑏𝑐𝑑 the singularity is benign; that is, the singularity is due to the degeneracy of the metric tensor, which is smooth; there are 1 𝑆 := 𝑅 − 𝑅𝑔 , important cases when we can obtain a complete description 𝑎𝑏 𝑎𝑏 𝑛 𝑎𝑏 of the fields and their evolution, in terms of finite quantities. But what can we do if the singularities are malign? This where ∘ denotes the Kulkarni-Nomizu product: case is important, since all black hole singularities are malign. In [42–44] we show that although the black hole singularities (ℎ∘𝑘)𝑎𝑏𝑐𝑑 := ℎ𝑎𝑐𝑘𝑏𝑑 −ℎ𝑎𝑑𝑘𝑏𝑐 +ℎ𝑏𝑑𝑘𝑎𝑐 −ℎ𝑏𝑐𝑘𝑎𝑑. (30) appear to be malign, we can make them benign, by a proper choice of coordinates. This is somewhat analog to the method If the Riemann curvature tensor on a semiregular man- used in [45, 46] to show that the event horizon singularity ifold (𝑀, 𝑔) admits such a decomposition so that all of its is not a true singularity, being due to coordinates. In the terms are smooth, (𝑀, 𝑔) is said to be quasi-regular. following sections, we will review these results. 6 Advances in High Energy Physics

5. Schwarzschild Singularity Is Semi-Regular where

2𝑆 𝑆 2 The Schwarzschild metric is given in Schwarzschild coordi- Δ:=𝜌 − 2𝑚𝜌 +𝑞 . (41) nates by To remove the infinity of the metric at 𝑟=0and ensure 2𝑚 2𝑚 −1 𝑠2 =−(1− ) 𝑡2 + (1− ) 𝑟2 +𝑟2 𝜎2, (34) analiticity, we have to choose d 𝑟 d 𝑟 d d 𝑆 ≥ 1, 𝑇 ≥ 𝑆 +1. (42) where 2 2 2 2 (𝑡,𝑟,𝜙,𝜃) d𝜎 = d𝜃 + sin 𝜃d𝜙 . (35) In the Reissner-Nordstrom¨ coordinates ,the electromagnetic potential is singular at 𝑟=0, Let us change the coordinates to 𝑞 2 4 𝐴=− d𝑡. (43) 𝑟=𝜏,𝑡=𝜉𝜏. (36) 𝑟 The four-metric becomes But in the new coordinates (𝜏,𝜌,𝜙,𝜃), the electromagnetic 4 potential is 2 4𝜏 2 2 4 2 4 2 d𝑠 =− d𝜏 + (2𝑚 − 𝜏 )𝜏 (4𝜉d𝜏+𝜏d𝜉) +𝜏 d𝜎 , 𝑇−𝑆−1 2𝑚 − 𝜏2 𝐴=−𝑞𝜌 (𝜌d𝜏+𝑇𝜏d𝜌) , (44) (37) and the electromagnetic field is which is analytic and semiregular at 𝑟=0[42]. 𝑇−𝑆−1 The problems were fixed by a coordinate change. Does not 𝐹=𝑞(2𝑇 −) 𝑆 𝜌 d𝜏∧d𝜌, (45) this mean that the singularity depends on the coordinates? Well, this deserves an explanation. Changing the coordinates and they are analytic everywhere, including at the singularity does not make a singularity appear or disappear, if the 𝜌=0[43]. coordinate transformation is a local diffeomorphism. But The proposed coordinates define a space + time foliation a regular tensor can become singular or a singular tensor only if 𝑇≥3𝑆[43]. can become regular, if the coordinate transformation itself is singular. This situation is very similar to that of the event horizon singularity 𝑟=2𝑚of the Schwarzschild metric, 7. Rotating Black Holes in Schwarzschild coordinates (34). This singularity vanishes Electrically neutral rotating black holes are represented by the when we go to the Eddington-Finkelstein coordinates. This Kerr solution. If they are also charged, they are described by proves that the Eddington-Finkelstein coordinates are from the very similar Kerr-Newman solution. thecorrectatlas,whiletheoriginalSchwarzschildcoordinates 3 Consider the space R × R ,whereR represents the time were in fact singular at 𝑟=2𝑚.Inourcase,thecoordinate 3 coordinate and R the space, parameterized by the spherical transformation (36)allowsustomovetoanatlasinwhich coordinates (𝑟, 𝜙, 𝜃). The rotation is characterized by the themetricisanalyticandsemiregular,showingthatthe parameter 𝑎≥0, 𝑚≥0is the mass, and 𝑞∈R the charge. Schwarzschild coordinates were in fact singular at 𝑟=0. The following notations are useful:

2 2 2 6. Charged and Nonrotating Black Holes Σ (𝑟,) 𝜃 := 𝑟 +𝑎 cos 𝜃, (46) Charged nonrotating black holes are described by the Δ (𝑟) := 𝑟2 − 2𝑚𝑟 +𝑎2 +𝑞2. Reissner-Nordstrom¨ metric: 2𝑚 𝑞2 The nonvanishing components of the Kerr-Newman metric 𝑠2 =−(1− + ) 𝑡2 d 𝑟 𝑟2 d are [64] 2 2 −1 (38) Δ (𝑟) −𝑎 sin 𝜃 2𝑚 𝑞2 𝑔 =− , +(1− + ) 𝑟2 +𝑟2 𝜎2. 𝑡𝑡 Σ (𝑟,) 𝜃 𝑟 𝑟2 d d Σ (𝑟,) 𝜃 𝑔 = , Tomakethesingularitybenign,wechoosethenew 𝑟𝑟 Δ (𝑟) coordinates 𝜌 and 𝜏 [43]; so that 𝑔𝜃𝜃 =Σ(𝑟,) 𝜃 , 𝑇 𝑆 (47) 𝑡=𝜏𝜌 ,𝑟=𝜌. (39) 2 2 2 2 2 (𝑟 +𝑎 ) −Δ(𝑟) 𝑎 sin 𝜃 In the new coordinates, the metric has the following form: 𝑔 = 2𝜃, 𝜙𝜙 Σ (𝑟,) 𝜃 sin 𝑆2 2 2𝑇−2𝑆−2 2 4𝑆−2 2 2𝑆 2 2 2 2 d𝑠 =−Δ𝜌 (𝜌d𝜏+𝑇𝜏d𝜌) + 𝜌 d𝜌 +𝜌 d𝜎 , 2𝑎 sin 𝜃(𝑟 +𝑎 −Δ(𝑟)) Δ 𝑔 =𝑔 =− . (40) 𝑡𝜙 𝜙𝑡 Σ (𝑟,) 𝜃 Advances in High Energy Physics 7

Future extended solution

Future benign singularity

Inside the black hole

Universe, outside Parallel universe the black hole

Inside the white hole

Past benign singularity

Past extended solution

Figure 1: Schwarzschild solution, analytically extended beyond the 𝑟=0singularity.

In [44]itwasshownthatinthecoordinates𝜏, 𝜌,and𝜇, 8. Global Hyperbolicity and Information Loss are defined by 8.1. Foliations with Cauchy Hypersurfaces. While Einstein’s equation describes the relation between geometry and matter 𝑡=𝜏𝜌T ,𝑟=𝜌S, in a block-world view of the universe, there are equivalent (48) 𝜙=𝜇𝜌M,𝜃=𝜃, formulations which express this relation from the perspective of the time evolution. Einstein’s equation can be expressed in termsofaCauchyproblem[65–70]. where S, T, M ∈ N are positive integers so that The standard black hole solutions pose two main prob- lemstotheCauchyproblem.First,thesolutionshavemalign S ≥1, singularities. Second, they have in general Cauchy horizons. Luckily,thereismorethanonewaytoskinablackhole. T ≥ S +1, (49) The evolution equations make sense at least locally, if the singularities are benign. The black hole singularities appear M ≥ S +1, to be malign in the coordinates used so far, but by removing the coordinate’s contribution to the singularity, they become and the metric is analytic. benign. Even so, to formulate initial value problems globally, Not only the metric becomes analytic in the proposed spacetime has to admit space + time foliations. The space- coordinates, but also the electromagnetic potential and elec- like hypersurfaces have to be Cauchy surfaces; in other tromagnetic field. The electromagnetic potential of the Kerr- words, the global hyperbolicity condition has to be true. Newman solution is, in the standard coordinates, the 1-form: The topology of the space-like hypersurfaces must remain independent on the time 𝑡, although the metric is allowed 𝑞𝑟 2 to become degenerate. This seems to be prevented in the 𝐴=− (d𝑡−𝑎sin 𝜃d𝜙) . Σ (𝑟,) 𝜃 (50) case of Reissner-Nordstrom¨ and Kerr-Newman black holes, by the existence of Cauchy horizons. As shown in [71], Intheproposedcoordinates the stationary black hole singularities admit such foliations and are therefore compatible with the condition of global hyperbolicity. 𝑞𝜌S 𝐴=− (𝜌T 𝜏+T𝜏𝜌T−1 𝜌−𝑎 2 𝜃𝜌M 𝜇) . (51) Σ (𝑟,) 𝜃 d d sin d 8.2. Schwarzschild Black Holes. In the proposed coordinates which is smooth [44]. The electromagnetic field 𝐹=d𝐴 is for the Schwarzschild black hole, the metric extends analyti- smooth too. cally beyond the 𝑟=0singularity (Figure 1). 8 Advances in High Energy Physics

Parallel universe, Parallel universe, outside the black outside the black hole hole (a) Singularity Parallel universe, Singularity outside the black hole Cauchy Inside the black hole horizons Singularity Parallel universe, Our universe, Our universe, outside the black outside the black outside the black hole hole hole

(b) (c)

2 2 2 2 2 2 Figure 2: Reissner-Nordstrom¨ black holes. (a) Naked solutions (𝑞 >𝑚). (b) Extremal solution (𝑞 =𝑚). (c) Solutions with 𝑞 <𝑚.

This solution can be foliated in space + time and therefore that Quantum Gravity would naturally cure this problem, but is globally hyperbolic. it has been suggested that in fact it would make the problem exist even in the absence of black holes [95]. 8.3.Space-LikeFoliationoftheReissner-Nordstrom¨ Solution. Since the extended Schwarzschild solution can be foliated Figure 2 shows the standard Penrose diagrams for the Reiss- in space + time (Sections 5 and 8.2),itcanbeusedto ner-Nordstrom¨ spacetimes [72]. represent evaporating electrically neutral nonrotating black 𝑟=0 The Penrose diagram 3 shows how our extensions beyond holes. The solution can be analytically extended beyond , the singularities allow the Reissner-Nordstrom¨ solutions to and hence the affirmation that the information is lost at the be foliated in Cauchy hypersurfaces. In Figures 3(b) and 3(c), singularity is no longer supported. In Figure 4,itcanbeseen in addition to extending the solution beyond the singularity, that our solution extends through the singularity and allows we cut out the spacetime along the Cauchy horizons. This is the existence of globally hyperbolic spacetimes containing justified if the black holes form by collapse at a finite timeand evaporating black holes. then evaporate after a finite lifetime [43, 71]. For the Kerr-Newman black holes, the foliations are 9. Possible Experimental Consequences and similartothosefortheReissner-Nordstrom¨ solutions [71], Quantum Gravity especially because the extension proposed in [44]canbe chosen so that the closed time-like curves disappear. 9.1. Can We Do Experiments with Singularities? We reviewed the foundations of Singular General Relativity (SGR) and 8.4. Black Hole Information Paradox. Bekenstein and Hawk- its applications to black hole singularities. SGR is a natural ingdiscoveredthatblackholesobeylawssimilartothose extension of GR, but, nevertheless, it would be great to be of thermodynamics and proposed that these laws are in fact able to submit it to experimental tests. We have seen that thermodynamics (see [73–75], also [76, 77], and references the solutions are the same as those predicted by Einstein’s therein). Hawking realized that black holes evaporate, and equation, as long as the metric is nondegenerate. The only the radiation is thermal. This led him to the idea that differences appear where the metric is degenerate, at singu- after evaporation, the information is lost [78–80]. Many larities. But how can we go to the singularities, or how can we solutions were proposed, such as [81–94]. It was proposed generate singularities, and test the results at the singularities? Advances in High Energy Physics 9

Benign singularity

Space-like hypersurface (b) (a)

Space-like hypersurface (c)

Figure 3: Reissner-Nordstrom¨ black hole solutions, extended beyond the singularities and restricted to globally hyperbolic regions. (a) Naked 2 2 2 2 2 2 solutions (𝑞 >𝑚). (b) Extremal solution (𝑞 =𝑚). (c) Solutions with 𝑞 <𝑚.

Malign singularity

Benign singularity

(a) (b)

Figure 4: (a) Standard evaporating black hole, whose singularity destroys the information. (b) Evaporating black hole extended through the singularity preserves information and admits a space + time foliation. 10 Advances in High Energy Physics

How could we design an experimental apparatus which is not did this, by modifying General Relativity. In this section destroyed by the singularity? It seems that a direct experiment we point that several types of dimensional reduction, which to test the predictions of SGR is not possible. were postulated by various authors, occur naturally at our What about indirect tests? For example, if information semiregular and quasi-regular singularities [96]. is preserved, this would be evidence in favor of SGR. But how can we test this? Can we monitor a black hole, from 9.3.1. Geometric Dimensional Reduction. First, at each point thetimewhenitisformedtothetimewhenitevaporates where the metric becomes degenerate, a geometric or metric completely, and check that the information is preserved reduction takes place, because the rank of the metric is during this entire process? The current knowledge predicts reduced: that this information will be anyway extremely scrambled. Evenifwewouldbeabletodothissomeday,theconservation ∙ dim 𝑇𝑝∙𝑀=dim 𝑇𝑝𝑀=rank 𝑔𝑝. (52) of information is predicted by a long list of other approaches to Hawking’s information loss paradox (see Section 8.4). In General Relativity, classical elementary particles can 9.3.2. Topological Dimensional Reduction. From the Kupeli be considered small black holes. If they are pointlike and theorem [47] follows that for constant signature, the manifold 𝑀=𝑃×𝑁 have definite trajectories, then they are singularities, like is locally a product 0 between a manifold of 𝑃 𝑁 0 the Schwarzschild, Reissner-Nordstrom,¨ and Kerr-Newman lower dimension and another manifold with metric . singularities. To go from classical to quantum, one applies In other words, from the viewpoint of geometry, a region path integrals over the classical trajectories. In this way, where the metric is degenerate and has constant signature can possible effects of the singularities may also be present at the be identified with a lower dimensional space. This suggests points where the metric is nonsingular. aconnectionwiththetopological dimensional reduction In [96] we suggested that the geometric and topological explored by Shirkov and Fiziev [110–114]. properties we identified at singularities have implications to Quantum Gravity (QG), as we shall see in the following. This 9.3.3. Vanishing of Gravitons. If the singularity is quasi regu- suggeststhatitmightbepossibletotestourapproachby lar, the Weyl tensor 𝐶𝑎𝑏𝑐𝑑 →0as approaching a quasi- QG effects. One feature that seems to be required by most, if regular singularity. This implies that the local degrees of free- not all approaches to QG, is dimensional reduction. Singular dom, that is, the gravitational waves for GR and the gravitons General Relativity shows that singularities are accompanied for QG, vanish, allowing by this the needed renormalizability in a natural way by dimensional reduction. [116].

9.2. Dimensional Reduction in QFT And QG. Various results 9.3.4. Anisotropy between Space and Time. In [43]we obtained in Quantum Field Theory (QFT) and in QG suggest obtained new coordinates, which make the Reissner-Nord- thatatsmallscalesadimensionalreductionshouldtakeplace. strom¨ metric analytic at the singularity. In these coordinates, The definition and the cause of this reduction differ from one the metric is given by (40). A charged particle with spin 0 can approach to another. Here is just a small part of the literature be viewed, at least classically, as a Reissner-Nordstrom¨ black using one form of dimensional reduction or another to obtain hole. The above metric reduces its dimension to dim =2. regularization in QFT and QG: To admit space + time foliation in these coordinates, we should take 𝑇≥3𝑆. An open research problem is whether (i) fractal universe [97, 98],basedonaLebesgue-Stieltjes this anisotropy is connected to the similar anisotropy from measure or a fractional measure [99], fractional Hoˇrava-Lifschitz gravity, introduced in [120]. calculus, and fractional action principles [100–109]; (ii) topological dimensional reduction [110–114]; 9.3.5. Measure Dimensional Reduction. In the fractal universe (iii) vanishing dimensions at LHC [115]; approach [97, 98, 127], one expresses the measure in the (iv) dimensional reduction in QG [116–118]; integral (v) asymptotic safety [119]; ̆ 𝑆=∫ d󰜚 (𝑥) L, (53) (vi) Horava-Lifschitz gravity [120]; M (vii) other approaches to Quantum Gravity based on 𝑓 (𝑥) dimensional reduction including [121–126]. in terms of some functions (𝜇) ,someofthemvanishing at low scales: Some of these types of dimensional reduction are very similar to those predicted by SGR to occur at benign singu- 𝐷−1 󰜚 (𝑥) = ∏𝑓 (𝑥) 𝑥𝜇. larities. d (𝜇) d (54) 𝜇=0

9.3. Is Dimensional Reduction due to the Benign Singularities? In Singular General Relativity, Quantum Gravity is perturbatively nonrenormalizable, but it canbemaderenormalizablebyassumingonekindoranother 𝐷 of dimensional reduction. The above mentioned approaches d󰜚 (𝑥) = √− det 𝑔d𝑥 . (55) Advances in High Energy Physics 11

𝜇 If the metric is diagonal in the coordinates (𝑥 ),thenwecan There is a rich literature concerning gravity, black holes, take and singularities in lower or higher dimensions (see e.g., [76, 128–130] and references therein). While the geometric 󵄨 󵄨 apparatus of Singular Semi-Riemannian Geometry reviewed 𝑓 (𝑥) = √󵄨𝑔 (𝑥)󵄨. (56) (𝜇) 󵄨 𝜇𝜇 󵄨 in Section 3 worksforotherdimensionstoo,inthisreview we focused only on four-dimensional spacetimes, and some This suggests that the results obtained by Calcagni by of the results do not work in more dimensions. considering the universe to be fractal follow naturally from thebenignmetrics. Conflict of Interests

9.4. Dimensional Reduction and Quantum Gravity. The Sin- The author declares that there is no conflict of interests gular General Relativity approach leads, as a side effect, section regarding the publication of this paper. to various types of dimensional reduction, which are similar to those proposed in the literature to make Quantum Acknowledgment Gravity perturbatively renormalizable. By investigating the nonrenormalizability problems appearing when quantizing The author thanks an anonymous referee for the valuable gravity, many researchers were led to the conclusion that the suggestions to improve the completeness of this review. problem would vanish if one kind of dimensional reduction or another is postulated (sometimes ad hoc). By contrary, our References approach led to this as a natural consequence of understanding the singularities. [1] V. K. Schwarzschild, “Uber¨ das gravitationsfeld einer kugel Ofcourse,inSGRthedimensionalreductionappearsat aus inkompressibler flussigkeit¨ nach der Einsteinschen theorie,” the singularity, while QG is expected to be perturbatively Sitzungsberichte der Preussischen Akademie der Wissenschaften, renormalizable everywhere. But if classical particles are sin- vol. K1, pp. 424–434, 1916. gularities, quantum particles behave like sums over histories [2] V. K. Schwarzschild, “Uber¨ das gravitationsfeld eines massen- of classical particles. Thus, at any point there will be virtual punktes nach der Einsteinschen theorie,” Sitzungsberichte der singularitiestocontributetotheFeynmanintegrals.This Preussischen Akademie der Wissenschaften,vol.K1,pp.189–196, means that the effects will be present everywhere. They are 1916. expected as a reduction of the determinant of the metric, [3] A. Friedman, “Uber die krummung¨ des raumes,” Zeitschriftur f¨ and of the Weyl curvature tensor, which allows the desired Physik,vol.10,no.1,pp.377–386,1922. regularization. Moreover, as the energy increases, the order [4] A. Friedmann, “On the curvature of space,” General Relativity of the Feynman diagrams in the same region increases, and and Gravitation,vol.31,no.12,pp.1991–2000,1999. we expect that the dimensional reduction effects induced [5] A. Friedmann, “Uber¨ die moglichkeit¨ einer welt mit konstanter by singularities become more significant too. It is an open negativer krummung¨ des raumes,” Zeitschriftur f¨ Physik,vol.21, question at this time whether this dimensional reduction is no. 1, pp. 326–332, 1924. enough to regularize gravity, but this research is just at the [6] G. Lemaˆıtre, “Un Univers homogene` de masse constante et beginning. derayoncroissantrendantcomptedelavitesseradialedes nebuleuses´ extra-galactiques,” Annales de la Societ´ e´ Scientifique de Bruxelles,vol.47,pp.49–59,1927. 10. Conclusions [7]H.P.Robertson,“Kinematicsandworld-structure,”The Astro- We reviewed some of our results of Singular General Rel- physical Journal,vol.82,p.284,1935. ativity [38], concerning the black hole singularities. Some [8] H. P. Robertson, “Kinematics and world-structure II,” The singularities allow the canonical and invariant construction Astrophysical Journal,vol.83,p.187,1936. of geometric objects which remain smooth and nonsingular. [9] H. P. Robertson, “Kinematics and world-structure III,” The By using these objects, one can write equations which are Astrophysical Journal,vol.83,p.257,1936. equivalent to Einstein’s equations outside singularities but [10] A. G. Walker, “On Milne’s theory of world-structure,” Proceed- in addition extend smoothly at singularities. The FLRW big ings of the London Mathematical Society,vol.s2-42,no.1,pp. bang singularities turn out to be of this type. The black 90–127, 1937. hole singularities can be made so by removing the coor- [11] R. Penrose, “Gravitational collapse and space-time singulari- dinate singularity for the charged black hole singularities, ties,” Physical Review Letters,vol.14,no.3,pp.57–59,1965. the electromagnetic potential and field become smooth. The [12] R. Penrose, “Gravitational collapse: the role of general relativity,” singularities of the black hole having a finite life span are com- Revista del Nuovo Cimento,vol.1,pp.252–276,1969. patible with global hyperbolicity and conservation of infor- [13] S. W. Hawking, “The occurrence of singularities in cosmology,” mation. Such singularities are accompanied by dimensional Proceedings of the Royal Society A,vol.294,no.1439,pp.511–521, reduction, a feature which is desired by many approaches to 1966. Quantum Gravity. While in these approaches dimensional [14] S. W. Hawking, “The occurrence of singularities in cosmology. reduction is obtained by modifying General Relativity, these II,” Proceedings of the Royal Society A,vol.295,no.1443,pp.490– singularities lead naturally to it, within the framework of GR. 493, 1966. 12 Advances in High Energy Physics

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Research Article Intermediate Mass Black Holes: Their Motion and Associated Energetics

C. Sivaram1 and Kenath Arun2 1 Indian Institute of Astrophysics, Bangalore 560 034, India 2 Christ Junior College, Bangalore 560 029, India

Correspondence should be addressed to Kenath Arun; [email protected]

Received 29 November 2013; Revised 17 January 2014; Accepted 20 January 2014; Published 27 February 2014

Academic Editor: Christian Corda

Copyright © 2014 C. Sivaram and K. Arun. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP3.

There is a lot of current astrophysical evidence and interest in intermediate mass black holes (IMBH), ranging from a few hundred to several thousand solar masses. The active galaxy M82 and the globular cluster G1 in M31, for example, are known to host such objects. Here, we discuss several aspects of IMBH such as their expected luminosity, spectral nature of radiation, and associated jets. We also discuss possible scenarios for their formation including the effects of dynamical friction, and gravitational radiation. We also consider their formation in the early universe and also discuss the possibility of supermassive black holes forming from mergers of several IMBH and compare the relevant time scales involved with other scenarios.

1. Introduction The Bondi accretion rate for a black hole (of mass M moving with a velocity V in a medium of density 𝜌=𝑛𝑚𝑃), Intermediate mass black holes (IMBH) are those black holes is given by [2]: having mass between that of stellar black holes and supermas- 4 2 sive black holes, that is, in the range of 500 to 10 MΘ.Recent 𝑚=4𝜋𝑅̇ 𝑛𝑚𝑃𝑉, (2) observations indicate an intermediate mass black hole in the 2 2 elliptical galaxy NGC 4472, with an X-ray luminosity of 4 × where 4𝜋𝑅 is the cross section and it is given by 4𝜋𝑅 = 32 2 2 10 J/s [1]. 4𝜋(𝐺𝑀/𝑉 ) . The Schwarzschild radius corresponding to these masses The velocity is given by is of the order of 2 2 2 𝑉 =𝑐𝑆 + V , (3) 2𝐺𝑀 𝑅 = ≈ 1.5 × 103 −3×104 . 𝑐 𝑆 𝑐2 km (1) where 𝑆 is the velocity of sound in the medium of density 𝜌 and it is given by 𝑐𝑆 = √𝛾𝑇𝑅,and𝛾=5/3is the ratio of specific heats and 𝑅 is the universal gas constant. Anaccretiondiscisformedbymaterialfallingintoa The equation describing the velocity of isothermal winds gravitational source. Conservation of angular momentum has many solutions depending on the initial conditions at the requires that, as a large cloud of material collapses inward, base of the wind. There is only one critical solution for which any small rotation it may have will increase. Centrifugal force thevelocityincreasesfromsubsonicatthebasetosupersonic causes the rotating cloud to collapse into a disc, and tidal far out. This velocity passes through the critical point and effects will tend to align this disc’s rotation with the rotation implies one particular value of the initial velocity at the lower of the gravitational source in the middle. boundary of the isothermal region. 2 Advances in High Energy Physics

If the density at this point is fixed, the mass loss rate Duetoalargenumberoffieldstarscontainedwithin 2 is fixed by (2)as𝑚=4𝜋𝑅̇ 0𝑛𝑚𝑃𝑉. The total energy the accretion radius the Bondi accretion by an IMBH is increases from negative at the base of the wind to positive complicated [4].Theaccretionradiusisgivenby in the supersonic region, so the flow requires the input 2𝐺𝑀 of energy into the wind. This energy input is needed to 𝑅 = BH ≈ 0.4 , 𝑀 =104 ; keeptheflowisothermalanditisthisenergythatis acc V2 pc for BH MΘ (7) transferred into kinetic energy of the wind through the gas V ≈15 / . pressure. km s As we will see in the next section, for an intermediate In the case of globular cluster G1 in M31, there are more mass black hole, the temperature is typically of the order of 5 4 than 10 stars within 0.4 pc of the centre [5]. The dynamical 10 K. In this case, the velocity of sound 𝑐𝑆 = √𝛾𝑇𝑅 is of the 4 effects of these stars should also be considered. order of 10 m/s. 𝑅 = 2𝐺𝑀/𝑉2 Many more such dense stellar clusters are known to exist. Also we have ; hence, the accretion rate is For instance, a recently recognised super star cluster in our given by own galaxy is the Westerlund I. Most of its estimated half a million stars are crowded in a region hardly 3 parsecs wide. 4𝜋𝑛𝑚 (𝐺𝑀)2 𝑚=̇ 𝑃 . (4) Several dozen of these stars are among the most massive and 𝑉3 luminous superhot Wolf-Rayet stars, LBV, red supergiants, yellow hypergiants, and so forth. Also, near the Milky Way 20 3 For a typical number density of 𝑛≈10/𝑚 ,and centre, we have the well-known Arches and Quintuplet 4 the IMBH mass of 𝑀=10MΘ, the accretion rate is: clusters. 17 𝑚=10̇ kg/s. In the centre of M15 there are approximately five million The above expression implies that the higher the density stars per cubic parsec, having hundred million times more of the medium through which the black hole is travelling, the stellar density than the solar neighbourhood. Also galaxies more the accretion rate. Also, the accretion rate is inversely like M31, M33, and the Milky Way itself have comparable related to the velocity with which the black hole is travelling in central stellar densities. However M32 (satellite galaxy of the medium. Although there is not much conclusive evidence Andromeda)hasthirtymillionstarsinacubicparsecatits for the IMBH, there is indirect evidence. core. Even HST cannot resolve individual stars in this region. More and more massive binary stars are being found, like, For a given mass of luminous object, there is a maximum for instance, WR20a (>80 MΘ each, with 3.7-day period). As value for its luminosity (Eddington luminosity), as the radia- discussed above, IMBH is likely to form in dense clusters tion pressure would tend to push the matter apart exceeding containing young massive stars. IMBH is believed to power the gravitational force supporting it. This is the luminosity ultraluminous X-ray sources (ULXs). IMBH can capture a body would have to have so that the force generated companion stars, which provide accreting material to sustain by radiation pressure exceeds the gravitational force. Thus, ULX. These stars may be blue giants, white dwarfs, andso observed luminosity can set a lower limit on the mass of an forth. accreting black hole. Tidalforcescanripgiantsandthematerialcanfallinto Chandra and XMM-Newton observations in the nearby the IMBH. Torip apart these stars the mass of the black hole is spiral galaxy have detected X-ray sources of luminosities of around the mass of the IMBH [6]. The energy released during 1033 the order of W, with the source away from the centre [3]. thisprocesswillbethebindingenergyofthestarswhichis In Section 4,wewilldiscussthedynamicsoftheIMBH. given by The X-ray luminosity corresponding to the accretion rate 17 2 31 of 𝑚=10̇ kg/s is given by 𝐿𝑋 =𝜀𝑚𝑐̇ ≈5×10 W. This 3 𝐺𝑀2 𝐸 = ≈1042 . matches with recent observed results [4]. BE J (8) 4 5 𝑅 The Eddington luminosity (for an IMBH of mass 10 MΘ) is given by: Several ULXs are identified in the Antenna, a pair of colliding galaxies, producing several stars in dense cluster. 4𝜋𝑐𝐺𝑀𝑚 𝐿 = 𝑃 ≈1035 , Stellar collisions lead to formation of so-called megastars Edd 𝜎 W (5) 3 𝑇 (∼10 MΘ), which collapses on short time scales to form IMBH. Clusters in M82 have such stellar densities. Binary and where, the Thomson cross section is given by: multipleIMBHcanalsoform.

8𝜋 𝑒2 2 𝜎 = ( ) ≈10−28 2. 𝑇 2 m (6) 2. Black Body Considerations of IMBH 3 𝑚𝑒𝑐 We have already obtained the Eddington luminosity as A possible candidate for the IMBH is the globular cluster 4𝜋𝑐𝐺𝑀𝑚 G1 in M31, which is the most massive stellar cluster in the local 𝐿 = 𝑃 . 107 Edd (9) group, with a mass of the order of MΘ. 𝜎𝑇 Advances in High Energy Physics 3

Byconsideringtheblackholeemissiontoobeyblackbody 3. Jets from the Black Hole radiation, we can use Stefan’s law to relate the luminosity to the temperature, 𝑇,as One of the manifestations of this accretion energy release is the production of so-called jets: the collimated beams 4𝜋𝑐𝐺𝑚 𝑀 of matter that are expelled from the innermost regions of 𝑃 =𝜎𝐴𝑇4 , 𝜎 max (10) accretion disks. These jets shine particularly brightly at radio 𝑇 frequencies. In rotating black holes, the matter forms a disk due to the mechanical forces present. In a Schwarzschild where 𝜎 is Stefan’s constant and area of the black hole and it blackhole,thematterwouldbedrawninequallyfromall is given by directions and thus would form an omnidirectional accretion cloud rather than disk. Jets form in Kerr black holes (rotating 𝐴=𝑓(4𝜋𝑅2), 𝑆 (11) black holes) that have an accretion disk [8]. Black holes convert a specific fraction of accretion energy 2 where 𝑅𝑆 =2𝐺𝑀/𝑐is the Schwarzschild radius and 𝑓 into radiation, which is traced by the X-ray luminosity and indicates the size of the ambient gas around the black hole. jet kinetic energy, which is traced by the radio-emission Making use of this in (10)weget luminosity [9]. The matter is funnelled into a disk-shaped torus by the black hole’s spin and magnetic fields, but, in 𝑐5 𝑚 the very narrow regions over the black hole’s poles, matter 𝑇4 =( 𝑃 )𝑀󳨐⇒𝑇 ∝𝑀−1/4. max 𝐺 𝑓𝜎 𝜎 max (12) can be energized to extremely high temperatures and speeds, 𝑇 escaping the black hole in the form of high-speed jets. Inferred jet velocities close to the speed of light suggest that This implies that the temperature of the IMBH (accretion 104 𝑓 jets are formed within a few gravitational radii of the event disk) of mass of the order of 500 to MΘ and ranging horizon of the black hole. from 10 to 100 is The horizon for the Kerr black hole is given by

6 √ 2 2 𝑇max ≈3×10 K. (13) 𝑟=𝑚± 𝑚 −𝑎 . (17)

2 Here, 𝑚=𝐺𝑀/𝑐is the geometric mass and 𝑎= From Wien’s law the corresponding wavelength is given 2 by 𝐽max/𝑀𝑐 is the geometric angular momentum. The corresponding wavelength is From the condition that 𝑟 should be real, the limiting case is given by 𝑚=𝑎. −3 (2.898 × 10 Km) From this, the maximum angular momentum is given by 𝜆= ≈10−9 . (14) 𝑇 m 𝑀2𝐺 𝐽 = . (18) max 𝑐 ThisliesinthesoftX-rayregionofthespectrum[7]. From the classical expression for the angular momentum During the earlier epochs of the universe, the density was 𝑙 much larger; hence, the ambient density is also larger by the associated with a jet of length , assuming the particles to be 3 travelling at near speed of light, the expression becomes same factor of (1 + 𝑧) . The present density of the universe is of the order of 𝐽=𝑚𝑐𝑙. (19) oneprotonpercubicmetre.AswewillseeinSection 5,the maximum redshift up to which we can detect supermassive Considering a conical jet with base radius r and density black hole is of the order of 𝑧=12. 𝜌,themassofthejetisgivenby The number density at this epoch is given by 1 𝑚= 𝜋𝑟2𝑙𝜌. 3 (20) 3 3 3 3 𝑛=1p/m (1+𝑧) ≈10 p/m . (15) Then the angular momentum becomes

The temperature corresponding to this redshift and this 1 2 2 𝐽= 𝜋𝑙 𝑟 𝑐𝜌. (21) number density is of the order of 3

7 Fromthegeometryofthejet,wecanrelatethelengthof 𝑇≈2×10 . ∘ K (16) the jet to the radius r as 𝑟=𝑙tan 5 . Here, we have assumed ∘ the small opening angle of the jet to be 5 . And the corresponding wavelength is of the order of −10 Length of the jet is 10 m. 1/4 This falls in the X-ray region of the spectrum. 3𝐺𝑀2 𝑙=( ) . (1 + 2 (22) This wavelength is further red-shifted by a factor of 𝜋𝜌𝑐2(tan 5) 𝑧). Hence, the observed wavelength will be of the order of −8 7×10 m. For typical densities of the ambient gas and for the IMBH 4 This lies in the UV region of the spectrum. of mass 10 MΘ, the length is of the order of 20 pc. 4 Advances in High Energy Physics

4. Evolution of a Star Cluster and Possible In this equation, 𝐶 is not a constant but depends on how V Scenario for IMBH Formation 𝑀 compares to the velocity dispersion of the surrounding matter. The greater is the density of the surrounding media, One of the possible models for the formation of an IMBH is thestrongerwillbetheforcefromdynamicalfriction. thecollapseofaclusterofstars[10]. The collapsed core can Similarly, the force is proportional to the square of the mass of accrete matter ejected during the formation. If a collection of the object. The force is also proportional to the inverse square a thousand 10 solar mass stars in a volume of a parsec cube of the velocity. This means the fractional rate of energy loss collapses, ejecting 30% of its mass, then the total mass of the drops rapidly at high velocities. ambient gas is given by Dynamical friction is, therefore, unimportant for objects that move relativistically, such as photons. Dynamical friction 𝑛𝑚 = 0.3 × 103 ×10 ≈6×1033 . 𝑃 MΘ kg (23) is particularly important in the formation of planetary sys- The accretion rate is given by (4)as tems and interactions between galaxies. During the formation of planetary systems, dynamical 16𝜋𝑛𝑚 𝐺2𝑀2 friction between the protoplanet and the protoplanetary disk 𝑚=̇ 𝑃 . (24) 𝑐3 causes energy to be transferred from the protoplanet to the disk. This results in the inward migration of the protoplanet. If 30% of the mass of each star is ejected, then this implies 7 When galaxies interact through collisions, dynamical an accretion rate of 𝑚≈4×10̇ kg/s. And the corresponding friction between stars causes matter to sink toward the centre (Eddington) luminosity is of the galaxy and for the orbits of stars to be randomised. The 𝐺𝑀𝑚̇ dynamical friction comes into effect in the evolution of cluster 𝐿 = ≈1024 . Edd 𝜎 𝑅2𝑛 W (25) between the ambient gas and dust and the central IMBH. 𝑇 𝑆 𝑀 V If an IMBH of mass BH is moving with a velocity of 𝑏 in Dynamical friction is related to loss of momentum and a uniform background of “fixed” lighter stars of equal masses 𝑚 kinetic energy of moving bodies through a gravitational inter- . Then as the IMBH moves, a star approaching with impact 𝑏 action with surrounding matter in space. The effect must exist parameter , will have a velocity change given by: if the principle of conservation of energy and momentum is ΔV ≈𝑎Δ𝑡, (27) valid since any gravitational interaction between two or more bodies corresponds to elastic collisions between those bodies. where Δ𝑡 is the encounter duration and 𝑎 is the acceleration. For example, when a heavy body moves through a cloud of They are given by lighter bodies, the gravitational interaction between this body 𝐺𝑚𝑀 𝑏 and the lighter bodies causes the lighter bodies to accelerate 𝑚𝑎 = BH ;Δ𝑡=. 2 (28) and gain momentum and kinetic energy. 𝑏 V𝑏 Since energy and momentum are conserved, this body has to lose a part of its momentum and energy equal to The change in velocity becomes the sums of all momenta and energies gained by the light 𝐺𝑀 𝑏 ΔV ≈𝑎Δ𝑡≈ BH . bodies. Because of the loss of momentum and kinetic energy 2 (29) 𝑏 V𝑏 of the body under consideration, the effect is called dynamical friction. Of course the mechanism works the same way for The kinetic energy gained by the star corresponding to all masses of interacting bodies and for any relative velocities this change in velocity is given by between them. 2 However,whileintheabovecasethemostprobableout- 1 2 1 𝐺𝑀 Δ𝐸 = 𝑚(ΔV) ≈ 𝑚( BH ) . (30) come is the loss of momentum and energy by the body under 2 2 𝑏V𝑏 consideration, in the general case it might be either loss or 𝑛 gain. In a case when the body under consideration is gaining If is the number density of the stars, then the number 𝑏+Δ𝑏 𝑏 momentum and energy, the same physical mechanism is of encounters with impact parameter between and is called sling effect. given by The full Chandrasekhar dynamical friction formula for Δ𝑁 ≈ 𝑛V ( Δ𝑡) Δ (𝜋𝑏2). the change in velocity of the object involves integrating over BH (31) the phase space density of the field of matter11 [ ]. By assuming The total change in velocity is given by aconstantdensity,though,asimplifiedequationfortheforce 2 𝑏 from dynamical friction, 𝑓𝑑,isgivenas 𝑑V 1 𝑑𝐸 𝜋𝐺 𝑛𝑀 2 𝑑𝑏 BH ≈ ∫ 𝑑𝑁 ≈ BH ∫ , (32) 2 𝑑𝑡 𝑀 V 𝑑𝑡 V2 𝑏 (𝐺𝑀) 𝜌 BH BH 𝑏1 𝑓 ≈𝐶 , BH 𝑑 V2 (26) 𝑀 where 𝑏1,thelowerboundon𝑏, is given for the case where the gravitational energy is of the order of the kinetic energy where 𝐺 is the gravitational constant, 𝑀 is the mass of the of the black hole. That is, moving object, 𝜌 is the density, and V𝑀 is the velocity of the 𝐺𝑚𝑀 1 object in the frame in which the surrounding matter was BH ≈ 𝑚V2 . BH (33) initially at rest. 𝑏1 2 Advances in High Energy Physics 5

And the upper limit 𝑏2 is the size of the system. Let The black hole undergoes a damped oscillation in the star 𝑏 ∫ 2 (𝑑𝑏/𝑏) = Λ cluster. The damping time corresponding to the system is 𝑏 ln . 1 given by Thenthetotalchangeinvelocityoftheblackholeisgiven by 𝑀 𝑡= BH , 𝛾 (42) 𝑑V 𝜋𝐺2𝑛𝑀 BH ≈ BH ln Λ. (34) 𝑑𝑡 V2 where BH 2 2 −3𝜋𝐺 𝑛𝑚𝑀 ln Λ In the above discussion we have assumed that the stars are 𝛾= BH . 3 (43) stationary. This need not be true. If the stars have a velocity (√2𝜎) dispersion of 𝜎 and the black hole is moving at a slower V ≪𝜎 velocity than this, that is, BH , then the dynamical In the case of the system M82, which harbours an IMBH 4 friction force is given by of mass in the range of 500 to 10 MΘ,theblackholeisnot foundatthecentrebutdisplacedbyaboutonekilo-parsec −3𝜋𝐺2𝑛𝑚𝑀2 Λ 𝐹 =𝑚𝑎 ≈ BH ln V =−𝛾V , from the centre. DF DF 3 BH BH (35) (√2𝜎) The period corresponding to the oscillation is given by

2𝜋 𝑘 −13 −1 where 𝑇= ;𝜔=√ ≈10 . (44) 𝜔 𝑀 s BH 2 2 −3𝜋𝐺 𝑛𝑚𝑀 ln Λ 𝛾= BH . From (37), we can work out the time taken for the BH to √ 3 (36) ( 2𝜎) shift by this distance. Using this equation along with32 ( )and 6 (35), we get the time of the order of 10 years. In the case of the velocity of the black hole being faster For the system under consideration, the number density 𝜎 4 3 than , the dynamical friction will be reduced and the black of the stars in the cluster is 𝑛≈10/(pc) ,typicalmassofthe hole will slide through the cluster. star in the cluster is about one solar mass, and the velocity Theblackholeisalsosubjectedtoagravitationalforcedue dispersion is of the order of 𝜎≈30km/s. to the star cluster, which is given by We get the damping time for the system as

2 3 ∇ 𝜙 (𝑟) =4𝜋𝐺𝜌(𝑟) =4𝜋𝐺𝑚𝑛(𝑟) , (37) (√2𝜎) 𝑡= ≈4×1012 . (45) 3𝜋𝐺2𝑛𝑚𝑀 Λ s where 𝑛(𝑟) is the density distribution of stars and 𝑚 is the BH ln typical mass of the star (assuming all stars are of the same 105 mass). This works out to be of the order of years. Taking the effects of dynamical friction into considera- For a constant density, 𝜌(𝑟)0 =𝜌 ,thepotentialisgivenby tion, the relaxation time for the system to form the IMBH is 2 1 2 given by 𝜙 (𝑟) =−2𝜋𝐺𝜌0 (𝑅 − 𝑟 ), (38) 3 3 V 7 𝑡= ≈10 years. (46) 𝑅 𝑛𝐺2𝑀2 𝑁 where is the radius of the cluster. BH ln The gravitational force on the black hole is given by 3 3 For a denser core, that is, about 10 stars/(0.01 pc) ,the 4 𝐹 =−𝑀 ∇𝜙 (𝑟) =− 𝜋𝐺𝜌 𝑀 𝑟=−𝑘𝑟. timetakentoformtheIMBHisgivenby 𝑔 BH 3 0 BH (39) V3 𝑡= ≈4×1012 =106 . (Theparticleinsideahomogenousgravitationalsystem 𝑛𝐺2𝑀2 𝑁 s years (47) BH ln performs simple harmonic motion!) The equation of motion of the black hole in the star cluster is given by 5. Formation of Supermassive Black Hole (SBH): Merger of IMBH 𝑑2𝑟 𝑑𝑟 𝑀 +𝑘𝑟+𝛾 =0. (40) Zwartetal.[13] propound that dense star densities near BH 𝑑𝑡2 𝑑𝑡 galactic centres can lead to runaway stellar mergers thus efficiently producing IMBH. Indeed they suggest that the This corresponds to a damped oscillator. And the solution innermosttenparsecsofourgalaxycancontainabout50 is given by [12] IMBH each of about thousand solar masses. These can sink towards the core as they interact with the stars. Every time an 𝛾𝑡 𝑘 𝑟=𝑀 (− ) ((√ )𝑡+𝛾). (41) IMBH has a stellar encounter, the stars’ velocity is boosted. BH exp 2𝑀 cos 𝑀 BH BH This reduces the potential energy making the IMBH sink to 6 Advances in High Energy Physics thegalacticcore.TheseIMBHcanthusultimatelymergewith 𝑓 is a numerical parameter, and integrating we get 𝑀= thegalaxy’ssupermassiveblackhole. 𝑀0 exp((4𝜋𝐺𝑀𝑚𝑝/𝑐𝜎𝑇)𝑡) =0 𝑀 exp(𝑡/𝑡0).Thistimecanbe However if there are not enough stars very near to the translated into corresponding redshift. galaxy’s SBH, then the IMBH may stop falling inward, halting According to the model suggested above, the time taken 8 at about 0.01 light years from the centre. However if there are for the formation of SBH is of the order of 10 years. The several IMBH near the galactic core, they can interact with corresponding redshift is of the order of 𝑧=12.Forredshifts oneanotherandonceeverymillionyearsanIMBHcanmerge above this limit, as per this model, we should not be able to with the SBH. detect any supermassive black holes. It has been suggested that the presence of several IMBH Other possible ways in which SBH can form are the nearthegalacticcentremightexplainwhytherearesomany following. (1) The first is by the merger of two or more IMBH. clusters of young stars (like S2, the B type star) in the galactic Moving masses like black holes produce gravitational waves core, where tidal forces should rip apart the gas clouds from in the fabric of space-time. A more massive moving object which the stars form. will produce more powerful waves, and objects that move These stars could have formed much further away and the very quickly will produce more waves over a certain time IMBH could have shepherded them inwards, much like the period. (2) The second is by accretion of matter by the IMBH thin rings of Uranus or Saturn that are kept in place by being in systems such as AGNs and quasars. “shepherded” by tiny satellites. Gravitational waves are usually produced in an interac- This IMBH can merge with the stars in the surrounding tion between two or more compact masses. Such interactions 6 3 volume (∼10 MΘ/(1 pc) ) to give a supermassive black hole include the binary orbit of two black holes orbiting each other. in the time scale given by As the black holes orbit each other, they send out waves of 3 gravitational radiation that reaches the Earth; however, once V 15 8 𝑡= ≈10 s =10 years. (48) the waves do get to the Earth, they are extremely weak. 𝑛𝐺2𝑀2 ln 𝑁 This is because gravitational waves decrease in strength as This is the time scale for relaxing in a cluster of 𝑁 objects they move away from the source. Even though they are weak, of average mass 𝑀 moving with a mean velocity of V.If𝜎 is the the waves can travel unobstructed within the fabric of space- 𝜎≈(𝐺𝑀/V2)2;(𝐺𝑀/V2) time. average cross section for interaction, From Kepler’s third law, the period is related to the givesthe“radius”ofthesphereofinfluencearoundagiven separation by individual object and the mean free path is 1/𝑛𝜎,and𝑛 is the number density of objects. Then the “collision” or 2 2 4𝜋 3 “interaction” time scale is given by 𝑃 = 𝑅 . (51) 𝐺(𝑀1 +𝑀2) 1 1 V3 𝑡≈ = = . (49) 𝑛𝜎V 𝑛(𝐺𝑀/V2)2V 𝑛𝐺2𝑀2 Knowing the period, we can determine the orbital velocity from V𝑃=2𝜋𝑅. The ln 𝑁 term comes from many body effects (i.e., Power lost by gravitational waves (quadrupole formula ∫(𝑑𝑁/𝑁), the so-called Coulomb logarithm in plasma for gravitational waves) is given by (for objects of mass 𝑀 3 𝑅 physics). This leads to (48). Note the sharp V dependence as with separation of ) 1/𝑛𝑀2 well as . 32 𝐺 The maximum redshift observed till now is of the order of 𝐸̇ = 𝑀2𝑅4𝜔6. GW 5 𝑐5 (52) 𝑧= 6.3 .Theageoftheuniversecorrespondingtothisredshift is given by This emission results in the objects coming closer. The −1/2 1 1−(1+𝑧) 9 merger time is obtained by integrating the rate of binding 𝑡= ( ) ≈10 yrs. (50) energy change as follows: 𝐻0 1+𝑧 The above equation is the standard formula for the age 𝐺𝑀2 𝑑𝑅 32 𝐺 = 𝑀2𝑅4𝜔6. (53) of the universe, corresponding to any redshift 𝑧.Higher𝑧 𝑅2 𝑑𝑡 5 𝑐5 9 implies younger objects. So 𝑧= 6.3 corresponds to ∼10 years. Now, recent supermassive black holes are claimed to This implies a merger time for two objects separated by 8 have been observed at 𝑧≈10,hardly∼5 × 10 years after the an initial distance 𝑅0 of bigbang[14]. 𝜏 =𝐾𝑅4. Weestimatethetimetoformtheblackholefromthe mer 0 (54) accretion time scale, corresponding to Eddington luminosity. 𝑡 ≈𝑐𝜎/4𝜋𝐺𝑚 ≈ 5 3 3 This gives a logarithmic time scale of 0 𝑇 𝑝 With the constant, 𝐾 = (5/256)(𝑐 /𝐺 )(1/𝑀 ),forequal 8 5×10 years. These are typical time scales for growth of mass objects. massiveblackholesintheearlyuniverse. So a larger initial separation would imply a longer merger 9 Essentiallywehave(with𝑀 as the initial mass) accretion time.Soforthemergertimetobe≈10 years or less, the initial ̇ 2 power ≈𝐺𝑀𝑀/𝑅 ≤ 4𝜋𝐺𝑀𝑚𝑝𝑐/𝜎𝑇,with𝑅≈𝑓(𝐺𝑀/𝑐), separationcanbepreciselycalculated. Advances in High Energy Physics 7

4 10 For two IMBH of mass 𝑀=10MΘ each, to merge in This implies that the mass will have to increase ≈𝑒 -fold 𝜏 ≈109 mer years, the distance of separation should be of the by accretion for the IMBH to become an SBH. Hence, even order of this model does not provide an efficient way of formation of SBH from an IMBH. 𝐺𝑀2 𝑅= , (55) IMBHwouldhaveformedintheearlyuniverse.Owing 𝜏 𝐸̇ mer GW to low metal content, the earliest stars would have been very massive, a few hundred solar masses [15]. Such stars would where end up in a pair-instability supernova (around oxygen-neon 128V10 2𝜋𝑅 burning temperature of 2 billion degrees) and would collapse 𝐸̇ = , V = , GW 5𝐺𝑐5 𝑃 into a black hole (if their mass exceeds 250 solar masses). ∼ × 9 (56) Oxygen-neon burning occurs at 2 10 K. At this stage, 4𝜋2 1/2 2𝐺𝑀 temperatures are high enough for the electron-positron pair 𝑃=( 𝑅3) 󳨐⇒ V = √ . + − 9 production processes (𝛾 → 𝑒 +𝑒).At∼6 × 10 K, it 𝐺(𝑀1 +𝑀2) 𝑅 is maximal. Now if the nature of the black body radiation And the energy loss due to the gravitational waves changes from photons to electron-positron pairs (which are 4 4 emission is given by fermions) the energy density is no longer 𝑎𝑇 but (7/8)𝑎𝑇 . So once the pairs are forming, the radiation pressure drops 128 2𝐺𝑀 5 (1/8)𝑎𝑇4 𝐸̇ = ( ) . (57) by . So there is less support against gravitational GW 5𝐺𝑐5 𝑅 collapse or gravitational contraction. The balance between the two (or the “stability condition”) Therefore, the distance of separation is given by can be expressed as 3 1/4 2 2 (𝐺𝑀) 𝜏 12 1 𝐺𝑀 𝑅≈(8×10 mer ) ≈2×10 m. (58) 𝑎𝑇4 ≥ . (64) 𝑐5 8 𝑅4 9 8 The corresponding orbital frequency is given by Now, 𝑇∼2×10 K, 𝑅∼2×10 m (appropriate density 𝑀 V for oxygen-neon burning); this implies a limiting mass 𝑓= . (59) of 250 solar masses, beyond which the gravitational energy 2𝜋𝑅 density is higher. This result in the context of population III The orbital velocity is of the order of supermassive stars is discussed by Heger and Woosley [16]. Signatures of such explosions of supermassive stars at 2𝐺𝑀 𝑧≈10(when the universe was only half a billion years old V = √ ≈106 / . (60) 𝑅 m s and eleven times smaller) could be sought with future space telescopes [17, 18]. −7 This implies that the frequency is 𝑓≈10 Hz, and hence Observations of galaxy core show correlation between 7 the period will be given by 𝑃=10 s =1year. black hole masses and the spheroidal (bulge) component of −4 For two IMBH with separation of about 10 parsecs, the the host galaxy. Thus, the primordial low mass galaxies (blue time taken to merge is of the order of Hubble time. For such a galaxies) would host low mass central black holes, which just 3 −5 −6 model to produce an SBH, we need about 10 IMBH merging correspond to the IMBH mass (about 10 to 10 the galaxy together. mass). There is a well-known relation between black hole Hence, the model discussed earlier, with the IMBH mass and spheroidal bulge component mass of the host galaxy [19]. merging with the surrounding stars, gives a much more −3 efficient way of generating a supermassive black hole. Typically, the black hole mass is 10 the bulge mass. InthecaseofaccretionofmatterbyIMBHtoformSBH, Primordial galaxies (so-called blue galaxies as seen, e.g., in 8 the increase in mass is exponential with time Hubble deep field) have a lower mass ∼10 solar mass. This 4 wouldgiveacentralblackholemassof∼10 solar mass, 𝑀=𝑀 (𝑘𝑡) , 0 exp (61) corresponding to a typical IMBH. Merger of these primeval galaxies would lead to larger 𝑘−1 =𝑡 ≈6×108 where 0 yearsisthecharacteristictime galaxies and cluster and the IMBH would also merge and sink required for the mass to increase 𝑒-fold, with the accreting 9 to the core forming an SBH (like, e.g., M87 has a 3×10 MΘ disk emitting at maximum luminosity. black hole). For the IMBH to accrete enough matter to become SBH 8 of mass say 10 MΘ,wehave 6. Concluding Remarks 𝑀 (𝑘𝑡) = =104. exp 𝑀 (62) We have discussed several aspects of the expected character- 0 istics of IMBH like their luminosity, accretion rate, formation, The corresponding time scale is of the order of andsoforth.Theyarelikelytohaveformedintheearly universe, around 𝑧=12. This redshift corresponds toa 9 𝑡≈5×10 years. (63) minimal time scale for formation (and growth) of a massive 8 Advances in High Energy Physics

black hole. Beyond 𝑧=12,themasswouldnotbelarge [17] T. di Matteo, J. Colberg, V.Springel, L. Hernquist, and D. Sijacki, enough, as not enough time would have elapsed for the “Direct cosmological simulations of the growth of black holes growth of black holes by accretion or merger. Supermassive and galaxies,” The Astrophysical Journal,vol.676,no.1,pp.33– black holes are not likely to form from merger of IMBH. 53, 2008. Other possible scenarios are also discussed. [18] M. J. Jee, H. C. Ford, G. D. Illingworth et al., “Discovery of a ringlike dark matter structure in the core of the galaxy cluster Cl 0024+17,” The Astrophysical Journal,vol.661,no.2,pp.728– Conflict of Interests 749, 2007. [19] J. Binney and S. Tremaine, Galactic Dynamics, Princeton Uni- The authors declare that there is no conflict of interests versity Press, Princeton, NJ, USA, 1987. regarding the publication of this paper.

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[1] T. J. Maccarone, A. Kundu, S. E. Zepf, and K. L. Rhode, “Ablack hole in a globular cluster,” Nature,vol.445,no.7124,pp.183–185, 2007. [2] H. Bondi, “On spherically symmetrical accretion,” Monthly Notices of the Royal Astronomical Society,vol.112,pp.195–204, 1952. [3]J.M.MillerandC.S.Reynolds,“Blackholesandtheirenviron- ments,” Physics Today,vol.60,no.8,pp.42–47,2007. [4] D. Pooley and S. Rappaport, “X-rays from the globular cluster G1: intermediate-mass black hole or low-mass X-ray binary?” The Astrophysical Journal Letters,vol.644,no.1,pp.L45–L48, 2006. [5]K.Gebhardt,R.M.Rich,andL.C.Ho,“Anintermediate-mass black hole in the globular cluster G1: improved significance from new keck and Hubble Space Telescope Observations,” The Astrophysical Journal,vol.634,no.2,pp.1093–1102,2005. [6] K. Arun, Aspects of black hole energetics [M.S. thesis],2005. [7]J.M.Miller,G.Fabbiano,M.C.Miller,andA.C.Fabian,“X- ray spectroscopic evidence for intermediate-mass black holes: cool accretion disks in two ultraluminous X-ray sources,” The Astrophysical Journal Letters,vol.585,no.1,pp.L37–L40,2003. [8] C. Misner, K. Thorne, and J. Wheeler, Gravitation,W.H. Freeman, New York, NY, USA, 1973. [9] A. Merloni, S. Heinz, and T. di Matteo, “Afundamental plane of black hole activity,” Monthly Notices of the Royal Astronomical Society,vol.345,no.4,pp.1057–1076,2003. [10] M. C. Miller and D. P. Hamilton, “Production of intermediate- mass black holes in globular clusters,” Monthly Notices of the Royal Astronomical Society,vol.330,no.1,pp.232–240,2002. [11] S. Chandrashekar, Introduction to Stellar Dynamics, University of Chicago Press, Chicago, Ill, USA, 1942. [12] D. Zwillinger, Handbook of Differential Equations, Academic Press, New York, NY, USA, 1997. [13]S.F.P.Zwart,H.Baumgardt,P.Hut,J.Makino,andS.L.W. McMillan, “Formation of massive black holes through runaway collisions in dense young star clusters,” Nature,vol.428,no. 6984, pp. 724–726, 2004. [14] D. J. Mortlock, S. J. Warren, B. P. Venemans et al., “A luminous quasar at a redshift of 𝑧= 7.085 ,” Nature,vol.474,no.7353,pp. 616–619, 2011. [15] P. Madau and M. J. Rees, “Massive black holes as population III remnants,” The Astrophysical Journal Letters,vol.551,no.1,pp. L27–L30, 2001. [16] A. Heger and S. E. Woosley, “The nucleosynthetic signature of population III,” The Astrophysical Journal,vol.567,no.1,pp. 532–543, 2002. Hindawi Publishing Corporation Advances in High Energy Physics Volume 2014, Article ID 146094, 6 pages http://dx.doi.org/10.1155/2014/146094

Research Article Electrostatics in the Surroundings of a Topologically Charged BlackHoleintheBrane

Alexis Larrañaga,1 Natalia Herrera,2 and Sara Ramirez2

1 National Astronomical Observatory, National University of Colombia, Bogota 11001000, Colombia 2 Department of Physics, National University of Colombia, Bogota 11001000, Colombia

Correspondence should be addressed to Alexis Larranaga;˜ [email protected]

Received 27 December 2013; Revised 21 January 2014; Accepted 23 January 2014; Published 27 February 2014

Academic Editor: Christian Corda

Copyright © 2014 Alexis Larranaga˜ et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP3.

We determine the expression for the electrostatic potential generated by a point charge held stationary in the topologically charged black hole spacetime arising from the Randall-Sundrum II braneworld model. We treat the static electric point charge as a linear perturbation on the black hole background and an expression for the electrostatic multipole solution is given: PACS: 04.70.-s, 04.50.Gh, 11.25.-w, 41.20.-q, 41.90.+e.

1. Introduction metric of a spherical compact object on the brane is no longer described by Schwarzschild metric. In fact, black hole The idea of our universe as a brane embedded in a higher solutions in the braneworld model are particularly interesting dimensional spacetime has recently attracted attention. because they have considerably richer physical aspects than According to the braneworld scenario, the physical fields black holes in general relativity [5–11]. The first solutions (electromagnetic, Yang-Mills, etc.) in our 4-dimensional describing static and spherically symmetric exterior vacuum universeareconfinedtothethree-braneandonlygravity solutions of the braneworld model were proposed by Dadhich propagates in the bulk spacetime. One of the most interesting et al. [12] and Germani and Maartens [13]andlatertheywere scenarios is Randall-Sundrum II model in which it is consid- generalised by Chamblin et al. [14]andrevisitedbySheykhi Z ered a 2-symmetric, 5-dimensional, asymptotically anti-de- and Wang [15]. This kind of solutions carries a topological Sitter bulk [1] and our brane is identified as a domain wall. charge arising from the bulk Weyl tensor and the line element The 5-dimensional metric can be written in the general form 2 −𝐹(𝑦) 𝜇 ] 2 resembles Reissner-Nordstrom¨ solution, with the tidal Weyl 𝑑𝑠 =𝑒 𝜂𝜇]𝑑𝑥 𝑑𝑥 +𝑑𝑦 andduetotheappearanceof parameter playing the role of the electric charge. In order the warp factor, it reproduces a large hierarchy between the to obtain this solution, there was the null energy condition scale of particle physics and gravity. Moreover, even if the imposedonthethree-braneforabulkhavingnonzeroWeyl fifth dimension is uncompactified, standard 4-dimensional curvature. gravity on the brane is reproduced. However, due to the cor- On the other hand, the generation of an electromagnetic rection terms coming from the extradimensions, significant field by static sources in black hole backgrounds has been deviations from general relativity may occur at high energies considered in several papers beginning with the studies of [2–4]. Copson [16], Cohen and Wald [17], and Hanni and Ruffini As is well known, in general relativity the exterior [18] where they discussed the electric field of a point charge spacetime of a spherical compact object is described by in Schwarzschild background. Afterwards, Petterson19 [ ] Schwarzschild solution. In the braneworld scenario, the and later Linet [20] studied the magnetic field of a current high energy corrections to the energy density together with loop surrounding a Schwarzschild black hole. More recently, Weyl stresses from bulk gravitons imply that the exterior similar studies have been performed in other background 2 Advances in High Energy Physics

geometries including charged black holes of general relativity terms of the 5-dimensional cosmological constant Λ 5 and the and black holes with Brans-Dicke modifications or with brane tension 𝜏 by conical defects [21–25]. In this paper we are interested in obtaining an expression 𝜅2 𝜅2 Λ= 5 (Λ + 5 𝜏2). for the electrostatic potential generated by a point charge 2 5 6 (3) held stationary in the region outside the event horizon of a braneworld black hole. There are several vacuum solutions of 𝑇𝜇] is the stress-energy tensor of matter confined on the the spherically symmetric static gravitational field equations brane, Π𝜇] is a quadratic tensor in the stress-energy tensor on the brane with arbitrary parameters which depend on given by properties of the bulk or that are simply put in by using general physical considerations. At the present, it is theoretically Π𝜇] not known whether these parameters should be universal 1 1 1 1 (4) over all braneworld black holes, or whether each separate = 𝑇𝑇 − 𝑇 𝑇𝜎 + 𝑔 (𝑇 𝑇𝛼𝛽 − 𝑇2) black hole may have different values of them. Similarly, 12 𝜇] 4 𝜇𝜎 ] 8 𝜇] 𝛼𝛽 3 thereisnotasinglecompletesolutioninthesensethatthe 𝜎 metric in the bulk is uniquely known. Since this situation with 𝑇=𝑇𝜎 ,and𝐸𝜇] is the projection of the 5-dimensional 𝐴 𝐵 is unsatisfactory from a theoretical point of view, it may be bulk Weyl tensor 𝐶𝐴𝐵𝐶𝐷 on the brane (𝐸𝜇] =𝛿𝜇 𝛿] 𝐶𝐴𝐵𝐶𝐷 useful to investigate more closely the observational effects 𝐴 𝐵 𝐴 𝑛 𝑛 with 𝑛 the unit normal to the brane). 𝐸𝜇] encompasses of the black hole properties. Specifically, we consider the 𝐸𝜎 =0 effects due to the projections of the Weyl tensor and how they the nonlocal bulk effect and it is traceless, 𝜎 . specify the corrections on the electrostatic potential. Since Considering the Randall-Sundrum scenario with the generic form of the Weyl tensor in the full 5-dimensional 2 𝜅5 2 spacetime is yet unknown, the effects of known solutions Λ 5 =− 𝜏 (5) must be studied on a case-by-case basis. Although one can, 6 in principle, constrain the projections, this only yields very mild constraints on the 5-dimensional Weyl tensor [26, 27]. which implies Therefore we decided to derive the electrostatic potential Λ=0, generated by a point charge held stationary in the outside (6) region of the event horizon of the particular solution in the Randall-Sundrum braneworld model obtained by Dadhich thefour-dimensionalGaussandCodazziequationsforan et al. [12] and revisited by Sheykhi and Wang [15]. In this arbitrary static spherically symmetric object have been completely solved on the brane, obtaining a black hole type case, tidal charge is arising via gravitational effects from 𝑇 =0 the fifth dimension; that is, it is arising from the projection solution of the field equations (1)with 𝜇] ,givenbythe ontothebraneoffreegravitationalfieldeffectsinthebulk, line element [12, 14, 15, 28] and is this term the one that will modify the electrostatic 2 2 2 𝑑𝑟 2 2 potential produced by the particle. We obtain the corrections, 𝑑𝑠 =ℎ(𝑟) 𝑑𝑡 − −𝑟 𝑑Ω , (7) ℎ (𝑟) duetothetopologicalchargearisingfromthebulkWeyl tensor, to the electrostatic potential and deduce the necessary 𝑑Ω2 =𝑑𝜃2 + 2𝜃𝑑𝜑2 correction to incorporate Gauss’s law. Finally we also present where sin and the solution of Maxwell equations in the form of series of 2𝐺𝑀 𝛽 multipoles. ℎ (𝑟) =1− + . (8) 𝑟 𝑟2

2. The Topologically Charged Black Hole in Forthismodel,theexpressionoftheprojectedWeyl the Braneworld tensor transmitting the tidal charge stresses from the bulk to the brane is ThegravitationalfieldonthebraneisdescribedbytheGauss 𝑡 𝑟 𝜃 𝜑 𝛽 and Codazzi equations of 5-dimensional gravity [2], 𝐸 =𝐸 =−𝐸 =−𝐸 = . (9) 𝑡 𝑟 𝜃 𝜑 𝑟4 𝐺 =−Λ𝑔 +8𝜋𝐺𝑇 +𝜅4Π −𝐸 , 𝜇] 𝜇] 𝜇] 5 𝜇] 𝜇] (1) This result shows that the parameter 𝛽 canbeinterpreted as a tidal charge associated with the bulk Weyl tensor and 𝐺 =𝑅 − (1/2)𝑔 𝑅 where 𝜇] 𝜇] 𝜇] is the 4-dimensional Einstein therefore, there is no restriction on it to take positive as 𝜅 tensor and 5 is the 5-dimensional gravity coupling constant, well as negative values (other interpretations consider 𝛽 as a five-dimensional mass parameter as discussed14 in[ ]). The 48𝜋𝐺 4 2 induced metric in the domain wall presents horizons at the 𝜅5 =(8𝜋𝐺5) = , (2) 𝜏 radii (taking 𝐺=1). Consider the following: with 𝐺5 the gravitational constant in five dimensions and Λ 𝑟 =𝑀±√𝑀2 −𝛽. (10) is the 4-dimensional cosmological constant that is given in ± Advances in High Energy Physics 3

For 𝛽≥0there is a direct analogy to the Reissner- From now on, we will assume that the charge 𝑞 is held Nordstrom¨ solution, showing two horizons that, as in general outside the black hole; that is,𝑟0 >𝑟+ in the case 𝛽≥0 relativity, both lie inside the Schwarzschild radius 2𝑀;thatis, and 𝑟0 >𝑟∗ in the case 𝛽<0. We will not consider here the possibility of having the charge inside the event 0≤𝑟− ≤𝑟+ ≤𝑟𝑠. (11) horizon because; as well known, this construction leads to the description of a charged black hole of the Reissner- 𝛽 Clearly, there is an upper limit on ,namely, Nordstrom¨ type [15]. 𝑖 2 The spatial components of the potential vanish, 𝐴 =0, 0≤𝛽≤𝑀. 0 (12) while the temporal component 𝐴 =𝜙is determined by the 𝜇=0component of (14), giving the differential equation: However, there is nothing to stop us choosing 𝛽 to be negative. This intriguing new possibility is impossible in 𝑟 𝜕 𝜕𝜙 Reissner-Nordstrom¨ case and leads to only one horizon, ∗, (𝑟2 ) lying outside the corresponding Schwarzschild radius, 𝜕𝑟 𝜕𝑟 −1 √ 2 󵄨 󵄨 2𝑀 𝛽 𝑟∗ =𝑀+ 𝑀 + 󵄨𝛽󵄨 > 2𝑀. (13) +(1− + ) 𝑟 𝑟2 (17) In this case, the single horizon has a greater area that 1 𝜕 𝜕𝜙 1 𝜕2𝜙 its Schwarzschild counterpart. Thus, one concludes that the ×[ ( 𝜃 )+ ] 𝜃 𝜕𝜃 sin 𝜕𝜃 2𝜃 𝜕𝜑2 effect of the bulk producing a negative 𝛽 is to strengthen the sin sin gravitational field outside the black hole (obviously it also =−𝑟24𝜋𝐽0. increases the entropy and decreases the Hawking temperature but these facts are not important in this paper). In order to solve this equation we will perform the 3. The Electrostatic Field of a Point Particle substitutions

Copson [16] and Linet [20]foundtheelectrostaticpotential 𝑟=𝑟+𝑟̃ −, inaclosedformofapointchargeatrestoutsidethehorizonof 𝑟−𝑟 (18) a Schwarzschild black hole and that the multipole expansion 𝜙 (𝑟, 𝜃, 𝜑)= − 𝜙(𝑟−𝑟̃ ,𝜃,𝜑) 𝑟 − of this potential coincides with the one given by Cohen and Wald [17] and by Hanni and Ruffini [18]. In this section we will investigate this problem in the background of the which turn the differential equation into topologically charged black hole (7). According to the braneworld scenario, we will consider ̃ 𝜕 2 𝜕𝜙 that the physical fields are confined to the three brane and (𝑟̃ ) 𝜕𝑟̃ 𝜕𝑟̃ if the electromagnetic field of the point particle is assumed to be sufficiently weak so its gravitational effect is negligible, 2𝑚 −1 1 𝜕 𝜕𝜙̃ 1 𝜕2𝜙̃ and the Einstein-Maxwell equations reduce to Maxwell’s +(1− ) [ ( 𝜃 )+ ] ̃ sin 2 2 equations confined in the curved background of the brane (7). 𝑟 sin 𝜃 𝜕𝜃 𝜕𝜃 sin 𝜃 𝜕𝜑 These are written as 4𝜋𝑞̃ =− 𝛿(𝑟−̃ 𝑟̃ )𝛿(𝜃−𝜃 )𝛿(𝜑−𝜑 ), 𝜌𝜇 𝜇 𝜃 0 0 0 ∇𝜌𝐹 =4𝜋𝐽 , (14) sin (19) where 𝑚=√𝑀2 −𝛽 𝑞=𝑞̃ 𝑟̃ /(𝑟̃ +𝑟 ) 𝐹𝜇] =𝜕𝜇𝐴] −𝜕]𝐴𝜇 (15) where and 0 0 − . Note that, in the 𝑚=0 𝑀2 =𝛽 𝜇 𝜇 extremal case ,orequivalently ,(19)becomes with 𝐴 the electromagnetic vector potential and 𝐽 the Laplace’s equation in Minkowski spacetime. On the other 2 current density. Considering that the test charge 𝑞 is held hand, when 𝑚 =0̸(including values with 0≤𝛽<𝑀 as well stationary at the point (𝑟0,𝜃0,𝜑0),theassociatedcurrent 𝛽<0 𝑖 0 as ), (19) is formally identical to the partial differential density 𝐽 vanishes while the charge density 𝐽 is given by equation for the electrostatic potential in the Schwarzschild 𝜃 =0𝜑 =0 𝑞 spacetime. Hence, assuming 0 , 0 and proceeding 𝐽0 = 𝛿(𝑟−𝑟 )𝛿(𝜃−𝜃 )𝛿(𝜑−𝜑 ). in analogy with Copson [16], we find that the solution of (17), 𝑟2 𝜃 0 0 0 (16) 0 sin after using the substitutions18 ( ), is

2 𝑞 (𝑟−𝑀) (𝑟0 −𝑀)−(𝑀 −𝛽)cos 𝜃 𝜙𝐶 (𝑟,) 𝜃 = . (20) 𝑟0𝑟 √ 2 2 2 2 (𝑟−𝑀) +(𝑟0 −𝑀) −2(𝑟−𝑀) (𝑟0 −𝑀)cos 𝜃−(𝑀 −𝛽)sin 𝜃 4 Advances in High Energy Physics

𝑞 𝑀 This solution describes, as stated before, the potential for a 𝜙 (𝑟,) 𝜃 =𝜙 (𝑟,) 𝜃 + . 𝐶 𝑟 𝑟 (22) charge 𝑞 situated at the point (𝑟0,0,0)and it is regular for any 0 value of 𝑟 outside the event horizon, except at the position of 𝑟 the point charge. However, as shown by Linet [20]andLeaut´ e´ The electrostatic solution can be analysed for all values . 𝑞 and Linet [21], it is easy to see that this solution includes Considering this equation, it is clear that when the charge 𝑟 𝛽≥0 another source. Note that, for 𝑟→∞,thepotential𝜙𝐶 takes is approaching the outer horizon + in the case or the 𝑟 𝛽<0 𝜙 the asymptotic form horizon ∗ in the case , the electrostatic potential tends to become spherically symmetric, recovering a charged black hole of the Reissner-Nordstrom¨ type as stated before. 𝑞 𝑀 𝑟 >2𝑀 𝑞 𝜙 𝑟, 𝜃 󳨀→ (1 − ), When 0 , there is only one charge in the case 𝐶 ( ) (21) 𝛽≥0 𝛽<0 𝑟 ≤𝑟 < 𝑟 𝑟0 as well as in the case .However,when + 0 2𝑀 in the case 𝛽≥0, there are also two other charges 𝑞1 = 𝑞(1−2𝑀/𝑟0) and 𝑞2 =−𝑞1 located at (𝑟 = 2𝑀−𝑟0,𝜃=𝜋)and and consequently, by virtue of Gauss’s theorem, there is a 𝑟=0, respectively. This behaviour is obviously not present for −𝑞𝑀/𝑟 second charge with value 0.Solution(20)hasonlythe 𝛽<0because 𝑟∗ >2𝑀as shown in Section 2 of this paper. source 𝑞 outside the horizon and thus the second charge must Finally, in order to write formally the electrostatic poten- lieinsidethehorizon.Moreover,sincetheonlyelectricfield tial of a charge 𝑞 located at the arbitrary point with coor- which is regular outside the horizon is spherical symmetric dinates (𝑟0,𝜃0,𝜑0), we simply replace the term cos 𝜃 by the [29], the electrostatic potential for our physical system will function 𝜆(𝜃, 𝜑) = cos 𝜃 cos 𝜃0 + sin 𝜃 sin 𝜃0 cos(𝜑− 𝜑 0).This be of the form gives the general solution

2 𝑞 (𝑟−𝑀) (𝑟0 −𝑀)−(𝑀 − 𝛽) 𝜆 (𝜃, 𝜑) 𝑞 𝑀 𝜙 (𝑟, 𝜃,) 𝜑 = + . (23) 𝑟0𝑟 √ 2 2 2 2 𝑟 𝑟0 (𝑟−𝑀) +(𝑟0 −𝑀) −2(𝑟−𝑀) (𝑟0 −𝑀)𝜆(𝜃,𝜑)−(𝑀 −𝛽)[1−𝜆 (𝜃, 𝜑)]

3.1. Multipole Expansion. If the angular part of the potential one may obtain the well-known electric multipole solutions. in (19) is expanded in terms of Legendre polynomials in cos 𝜃, Using again the substitutions (18), the radial parts of the two linearly independent multipole solutions of (17)are

{1 for 𝑙=0 { { 𝑙 2 𝑙/2 𝑔 (𝑟) = 2 𝑙! (𝑙−1)!(𝑀 −𝛽) 𝑟2 − 2𝑀𝑟 +𝛽 𝑑𝑃 𝑟−𝑀 𝑙 { 𝑙 ( ) 𝑙=1,2,..., (24) { for (2𝑙)! 𝑟 𝑑𝑟 √𝑀2 −𝛽 {

(2𝑙 + 1)! 𝑟2 − 2𝑀𝑟 +𝛽 𝑑𝑄 𝑟−𝑀 𝑓 (𝑟) =− 𝑙 ( ) 𝑙=0,1,2,..., 𝑙 (𝑙+1)/2 for (25) 2𝑙 (𝑙+1)!𝑙!(𝑀2 −𝛽) 𝑟 𝑑𝑟 √𝑀2 −𝛽

where 𝑃𝑙 and 𝑄𝑙 are the two types of Legendre functions. As is well known, the gravitational field modifies the These functions satisfy the following: electrostatic interaction of a charged particle in such a way that the particle experiences a finite self-force22 [ , 23, 30–34] (1) for 𝑙=0, 𝑔0(𝑟) = 1 and 𝑓0(𝑟) = 1/𝑟; whose origin comes from the spacetime curvature associated (2) for all values of 𝑙,as𝑟→∞, the leading term of 𝑔𝑙(𝑟) with the gravitational field. However, even in the absence of 𝑙 −(𝑙+1) is 𝑟 while the leading term of 𝑓𝑙(𝑟) is 𝑟 ; curvature, it was shown that a charged point particle [35, 36] or a linear charge distribution [37] placed at rest may become (3) as 𝑟→𝑟+, 𝑔𝑙(𝑟) → 0 (for 𝑙 =0̸) while 𝑓𝑙(𝑟) → subject to a finite repulsive electrostatic self-force (see also finite constant. However, 𝑑𝑓𝑙/𝑑𝑟 blows up for 𝑙 =0̸ [24]). In these references, the origin of the force is the distor- when 𝑟→𝑟+. tion in the particle field caused by the lack of global flatness The above properties let us infer that only the set of solutions of the spacetime of a cosmic string. Therefore, one may (25) has the correct behaviour at infinity and only the conclude that the modifications of the electrostatic potential multipole term 𝑙=0does not produce a divergent field at the come from two contributions: one of geometric origin and horizon 𝑟=𝑟+. This set of solutions reproduces Israel’s result the other of a topological one. In a forthcoming paper we will [29]when𝛽=0(i.e., for the Schwarzschild black hole). show that, considering the topologically charged black hole Advances in High Energy Physics 5

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Research Article Energy Loss of a Heavy Particle Near 3D Rotating Hairy Black Hole

Jalil Naji1 and Hassan Saadat2

1 Physics Department, Ilam University, P.O. Box 69315-51, Ilam, Iran 2 Department of Physics, Shiraz Branch, Islamic Azad University, P.O. Box 71555-477, Shiraz, Iran

Correspondence should be addressed to Jalil Naji; [email protected]

Received 6 November 2013; Accepted 9 January 2014; Published 16 February 2014

Academic Editor: Xiaoxiong Zeng

Copyright © 2014 J. Naji and H. Saadat. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP3.

We consider rotating black hole in 3 dimensions with a scalar charge and discuss energy loss of heavy particle moving near the black hole horizon. We find that drag force was increased by scalar charge while it was decreased due to the rotation of black hole. We also study quasnormal modes.

1. Introduction 𝑃=𝐹−𝜁𝑃̇ , where in the nonrelativistic motion 𝑃=𝑚V, andintherelativisticmotion𝑃=𝑚V/√1−V2;also𝜁 is called The lower dimensional theories may be used as toy models to the friction coefficient. In order to obtain drag force, one study some fundamental ideas which yield to better under- can consider two special cases. The first case is the constant standing of higher dimensional theories, because they are eas- momentum which yields to obtain 𝐹=(𝜁𝑚)V for the norela- ier to study [1].Moreover,theseareusefulforapplicationof tivistic case. In this case, the drag force coefficient (𝜁𝑚) will be AdS/CFT correspondence [2–5]. This paper is indeed an obtained. In the second case, the external force is zero, so one application of AdS/CFT correspondence to probe moving can find 𝑃(𝑡) = 𝑃(0) exp(−𝜁𝑡).Inotherwords,bymeasuring charged particle near the three-dimensional black holes the ratio 𝑃/𝑃̇ or V̇/V, one can determine friction coefficient 𝜁 which are recently introduced by Xu et al. [6, 7]where without any dependence on mass 𝑚.Thesemethodsleadusto charged black holes with a scalar hair in (2 + 1) dimensions obtain the drag force for a moving heavy particle. The moving and rotating hairy black hole in (2 + 1) dimensions are con- heavy particle in context of QCD has dual picture in the string structed, respectively. Here, we are interested in the case of theory in which an open string is attached to the D-brane and rotating black hole with a scalar hair in (2 + 1) dimensions. stretched to the horizon of the black hole. Therefore, we can Recently, a charged rotating hairy black hole in 3 dimensions apply AdS/CFT correspondence to probe a charged particle corresponding to infinitesimal black hole parameters was (such as a quark) moving through 3D hairy black hole constructed [8]. Also, thermodynamics of such systems is background. recently studied in [9, 10]. We consider this background in Similar studies are already performed in several back- AdSsideasadualpictureofaQCDmodelasCFTside. grounds [11–22]. Most of them considered N =2and N =4 In this paper, we would like to study the motion of a heavy super Yang-Mills plasma with asymptotically AdS geome- charged particle near the black hole horizon and calculate the tries. Also [20] considered 4D Kerr-AdS black holes. All of the energy loss. The energy loss of moving heavy charged particle mentioned studies used AdS5/CFT4 correspondence. Now, through a thermal medium is known as the drag force. One we are going to consider the same problem in a rotating hairy can consider a moving heavy particle (such as charm and bot- 3D background and use AdS3/CFT2 correspondence [23–25]. tom quarks) near the black hole horizon with the momentum This paper is organized as follows. In the next section, 𝑃,mass𝑚, and constant velocity V,whichisinfluencedbyan we review rotating hairy black hole in (2 + 1) dimensions. In external force 𝐹. So, one can write the equation of motion as Section 3, we obtain equation of motion and in Section 4, 2 Advances in High Energy Physics we try to obtain solution and discuss about drag force. In 3. The Equations of Motion Section 5, we give linear analysis and discuss quasinormal modes. Finally, in Section 6, we summarized our results. The moving heavy particle near the black hole may be described by the following Nambu-Goto action: 1 2. Rotating Hairy Black Hole in 𝑆=− ∫ 𝑑𝜏√ 𝑑𝜎 −𝐺, (10) (2 + 1) Dimensions 2𝜋𝛼󸀠 𝑇 =1/2𝜋𝛼󸀠 𝜏 Rotating hairy black hole in (2 + 1) dimensions is described by where 0 isthestringtension.Thecoordinates 𝜎 𝐺 the following action: and are corresponding to the string world-sheet. Also, 𝑎𝑏 is the induced metric on the string world-sheet with determi- 1 3 𝜇] 𝑅 2 nant 𝐺 obtained as follows: 𝑆= ∫ 𝑑 𝑥√−𝑔𝑅−𝑔 [ ∇𝜇𝜙∇]𝜙− 𝜙 −2𝑉(𝜙)] , (1) 2 8 2 2 󸀠 2 𝑟 2 𝐺=−1−𝑟𝑓 (𝑟) (𝑥 ) + (𝑥̇) , (11) which yields to the following line element [1]: 𝑓 (𝑟) 1 where we used static gauge in which 𝜏=𝑡, 𝜎=𝑟,andthe 𝑑𝑠2 =−𝑓(𝑟) 𝑑𝑡2 + 𝑑𝑟2 +𝑟2(𝑑𝜓 +𝜔 (𝑟) 𝑑𝑡)2, 𝑥(𝑟, 𝑡) 𝑓 (𝑟) (2) string only extends in one direction .Then,theequation of motion is obtained as follows: where 𝑟2𝑓 (𝑟) 𝑥󸀠 𝑟2 𝑥̇ 𝜕𝑟 ( )− 𝜕𝑡 ( )=0. (12) √−𝐺 𝑓 (𝑟) √−𝐺 2𝛽𝐵 (3𝑟 + 2𝐵)2𝑎2 𝑟2 𝑓 (𝑟) =3𝛽+ + + , (3) 𝑟 𝑟4 𝑙2 We should obtain canonical momentum densities associated with the string as follows: where 𝑎 is a rotation parameter related to the angular momen- 𝑙 2 tum of the solution and is related to the cosmological con- 0 1 𝑟 2 𝜋 = 𝑥,̇ stant via Λ=−1/𝑙. 𝛽 is integration constants depending on 𝜓 2𝜋𝛼󸀠√−𝐺 𝑓 (𝑟) the black hole mass: 1 𝑟2 𝑀 𝜋0 =− 𝑥𝑥̇󸀠, 𝛽=− , 𝑟 2𝜋𝛼󸀠√−𝐺 𝑓 (𝑟) 3 (4) 0 1 2 󸀠 2 𝐵 𝜋 =− (1 + 𝑟 𝑓 (𝑟) (𝑥 ) ), and scalar charge is related to the scalar field as 𝑡 2𝜋𝛼󸀠√−𝐺 (13) 8𝐵 1 1 2 󸀠 𝜙 (𝑟) =±√ . (5) 𝜋𝜓 = 𝑟 𝑓 (𝑟) 𝑥 , 𝑟+𝐵 2𝜋𝛼󸀠√−𝐺 1 𝑟2 Also, one can obtain 𝜋1 =− (1 − 𝑥̇2), 𝑟 2𝜋𝛼󸀠√−𝐺 𝑓 (𝑟) (3𝑟 + 2𝐵) 𝑎 𝜔 (𝑟) =− , 1 𝑟3 𝜋1 = 𝑟2𝑓 (𝑟) 𝑥𝑥̇󸀠. (6) 𝑡 2𝜋𝛼󸀠√−𝐺 2 1 1 𝛽 𝑉(𝜙)= + [ + ]𝜙6. 𝑙2 512 𝑙2 𝐵2 The simplest solution of the equation of motion is static string described by 𝑥 = constant with total energy of the form, Ricci scalar of this model is given by 𝑟 𝑚 1 𝐸=−∫ 𝜋0𝑑𝑟 = (𝑟 −𝑟 )=𝑀 , 𝑡 󸀠 ℎ 𝑚 rest (14) 6𝑟6 + 36𝐵𝑙2𝑎2𝑟 + 30𝑙2𝑎2𝐵2 𝑟 2𝜋𝛼 𝑅=− . (7) ℎ 𝑙2𝑟6 where 𝑟𝑚 is an arbitrary location of D-brane. As we expected, We can see that Ricci scalar is singular at the origin. the energy of static particle is interpreted as the remaining Black hole horizon, which is obtained by 𝑓(𝑟),maybe =0 mass. written as follows: 4. Time Dependent Solution 4𝑙 𝐵𝐶 𝑟 = (1+√1− ) , ℎ 2𝐶 3𝑙2 (8) In the general case, we can assume that the particle moves with constant speed 𝑥=̇ V;inthatcase,theequationof where we defined motion (12)reducesto 𝑟2𝑓 (𝑟) 𝑥󸀠 2𝐵𝑀 3𝑙 𝜕 ( )=0, 𝐶≡ − . (9) 𝑟 (15) 27𝑎2 𝐵 √−𝐺 Advances in High Energy Physics 3 where 0 2 2 𝑟 𝐺=−1−𝑟2𝑓 (𝑟) (𝑥󸀠) + V2. 𝑓 (𝑟) (16)

Equation (15) gives the following expression: 𝐶2 (𝑟2V2 −𝑓(𝑟)) 󸀠 2 (𝑥 ) = , (17) −20 𝑟2𝑓(𝑟)2 (𝐶2 −𝑟2𝑓 (𝑟)) where 𝐶 is an integration constant which will be determined by using reality condition of √−𝐺. Therefore, we yield to the dP/dt following canonical momentum densities: 1 𝜋1 =− 𝐶, 𝜓 󸀠 2𝜋𝛼 −40 (18) 1 𝜋1 = 𝐶V. 𝑡 2𝜋𝛼󸀠 These give us loosing energy and momentum through an endpoint of string: 0 10 B 𝑑𝑃 1 1 =𝜋 |𝑟=𝑟 =− 𝐶, 𝐵 𝑀=1𝑙=1 V = 0.1 𝑑𝑡 𝜓 ℎ 2𝜋𝛼󸀠 Figure 1: Drag force in terms of for , ,and ; (19) 𝑎 = 1.8 (dotted line), 𝑎=3(solid line), and 𝑎 = 4.2 (dashed line). 𝑑𝐸 1 1 =𝜋|𝑟=𝑟 = 𝐶V. 𝑑𝑡 𝑡 ℎ 2𝜋𝛼󸀠 We assume outgoing boundary conditions near the black hole √ As we mentioned before, reality condition of −𝐺 gives us horizon and use the following approximation: √ constant 𝐶. The expression −𝐺 is real for 𝑟=𝑟𝑐 >𝑟ℎ.Inthe (4𝜋𝑇)2 (𝑟 − 𝑟 )𝜕 (𝑟 − 𝑟 )𝑥󸀠 =𝜇2𝑥, case of small V,onecanobtain ℎ 𝑟 ℎ (24) 2 2 which suggests the following solutions: 𝑟 V 4 𝑟𝑐 =𝑟ℎ + |𝑟=𝑟 + O (V ), (20) 𝑓(𝑟)󸀠 ℎ −𝜇/4𝜋𝑇 𝑥=𝑐(𝑟−𝑟ℎ) , (25) which yields to where 𝑇 is the black hole temperature. In the case of infinitesimal 𝜇, we can use the following expansion: 𝐶=V𝑟2 + O (V3). ℎ (21) 2 𝑥=𝑥0 +𝜇 𝑥1 +⋅⋅⋅ . (26) Therefore, we can write drag force as follows: Inserting this equation in the relation (24)gives𝑥0 =constant, 2 and 𝑑𝑃 V𝑟ℎ 3 =− + O (V ). 𝑟 2 󸀠 (22) 𝐴 𝑚 𝑟 𝑑𝑡 2𝜋𝛼 𝑥󸀠 = ∫ 𝑑𝑟, 1 2 (27) 𝑟 𝑓 (𝑟) 𝑟 𝑓 (𝑟) Wedraw drag force in terms of velocity and in agreement with ℎ the previous works such as [11–22]; the value of drag force where 𝐴 is a constant. Assuming near horizon limit enables us increased by V.InFigure 1,wecanseebehaviorofdragforce to obtain the following solution: with rotation parameter and scalar charge. It is shown that the 𝐴 𝑟2 scalar charge increases the value of drag force but the increas- 𝑥 ≈ (−𝑟 + ℎ (𝑟 − 𝑟 )) . 1 4𝜋𝑇𝑟2 (𝑟 − 𝑟 ) 𝑚 4𝜋𝑇 ln ℎ (28) ing rotational parameter decreases the value of the drag force. ℎ ℎ Comparing (25)and(27)givesthefollowingquasinormal 5. Linear Analysis mode condition: 𝑟2 Motion of string yields to small perturbation after late time 𝜇= ℎ . (29) due to the drag force. In that case, the speed of particle is 𝑟𝑚 infinitesimal and one can write 𝐺≈−1. Also, we assume that −𝜇𝑡 𝑥=𝑒 ,where𝜇 is the friction coefficient. Therefore, one It is interesting to note that these results recover drag force can rewrite the equation of motion as follows: (22) for infinitesimal speed. In Figure 2,wecanseebehavior of 𝜇 with rotational parameter and scalar charge. We find that 𝑓 (𝑟) 2 󸀠 2 scalar charge increases the value of friction coefficient, but the 𝜕 (𝑟 𝑓 (𝑟) 𝑥 )=𝜇𝑥. (23) 𝑟2 𝑟 effect of rotation decreases 𝜇. 4 Advances in High Energy Physics

𝜇

c 0 𝜇

−20

0 0 5

rm 0 6 B Figure 2: 𝜇 in terms of 𝑟𝑚: 𝐵 = 0.5 and 𝑎=2(blue dashed line), 𝐵=1 𝑎=2 𝐵=2 𝑎=2 and (blue solid line), and (blue dotted line), Figure 3: 𝜇𝑐 in terms of 𝐵 for 𝑀=1and 𝑙=1; 𝑎=1(blue line), 𝐵=1 𝑎 = 0.2 𝐵=1 𝑎 = 0.4 and and (green dashed line), and and 𝑎=2(black line), and 𝑎=4(red line). (green solid line).

5.1. Low Mass Limit. Lowmasslimitmeansthat𝑟𝑚 → rℎ, particle is calculated. Numerically, we found that the scalar andweusethefollowingassumptions: charge increases the value of drag force but rotational parameter decreases the value of the drag force. Therefore, in order 𝑓 (𝑟) ≈ 4𝜋𝑇 (𝑟ℎ −𝑟 ), tohavethemostfreemotionweneedtoincrease𝑎 and 𝐵 𝑎 𝐵 2 2 (30) decrease .Itmeansthat and may cancel the effect of each 𝑟 =𝑟ℎ +2𝑟ℎ (𝑟 −ℎ 𝑟 )+⋅⋅⋅ , other on the drag force. We can find critical values of scalar charge and rotational parameters in which the value of drag so, by using relation (23)wecanwrite force will be infinite as −𝜇/4𝜋𝑇 𝑥 (𝑟) =(𝑟−𝑟) (1 + (𝑟 − 𝑟 )𝐴+⋅⋅⋅). (31) 2𝑀 ℎ ℎ 𝑎 = √ 𝐵 . (34) 𝑐 81𝑙 𝑐 Then, we can obtain constant 𝐴 as follows: 𝜇 Then, we studied quasinormal modes and obtained friction 𝐴= . (32) coefficient 𝜇 which was enhanced by the black hole charge 2𝜋𝑇𝑟ℎ −𝜇𝑟ℎ and reduced by rotation. Quasinormal mode analysis also It tells that 𝜇=2𝜋𝑇yields to divergence; therefore we called reproduced drag force at slow velocities. It is also possible this a critical behavior of the friction coefficient and found to study dispersion relations which again reproduce the drag that force which was obtained in (22). For the future work, we will consider charged rotating 3D hairy black hole and study drag 3𝑟6 +𝐵𝑀𝑙2𝑟3 − 27𝑎2𝑙2𝑟2 − 54𝐵𝑎2𝑙2𝑟 − 24𝐵2𝑎2𝑙2 𝜇 = ℎ ℎ ℎ ℎ . force. 𝑐 5 𝑟ℎ (33) Conflict of Interests Figure 3 shows behavior of critical friction coefficient with The authors declare that there is no conflict of interests the black hole parameters. regarding the publication of this paper.

6. Conclusions References

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[23]P.Kraus,“Lecturesonblackholesandthe𝐴𝑑𝑆3/𝐶𝐹𝑇2 correspondence,”in Supersymmetric Mechanics. Vol. 3,vol.755ofLec- ture Notes in Physics, pp. 193–247, Springer, Berlin, Germany, 2008. Hindawi Publishing Corporation Advances in High Energy Physics Volume 2014, Article ID 527874, 9 pages http://dx.doi.org/10.1155/2014/527874

Research Article Hawking Radiation-Quasi-Normal Modes Correspondence and Effective States for Nonextremal Reissner-Nordström Black Holes

C. Corda,1,2,3 S. H. Hendi,4,5 R. Katebi,6 and N. O. Schmidt7

1 Dipartimento di Fisica e Chimica, Universita` Santa Rita and Institute for Theoretical Physics and Advanced Mathematics Einstein-Galilei (IFM), 59100 Prato, Italy 2 International Institute for Applicable Mathematics & Information Sciences (IIAMIS), Hyderabad, India 3 International Institute for Applicable Mathematics & Information Sciences (IIAMIS), Udine, Italy 4 Physics Department and Biruni Observatory, College of Sciences, Shiraz University, Shiraz 71454, Iran 5 Research Institute for Astrophysics and Astronomy of Maragha (RIAAM), P.O. Box 55134-441, Maragha, Iran 6 Department of Physics, California State University Fullerton, 800 North State College Boulevard, Fullerton, CA 92831, USA 7 DepartmentofMathematics,BoiseStateUniversity,1910UniversityDrive,Boise,ID83725,USA

Correspondence should be addressed to C. Corda; [email protected]

Received 8 December 2013; Accepted 21 December 2013; Published 11 February 2014

Academic Editor: Xiaoxiong Zeng

Copyright © 2014 C. Corda et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP3.

It is known that the nonstrictly thermal character of the Hawking radiation spectrum harmonizes Hawking radiation with black hole (BH) quasi-normal modes (QNM). This paramount issue has been recently analyzed in the framework of both Schwarzschild BHs (SBH) and Kerr BHs (KBH). In this assignment, we generalize the analysis to the framework of nonextremal Reissner-Nordstrom¨ BHs (RNBH). Such a generalization is important because in both Schwarzschild and Kerr BHs an absorbed (emitted) particle has only mass. Instead, in RNBH the particle has charge as well as mass. In doing so, we expose that, for the RNBH, QNMs can be naturally interpreted in terms of quantum levels for both particle emission and absorption. Conjointly, we generalize some concepts concerning the RNBH’s “effective states.”

1. Introduction Energy conservation plays a fundamental role in BH radiance [2]becausetheemissionorabsorptionofHawkingquanta 𝑀 𝑀 A RNBH of mass is identical to a SBH of mass except with mass 𝑚 and energy-frequency 𝜔 causes a BH of mass 𝑄 that a RNBH has the nonzero charge quantity .Inthispaper, 𝑀 to undergo a transition between discrete energy spectrum we are interested in RNBHs with the nonextremal constraint levels [3–7], where 𝑀>𝑄[1]. The quantity 𝑄 is the physical mechanism for the RNBH’s dual horizons from (1) in [1]: 𝐸=𝑚=𝜔=Δ𝑀 (3)

√ 2 2 for 𝐺=𝑐=𝑘𝐵 =ℎ=1/4𝜋𝜖0 =1(Planck units). 𝑟± =𝑅± (𝑀, 𝑄) =𝑀± 𝑀 −𝑄 , (1) RNBH Given that emission and absorption are reverse processes because the RNBH outer (event) horizon radius 𝑅+ (𝑀, 𝑄) for the quantized energy spectrum conservation [3–7], we RNBH and the RNBH inner (Cauchy) horizon radius 𝑅− (𝑀, 𝑄) consider this pair of transitions as being equal in magnitude RNBH are clearly functions of both 𝑀 and 𝑄,notjust𝑀,asinthe but opposite in direction from the neutral radius perspective 𝑟 =(𝑟 +𝑟 )/2 wellknowncaseoftheSBH horizon radius of 0 + − . It is known that the countable character of successive 𝑟 =𝑅 (𝑀) = 2𝑀. 𝑠 SBH (2) emissionsofHawkingquantawhichisaconsequenceof 2 Advances in High Energy Physics the nonstrictly thermal character of the Hawking radiation respectively, for the SBHareaquantanumber spectrum (see [3–12]) generates a natural correspondence between Hawking radiation and BH QNMs [3–7]. Moreover, 𝐴 (𝑀) 𝑁 (𝑀, 𝜔) = 󵄨 SBH 󵄨, it has also been shown that QNMs can be naturally inter- SBH 󵄨Δ𝐴 (𝑀, 𝜔)󵄨 (5) 󵄨 SBH 󵄨 preted in terms of quantum levels, where the emission or absorption of a particle is interpreted as a transition between such that the SBH horizon area change for the corresponding two distinct levels on the discrete energy spectrum [3–7]. The mass change Δ𝑀 is thermal spectrum correction is an imperative adjustment to the physical interpretation of BH QNMs because these results Δ𝐴 (𝑀, 𝜔) =𝐴 (𝑀±𝜔) −𝐴 (𝑀) are important to realize the underlying unitary quantum SBH SBH SBH 2 gravitytheory[3–7]. Hod’sintriguing works [13, 14]suggested = 32𝜋𝑀𝜔 + 𝑂(𝜔 ) ∼ 32𝜋𝑀Δ𝑀 (6) that BH QNMs carry principle information regarding a BH’s horizon area quantization. Hod’s influential conjecture was = 32𝜋𝑀Δ𝐸, later refined and clarified by Maggiore [15]. Moreover, it is also believed that QNMs delve into the microstructure of because the transition’s minus (−) and plus (+) signs depend spacetime [16]. on emission and absorption, respectively. Next, in [3–5], the Tomakesenseofthestatespacefortheenergyspectrum Bekenstein-Hawking SBH initial and final entropy are states and the underlying BH perturbation field states, an effective framework based on the nonstrictly thermal behavior 𝐴 (𝑀) 𝑆 (𝑀) = SBH , of Hawking’s framework began to emerge [3–7]. In the midst SBH 4 of this superceding BH effective framework [3–7], the BH (7) 𝐴 (𝑀±𝜔) effective state concept was originally introduced for KBHs 𝑆 (𝑀±𝜔) = SBH , in [6] and subsequently applied through Hawking’s period- SBH 4 icity arguments [17, 18] to the BH tunneling mechanism’s nonstrictly black body spectrum [7]. The effective state is respectively, where the corresponding SBH entropy change is meaningful to BH physics and thermodynamics research because one needs additional features and knowledge to Δ𝐴 (𝑀, 𝜔) Δ𝑆 (𝑀, 𝜔) = SBH . (8) consider in future experiments and observations. SBH 4 In this paper, our objective is to apply the nonstrictly thermal BH effective framework of3 [ –7] to nonextremal Subsequently, the SBHinitialandfinaltotalentropyare [3–5] RNBHs. Thus, upon recalling that a RNBH of mass 𝑀 is identical to a SBH of mass 𝑀 except that a RNBH has the 𝑆 (𝑀) =𝑆 (𝑀) − 𝑆 (𝑀) SBH−total SBH ln SBH charge 𝑄,weprepareforourBHQNMinvestigationby 3 reviewing relevant portions of the SBH effective framework + , 2𝐴 (𝑀) [3–7] for quantities related to SBH states and transitions in SBH Section 2.TheninSection 3, we launch our RNBH QNM (9) 𝑆 (𝑀±𝜔) =𝑆 (𝑀±𝜔) − 𝑆 (𝑀±𝜔) exploration by introducing a RNBH effective framework SBH−total SBH ln SBH for quantities pertaining to RNBH states and transitions. 3 + , Finally, we conclude with a brief comparison between the 2𝐴 (𝑀±𝜔) fundamental SBH and RNBH results in Section 4 followed by SBH the recapitulation in Section 5. respectively. Additionally, the SBH initial and final Hawking temperature are [3–5] 2. Schwarzschild Black Hole Framework: 1 Background and Review 𝑇𝐻 (𝑀) = , SBH 8𝜋𝑀 2.1. Schwarzschild Black Hole States and Transitions. Here, we 1 (10) recall some quantities that characterize the SBH. 𝑇𝐻 (𝑀±𝜔) = , SBH 8𝜋 (𝑀±𝜔) First, consider a SBH of initial mass 𝑀, when the SBH emits or absorbs a quantum of energy-frequency 𝜔 (for particle mass 𝑚 and SBH mass change Δ𝑀,suchthat𝑚=𝜔= respectively. Therefore, the quantum transition’s SBH emis- Δ𝑀)toachieveafinalmassof𝑀−𝜔 or 𝑀+𝜔,respectively,for sion tunneling rate is [3–5] the SBH mass-energy transition between states in state space. 𝜔 Thus, we follow [3–5], where the SBH initial and final horizon Γ (𝑀, 𝜔) ∼ [−8𝜋𝑀𝜔 (1− )] SBH exp 2𝑀 area are 𝜔 𝜔 ∼ [− (1− )] (11) 𝐴 (𝑀) = 16𝜋𝑀2 =4𝜋𝑅2 (𝑀) , exp 𝑇 (𝑀) 𝑅 (𝑀) SBH SBH 𝐻 SBH (4) SBH 𝐴 (𝑀±𝜔) = 16𝜋(𝑀±𝜔)2 =4𝜋𝑅2 (𝑀±𝜔) , ∼ [+Δ𝑆 (𝑀, 𝜔)]. SBH SBH exp SBH Advances in High Energy Physics 3

2.2. Schwarzschild Black Hole Effective States and Transitions. such that (14) defines the SBH effective entropy change as Here, we recall some effective quantities that characterize the Δ𝐴𝐸 (𝑀, 𝜔) SBH. SBH Δ𝑆𝐸 (𝑀, 𝜔) =𝑆 (𝑀±𝜔) −𝑆 (𝑀) = Given that 𝑀 is the mass state before and 𝑀± 𝜔is the SBH SBH SBH 4 mass state after the quantum transition, the SBH effective mass (18) and SBH effective horizon are, respectively, identified in3 [ –5] as because the SBH effective horizon area change is

𝑀+(𝑀±𝜔) 𝜔 Δ𝐴𝐸 (𝑀, 𝜔) = 16𝜋𝑀𝐸 (𝑀, 𝜔) 𝜔 (19) 𝑀 (𝑀, 𝜔) = =𝑀± , SBH 𝐸 2 2 (12) and the SBH effective area quanta number is 𝑅𝐸 (𝑀, 𝜔) =2𝑀𝐸 (𝑀, 𝜔) , SBH 𝐴𝐸 (𝑀, 𝜔) SBH 𝑁𝐸 (𝑀, 𝜔) = . (20) which are average quantities between the two states before and SBH Δ𝐴𝐸 (𝑀, 𝜔) after the process [3–5]. Consequently, using (4)and(12)we SBH define the SBH effective horizon area as 2.3. Effective Application of Quasi-Normal Modes to the 𝐴 (𝑀) +𝐴 (𝑀±𝜔) Schwarzschild Black Hole. Here, we recall how the SBH SBH SBH 𝐴𝐸 (𝑀, 𝜔) ≡ perturbation field QNM states can be applied to the SBH SBH 2 (13) effective framework. 2 2 = 16𝜋𝑀𝐸 (𝑀, 𝜔) =4𝜋𝑅𝐸 (𝑀, 𝜔) , The quasi-normal frequencies (QNFs) are typically SBH labeled as 𝜔𝑛𝑙,where𝑙 is the angular momentum quantum 𝑙 𝑙≥2 which is the average of the SBH’s initial and final horizon number [3–5, 15, 19]. Thus, for each ,suchthat for gravitational perturbations, there is a countable sequence of areas. Subsequently, utilizing (7), the Bekenstein-Hawking 𝑛 SBH effective entropy is defined as QNMs labeled by the overtone number ,whichisanatural number [3–5, 15]. 𝑆 (𝑀) +𝑆 (𝑀±𝜔) Now |𝜔𝑛| is the damped harmonic oscillator’s proper SBH SBH 𝑆𝐸 (𝑀, 𝜔) ≡ , (14) frequency that is defined as [3–5, 15] SBH 2 󵄨 󵄨 󵄨𝜔 󵄨 = (𝜔 ) = √𝜔2 +𝜔2 . and consequently employs (13)and(14) to define the SBH 󵄨 𝑛󵄨 0 𝑛 𝑛R 𝑛I (21) effective total entropy as |𝜔 |=𝜔 Maggiore [15] articulated that the establishment 𝑛 𝑛R is 𝑆𝐸 (𝑀, 𝜔) only correct for the very long-lived and lowly excited QNMs SBH−total approximation |𝜔𝑛|≫𝜔𝑛 ,whereasforalotofBHQNMs, 3 I such as those that are highly excited, the opposite limit is ≡𝑆𝐸 (𝑀, 𝜔) − ln 𝑆𝐸 (𝑀, 𝜔) + . SBH SBH 2𝐴𝐸 (𝑀, 𝜔) 𝜔 SBH correct [3–5, 15]. Therefore, the parameter in (12)–(20)is (15) substituted for the |𝜔𝑛| parameter [3–5]becausewewishto employ BH QNFs. When 𝑛 is large, the SBH QNFs become Thus, employing (3)and(10), the SBH effective temperature is independent of 𝑙 and thereby exhibit the nonstrictly thermal [3–5] structure [3–5]

−1 󵄨 󵄨 1 󵄨 󵄨 −1 −1 𝜔 = 3×𝑇 (𝑀, 󵄨𝜔 󵄨)+2𝜋𝑖(𝑛+ )×𝑇 (𝑀, 󵄨𝜔 󵄨) 𝑇𝐻 (𝑀) +𝐻 𝑇 (𝑀 ± 𝜔) 𝑛 ln 𝐸 󵄨 𝑛󵄨 𝐸 󵄨 𝑛󵄨 SBH SBH SBH 2 SBH 𝑇𝐸 (𝑀, 𝜔) =( ) SBH 2 3 2𝜋𝑖 + O (𝑛−1/2)= ln + −1 󵄨 󵄨 󵄨 󵄨 𝑀+𝑀±𝜔 4𝜋 [2𝑀 − 󵄨𝜔𝑛󵄨] 4𝜋 [2𝑀 − 󵄨𝜔𝑛󵄨] =(8𝜋[ ]) (16) 2 1 −1/2 ln 3 ×(𝑛+ )+O (𝑛 )= 󵄨 󵄨 1 1 2 8𝜋𝑀 (𝑀, 󵄨𝜔 󵄨) = = , 𝐸 󵄨 𝑛󵄨 4𝜋 (2𝑀 ± 𝜔) 8𝜋𝑀𝐸 (𝑀, 𝜔) 2𝜋 (𝑛+1/2) −1/2 + 󵄨 󵄨 𝑖+O (𝑛 ), which is the inverse of the average value of the inverses of 8𝜋𝑀𝐸 (𝑀, 󵄨𝜔𝑛󵄨) the initial and final Hawking temperatures. Consequently, (22) (16)letsonerewrite(11) to define the SBH effective emission tunneling rate (in the Boltzmann-like form) as [3–5] where ln 3 𝜔 𝑚𝑛 ≡𝜔𝑛 = 󵄨 󵄨 , R 8𝜋𝑀 (𝑀, 󵄨𝜔 󵄨) Γ𝐸 (𝑀, 𝜔) ∼ exp [− ] 𝐸 󵄨 𝑛󵄨 SBH 𝑇𝐸 (𝑀, 𝜔) (23) SBH (17) 2𝜋 1 𝑝𝑛 ≡𝜔𝑛 = 󵄨 󵄨 (𝑛 + ). = exp [+Δ𝑆𝐸 (𝑀, 𝜔)], I 8𝜋𝑀 (𝑀, 󵄨𝜔 󵄨) 2 SBH 𝐸 󵄨 𝑛󵄨 4 Advances in High Energy Physics

Thus, when referring to highly excited QNMs one gets mass of 𝑀−𝜔or 𝑀+𝜔and a final charge of 𝑄−𝑞or 𝑄+𝑞, |𝜔𝑛|≈𝑝𝑛 [3–5], where the quantized levels differ from [15] respectively, for the RNBH mass-energy transition between because they are not equally spaced in exact form. Therefore, states in state space. For this, all we need to do is replace the according to [3–5], we have RNBH’smassandchargeparametersin(26)and(29). Thus, (26) establishes the RNBH final event horizon area as 2 2 √(ln 3) +4𝜋2(𝑛+1/2) 󵄨 󵄨 𝐴+ (𝑀±𝜔,𝑄±𝑞) 󵄨𝜔 󵄨 = 󵄨 󵄨 RNBH 󵄨 𝑛󵄨 8𝜋𝑀 (𝑀, 󵄨𝜔 󵄨) 𝐸 󵄨 𝑛󵄨 2 (24) =4𝜋𝑅+ (𝑀 ± 𝜔, 𝑄 ±𝑞) RNBH (30) 1 2 󵄨 󵄨 √ 2 2 2 =𝑇𝐸 (𝑀, 󵄨𝜔𝑛󵄨) (ln 3) +4𝜋 (𝑛 + ) , 2 2 SBH 2 =4𝜋((𝑀±𝜔) + √(𝑀±𝜔) −(𝑄±𝑞) ) . which is solved to yield Equation (27)presentstheBekenstein-Hawking RNBH final entropy as √( 3)2 +4𝜋2(𝑛+1/2)2 󵄨 󵄨 √ 2 ln (25) 𝐴+ (𝑀±𝜔,𝑄±𝑞) 󵄨𝜔 󵄨 =𝑀− 𝑀 − RNBH 󵄨 𝑛󵄨 𝑆+ (𝑀 ± 𝜔, 𝑄 ± 𝑞)= , (31) 4𝜋 RNBH 4 when we obey |𝜔𝑛|<𝑀because a BH cannot emit more and (28) defines the RNBH final electrostatic potential as energy than its total mass. Φ+ (𝑀 ± 𝜔, 𝑄 ±𝑞) 𝑄 3. Reissner-Nordström Black Hole = 4𝜋𝑅+ (𝑀 ± 𝜔, 𝑄 ±𝑞) Framework: An Introduction RNBH (32) 𝑄 We note that for this framework we consider the RNBH event = 2 2 horizon features, which are derived from the 𝑅+ (𝑀, 𝑄) in 4𝜋 ( 𝑀±𝜔) + √(𝑀±𝜔) −(𝑄±𝑞) ) RNBH (1). for usage in (29) of [20],whereitisproposedthattheRNBH 3.1. Reissner-Nordstrom¨ Black Hole States and Transitions. adiabatic invariant is Here, we recall some quantities that characterize the RNBH. 𝜔−Φ+ (𝑀±𝜔,𝑄) 𝑞 𝑀 𝐼+ (𝑀,𝜔,𝑄,𝑞)=∫ First, consider a RNBH of initial mass and initial RNBH 𝜔 charge 𝑄.Using(1), we define the RNBH initial event horizon (33) Δ𝑀 − Φ (𝑀±Δ𝑀,𝑄) Δ𝑄 area as = ∫ + 2 Δ𝑀 √ 2 2 2 𝐴+ (𝑀, 𝑄) =4𝜋(𝑀+ 𝑀 −𝑄 ) =4𝜋𝑅+ (𝑀, 𝑄) , Δ𝑄 = 𝑞 RNBH RNBH because .Hence,(29) identifies the RNBH final (26) Hawking temperature as

𝑇+𝐻 (𝑀±𝜔,𝑄±𝑞) the Bekenstein-Hawking RNBH initial entropy as RNBH

𝑅+ (𝑀 ± 𝜔, 𝑄 ± 𝑞)− 𝑅 (𝑀±𝜔,𝑄±𝑞) 𝐴+ (𝑀, 𝑄) RNBH RNBH RNBH = 𝑆+ (𝑀, 𝑄) = , (27) 𝐴 (𝑀 ± 𝜔, 𝑄 ±𝑞) RNBH 4 + RNBH (34) and the RNBH initial electrostatic potential as √(𝑀±𝜔)2 −(𝑄±𝑞)2 = . 𝑄 𝑄 2 Φ (𝑀, 𝑄) = = . √ 2 2 + 2 2 2𝜋((𝑀±𝜔) + (𝑀±𝜔) −(𝑄±𝑞) ) 4𝜋𝑅+ (𝑀, 𝑄) 4𝜋 (𝑀 + √𝑀 −𝑄 ) RNBH (28) Next, upon generalizing (16) in [2]andthework[21], we define the RNBH tunneling rate as Consequently, (17) of [2] identifies the RNBH initial Hawking Γ+ (𝑀, 𝜔, 𝑄, 𝑞) temperature as RNBH √𝑀2 −𝑄2 𝜔 𝑇 (𝑀, 𝑄) = ∼ exp [−4𝜋 (2𝜔 (𝑀± ) +𝐻 2 2 RNBH 2𝜋(𝑀 + √𝑀2 −𝑄2) (29) 2 2 − (𝑀±𝜔) √(𝑀±𝜔) −(𝑄±𝑞) (35) 𝑅+ (𝑀, 𝑄) −𝑅− (𝑀, 𝑄) = RNBH RNBH . 𝐴+ (𝑀, 𝑄) 2 2 RNBH +𝑀√𝑀 −𝑄 )] Second, consider when the RNBH emits or absorbs a quan- 𝜔 𝑞 ∼ exp [Δ𝑆+ (𝑀,𝜔,𝑄,𝑞)], tum of energy-frequency with charge to achieve a final RNBH Advances in High Energy Physics 5 whereweutilize(30) to define the Bekenstein-Hawking RNBH Afterwards, we use (28)and(42) to define the RNBH effective entropy change as electrostatic potential as

Φ+𝐸 (𝑀,𝜔,𝑄,𝑞) Δ𝐴+ (𝑀,𝜔,𝑄,𝑞) RNBH Δ𝑆+ (𝑀,𝜔,𝑄,𝑞)= , (36) RNBH 4 𝑄 (𝑄, 𝑞) ≡ 𝐸 4𝜋𝑅+𝐸 (𝑀,𝜔,𝑄,𝑞) such that the RNBH event horizon area change is RNBH (44) 𝑄 (𝑄, 𝑞) ≡ 𝐸 Δ𝐴+ (𝑀,𝜔,𝑄,𝑞)=𝐴+ (𝑀±𝜔,𝑄±𝑞) RNBH RNBH 4𝜋 (𝑀 (𝑀, 𝜔) + √𝑀2 (𝑀, 𝜔) −𝑄2 (𝑄, 𝑞)) (37) 𝐸 𝐸 𝐸 −𝐴+ (𝑀, 𝑄) RNBH so we can utilize the 𝑇𝐸 (𝑀, 𝜔) in (16)alongwith(39), (40), SBH so we can define the RNBH event horizon area quanta number and (44) to define the RNBH effective adiabatic invariant as as 𝐼+𝐸 (𝑀,𝜔,𝑄,𝑞) RNBH 𝐴+ (𝑀, 𝑄) RNBH 𝑑𝑀 (𝑀, 𝜔) −Φ (𝑀,𝜔,𝑄,𝑞)𝑑𝑄 (𝑄, 𝑞) (45) 𝑁+ (𝑀, 𝜔, 𝑄, 𝑞)= 󵄨 󵄨. 𝐸 +𝐸 𝐸 RNBH 󵄨 󵄨 (38) ≡ ∫ . 󵄨Δ𝐴+ (𝑀,𝜔,𝑄,𝑞)󵄨 󵄨 RNBH 󵄨 𝑇𝐸 (𝑀, 𝜔) SBH

3.2. Reissner-NordstromBlackHoleEffectiveStatesandTran-¨ At this point, (16)and(35) let us introduce and define the sitions. Here, we define some effective quantities that charac- RNBH effective temperature as terize the RNBH. 𝑇+𝐸 (𝑀,𝜔,𝑄,𝑞) The RNBH effective mass is equivalent to the SBH effective RNBH mass component of (12), which is √(𝑀±𝜔/2)2 −(𝑄±𝑞/2)2 ≡ 𝑀+(𝑀±𝜔) 2 𝑀 (𝑀, 𝜔) ≡ . (39) 2𝜋[(𝑀±𝜔/2) + √(𝑀±𝜔/2)2 −(𝑄±𝑞/2)2] 𝐸 2

√ 2 2 Next, we define the RNBH effective charge as 𝑀𝐸 (𝑀, 𝜔) −𝑄𝐸 (𝑄, 𝑞) ≡ 2 2 2 𝑄+(𝑄±𝑞) 2𝜋(𝑀𝐸(𝑀, 𝜔) + √𝑀 (𝑀, 𝜔) −𝑄 (𝑄, 𝑞)) 𝑄 (𝑄, 𝑞) ≡ , (40) 𝐸 𝐸 𝐸 2 𝑅+𝐸 (𝑀, 𝜔, 𝑄, 𝑞)−𝐸 −𝑅 (𝑀,𝜔,𝑄,𝑞) ≡ RNBH RNBH , 𝑄 which is the average of the RNBH’s initial charge and final 𝐴+𝐸 (𝑀, 𝜔, 𝑄, 𝑞) 𝑄±𝑞 RNBH charge .Fromthis,(1), (39), and (40) are used to define (46) the corresponding RNBH effective event horizon and RNBH effective Cauchy horizon as which authorizes us to exercise (36)and(46)torewrite(35) to define the RNBH effective tunneling rate as 𝑟±𝐸 ≡𝑅±𝐸 (𝑀,𝜔,𝑄,𝑞)≡𝑀𝐸 (𝑀, 𝜔) RNBH ±𝜔 (41) Γ+𝐸 (𝑀, 𝜔, 𝑄, 𝑞)∼ exp [ ] ± √𝑀2 (𝑀, 𝜔) −𝑄2 (𝑄, 𝑞), RNBH 𝑇+𝐸 (𝑀, 𝜔, 𝑄, 𝑞) 𝐸 𝐸 RNBH (47)

∼ exp [Δ𝑆+ (𝑀,𝜔,𝑄,𝑞)], with respect to the energy conservation and pair production RNBH neutrality of (39). Next, we employ (26), (39), and (41)to such that the RNBH effective entropy change is defined as define the RNBH effective event horizon area as Δ𝐴+𝐸 (𝑀,𝜔,𝑄,𝑞) RNBH Δ𝑆+𝐸 (𝑀,𝜔,𝑄,𝑞)≡ (48) 𝐴+𝐸 (𝑀,𝜔,𝑄,𝑞) RNBH RNBH 4 2 ≡4𝜋𝑅+𝐸 (𝑀, 𝜔, 𝑄, 𝑞) for the RNBH effective event horizon area change RNBH (42) 3 2 2𝜔𝑞 +𝑄 𝜋 2 2 ≡4𝜋(𝑀 (𝑀, 𝜔) + √𝑀 (𝑀, 𝜔) −𝑄 (𝑄, 𝑞)) , Δ𝐴+𝐸 (𝑀,𝜔,𝑄,𝑞)≡ (49) 𝐸 𝐸 𝐸 RNBH (𝑀2 −𝑄2)3/2 whichisthenusedtodefinetheRNBH effective entropy as and the RNBH effective event horizon area quanta number

𝐴+𝐸 (𝑀, 𝜔, 𝑄, 𝑞) 𝐴+𝐸 (𝑀,𝜔,𝑄,𝑞) 𝑁 (𝑀,𝜔,𝑄,𝑞)≡ RNBH . RNBH +𝐸 󵄨 󵄨 (50) 𝑆+𝐸 (𝑀,𝜔,𝑄,𝑞)≡ . (43) RNBH 󵄨Δ𝐴 (𝑀, 𝜔, 𝑄,󵄨 𝑞) RNBH 󵄨 +𝐸 󵄨 4 󵄨 RNBH 󵄨 6 Advances in High Energy Physics

4. Effective Application of Quasi-Normal From (39), (41), and (46)wedefinetheeffectivequantities Modes to the Reissner-Nordström associated with the QNMs as Black Hole Here, we explain how the RNBH perturbation field QNM 󵄨 󵄨 states can be applied to the RNBH effective framework. 󵄨 󵄨 𝑀+(𝑀−󵄨𝜔𝑛󵄨) 𝑀 (𝑀, 󵄨𝜔 󵄨) ≡ , (54) Similarly to SBH QNFs, the RNBH QNFs become inde- 𝐸 󵄨 𝑛󵄨 2 pendent of 𝑙 for large 𝑛 [22]. Thus, for large 𝑛,wehavetwo 󵄨 󵄨 families of the QNM: 𝑟±𝐸 ≡𝑅±𝐸 (𝑀, 󵄨𝜔𝑛󵄨 ,𝑄,𝑞)= RNBH 󵄨 󵄨 (55) 1 󵄨 󵄨 √ 2 󵄨 󵄨 2 =𝑀𝐸 (𝑀, 󵄨𝜔𝑛󵄨)± 𝑀𝐸 (𝑀, 󵄨𝜔𝑛󵄨)−𝑄𝐸 (𝑄, 𝑞), 𝜔𝑛 = ln 3×𝑇+𝐻 (𝑀, 𝑄) −2𝜋(𝑛+ )𝑖×𝑇+𝐻 (𝑀, 𝑄) SBH 2 SBH 𝑞𝑄 󵄨 󵄨 𝑇+𝐸 (𝑀, 󵄨𝜔𝑛󵄨 ,𝑄,𝑞) + , RNBH 󵄨 󵄨 𝑅+ (𝑀, 𝑄) SBH (51) √ 󵄨 󵄨 2 2 (𝑀 − 󵄨𝜔𝑛󵄨 /2) −(𝑄−𝑞/2) 1 ≡ 𝜔𝑛 = ln 2×𝑇+𝐻 (𝑀, 𝑄) −2𝜋(𝑛+ )𝑖×𝑇+𝐻 (𝑀, 𝑄) 2 RNBH 2 RNBH 󵄨 󵄨 √ 󵄨 󵄨 2 2 2𝜋[(𝑀 − 󵄨𝜔𝑛󵄨 /2)+ (𝑀 − 󵄨𝜔𝑛󵄨 /2) −(𝑄−𝑞/2) ] 𝑞𝑄 2√𝑀2 −𝑄2 + = ln 2 𝑅+ (𝑀, 𝑄) 2𝜋(𝑀 + √𝑀2 −𝑄2) 󵄨 󵄨 RNBH √ 2 󵄨 󵄨 2 𝑀𝐸 (𝑀, 󵄨𝜔𝑛󵄨)−𝑄𝐸 (𝑄, 𝑞) = (𝑛+1/2) √𝑀2 −𝑄2 𝑞𝑄 2 − 𝑖+ . 󵄨 󵄨 2 󵄨 󵄨 2 2 2𝜋(𝑀𝐸 (𝑀, 󵄨𝜔𝑛󵄨)+√𝑀 (𝑀, 󵄨𝜔𝑛󵄨)−𝑄 (𝑄, 𝑞)) √ 2 2 𝑅+ (𝑀, 𝑄) 󵄨 󵄨 𝐸 󵄨 󵄨 𝐸 (𝑀 + 𝑀 −𝑄 ) RNBH (52) 󵄨 󵄨 󵄨 󵄨 𝑅+𝐸 (𝑀, 󵄨𝜔𝑛󵄨 ,𝑄,𝑞)−𝑅−𝐸 (𝑀, 󵄨𝜔𝑛󵄨 ,𝑄,𝑞) RNBH 󵄨 󵄨 RNBH 󵄨 󵄨 = 󵄨 󵄨 , 𝐴+𝐸 (𝑀, 󵄨𝜔𝑛󵄨 ,𝑄,𝑞) RNBH 󵄨 󵄨 Now the approximation of (51)and(52)isonlyrelevant (56) under the assumption that the BH radiation spectrum is strictly thermal [3–5] because they both use the Hawking temperature 𝑇+𝐻 in (29).Hence,tooperateincompliance RNBH with [3–5] and thereby account for the thermal spectrum respectively, for the quantum overtone number 𝑛 in (53). deviation of (35), we opt to select the (52) case and upgrade Hence, (53)letsusrewritetheSBHcaseof(23)topresent it by effectively replacing its 𝑇𝐻 in (29)withthe𝑇+𝐸 in RNBH RNBH the RNBH case (46). Therefore, the corrected expression for the RNBH QNFs of (52) which encodes the nonstrictly thermal behavior of the radiation spectrum is defined as 󵄨 󵄨 𝑚𝑛 ≡ ln 2×𝑇+𝐸 (𝑀, 󵄨𝜔𝑛󵄨 ,𝑄,𝑞) RNBH 󵄨 󵄨 󵄨 󵄨 𝜔𝑛 ≡ ln 2×𝑇+𝐸 (𝑀, 󵄨𝜔𝑛󵄨 ,𝑄,𝑞) RNBH 𝑒𝑄 (𝑄, 𝑞) 1 󵄨 󵄨 + 𝐸 −2𝜋(𝑛+ )𝑖×𝑇 (𝑀, 󵄨𝜔 󵄨 ,𝑄,𝑞) 󵄨 󵄨 +𝐸 󵄨 𝑛󵄨 𝑅+𝐸 (𝑀, 󵄨𝜔𝑛󵄨 ,𝑄,𝑞) 2 RNBH RNBH 𝑞𝑄 (𝑄, 𝑞) 𝐸 󵄨 󵄨 + 󵄨 󵄨 2√𝑀2 (𝑀, 󵄨𝜔 󵄨)−𝑄2 (𝑄, 𝑞) 𝑅+𝐸 (𝑀, 󵄨𝜔𝑛󵄨 ,𝑄,𝑞) ln 𝐸 󵄨 𝑛󵄨 𝐸 RNBH = 2 √ 2 󵄨 󵄨 2 󵄨 󵄨 √ 2 󵄨 󵄨 2 ln 2 𝑀𝐸 (𝑀, 󵄨𝜔𝑛󵄨)−𝑄𝐸 (𝑄, 𝑞) 2𝜋(𝑀𝐸(𝑀, 󵄨𝜔𝑛󵄨)+ 𝑀𝐸(𝑀, 󵄨𝜔𝑛󵄨)−𝑄𝐸(𝑄, 𝑞)) ≡ 2 󵄨 󵄨 √ 2 󵄨 󵄨 2 (57) 2𝜋(𝑀𝐸 (𝑀, 󵄨𝜔𝑛󵄨)+ 𝑀𝐸 (𝑀, 󵄨𝜔𝑛󵄨)−𝑄𝐸 (𝑄, 𝑞)) 𝑞𝑄𝐸 (𝑄, 𝑞) 󵄨 󵄨 + 󵄨 󵄨 , (𝑛+1/2) √𝑀2 (𝑀, 󵄨𝜔 󵄨)−𝑄2 (𝑄, 𝑞) 𝑅+𝐸 (𝑀, 󵄨𝜔𝑛󵄨 ,𝑄,𝑞) 𝐸 󵄨 𝑛󵄨 𝐸 RNBH 󵄨 󵄨 − 𝑖 2 󵄨 󵄨 √ 2 󵄨 󵄨 2 1 (𝑀𝐸 (𝑀, 󵄨𝜔𝑛󵄨)+ 𝑀𝐸 (𝑀, 󵄨𝜔𝑛󵄨)−𝑄𝐸 (𝑄, 𝑞)) 󵄨 󵄨 𝑝𝑛 ≡−2𝜋(𝑛+ )×𝑇+𝐸 (𝑀, 󵄨𝜔𝑛󵄨 ,𝑄,𝑞) 2 RNBH 󵄨 󵄨 𝑞𝑄𝐸 (𝑄, 𝑞) √ 2 󵄨 󵄨 2 + . (𝑛+1/2) 𝑀𝐸 (𝑀, 󵄨𝜔𝑛󵄨)−𝑄𝐸 (𝑄, 𝑞) 󵄨 󵄨 =− . 𝑅+𝐸 (𝑀, 󵄨𝜔𝑛󵄨 ,𝑄,𝑞) 2 RNBH 󵄨 󵄨 󵄨 󵄨 (𝑀 (𝑀, 󵄨𝜔 󵄨)+√𝑀2 (𝑀, 󵄨𝜔 󵄨)−𝑄2 (𝑄, 𝑞)) (53) 𝐸 󵄨 𝑛󵄨 𝐸 󵄨 𝑛󵄨 𝐸 Advances in High Energy Physics 7

󵄨 󵄨 |𝜔 |≈𝑝 𝐴+𝐸 (𝑀, 󵄨Δ𝜔𝑛,𝑛−1󵄨 ,𝑄,𝑞) Thus, we recall that if 𝑛 𝑛, then we are referring to 󵄨 󵄨 RNBH 󵄨 󵄨 𝑆+𝐸 (𝑀, 󵄨Δ𝜔𝑛,𝑛−1󵄨 ,𝑄,𝑞)≡ , highly excited QNMs [3–5]. Therefore, the SBH case of (24) RNBH 4 becomes the RNBH case 󵄨 󵄨 Φ+𝐸 (𝑀, 󵄨Δ𝜔𝑛,𝑛−1󵄨 ,𝑄,𝑞)

𝑄𝐸 (𝑄, 𝑞) 󵄨 󵄨 ≡ 󵄨 󵄨 󵄨𝜔 󵄨 4𝜋𝑅+𝐸 (𝑀, 󵄨Δ𝜔𝑛,𝑛−1󵄨 ,𝑄,𝑞) 󵄨 𝑛󵄨 RNBH

2 󵄨 󵄨 2 2 2 𝑄 (𝑄, 𝑞) √𝑀 (𝑀, 󵄨𝜔 󵄨)−𝑄 (𝑄, 𝑞)√( 2) −4𝜋2(𝑛+1/2) ≡ 𝐸 , 𝐸 󵄨 𝑛󵄨 𝐸 ln 󵄨 󵄨 󵄨 󵄨 ≡ 4𝜋 (𝑀 (𝑀, 󵄨Δ𝜔 󵄨)+√𝑀2 (𝑀, 󵄨Δ𝜔 󵄨)−𝑄2 (𝑄, 𝑞)) 󵄨 󵄨 󵄨 󵄨 2 𝐸 󵄨 𝑛,𝑛−1󵄨 𝐸 󵄨 𝑛,𝑛−1󵄨 𝐸 2𝜋(𝑀 (𝑀, 󵄨𝜔 󵄨)+√𝑀2 (𝑀, 󵄨𝜔 󵄨)−𝑄2 (𝑄, 𝑞)) 𝐸 󵄨 𝑛󵄨 𝐸 󵄨 𝑛󵄨 𝐸 󵄨 󵄨 𝐼+𝐸 (𝑀, 󵄨Δ𝜔𝑛,𝑛−1󵄨 ,𝑄,𝑞) RNBH 󵄨 󵄨 𝑞𝑄 (𝑄, 𝑞) 𝐸 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 + 󵄨 󵄨 =𝑇+𝐸 (𝑀, 󵄨𝜔𝑛󵄨 ,𝑄,𝑞) 𝑑𝑀𝐸 (𝑀, 󵄨Δ𝜔𝑛,𝑛−1󵄨)−Φ+𝐸 (𝑀, 󵄨Δ𝜔𝑛,𝑛−1󵄨 ,𝑄,𝑞)𝑑𝑄𝐸 (𝑄, 𝑞) 󵄨 󵄨 RNBH 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 𝑅+𝐸 (𝑀, 󵄨𝜔𝑛󵄨 ,𝑄,𝑞) =∫ 󵄨 󵄨 , RNBH 𝑇𝐸 (𝑀, 󵄨Δ𝜔𝑛,𝑛−1󵄨) SBH 󵄨 󵄨 1 2 𝑞𝑄 (𝑄, 𝑞) (60) √ 2 2 𝐸 × (ln 2) −4𝜋 (𝑛 + ) + 󵄨 󵄨 . 2 𝑅+𝐸 (𝑀, 󵄨𝜔𝑛󵄨 ,𝑄,𝑞) RNBH 󵄨 󵄨 (58) and (47)–(50)become

󵄨 󵄨 Γ+𝐸 (𝑀, 󵄨Δ𝜔𝑛,𝑛−1󵄨 ,𝑄,𝑞) RNBH 󵄨 󵄨 󵄨 󵄨 Hence, upon considering (40)and(54), one can rewrite (58) ± 󵄨Δ𝜔 󵄨 ∼ [ 󵄨 𝑛,𝑛−1󵄨 ] as exp 󵄨 󵄨 𝑇+𝐸 (𝑀, 󵄨Δ𝜔𝑛,𝑛−1󵄨 ,𝑄,𝑞) RNBH 󵄨 󵄨 󵄨 󵄨 ∼ exp [Δ𝑆+ (𝑀, 󵄨Δ𝜔𝑛,𝑛−1󵄨 ,𝑄,𝑞)], RNBH 󵄨 󵄨 √ 󵄨 󵄨 2 2√ 2 2 2 󵄨 󵄨 󵄨 󵄨 (𝑀 − 󵄨𝜔𝑛󵄨 /2) − (𝑄 − 𝑞/2) (ln 2) −4𝜋 (𝑛+1/2) Δ𝑆+𝐸 (𝑀, 󵄨Δ𝜔𝑛,𝑛−1󵄨 ,𝑄,𝑞) 󵄨𝜔 󵄨 ≡ RNBH 󵄨 𝑛󵄨 2 󵄨 󵄨 √ 󵄨 󵄨 2 2 󵄨 󵄨 2𝜋[(𝑀 − 󵄨𝜔𝑛󵄨 /2) + (𝑀 − 󵄨𝜔𝑛󵄨 /2) − (𝑄 − 𝑞/2) ] 󵄨 󵄨 Δ𝐴+𝐸 (𝑀, 󵄨Δ𝜔𝑛,𝑛−1󵄨 ,𝑄,𝑞) ≡ RNBH , (61) 4 𝑞 (𝑄 − 𝑞/2) + , 󵄨 󵄨 3 󵄨 󵄨 √ 2 2 󵄨 󵄨 2 󵄨Δ𝜔𝑛,𝑛−1󵄨 𝑞+𝜋𝑄 (𝑀 − 󵄨𝜔𝑛󵄨 /2) + (𝑀 − |𝜔𝑛|/2) − (𝑄 − 𝑞/2) 󵄨 󵄨 Δ𝐴+𝐸 (𝑀, 󵄨Δ𝜔𝑛,𝑛−1󵄨 ,𝑄,𝑞)≡ , (59) RNBH (𝑀2 −𝑄2)3/2 󵄨 󵄨 𝑁+𝐸 (𝑀, 󵄨Δ𝜔𝑛,𝑛−1󵄨 ,𝑄,𝑞) RNBH 󵄨 󵄨 󵄨 󵄨 |𝜔 | 𝐴+𝐸 (𝑀, 󵄨Δ𝜔𝑛,𝑛−1󵄨 ,𝑄,𝑞) where the solution of (59)intermsof 𝑛 will be the answer ≡ RNBH , |𝜔 | 󵄨 󵄨 󵄨 󵄨 of 𝑛 . Therefore, given a quantum transition between the 󵄨Δ𝐴+𝐸 (𝑀, 󵄨Δ𝜔𝑛,𝑛−1󵄨 ,𝑄,𝑞)󵄨 󵄨 RNBH 󵄨 󵄨 󵄨 levels 𝑛 and 𝑛−1, we define |Δ𝜔𝑛,𝑛−1|≡|𝜔𝑛 −𝜔𝑛−1| where (41)–(45) are rewritten as respectively.

󵄨 󵄨 5. A Brief Comparison 𝑟±𝐸 ≡𝑅±𝐸 (𝑀, 󵄨Δ𝜔𝑛,𝑛−1󵄨 ,𝑄,𝑞) RNBH 󵄨 󵄨 󵄨 󵄨 Here, we will show that the SBH results of Section 2 are in ≡𝑀𝐸 (𝑀, 󵄨Δ𝜔𝑛,𝑛−1󵄨) fundamental agreement with the RNBH results of Section 3 󵄨 󵄨 for small 𝑄, where we recall that the RNBH of mass 𝑀 is ± √𝑀2 (𝑀, 󵄨Δ𝜔 󵄨)−𝑄2 (𝑄, 𝑞), 𝐸 󵄨 𝑛,𝑛−1󵄨 𝐸 identical to a SBH of mass 𝑀 except that a RNBH has the 󵄨 󵄨 nonzero charge quantity 𝑄. 𝐴+𝐸 (𝑀, 󵄨Δ𝜔𝑛,𝑛−1󵄨 ,𝑄,𝑞) RNBH First, for small 𝑄,theSBH’s𝑇𝐸 (𝑀, 𝜔) of (16)isrelated SBH 2 󵄨 󵄨 to the RNBH’s 𝑇+𝐸 (𝑀, 𝜔, 𝑄, 𝑞) of (46)as ≡4𝜋𝑅+𝐸 (𝑀, 󵄨Δ𝜔𝑛,𝑛−1󵄨 ,𝑄,𝑞) RNBH RNBH 󵄨 󵄨 ≡4𝜋(𝑀 (𝑀, 󵄨Δ𝜔 󵄨) 𝑇+𝐸 (𝑀,𝜔,𝑄,𝑞) 𝐸 󵄨 𝑛,𝑛−1󵄨 RNBH 2 2 (62) 2 3𝑞 𝑄 4 4 2 󵄨 󵄨 2 ≡𝑇 𝑀, 𝜔 − + O (𝑄 ,𝑞 ). + √𝑀 (𝑀, 󵄨Δ𝜔 󵄨)−𝑄 (𝑄, 𝑞)) , 𝐸 ( ) 5 𝐸 󵄨 𝑛,𝑛−1󵄨 𝐸 SBH 8(2𝑚 ± 𝜔) 𝜋 8 Advances in High Energy Physics

Second, for small 𝑄,theSBH’s𝐴𝐸 (𝑀, 𝜔) of (13)complies for an emission involving quantum levels 𝑛 and 𝑛−1,which SBH with the RNBH’s 𝐴+𝐸 (𝑀,𝜔,𝑄,𝑞)of (42)as becomes RNBH 2 2 4 𝑞 1 1 𝐴+𝐸 (𝑀,𝜔,𝑄,𝑞)≡𝐴𝐸 (𝑀, 𝜔) −8𝜋𝑄 + O (𝑄 ). Δ𝑀 ≈ √𝑀2 + −𝑄𝑞− (𝑛 + ) RNBH SBH 𝑛 2 2 2 (63) (68) 𝑞2 1 1 Third, for small 𝑄,theSBH’s𝑆𝐸 (𝑀, 𝜔) of (14) corresponds √ 2 SBH − 𝑀 + −𝑄𝑞− (𝑛 − ) with the RNBH’s 𝑆+𝐸 (𝑀,𝜔,𝑄,𝑞)of (43)as 2 2 2 RNBH 𝑛 2 4 for large . 𝑆+𝐸 (𝑀,𝜔,𝑄,𝑞)≡𝑆𝐸 (𝑀, 𝜔) −2𝜋𝑄 + O (𝑄 ). (64) RNBH SBH 6. Conclusion Remarks Fourth, for small 𝑄,theSBH’sQNF|𝜔𝑛| of (24)isconsistent |𝜔 | with the RNBH’s QNF 𝑛 of (58)and(59)as We began our paper by summarizing some basic similarities and differences between SBHs and RNBHs in terms of charge √ 2 2 2 󵄨 󵄨 ln 2 −4𝜋 (𝑛+1/2) 𝑞𝑄 and horizon radii. Moreover, we briefly explored the Parikh- 󵄨𝜔 󵄨 ≡ 󵄨 󵄨 + 󵄨 󵄨 󵄨 𝑛󵄨 4(2𝑀−󵄨𝜔 󵄨)𝜋 2𝑀 − 󵄨𝜔 󵄨 Wilczek statement that explains how energy conservation and 󵄨 𝑛󵄨 󵄨 𝑛󵄨 pair production [2, 23]arefundamentallyrelatedtosuchBHs. 2 󵄨 󵄨 󵄨 󵄨2 For a BH’s discrete energy spectrum, the emission or absorp- = ( (3 (16𝜋𝑀 − 16𝜋 󵄨𝜔 󵄨 𝑀+4𝜋󵄨𝜔 󵄨 󵄨 𝑛󵄨 󵄨 𝑛󵄨 tion of a particle yields a transition between two distinct levels, where particle emission and absorption are reverse √ (65) + − (ln 2+𝜋+2𝜋𝑛)(− ln 2+𝜋+2𝜋𝑛))) processes [3–7].Forthis,wetouchedontheimportantissue that the nonstrictly thermal character of Hawking’s radia- 󵄨 󵄨 5 −1 2 2 ×(8(2𝑀 − 󵄨𝜔𝑛󵄨) 𝜋) )𝑄 𝑞 tion spectrum generates a natural correspondence between Hawking’s radiation and BH QNMs, because these structures 𝑞2 exemplify features of the BH’s energy spectrum [3–5], which = + O (𝑄4,𝑞4), 󵄨 󵄨 has been recently generalized to the emerging concept of a 4𝑀 − 󵄨𝜔𝑛󵄨 BH’s effective state [6, 7]. which can be applied to (62)-(63)byreplacingthe𝜔 param- Next, we prepared for our nonextremal RNBH QNM eter with the pertinent |𝜔𝑛|.Hence,(62)–(65)indicatethat investigation by first reviewing relevant portions of the SBH in general the SBH results of Section 2 are fundamentally effective framework [3–5]inSection 2. There, we listed the consistent with the RNBH results of Section 3 for small 𝑄. noneffective and effective quantities for SBH states and tran- Moreover, in (65)forlarge𝑛, the result is consistent with sitions, with direct application to the QNM characterization the SBH because ln 2 is negligible, but for small 𝑛 there is an and framework of [3–5]. Subsequently, in Section 3, we iden- argument between scientists regarding ln 2 and ln 3 because tified some existing noneffective quantities and introduced these refer to the two distinct QNM families of (51)and(52). new effective quantities for RNBH states and transitions so Here,weprovidethephysicalanswerof(65)forthecase we could apply the BH framework of [3–5]toimplement of emission by using the fact that 𝑄 is small, so the term which a RNBH framework. These results are crucial because the 2 includes 𝑄 is also very small and therefore negligible: effective quantities in3 [ –5]havebeenachievedforthe stable four-dimensional RNBH solution in Einstein’s general 󵄨 󵄨 relativity—now effective frameworks exist for the SBH, KBH, (𝜔0) ≡ 󵄨𝜔𝑛󵄨 ≈𝑀 𝑛 and (nonextremal) RNBH solutions. Ultimately, the RNBH effective quantities permitted us to 2 2 √ 𝑞 1 1 utilize both the KBH’s effective state concept [6, 7]andthe − 𝑀2 + −𝑄𝑞− √ln 22 −4𝜋2(𝑛 + ) . 2 4𝜋 2 BH QNMs [3–5] to construct a foundation for the RNBH’s (66) effectivestateinthisdevelopingBHeffectiveframework. The RNBH effective state concept is meaningful because,

Thus, by setting (𝜔0)𝑛 ≡|𝜔𝑛| we obtain as scientists who wish to demystify the BH paradigm, we need additional features and knowledge to consider in future experiments and observations. Δ𝑀𝑛 ≡−Δ𝜔𝑛,𝑛−1 =(𝜔0)𝑛−1 −(𝜔0)𝑛 Finally, we stress that the nonstrictly thermal behavior of the Hawking radiation spectrum has been recently used 𝑞2 1 1 2 √ 2 2 2 to construct two very intriguing proposals to solve the ≡ 𝑀 + −𝑄𝑞− √(ln 2) +4𝜋 (𝑛 + ) 2 4𝜋 2 BH information loss paradox. The first one received the First Award in the 2013 Gravity Research Foundation Essay 2 2 Competition [12]. The latter won the Community Rating at √ 𝑞 1 2 1 − 𝑀2 + −𝑄𝑞− √(ln 2) +4𝜋2(𝑛 − ) the 2013 FQXi Essay Contest—It from Bit or Bit from It [24]. 2 4𝜋 2 We are working to extend this second approach to the RNBH (67) framework [25]. Advances in High Energy Physics 9

Conflict of Interests [16] M. Casals and A. C. Ottewill, “Branch cut and quasinormal modes at large imaginary frequency in Schwarzschild space- The authors declare that there is no conflict of interests time ,” Physical Review D,vol.86,no.2,ArticleID024021,2012. regarding the publication of this paper. [17] R. Banerjee and B. R. Majhi, “Quantum tunneling and trace anomaly,” Physics Letters B,vol.674,no.3,pp.218–222,2009. Acknowledgments [18] S. W. Hawking, “The path integral approach to quantum gravity,” in General Relativity: An Einstein Centenary Survey,S. S. H. Hendi wishes to thank the Shiraz University Research W. Hawking and W. Israel, Eds., Cambridge University Press, Council. The work of S. H. Hendi has been supported Cambridge, UK, 1979. financially by the Research Institute for Astronomy and [19] T. Padmanabhan, “Quasi-normal modes: a simple derivation Astrophysics of Maragha (RIAAM), Iran. of the level spacing of the frequencies,” Classical and Quantum Gravity,vol.21,no.1,p.L1,2004. [20] A. Lopez-Ortega, “Area spectrum of the d-dimensional References Reissner-Nordstrom¨ black hole in the small charge limit ,” Classical and Quantum Gravity,vol.28,no.3,ArticleID [1] B. Wang, R. K. Su, P. K. N. Yu, and E. C. M. Young, “Can a 035009, 2011. nonextremal Reissner-Nordstrom¨ black hole become extremal [21] J. Zhang and Z. Zhao, “Hawking radiation of charged particles by assimilating an infalling charged particle and shell?” Physical via tunneling from the Reissner-Nordstrom¨ black hole,” Journal Review D, vol. 57, no. 8, pp. 5284–5286, 1998. of High Energy Physics,vol.10,p.55,2005. [2] M. K. Parikh and F. Wilczek, “Hawking radiation as tunneling,” [22] S. Hod, “Quasinormal spectrum and quantization of charged Physical Review Letters,vol.85,no.24,pp.5042–5045,2000. black holes,” Classical and Quantum Gravity,vol.23,no.4,p. [3] C. Corda, “Effective temperature for black holes,” Journal of L23, 2006. High Energy Physics, vol. 2011, p. 101, 2011. [23] M. K. Parikh, “A secret tunnel through the horizon,” General [4] C. Corda, “Black hole quantum spectrum,” The European Relativity and Gravitation, vol. 36, no. 11, pp. 2419–2422, 2004. Physical Journal C, vol. 73, p. 2665, 2013. [24] C. Corda, “Time-Dependent Schrodinger Equation [5] C. Corda, “Effective temperature, hawking radiation and quasi- for Black Hole Evaporation: no Information Loss,” normal modes,” International Journal of Modern Physics D,vol. http://fqxi.org/community/forum/topic/1856. 21, no. 11, Article ID 1242023, 2012. [25] C. Corda, S. H. Hendi, R. Katebi, and N. O. Schmidt, inprepa- [6] C. Corda, S. H. Hendi, R. Katebi, and N. O. Schmidt, “Effective ration. state, Hawking radiation and quasi-normal modes for Kerr black holes,” JournalofHighEnergyPhysics,vol.2013,p.8,2013. [7] C. Corda, “Non-strictly black body spectrum from the tunnelling mechanism,” Annals of Physics,vol.337,pp.49–54,2013. [8] B. Zhang, Q.-Y.Cai, L. You, and M. S. Zhan, “Hidden messenger revealed in Hawking radiation: a resolution to the paradox of black hole information loss,” Physics Letters B,vol.675,no.1, pp. 98–101, 2009. [9]B.Zhang,Q.-Y.Cai,M.S.Zhan,andL.You,“Entropyis conserved in Hawking radiation as tunneling: a revisit of the black hole information loss paradox,” Annals of Physics,vol.326, no. 2, pp. 350–363, 2011. [10] B. Zhang, Q.-Y. Cai, M. S. Zhan, and L. You, “Noncommutative information is revealed from Hawking radiation as tunneling,” EPL (Europhysics Letters), vol. 94, no. 2, Article ID 20002, 2011. [11] B. Zhang, Q.-Y. Cai, M. S. Zhan, and L. You, “Comment on ‘What the information loss is not’,” http://arxiv.org/abs/ 1210.2048.Inpress. [12] B. Zhang, Q. Y. Cai, M. S. Zhan, and L. You, “Information conservation is fundamental: recovering the lost information in Hawking radiation,” International Journal of Modern Physics D, vol. 22, Article ID 1341014, 8 pages, 2013. [13] S. Hod, “Gravitation, the quantum and Bohr’s correspondence principle,” General Relativity and Gravitation,vol.31,no.11,pp. 1639–1644, 1999. [14] S. Hod, “Bohr’s correspondence principle and the area spectrum of quantum black holes,” Physical Review Letters,vol.81,no.20, pp. 4293–4296, 1998. [15] M. Maggiore, “Physical interpretation of the spectrum of black hole quasinormal modes,” Physical Review Letters,vol.100,no. 14, Article ID 141301, 2008. Hindawi Publishing Corporation Advances in High Energy Physics Volume 2013, Article ID 236974, 4 pages http://dx.doi.org/10.1155/2013/236974

Research Article A Little Quantum Help for Cosmic Censorship and a Step Beyond All That

Nikolaos Pappas

Division of Theoretical Physics, Department of Physics, University of Ioannina, 451 10 Ioannina, Greece

Correspondence should be addressed to Nikolaos Pappas; [email protected]

Received 27 September 2013; Revised 26 November 2013; Accepted 29 November 2013

Academic Editor: Christian Corda

Copyright © 2013 Nikolaos Pappas. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The hypothesis of cosmic censorship (CCH) plays a crucial role in classical general relativity, namely, to ensure that naked singularities would never emerge, since it predicts that whenever a singularity is formed an event horizon would always develop around it as well, to prevent the former from interacting directly with the rest of the Universe. Should this not be so, naked singularities could eventually form, in which case phenomena beyond our understanding and ability to predict could occur, since at the vicinity of the singularity both predictability and determinism break down even at the classical (e.g., nonquantum) level. More than 40 years after it was proposed, the validity of the hypothesis remains an open question. We reconsider CCH in both itsweak and strong versions, concerning point-like singularities, with respect to the provisions of Heisenberg’s uncertainty principle. We argue that the shielding of the singularities from observers at infinity by an event horizon is also quantum mechanically favored, but ultimately it seems more appropriate to accept that singularities never actually form in the usual sense; thus no naked singularity danger exists in the first place.

1. Introduction satisfied (the theorem actually goes as follows: let 𝑀, 𝑔𝑎𝑏 be a time-oriented spacetime satisfying the following conditions. 𝑎 𝑏 𝑎 Singularities, conceived as spacetime regions, where cur- (A) 𝑅𝑎𝑏𝑉 𝑉 ≥0for any nonspace-like 𝑉 . (B) The time- vature (as described by scalar invariant quantities like 𝜇]𝜌𝜎 like and null generic conditions are fulfilled. (C) There isno 𝑅𝜇]𝜌𝜎𝑅 ) blows up to exceed any possible upper bound, closed time-like curve. (D) At least one of the following holds: are one of the most problematic notions in Physics. After (Da) there exists a compact achronal set without edge; (Db) all, strictly speaking, if the spacetime metric is ill-behaved there exists a trapped surface; (Dc) there is a 𝑝∈𝑀such that at a certain point, then the latter should not be considered the expansion of the future directed null geodesics through 𝑝 as part of that spacetime in the first place. Nevertheless, becomes negative along each geodesic. Then 𝑀, 𝑔𝑎𝑏 contains it is this metric that we rely on to try to describe the at least one incomplete time-like or null geodesic) [1]. Since properties of that point. We overcome this incoherence by all of them are redeemed in our Universe too, singularities considering an augmented spacetime that contains such are expected with certainty to form in the latter as well. In singular points as ideal boundary points attached to the order to deal with these “monsters,” Penrose proposed the ordinary, well-behaved manifold. Since the spacetime struc- famous cosmic censorship hypothesis (CCH) [2]. The weak ture breaks down at singularities while, at the same time, version of the hypothesis (w-CCH) suggests that observers at physical laws presuppose space and time to develop and infinity can never directly see a singularity, the latter being manifest themselves, naked singularities would be sources atalltimesclothedbyanabsoluteeventhorizon,whereasits of lawlessness, absurdity, and uncontrollable information, strong version (s-CCH) states that an observer cannot have therefore an anathema for our perception of the Universe. any direct interaction with a singularity at any time or place Even worse, Hawking and Penrose have shown that the [3]. Because of cosmic censorship, then, a naked singularity emergence of singularities is inevitable in a very large class of should never occur except, conceivably, for some special universe types, where sufficiently reasonable conditions are configurations, which are not expected to occur in an actual 2 Advances in High Energy Physics astrophysical circumstance (for a thorough and enlightening 2. Weak Censorship Revisited presentation of the issues concerning singularities and CCH see [4–6]). Point-like singularities are expected to form because of the It should be noted here that CCH does not stem from unstoppablecollapseofmatterthatoccurswhenatoolarge some well-established physical law or mathematical theorem. mass is concentrated in a too small volume. The volume of Rather, it is a convenient hypothesis that, considering the these singularities would effectively tend to zero by definition; catastrophic impact of the alternative, we gladly accept as thus they should occupy a single point of spacetime. In (probably) true. Soon after it was proposed, it was declared as thecaseofanakedsingularity,anobserveratinfinity one of the most important open questions in classical general (i.e., at sufficiently large distance away from it so as to relativity [3], whose derivation remains obscure until now be in an asymptotically flat region of spacetime) would in (see [7–13] for reviews on the work done sofar). Initially, argu- principle be able to determine its position with arbitrarily high accuracy by, for example, direct observation. When we ments supporting the idea were based largely on geometry Δ𝑥 → 0 and issues concerning causality, usually expressed in terms make a measurement with uncertainty concerning of TIFs and TIPs (terminal indecomposable futures/pasts, the position of a quantum system, however, the uncertainty principle states that we have to end up with complete lack of resp.) [14] (see again [6]). Penrose was able to derive Δ𝑝 → ∞ inequalities involving black hole masses and horizon radii knowledge about its momentum (i.e., ), therefore [15] in support of his hypothesis, which interestingly enough about its energy as well. We argue, though, that this is not the wereshowntoholdtrueinaseriesofdifferentsituations case with naked singularities. Since in principle they can be of [16–21]. Moreover, CCH was proven to be valid in various arbitrarily large mass, one reasonably expects that the actual specific spacetimes [22–28].Atthesametime,sceptics procedure of determining their position could not change were trying to construct counterexamples in which naked their momentum significantly. Furthermore, even though singularities could emerge [29–37]. However, the majority quantum gravity is necessary to describe the singularity of those examples presupposed very special and idealized per se, it is legitimate to anticipate that general relativity is conditions to hold (thus least possible to occur in a realistic sufficiently accurate to describe spacetime at macroscopic universe) to hold, so the credibility of CCH was far from distances away from it. Then, it would be possible to “weigh” being fatally undermined by them. Soon it was evident that the singularity by observing potential gravitational lensing a quantum treatment was necessary. Besides, the problematic effects or through measuring the trajectory, speed, and way we describe singularities represents much more our acceleration of test bodies that get attracted by it and so forth. lack of understanding their true nature, namely, the laws This way the mass/energy of the singularity would be known withuncertaintyatmostoftheorderofthemassitself(Δ𝑀 ∼ of quantum gravity that presumably take over when radii 𝑀). All these mean that the existence of a naked singularity, of spacetime curvature of the order of Planck length are apart from all other undesired consequences, would also attained, rather than their actual behavior. It was proposed violate the uncertainty principle. The conundrum gets settled (and hoped) by many scientists that the inclusion of quantum when the provisions of the w-CCH are taken into account. phenomena in the picture of gravitational collapse would be The existence of an event horizon of radius 𝑟ℎ ∼𝑀,which the answer to all our difficulties to cope with singularities. In emerges because of the warping of the spacetime continuum fact, quantum mechanics has been proven very successful in by the singularity mass itself and thus exists in every kind resolving many of the counterexample gedankenexperiments of black hole type, means that the actual position of the in favor of CCH. More specifically it was used to show that singularity can be determined with uncertainty at least Δ𝑥 ∼ it is impossible to overspin or overcharge a maximal Kerr 𝑟ℎ.Thenwegetfrom(1)thatΔ𝑝 ≳ 1/𝑀 and consequently 3 black hole to produce a naked singularity (a procedure first we find for the singularity energy the inequality 𝐸≳1/𝑀, considered in [38]) [39–41]. In this framework, recent results which obviously is perfectly compatible with measuring its on the correspondence between Hawking radiation and black mass/energy with Δ𝑀 ∼.Inthissensew-CCHnotonlyis 𝑀 holes quasinormal modes [42–45]looktobeparticularly necessary to make general relativity self-consistent, but has a interesting since they stress too the need for a quantum strong quantum support as well. mechanical approach to the black hole properties if we are to gain a deeper understanding of the latter. 3. Strong Censorship Revisited Following this line of thinking we try here to engage quantum mechanics in the treatment of point-like singulari- What about s-CCH then? It is not hard to imagine a situation ties lying at a finite distance (as opposed to singularities lying whereaverylargeandmassivesystemisinquestion(e.g., at infinity or thunderbolts). The key idea proposed is to make the central region of a galaxy); a trapped surface has already appeal to Heisenberg’s uncertainty principle, formed while observers living on a planet within the trapped region exist and expect quantum mechanics to hold at all Δ𝑥 ⋅ Δ𝑝 ≥1 ( 𝐺=𝑐=ℎ=1) times until they crash into the singularity that will develop in natural units where (1) some time in their future. Even though the soon-to-form singularity would remain unseen by observers at infinity whichweconsiderthemostfundamentalfeatureofeverynat- (so w-CCH is satisfied), an observer inside the horizon ural system, and check the constraints it imposes, concerning would actually encounter a naked singularity (being at the the properties of systems that involve singularities. same time at a significantly large distance away from it). Advances in High Energy Physics 3

All arguments presented in the paragraphs above hold true [7] R. M. Wald, “Gravitational collapse and cosmic censorship,” for this observer too, so a paradox a rises. The s-CCH is http://arxiv.org/abs/gr-qc/9710068. established to resolve the paradox by predicting that an [8] T. P. Singh, “Gravitational collapse, black holes and naked observer would never actually see the singularity, but since singularities,” Journal of Astrophysics & Astronomy,vol.20,pp. it does not provide us with a mechanism capable of deterring 221–232, 1999. this interaction, it looks more like the expression of a hope [9] W. A. Hiscock, “Magnetic charge, black holes, and cosmic than a constraint imposed by some physical law. The only censorship,” Annals of Physics,vol.131,no.2,pp.245–268,1981. way out, then, is to admit that the notion of unstoppable [10] J. D. Bekenstein and C. 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