DOCSLIB.ORG

Counting Black Hole Microstates in String Theory

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Counting Black Hole Microstates in String Theory Counting Black Hole Microstates in String Theory Jorge Laraña Aragón1 1Theoretical Physics Division, Department of Physics and Astronomy, Uppsala University, Uppsala, Sweden. [email protected] Advisor: Giuseppe Dibitetto Subject Reader: Ulf Danielsson In this project, we explore the area of black holes in String Theory. String Theory has had several successes in describing properties of black holes. Recent progress in String Theory points towards the possibility that black holes should not be thought of as fundamental objects, but rather as statistical descriptions of a huge number of smooth horizonless microstate geometries. We study this deep connection between the microscopic description of black holes and String Theory. The main goal is to understand and learn how some of the modern techniques in String Theory can be applied to model black holes, in particular, calculating an expression for the entropy. The main idea is to construct black hole solutions from intersecting certain types of branes, in the context of Supergravity theories that emerge as low-energy limits of Superstring theories. With this microscopic approach, the aim is to count the number of microstates and then compare with the macroscopic Bekenstein-Hawking entropies. We plan to construct such solutions, for both supersymmetric and non-supersymmetric black holes. Also, this microscopic origin of the black hole entropy could provide new insights to the black hole information paradox. Sammanfattning I detta projekt behandlar vi svarta hål i strängteori. Svarta hål är några av naturens mest fascinerande objekt. De är indirekt observerade, som mörka objekt av stjärnstorlek, vilka finns i binära system, eller som supermassiva objekt i galaxers centra. Svarta hål är vanligare förekom- mande i universum än vad vi tidigare har trott. Uppskattningsvis finns 100 miljoner mörka objekt enbart i Vintergatan. Svarta hål dyker upp i allmän relativitetsteori som klassiska materielösningar som gravitationellt har kollapsat till en punkt, en singularitet. Från en teoretisk synvinkel förser svarta hål oss med laboratorier för tankeexperiment, där nya teoretiska ideér kan testas. 1976 upptäckte Stephen Hawking att svarta hål inte är så svarta - de sänder ut värmestrålning motsvarande en karaktäristisk temperatur, Hawking-temperaturen. Ter- miska egenskaper, såsom Bekenstein-Hawking-entropi, associerades nu med svarta hål. 1996 härled- des denna entropi med strängteoretiska metoder. Man bestämde de mikroskopiska frihetsgraderna för ett speciellt slags svarta hål, och sedan dess har strängteori skördat många framgångar inom forskningsområdet. Här studerar vi detta djupa samband mellan strängteori och den mikroskopiska beskrivningen av svarta hål. Huvudmålet är att förstå hur vissa moderna strängteoretiska tekniker kan appliceras på svarta hålmodeller, framför allt när entropin ska beräknas. 2 CONTENTS I. Introduction 4 II. Black Holes are not that black 5 A. Black Hole Thermodynamics 5 1. Zeroth Law 5 2. First Law 5 3. Second Law 6 4. Third Law 6 B. Temperature as cyclic imaginary time 7 C. Macroscopic Black Hole Temperatures and Entropies 8 1. Schwarzschild black hole 8 2. D-dimensional Schwarzschild black hole 10 3. Reissner-Nordström black hole 12 4. D-dimensional Reissner-Nordström black hole 14 III. String Thermodynamics 16 A. The Hagedorn temperature 16 B. Counting partitions 17 1. Bosonic partitions 17 2. Fermionic partitions and generalisation 19 C. Entropy and temperature of a relativistic string at rest 20 1. Bosonic string 20 2. Fermionic string 22 3. Open superstring 22 4. Closed superstring 23 D. The string partition function 24 E. The Hagedorn temperature, the black hole entropy and the stretched horizon 26 IV. The Long String construction and the Random Walk model 28 A. Long strings are entropically favoured 28 B. The size of a string state and the Random Walk model 30 C. Estimation of the Schwarzschild black hole entropy 31 D. Recent developments 33 V. Black Holes from Brane intersections 33 A. D=5 3-charge Extremal Reissner-Nordström black holes 34 1. Construction in the Supergravity approximation 34 2. Microstate counting 37 B. D=4 4-charge Extremal Reissner-Nordström black holes 38 1. Construction in the Supergravity approximation 38 2. Microstate counting 41 C. Dual brane configurations 41 1. From D1-D5-W to D0-D4-F1 42 2. From D2-D6-NS5-W to D2-D2-D2-D6 43 D. Counting in the dual CFT 43 1. Generating functions 44 2. Saddle point approximation 45 3. Bosonic strings 46 4. Type II superstrings 47 E. Non-extremal Black Holes 49 1. D=5 3-charge Non-extremal Reissner-Nordström black holes 49 2. D=4 4-charge Non-extremal Reissner-Nordström black holes 53 VI. The Black Hole Information Paradox and the Microscopic Interpretation 56 VII. Conclusions 56 3 Acknowledgments 58 References 58 4 I. INTRODUCTION Black holes are among the most fascinating objects in Nature. They are indirectly observed, either as stellar mass dark objects in binary systems or as supermassive ones at the centres of galaxies and some of the most energetic emissions in the universe are associated with matter accreting onto black holes or black hole mergers. Black holes are more common in the universe than we used to think. It has been estimated that the Milky Way contains around 100 million black holes. From a theoretical point of view, black holes provide laboratories for gedanken experiments where we can generate and test new theoretical ideas. They appear in General Relativity as classical solutions to matter that has gravitationally collapsed to a point, known as a singularity. They are one of the most interesting scenarios to test physical theories because they have extreme conditions and it is where both physical descriptions of General Relativity and Quantum Mechanics clash apparently leading to a contradiction. That is why the real puzzles regarding black holes arise at the quantum level. In 1976, Hawking found out black holes are not that black, instead, they emit thermal radiation at a characteristic temperature called the Hawking temperature [1]. This was the beginning of Black Hole Thermodynamics. Thermal properties were associated with black holes, such as the so called Bekenstein-Hawking entropy. This later led to the famous black hole information paradox [2]. The thermal properties of black holes are difficult to understand at the fundamental level of statistical mechanics. However, it is natural to consider the possibility of calculating the black hole entropy by using this standard statistical mechanics framework provided that we are able to count the number of black hole microstates that a black hole can be in. This is sometimes referred as the black hole entropy problem. Together with the black hole information paradox they point out the necessity of a quantum theory of gravity. String Theory has recently turned out to be very interesting for black hole physics, in particular, in understanding the microscopic origin of the black hole entropy since in 1996, the Bekenstein-Hawking entropy expression was derived by counting microstates for extremal and charged black holes [3], confirming the relation between the macroscopic en- tropy with the microscopic construction from string theory. String theory then allows us to calculate the temperature and entropy of black holes from the density of the associated microstates, at least for certain types of black holes. For those, the string statistical approach makes it possible to compute the entropy, at least, up to numerical factors. The results agree with the Bekenstein-Hawking entropy. Furthermore, for a few cases where the black holes are invariant under a certain amount of supersymmetry (BPS black holes) the calculations give the exact numerical coefficients. To some extent, string theory is capable to solve the problems of entropy and information considering black holes as ordinary quantum mechanical systems (which respect unitarity). This could revolutionise our understanding of black holes and quantum gravity in general: the classical black hole solution would be the analogue of the thermodynamic description of a gas, while the horizonless microstates would be the analogue of the statistical description of this gas. Interestingly, the study of black holes has provided key insights into the nature of gravity and its interplay with quantum mechanics. Black holes and their higher-dimensional generalisations have played a crucial role in the development of modern string theory, in advancing our understanding of non-perturbative string dynamics and con- tributing to paradigm shifts such as string duality and the holographic principle. Indeed, it was the scaling properties of black hole entropy that revealed the holographic nature of gravity, in other words, its fundamental degrees of freedom live in fewer dimensions than the classical limit indicates [4, 5]. The non-extensive character of the black hole entropy (scaling with the area) and the large number of degrees of freedom led to the development of a holographic representation of gravity, called holographic duality and also known as gauge/gravity correspondence, which relates a gravitational theory in d+1 space-time dimensions with a negative cosmological constant to a strongly coupled quantum field theory description of a non-gravitational system in d dimensions. This project has the following structure. In the first chapter, section II, we will review the theory of Black Hole Thermodynamics consisting of four laws analogous to Classical Thermodynamics and we will perform semiclassical calculations of the temperature and entropy for different types of black holes, not using the original Hawking’s approach (the Bogoliubov formalism in the context of QFT in curved spacetime) but instead, a more modern approach based on path integral arguments. We try to justify the method and then apply it to two specific types of black holes: the Schwarzschild and the Reissner-Nordström solutions. We do this for both the realistic case of four spacetime dimensions as well as for the general case of an arbitrary number of spacetime dimensions. Later, in section III, we address thermodynamical aspects of String Theory and the so called concept of the Hagedorn temperature.
Load more

Recommended publications
  • Limits on New Physics from Black Holes Arxiv:1309.0530V1
    Limits on New Physics from Black Holes Clifford Cheung and Stefan Leichenauer California Institute of Technology, Pasadena, CA 91125 Abstract Black holes emit high energy particles which induce a finite density potential for any scalar field φ coupling to the emitted quanta. Due to energetic considerations, φ evolves locally to minimize the effective masses of the outgoing states. In theories where φ resides at a metastable minimum, this effect can drive φ over its potential barrier and classically catalyze the decay of the vacuum. Because this is not a tunneling process, the decay rate is not exponentially suppressed and a single black hole in our past light cone may be sufficient to activate the decay. Moreover, decaying black holes radiate at ever higher temperatures, so they eventually probe the full spectrum of particles coupling to φ. We present a detailed analysis of vacuum decay catalyzed by a single particle, as well as by a black hole. The former is possible provided large couplings or a weak potential barrier. In contrast, the latter occurs much more easily and places new stringent limits on theories with hierarchical spectra. Finally, we comment on how these constraints apply to the standard model and its extensions, e.g. metastable supersymmetry breaking. arXiv:1309.0530v1 [hep-ph] 2 Sep 2013 1 Contents 1 Introduction3 2 Finite Density Potential4 2.1 Hawking Radiation Distribution . .4 2.2 Classical Derivation . .6 2.3 Quantum Derivation . .8 3 Catalyzed Vacuum Decay9 3.1 Scalar Potential . .9 3.2 Point Particle Instability . 10 3.3 Black Hole Instability . 11 3.3.1 Tadpole Instability .
    [Show full text]
  • Spacetime Geometry from Graviton Condensation: a New Perspective on Black Holes
    Spacetime Geometry from Graviton Condensation: A new Perspective on Black Holes Sophia Zielinski née Müller München 2015 Spacetime Geometry from Graviton Condensation: A new Perspective on Black Holes Sophia Zielinski née Müller Dissertation an der Fakultät für Physik der Ludwig–Maximilians–Universität München vorgelegt von Sophia Zielinski geb. Müller aus Stuttgart München, den 18. Dezember 2015 Erstgutachter: Prof. Dr. Stefan Hofmann Zweitgutachter: Prof. Dr. Georgi Dvali Tag der mündlichen Prüfung: 13. April 2016 Contents Zusammenfassung ix Abstract xi Introduction 1 Naturalness problems . .1 The hierarchy problem . .1 The strong CP problem . .2 The cosmological constant problem . .3 Problems of gravity ... .3 ... in the UV . .4 ... in the IR and in general . .5 Outline . .7 I The classical description of spacetime geometry 9 1 The problem of singularities 11 1.1 Singularities in GR vs. other gauge theories . 11 1.2 Defining spacetime singularities . 12 1.3 On the singularity theorems . 13 1.3.1 Energy conditions and the Raychaudhuri equation . 13 1.3.2 Causality conditions . 15 1.3.3 Initial and boundary conditions . 16 1.3.4 Outlining the proof of the Hawking-Penrose theorem . 16 1.3.5 Discussion on the Hawking-Penrose theorem . 17 1.4 Limitations of singularity forecasts . 17 2 Towards a quantum theoretical probing of classical black holes 19 2.1 Defining quantum mechanical singularities . 19 2.1.1 Checking for quantum mechanical singularities in an example spacetime . 21 2.2 Extending the singularity analysis to quantum field theory . 22 2.2.1 Schrödinger representation of quantum field theory . 23 2.2.2 Quantum field probes of black hole singularities .
    [Show full text]
  • Black Hole Production and Graviton Emission in Models with Large Extra Dimensions
    Black hole production and graviton emission in models with large extra dimensions Dissertation zur Erlangung des Doktorgrades der Naturwissenschaften vorgelegt beim Fachbereich Physik der Johann Wolfgang Goethe–Universit¨at in Frankfurt am Main von Benjamin Koch aus N¨ordlingen Frankfurt am Main 2007 (D 30) vom Fachbereich Physik der Johann Wolfgang Goethe–Universit¨at als Dissertation angenommen Dekan ........................................ Gutachter ........................................ Datum der Disputation ................................ ........ Zusammenfassung In dieser Arbeit wird die m¨ogliche Produktion von mikroskopisch kleinen Schwarzen L¨ochern und die Emission von Gravitationsstrahlung in Modellen mit großen Extra-Dimensionen untersucht. Zun¨achst werden der theoretisch-physikalische Hintergrund und die speziel- len Modelle des behandelten Themas skizziert. Anschließend wird auf die durchgefuhrten¨ Untersuchungen zur Erzeugung und zum Zerfall mikrosko- pisch kleiner Schwarzer L¨ocher in modernen Beschleunigerexperimenten ein- gegangen und die wichtigsten Ergebnisse zusammengefasst. Im Anschluss daran wird die Produktion von Gravitationsstrahlung durch Teilchenkollisio- nen diskutiert. Die daraus resultierenden analytischen Ergebnisse werden auf hochenergetische kosmische Strahlung angewandt. Die Suche nach einer einheitlichen Theorie der Naturkr¨afte Eines der großen Ziele der theoretischen Physik seit Einstein ist es, eine einheitliche und m¨oglichst einfache Theorie zu entwickeln, die alle bekannten Naturkr¨afte beschreibt.
    [Show full text]
  • Light Rays, Singularities, and All That
    Light Rays, Singularities, and All That Edward Witten School of Natural Sciences, Institute for Advanced Study Einstein Drive, Princeton, NJ 08540 USA Abstract This article is an introduction to causal properties of General Relativity. Topics include the Raychaudhuri equation, singularity theorems of Penrose and Hawking, the black hole area theorem, topological censorship, and the Gao-Wald theorem. The article is based on lectures at the 2018 summer program Prospects in Theoretical Physics at the Institute for Advanced Study, and also at the 2020 New Zealand Mathematical Research Institute summer school in Nelson, New Zealand. Contents 1 Introduction 3 2 Causal Paths 4 3 Globally Hyperbolic Spacetimes 11 3.1 Definition . 11 3.2 Some Properties of Globally Hyperbolic Spacetimes . 15 3.3 More On Compactness . 18 3.4 Cauchy Horizons . 21 3.5 Causality Conditions . 23 3.6 Maximal Extensions . 24 4 Geodesics and Focal Points 25 4.1 The Riemannian Case . 25 4.2 Lorentz Signature Analog . 28 4.3 Raychaudhuri’s Equation . 31 4.4 Hawking’s Big Bang Singularity Theorem . 35 5 Null Geodesics and Penrose’s Theorem 37 5.1 Promptness . 37 5.2 Promptness And Focal Points . 40 5.3 More On The Boundary Of The Future . 46 1 5.4 The Null Raychaudhuri Equation . 47 5.5 Trapped Surfaces . 52 5.6 Penrose’s Theorem . 54 6 Black Holes 58 6.1 Cosmic Censorship . 58 6.2 The Black Hole Region . 60 6.3 The Horizon And Its Generators . 63 7 Some Additional Topics 66 7.1 Topological Censorship . 67 7.2 The Averaged Null Energy Condition .
    [Show full text]
  • Plasma Modes in Surrounding Media of Black Holes and Vacuum Structure - Quantum Processes with Considerations of Spacetime Torque and Coriolis Forces
    COLLECTIVE COHERENT OSCILLATION PLASMA MODES IN SURROUNDING MEDIA OF BLACK HOLES AND VACUUM STRUCTURE - QUANTUM PROCESSES WITH CONSIDERATIONS OF SPACETIME TORQUE AND CORIOLIS FORCES N. Haramein¶ and E.A. Rauscher§ ¶The Resonance Project Foundation, [email protected] §Tecnic Research Laboratory, 3500 S. Tomahawk Rd., Bldg. 188, Apache Junction, AZ 85219 USA Abstract. The main forces driving black holes, neutron stars, pulsars, quasars, and supernovae dynamics have certain commonality to the mechanisms of less tumultuous systems such as galaxies, stellar and planetary dynamics. They involve gravity, electromagnetic, and single and collective particle processes. We examine the collective coherent structures of plasma and their interactions with the vacuum. In this paper we present a balance equation and, in particular, the balance between extremely collapsing gravitational systems and their surrounding energetic plasma media. Of particular interest is the dynamics of the plasma media, the structure of the vacuum, and the coupling of electromagnetic and gravitational forces with the inclusion of torque and Coriolis phenomena as described by the Haramein-Rauscher solution to Einstein’s field equations. The exotic nature of complex black holes involves not only the black hole itself but the surrounding plasma media. The main forces involved are intense gravitational collapsing forces, powerful electromagnetic fields, charge, and spin angular momentum. We find soliton or magneto-acoustic plasma solutions to the relativistic Vlasov equations solved in the vicinity of black hole ergospheres. Collective phonon or plasmon states of plasma fields are given. We utilize the Hamiltonian formalism to describe the collective states of matter and the dynamic processes within plasma allowing us to deduce a possible polarized vacuum structure and a unified physics.
    [Show full text]
  • Black Holes and Qubits
    Subnuclear Physics: Past, Present and Future Pontifical Academy of Sciences, Scripta Varia 119, Vatican City 2014 www.pas.va/content/dam/accademia/pdf/sv119/sv119-duff.pdf Black Holes and Qubits MICHAEL J. D UFF Blackett Labo ratory, Imperial C ollege London Abstract Quantum entanglement lies at the heart of quantum information theory, with applications to quantum computing, teleportation, cryptography and communication. In the apparently separate world of quantum gravity, the Hawking effect of radiating black holes has also occupied centre stage. Despite their apparent differences, it turns out that there is a correspondence between the two. Introduction Whenever two very different areas of theoretical physics are found to share the same mathematics, it frequently leads to new insights on both sides. Here we describe how knowledge of string theory and M-theory leads to new discoveries about Quantum Information Theory (QIT) and vice-versa (Duff 2007; Kallosh and Linde 2006; Levay 2006). Bekenstein-Hawking entropy Every object, such as a star, has a critical size determined by its mass, which is called the Schwarzschild radius. A black hole is any object smaller than this. Once something falls inside the Schwarzschild radius, it can never escape. This boundary in spacetime is called the event horizon. So the classical picture of a black hole is that of a compact object whose gravitational field is so strong that nothing, not even light, can escape. Yet in 1974 Stephen Hawking showed that quantum black holes are not entirely black but may radiate energy, due to quantum mechanical effects in curved spacetime. In that case, they must possess the thermodynamic quantity called entropy.
    [Show full text]
  • Firewalls and the Quantum Properties of Black Holes
    Firewalls and the Quantum Properties of Black Holes A thesis submitted in partial fulfillment of the requirements for the degree of Bachelor of Science degree in Physics from the College of William and Mary by Dylan Louis Veyrat Advisor: Marc Sher Senior Research Coordinator: Gina Hoatson Date: May 10, 2015 1 Abstract With the proposal of black hole complementarity as a solution to the information paradox resulting from the existence of black holes, a new problem has become apparent. Complementarity requires a vio- lation of monogamy of entanglement that can be avoided in one of two ways: a violation of Einstein’s equivalence principle, or a reworking of Quantum Field Theory [1]. The existence of a barrier of high-energy quanta - or “firewall” - at the event horizon is the first of these two resolutions, and this paper aims to discuss it, for Schwarzschild as well as Kerr and Reissner-Nordstr¨omblack holes, and to compare it to alternate proposals. 1 Introduction, Hawking Radiation While black holes continue to present problems for the physical theories of today, quite a few steps have been made in the direction of understanding the physics describing them, and, consequently, in the direction of a consistent theory of quantum gravity. Two of the most central concepts in the effort to understand black holes are the black hole information paradox and the existence of Hawking radiation [2]. Perhaps the most apparent result of black holes (which are a consequence of general relativity) that disagrees with quantum principles is the possibility of information loss. Since the only possible direction in which to pass through the event horizon is in, toward the singularity, it would seem that information 2 entering a black hole could never be retrieved.
    [Show full text]
  • Introduction to Black Hole Entropy and Supersymmetry
    Introduction to black hole entropy and supersymmetry Bernard de Wit∗ Yukawa Institute for Theoretical Physics, Kyoto University, Japan Institute for Theoretical Physics & Spinoza Institute, Utrecht University, The Netherlands Abstract In these lectures we introduce some of the principles and techniques that are relevant for the determination of the entropy of extremal black holes by either string theory or supergravity. We consider such black holes with N =2andN =4 supersymmetry, explaining how agreement is obtained for both the terms that are leading and those that are subleading in the limit of large charges. We discuss the relevance of these results in the context of the more recent developments. 1. Introduction The aim of these lectures is to present a pedagogical introduction to re- cent developments in string theory and supergravity with regard to black hole entropy. In the last decade there have been many advances which have thoroughly changed our thinking about black hole entropy. Supersymmetry has been an indispensable ingredient in all of this, not just because it is an integral part of the theories we consider, but also because it serves as a tool to keep the calculations tractable. Comparisons between the macroscopic and the microscopic entropy, where the latter is defined as the logarithm of the degeneracy of microstates of a certain brane or string configuration, were first carried out in the limit of large charges, assuming the Bekenstein- Hawking area law on the macroscopic (supergravity) side. Later on also subleading corrections could be evaluated on both sides and were shown to be in agreement, although the area law ceases to hold.
    [Show full text]
  • Jhep02(2019)160
    Published for SISSA by Springer Received: January 18, 2019 Accepted: February 10, 2019 Published: February 25, 2019 Holographic complexity equals which action? JHEP02(2019)160 Kanato Goto,a Hugo Marrochio,b;c Robert C. Myers,b Leonel Queimadab;c and Beni Yoshidab aThe University of Tokyo, Komaba, Meguro-ku, Tokyo 153-8902, Japan bPerimeter Institute for Theoretical Physics, Waterloo, ON N2L 2Y5, Canada cDepartment of Physics & Astronomy, University of Waterloo, Waterloo, ON N2L 3G1, Canada E-mail: [email protected], [email protected], [email protected], [email protected], [email protected] Abstract: We revisit the complexity = action proposal for charged black holes. We in- vestigate the complexity for a dyonic black hole, and we find the surprising feature that the late-time growth is sensitive to the ratio between electric and magnetic charges. In particular, the late-time growth rate vanishes when the black hole carries only a magnetic charge. If the dyonic black hole is perturbed by a light shock wave, a similar feature ap- pears for the switchback effect, e.g. it is absent for purely magnetic black holes. We then show how the inclusion of a surface term to the action can put the electric and magnetic charges on an equal footing, or more generally change the value of the late-time growth rate. Next, we investigate how the causal structure influences the late-time growth with and without the surface term for charged black holes in a family of Einstein-Maxwell-Dilaton theories. Finally, we connect the previous discussion to the complexity=action proposal for the two-dimensional Jackiw-Teitelboim theory.
    [Show full text]
  • Spin Rate of Black Holes Pinned Down
    IN FOCUS NEWS 449–451; 2013) that used data from NASA’s NuSTAR mission, which was launched last year (see Nature 483, 255; 2012). Study leader Guido Risaliti, an astronomer at the Harvard-Smith- JPL-CALTECH/NASA sonian Center for Astrophysics in Cambridge, Massachusetts, says that NuSTAR provides access to higher-energy X-rays, which has allowed researchers to clarify the influence of the black hole’s gravity on the iron line. These rays are less susceptible than lower-energy X-rays to absorption by clouds of gas between the black hole and Earth, which some had speculated was the real cause of the distortion. In the latest study, astronomers calculated spin more directly (C. Done et al. Mon. Not. R. Astron. Soc. http://doi.org/nc2; 2013). They found a black hole some 150 million parsecs away with a mass 10 million times that of the Sun. Using the European Space Agency’s X-rays emitted by swirling disks of matter hint at the speed at which a supermassive black hole spins. XMM-Newton satellite, they focused not on the iron line but on fainter, lower-energy ASTROPHYSICS X-rays emitted directly from the accretion disk. The spectral shape of these X-rays offers indirect information about the temperature of the innermost part of the disk — and the Spin rate of black temperature of this material is, in turn, related to the distance from the event horizon and the speed at which the black hole is spinning. The calculations suggest that, at most, the black holes pinned down hole is spinning at 86% of the speed of light.
    [Show full text]
  • Kaluza-Klein Theory, Ads/CFT Correspondence and Black Hole Entropy
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server IFIC–01–37 Kaluza-Klein theory, AdS/CFT correspondence and black hole entropy J.M. Izquierdoa1, J. Navarro-Salasb1 and P. Navarrob 1 a Departamento de F´ısica Te´orica, Universidad de Valladolid E{47011 Valladolid, Spain b Departamento de F´ısica Te´orica, Universidad de Valencia and IFIC, Centro Mixto Universidad de Valencia{CSIC E{46100 Burjassot (Valencia), Spain Abstract The asymptotic symmetries of the near-horizon geometry of a lifted (near-extremal) Reissner-Nordstrom black hole, obtained by inverting the Kaluza-Klein reduction, explain the deviation of the Bekenstein- Hawking entropy from extremality. We point out the fact that the extra dimension allows us to justify the use of a Virasoro mode decom- position along the time-like boundary of the near-horizon geometry, AdS Sn, of the lower-dimensional (Reissner-Nordstrom) spacetime. 2× 1e-mails: [email protected], [email protected]fic.uv.es, [email protected]fic.uv.es 1 1 Introduction The universality of the Bekenstein-Hawking area law of black holes could be explained if the density of the microscopic states is controlled by the con- formal symmetry [1]. A nice example of this philosophy is provided by the BTZ black holes [2]. The Bekenstein-Hawking entropy can be derived [3], via Cardy’s formula, from the two-dimensional conformal symmetry arising at spatial infinity of three-dimensional gravity with a negative cosmological constant [4]. Moreover, one can look directly at the black hole horizon and treat it as a boundary.
    [Show full text]
  • Black Holes and String Theory
    Black Holes and String Theory Hussain Ali Termezy Submitted in partial fulfilment of the requirements for the degree of Master of Science of Imperial College London September 2012 Contents 1 Black Holes in General Relativity 2 1.1 Black Hole Solutions . 2 1.2 Black Hole Thermodynamics . 5 2 String Theory Background 19 2.1 Strings . 19 2.2 Supergravity . 23 3 Type IIB and Dp-brane solutions 25 4 Black Holes in String Theory 33 4.1 Entropy Counting . 33 Introduction The study of black holes has been an intense area of research for many decades now, as they are a very useful theoretical construct where theories of quantum gravity become relevant. There are many curiosities associated with black holes, and the resolution of some of the more pertinent problems seem to require a quantum theory of gravity to resolve. With the advent of string theory, which purports to be a unified quantum theory of gravity, attention has naturally turned to these questions, and have remarkably shown signs of progress. In this project we will first review black hole solutions in GR, and then look at how a thermodynamic description of black holes is made possible. We then turn to introduce string theory and in particular review the black Dp-brane solutions of type IIB supergravity. Lastly we see how to compute a microscopic account of the Bekenstein entropy is given in string theory. 1 Chapter 1 Black Holes in General Relativity 1.1 Black Hole Solutions We begin by reviewing some the basics of black holes as they arise in the study of general relativity.
    [Show full text]