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Counting Black Hole Microstates in String Theory

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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.
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