American University in Cairo AUC Knowledge Fountain Theses and Dissertations Fall 7-26-2020 Study of the thermodynamics of charged rotating black holes in five-dimensional anti-de sitter spacetimes Hassan El-Sayed The American University in Cairo Follow this and additional works at: https://fount.aucegypt.edu/etds Recommended Citation APA Citation El-Sayed, H. (2020).Study of the thermodynamics of charged rotating black holes in five-dimensional anti- de sitter spacetimes [Master’s thesis, the American University in Cairo]. AUC Knowledge Fountain. https://fount.aucegypt.edu/etds/1475 MLA Citation El-Sayed, Hassan. Study of the thermodynamics of charged rotating black holes in five-dimensional anti- de sitter spacetimes. 2020. American University in Cairo, Master's thesis. AUC Knowledge Fountain. https://fount.aucegypt.edu/etds/1475 This Master's Thesis is brought to you for free and open access by AUC Knowledge Fountain. 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The American University in Cairo School of Sciences and Engineering Study of the Thermodynamics of Charged Rotating Black Holes in Five-Dimensional Anti-de Sitter Spacetimes A Thesis Presented in Partial Fulfillment of the Requirements for the Degree of Master of Science in Physics Author: Advisor: Hassan El-Sayed Dr. Adel Awad June 2020 This thesis was presented to the following thesis committee members: Dr. Adel Awad Professor Physics Department, School of Sciences and Engineering The American University in Cairo Thesis Advisor Dr. Mohammad AlFiky Assistant Professor Physics Department, School of Sciences and Engineering The American University in Cairo Internal Examiner Dr. Elsayed Lashin Professor Physics Department, Faculty of Science Ain Shams University External Examiner Dr. Ahmed Hamed Assistant Professor Physics Department, School of Sciences and Engineering The American University in Cairo Thesis Committee Moderator and Graduate Director i Table of Contents Acknowledgements v Preface vi Conventions and Notations viii Introduction 1 1 Black Holes 4 1.1 Preliminaries . 4 1.2 Schwarzschild Solution . 7 1.2.1 The Schwarzschild Metric . 7 1.2.2 The Schwarzschild Black Hole . 12 1.3 Event Horizons . 13 1.3.1 The Causal Structure . 13 1.3.2 Event Horizons . 15 1.3.3 Killing Horizons . 15 1.4 Charged, Rotating Black Holes . 17 1.5 Conserved Charges . 21 1.5.1 Electric Charge . 21 1.5.2 Komar Integrals . 22 ii 1.5.3 Brown-York Quasi-local Charges . 28 2 Black Hole Thermodynamics 38 2.1 Cosmic Censorship Conjecture . 38 2.2 The Penrose Process and Black Hole Mechanics . 39 2.3 Black Hole Thermodynamics . 45 3 Anti-de Sitter Spacetime 50 3.1 The Anti-de Sitter Spacetime . 50 3.2 The AdS/CFT Correspondence . 54 3.2.1 Background . 54 3.2.2 Presentation of the Conjecture . 56 3.2.3 Conformal Anomaly . 57 3.3 Divergences in AdS Spacetimes . 58 3.3.1 Introduction . 58 3.3.2 The Background Subtraction Method . 62 3.3.3 The Counterterms Subtraction Method . 63 3.3.4 Discussion of the Thermodynamics in Asymptotically AdS4 Spacetimes 66 3.4 Casimir Energy and the AdS/CFT Correspondence . 67 4 Thermodynamics of Charged Rotating Black Holes in AdS5 70 4.1 Presentation and Discussion of the Solution . 70 4.2 Verification of the Thermodynamical Relations of the Original Solution . 73 4.3 Calculations with the Counterterms Method . 78 4.3.1 Motivation, Action and Mass Calculations . 78 4.3.2 Conformal Anomaly Calculations . 86 4.3.3 Casimir Energy Calculations . 88 iii 4.3.4 Angular Momenta Calculation and First Law Verification . 90 Conclusion 92 A Mathematical Operations 94 B Hypersurfaces, Extrinsic Curvature and Foliation 100 B.1 Hypersurfaces . 100 B.2 Extrinsic Curvature . 102 B.3 Geometry of Foliations and Metric Decomposition . 104 C Symmetries and Killing Vectors 108 C.1 Symmetry Transformations . 108 C.1.1 Symmetries and Conservation Laws . 108 C.1.2 Symmetry Groups . 111 C.2 Stokes' Theorem and Conserved Charges in Curved Spacetimes . 114 C.3 Killing Vectors . 117 D Action Calculation and the Energy-Momentum Tensor 124 D.1 Action and Energy-Momentum Tensor Definitions . 124 D.2 Action Calculation . 131 E Chern-Simons Term in Electrodynamics 133 iv Acknowledgements For me to finish this thesis would not have been possible without many people to whom I will be eternally grateful. In particular I would like to thank my advisor, Professor Adel Awad, who picked this inter- esting and challenging research topic, which presented me with a tremendous opportunity to learn about black hole thermodynamics as well as a few things about the AdS/CFT correspondence. Moreover I am deeply thankful for him for having suggested to me that I study and do my research in General Relativity { a suggestion that completely reshaped my future plans in physics. I would also like to thank Professors Mohammad AlFiky and Ashraf ElFiqi for the invaluable education that they have provided me with, for constantly inciting my scientific curiosity, and for many moments of moral and academic support through the past 3 years. I also have to thank my friends Mohamed Hany, Ahmed Hemdan and Summer Kassem for the numerous interesting scientific discussions that we have had during the past couple of years. I consider these discussions an important part of the education that I have received at AUC. Finally, I would like to thank Ahmed Hemdan and Ahmed Kora¨ıemfor reading and pro- viding me with amazing feedback for this thesis. Hassan ElSayed, Cairo, Egypt 3 June 2020 v Preface This thesis is the culmination of the past two years of my life which I have dedicated, for the most part, to studying the general theory of relativity and doing research in black hole thermodynamics. While I have worked on two other small research projects in GR during that time, I consider the research presented here as the most valuable of this period. One of the most challenging parts of conducting my first research project in theoretical physics was finding good resources to learn about certain topics that I needed to understand. Many of the references I had to go through were { as you can imagine { quite challenging to follow. Nevertheless, I have come across several theses that were extremely helpful and aided me in understanding certain topics that I was trying to learn. Some were useful in explaining certain concepts and others were helpful in explaining the detailed procedure of certain calculations. My aim in this thesis is to simultaneously present to a defense committee my own research along with my understanding of the theoretical concepts that underlie it, and to offer as clear and detailed an explanation as possible of the topics that I discuss. In doing the latter I hope that maybe someday I could benefit someone in the way that many others have benefited myself. With that in mind, I have made a few decisions which I hope will improve the readability of this thesis. First, I have tried to do two things that can be at odd with each other: give a complete discussion of the elements that I am presenting, while keeping each discussion very compact. In order to do this I have moved many parts of the discussion to appendices. This helps make the discussions of the core chapters concise, while also presenting the information in the appendices in stand-alone discussions for anyone interested only in reading these parts. I have also attempted to give detailed proofs and derivations in all my discussions. Some of the proofs that I found for certain formulae were too complicated and I have attempted to come up with shorter proofs myself to present them here. Because reading a long proof can make the reader feel disconnected from the big picture, many proofs are placed in individual subsections. The uninterested reader can simply jump to the end of any proof (denoted by the conventional end-of-proof symbol \"). Aside from proofs, I have tried, to the best of my judgement, to present the details of every vi relevant calculation in ways to help anyone who is trying to reproduce these calculations. Here I include both previously-available calculations as well as my own calculations in Chapter 4. vii Conventions and Notations Units We work in the Plank natural units system. In this system we have: • Speed of light c = 1, • Gravitational constant G = 1, • Reduced Planck constant ~ = 1, • The Boltzmann constant kB = 1, • The Coulomb constant 1 = 1. 4π"0 Relativity and Tensors Unless otherwise stated, all metrics are assumed to have Lorentzian signatures (−; +; +; :::+). All mathematical results will be expressed in light of this assumption. We mostly use the abstract index notation of Roger Penrose wherein indices denoted by Latin letters fa; b; c:::gg indicate a tensors type rather than its components in a particular basis. Nevertheless, Greek letters are occasionally used to emphasize the fact that we are referring to a specific basis. On a D-dimensional manifold, Greek indices run from 0 to D − 1. The specific Latin letters fi; j; kg denote only spatial components in a particular basis and hence run from 1 to D − 1. The Einstein summation convention is used by default whenever an index of any of the above types is raised an lowered. When the coordinates are given by xa = (x0; :::xD−1), the partial derivative operator nota- tion that follows is viii @ @ ≡ : µ @xµ We also sometimes do deal with spatial vectors in 3 dimensions. These vectors are denoted by upper arrows, ~x = (x1; x2; x3): General Symbols