Holography for Black Holes in General Relativity and Beyond A Thesis Submitted to the College of Graduate and Postdoctoral Studies in Partial Fulfillment of the Requirements for the degree of Master of Science in the Department of Physics & Engineering Physics University of Saskatchewan Saskatoon By Canisius Bernard c Canisius Bernard, August 2020. All rights reserved. Permission to Use In presenting this thesis in partial fulfilment of the requirements for a Postgraduate degree from the University of Saskatchewan, I agree that the Libraries of this University may make it freely available for inspection. I further agree that permission for copying of this thesis in any manner, in whole or in part, for scholarly purposes may be granted by the professor or professors who supervised my thesis work or, in their absence, by the Head of the Department or the Dean of the College in which my thesis work was done. It is understood that any copying or publication or use of this thesis or parts thereof for financial gain shall not be allowed without my written permission. It is also understood that due recognition shall be given to me and to the University of Saskatchewan in any scholarly use which may be made of any material in my thesis. Requests for permission to copy or to make other use of material in this thesis in whole or part should be addressed to: Head of the Department Department of Physics and Engineering Physics University of Saskatchewan 163 Physics Building, 116 Science Place Saskatoon, Saskatchewan S7N 5E2 Canada or Dean College of Graduate and Postdoctoral Studies University of Saskatchewan 116 Thorvaldson Building, 110 Science Place Saskatoon, Saskatchewan S7N 5C9 Canada i Abstract Both extremal and non-extremal Kerr black holes have been considered to be holographically dual to two-dimensional (2D) conformal field theories (CFTs). In this thesis, we study the holography to the case of a rotating Janis-Newman-Winicour (JNW) black holes, a rotating Brans-Dicke-Kerr (BDK) black hole, and an asymptotically anti-de Sitter (AdS) rotating charged black holes in f(T ) gravity, where f(T ) = T + αT 2, where α is a constant. Firstly, we find that the rotating JNW solution does not satisfy the Einstein field equation. Thus, we could not establish a well-defined Kerr/CFT correspondence in this theory. Secondly, we find that the scalar wave equation in the background of BDK black hole is not separable. The existence of the SL(2;R)L × SL(2;R)R symmetry can be found in the radial equation of the scalar probe around the non-extremal black hole. Therefore, the inseparability of the scalar wave equation eliminates the possibility of any holography aspect for BDK black hole. Thirdly, we find that the scalar wave radial equation at the near-horizon region implies the existence of the 2D conformal symmetries. We note that the 2π identification of the azimuthal angle φ in the black hole line element, corresponds to a spontaneous breaking of the conformal symmetry by left and right temperatures TL and TR, respectively. We show that choosing proper central charges for the dual CFT, we produce exactly the macroscopic Bekenstein-Hawking entropy from the microscopic Cardy entropy for the dual CFT. These observations suggest that the rotating charged AdS black hole in f(T ) gravity is dual to a 2D CFT at finite temperatures TL and TR for a specific value of mass M, rotational, charge, and f(T ) parameters, Ω, Q, and jαj, respectively. ii Acknowledgements I am deeply grateful to my academic supervisor, Prof. Masoud Ghezelbash, for his invaluable help, mo- tivation, enthusiasm, patience, and immense knowledge he has provided throughout my time as his student. I would like to thank my advisory committee members, Prof. Rainer Dick, Prof. Rob Pywell, and Prof. Robert Green for their guidance and insightful comments. I would like to thank Prof. Steven Rayan, who acted as the external examiner in my defence. I also thank the Department of Physics and Engineering Physics secretaries, Deborah Gjertsen and Marjorie Granrude, for their administrative supports and con- stant attention. I am indebted to my mentor, Dr. Haryanto Siahaan, for his tremendous help, advice, and continuing encouragement. My journey to Canada would not have been possible without the support of my parents, Margaretha C. Sugiharti and Suryamin Maryadi, my aunts Maria E. M. Hegarwati and Miranda Maryadi, my uncles Anto- nius Yuliono and Erwin Budiman, my grandparents Paulus Kutomiharso and Veronica Kang, Andi Maryadi and Sisiliawati. No thanks can be adequate for the decades of their loving support and understanding. My special thank goes to numerous friends in the Department of Physics and Engineering Physics, among them Jeffery Zielinski, William Elcock, Patrick Braun, Alex Chang, Bardia Fahim, Niyusha Hoseiny, Ayub Khodaee, Marilyn Jimenez, Oleksandr Chapurin, Barbara Cid Mora, and Siyuan Li. I would like to thank Jeffery Zielinski for his constructive criticism of the thesis, help, advice, and useful discussion. Some special words of gratitude also go to my friends Saga Bergqvist, Len Song, and Mengying Liu. They have contributed in many different ways during the writing up of this thesis. Last but not least, I would like to thank the Indonesian community in Saskatoon for their warm welcome, hospitality, constant support, and attention. During the course of this work, I was supported by the Natural Sciences and Engineering Research Council of Canada, University of Saskatchewan Graduate Teaching Fellowships, and William Rowles Fellowship in Physics and Engineering Physics. I am very grateful to all these organizations. Canisius Bernard Saskatoon, Canada iii To my parents \Ipsa scientia potestas est." | Sir Francis Bacon, Meditationes sacrae iv Contents Permission to Use i Abstract ii Acknowledgements iii Dedication iv Contents v List of Figures vii List of Abbreviations viii 1 Introduction 1 2 Black Holes in Einstein Gravity and Beyond 5 2.1 Einstein gravity . 5 2.1.1 A variational approach to general relativity . 6 2.1.2 Kerr black holes . 10 2.1.3 The law of thermodynamics and black holes mechanics . 16 2.2 Gravity with torsion . 18 2.2.1 Torsion tensor . 19 2.2.2 Teleparallel equivalent of general relativity and the f(T ) extension . 20 2.2.3 Black holes in quadratic f(T ) gravity . 25 3 Introducing Conformal Field Theory 28 3.1 Conformal group in N > 2 dimensions . 28 3.2 Conformal group in N = 2 dimensions . 30 3.2.1 The central charge and the Virasoro algebra . 33 3.2.2 Energy-momentum tensor and Noether currents . 35 3.2.3 Primary fields . 38 3.2.4 Conformal field theory properties on an infinite cylinder . 39 3.2.5 Conformal field theory on the torus . 40 3.2.6 Cardy entropy . 42 4 Holographic Aspects of Kerr Black Holes 45 4.1 Extremal Kerr black hole and holography . 45 4.1.1 AdS3 and SL(2;R)L × SL(2;R)R .............................. 45 4.1.2 SL(2;R) Lie algebra . 46 4.1.3 AdS3/CFT2 correspondence . 47 4.1.4 Kerr/CFT correspondence . 47 4.2 Non-extremal Kerr black hole and holography . 49 4.2.1 Scalar scattering off a Kerr black hole . 49 4.2.2 Hidden conformal symmetry . 50 4.2.3 Temperature and entropy in conformal field theory . 52 5 Holographic Aspects of Black Holes in Modified Gravity 54 5.1 Is there a hidden conformal symmetry for black holes in scalar-tensor gravity? . 54 5.1.1 Scalar scattering off a Janis-Newman-Winicour black hole . 54 v 5.1.2 Validity of rotating Janis-Newman-Winicour spacetime . 57 5.1.3 Scalar scattering off a Brans-Dicke-Kerr black hole . 58 5.2 Hidden conformal symmetry of black holes in f(T ) gravity . 60 5.2.1 Rotating charged AdS black holes . 60 5.2.2 The first law of black hole thermodynamics . 61 5.2.3 Scalar scattering off a rotating charged AdS black hole . 65 5.2.4 Hidden conformal symmetry . 67 5.2.5 Cardy entropy . 72 6 Conclusion 75 References 78 vi List of Figures 1.1 Event Horizon Telescope image of the Messier 87 black hole shadow. This figure is taken from Ref. [1]. 1 1.2 Extremal Kerr black hole illustration. 3 2.1 Illustration of a Kerr black hole. 15 µ 2.2 Four-dimensional spacetime manifold M with a metric gµν where tetrad fields eA = eA@µ are defined at point p on the tangent space TP ............................. 19 3.1 Projecting S2 to C2 illustration. 32 3.2 Cylinder to complex plane mapping illustration. 39 3.3 Lattice of a torus. 40 3.4 Modular S-transformation on lattice of a torus. 42 5.1 φ~ as a function of (a) r with θ = π, (b) θ with r = 1, both with a = 1, M = 0:1, and several numerical values of γ. ........................................ 56 5.2 A(r) as a function of r with M = 5;Q = 0:1; and several numerical values of jαj. 61 5.3 S as a function of M and Q. In these plots, we set Ω = 1; l = 1; jαj = 0:05;L = 1; and r+ = 1. 64 5.4 S as a function of (a) Ω with l = 1, α = 0:05, and several numerical values of M. (b) jαj with M = 3:5, Ω = 1, and several numerical values of l, both with Q = 1, L = 1, and r+ = 1. 65 5.5 Equations (5.80) and (5.81), respectively, as a function of Q. In this plot, we set, M = 5; Ω = 1; l = 1; jαj = 0:05; and r+ = 2:08.