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Quantum Field Theory on Rotating Black Hole Spacetimes Quantum field theory on rotating black hole spacetimes Hugo Ricardo Cola¸coFerreira, MASt Thesis submitted to The University of Nottingham for the degree of Doctor of Philosophy December 2015 Abstract This thesis is concerned with the development of a general method to compute renormalised local observables for quantum matter fields, in a given quantum state, on a rotating black hole spacetime. The rotating black hole may be surrounded by a Dirichlet mirror, if necessary, such that a regular, isometry-invariant vacuum state can be defined. We focus on the case of a massive scalar field on a (2+1)-dimensional rotating black hole, but the method can be extended to other types of matter fields and higher-dimensional rotating black holes. The Feynman propagator of the matter field in the regular, isometry-invariant state is written as a sum over mode solutions on the complex Riemannian section of the black hole. A Hadamard renormalisation procedure is implemented at the level of the Feynman propagator by expressing its singular part as a sum over mode solutions on the complex Riemannian section of rotating Minkowski spacetime. This allows us to explicitly renormalise local observables such as the vacuum polarisation of the quantum field. The method is applied to the vacuum polarisation of a real massive scalar field on a (2+1)-dimensional warped AdS3 black hole surrounded by a mirror. Selected numerical results are presented, demonstrating the numerical efficacy of the method. The existence of classical superradiance and the classical linear mode stability of the warped AdS3 black hole to massive scalar field perturbations are also analysed. iii iv Acknowledgements First and foremost, I would like to thank my supervisor Dr. Jorma Louko for all his help during the last four years and for reading this manuscript. I am very grateful for having had the opportunity to work with you! I thank the Universitas 21 Network for awarding me a Universitas 21 Prize Schol- arship for a month-long visit to the Gravitational Theory Group of the University of Maryland. I especially thank Prof. Bei-Lok Hu for his hospitality. Many thanks to Dr. Vitor Cardoso, Dr. Sam Dolan, Prof. Christopher Fewster, Dr. Carlos Herdeiro, Prof. Bernard Kay, Prof. Elizabeth Winstanley and Dr. Helvi Witek for helpful discussions and comments during my PhD. This thesis marks the end of a five-year adventure in the UK, which started in Cambridge and ended in Nottingham. I would like to thank Adam Fraser and Sagi Elster for a wonderful time in Cambridge, and also Emily Kirk, who has successfully endured my company for five years! I thank my fellow physicists and mathematicians Benito Aubry, Johnny Espin, James Gaunt, Sara Tavares and Carlos Scarinci for the great time discussing Physics (and sometimes other things) in Nottingham. And I also thank Andrew Yiakoumetti, Anja Andrejeva, Jyothika Kumar, Iker P´erezand Jennifer Kiefer for making my life in Nottingham much more enjoyable! Finally, I wish to thank my parents Carlos and L´ıgiaFerreira for always being there and helping me achieve this important step in my life; my brother Andr´e Ferreira for his friendship all these years; and my cousin Maria Em´ıliaFernandes, with an academic greeting! The author acknowledges financial support from Funda¸c~aopara a Ci^enciae Tec- nologia (FCT)-Portugal through Grant No. SFRH/BD/69178/2010. v vi Contents Preface xiii Notation xiv Introduction 1 I Fundamentals 13 1 Mathematical preliminaries 15 1.1 Spacetime and causal structure . 15 1.2 Stationary spacetimes . 20 1.2.1 Globally and locally stationary spacetimes . 20 1.2.2 Riemannian sections of stationary spacetimes . 21 1.3 Bi-tensors . 23 1.3.1 Synge's world function . 24 1.3.2 Parallel propagator . 25 1.3.3 Covariant and non-covariant Taylor expansion . 26 1.3.4 The case of complex Riemannian manifolds . 28 1.4 Symplectic and Hilbert spaces . 30 1.4.1 Symplectic spaces . 31 1.4.2 Charged symplectic spaces . 31 1.4.3 Hilbert spaces . 32 1.4.4 Operators on Hilbert spaces . 35 vii 1.5 Complexification of real vector spaces . 38 1.6 Distributions . 39 1.7 Hyperbolic differential operators and Green operators . 42 1.7.1 Normally hyperbolic operators . 42 1.7.2 Cauchy problem . 44 1.7.3 Green operators . 45 2 Quantum field theory on curved spacetimes 49 2.1 Real scalar field . 50 2.1.1 Classical field theory . 51 2.1.2 Quantum field theory . 57 2.2 The case of stationary spacetimes . 64 2.2.1 Positive frequency solutions . 65 2.2.2 Green's distributions . 70 2.2.3 Rotating black hole spacetimes . 77 2.3 The case of spacetimes with boundaries . 79 2.4 Hadamard renormalisation . 82 2.4.1 The case of globally hyperbolic spacetimes . 82 2.4.2 The case of spacetimes with timelike boundaries . 87 3 Renormalised local observables in rotating black hole spacetimes 89 3.1 Scalar field on a rotating black hole . 89 3.1.1 (2+1)-dimensional rotating black hole . 90 3.1.2 Scalar field and Hartle-Hawking state . 91 3.1.3 Hadamard renormalisation . 97 3.2 Quasi-Euclidean method . 99 3.2.1 Complex Riemannian section . 100 3.2.2 Green's distribution in the Riemannian section . 101 3.3 Renormalisation procedure . 103 3.3.1 Subtraction of the Hadamard singular part . 104 viii 3.3.2 Large quantum number behaviour . 107 3.3.3 Fixing of the Minkowski free parameters . 108 3.4 Expectation value of the stress-energy tensor . 112 3.4.1 Derivatives of the Hadamard singular part . 113 3.4.2 Attempt to renormalise rθ~rθ~G ................. 114 II Application 119 4 Warped AdS3 black holes 121 4.1 2+1 gravity and topologically massive gravity . 121 4.1.1 2+1 gravity . 122 4.1.2 Topologically massive gravity . 123 4.2 Warped AdS3 solutions . 124 4.2.1 Warped AdS3 solutions . 124 4.2.2 Warped AdS3 black hole solutions . 129 5 Classical linear mode stability of the WAdS3 black holes 135 5.1 Classical superradiance . 135 5.1.1 Klein-Gordon field equation . 136 5.1.2 Asymptotic mode solutions . 137 5.1.3 Basis of mode solutions . 141 5.1.4 Existence of classical superradiance . 142 5.2 Quasinormal and bound state modes and classical linear mode stability147 5.2.1 Quasinormal and bound state modes . 147 5.2.2 Classical linear mode stability . 152 5.3 Case with a mirror-like boundary . 155 5.4 Conclusions on the classical stability . 164 2 6 Computation of hΦ (x)i on a WAdS3 black hole 167 6.1 Vacuum polarisation on a WAdS3 black hole . 167 ix 6.1.1 Scalar field and the Hartle-Hawking state . 168 6.1.2 Complex Riemannian section . 169 6.1.3 Green's distribution . 169 6.1.4 Hadamard renormalisation . 171 6.2 Numerical results . 173 Conclusions 176 III Appendices 181 A Complex Riemannian section of the Minkowski spacetime 183 B WKB expansions 189 C Hypergeometric functions 193 D Classical black hole superradiance 197 Bibliography 199 x List of Figures 1.1 A set and its causal future and causal past. 17 1.2 Normal and shift vectors to a spacelike surface. 19 1.3 Timelike and spacelike compact sets. 20 1 1.4 The causal future and causal past of supp f for f 2 C0 (M). 46 2.1 Cut complex plane for the holomorphic function G ........... 73 2.2 Cut complex plane for the holomorphic function Gβ........... 75 2.3 Carter-Penrose diagram of manifold with boundaries. 78 2.4 Carter-Penrose diagram of manifold with boundaries and \initial- value surface". 81 4.1 Carter-Penrose diagrams of the spacelike stretched black hole. 134 5.1 Effective potential of the scalar field for selected values of the squared mass. 138 5.2 In and up modes in the exterior region of the spacetime. 142 5.3 Eingenfrequencies in the complex plane for a spacelike stretched black hole. 154 5.4 \Right" frequencies as functions of the mirror's location for selected values of r+ and r−............................. 157 5.5 \Right" frequencies as functions of the mirror's location for selected values of m................................. 158 xi 5.6 \Right" frequencies as functions of the mirror's location for selected values of ν................................. 159 5.7 \Right" frequencies as functions of the mirror's location for selected values of k................................. 160 5.8 \Right" frequencies as functions of the mirror's location for selected values of n................................. 161 5.9 \Left" frequencies as functions of the mirror's location. 164 6.1 Carter-Penrose diagram of a non-extremal spacelike stretched black hole surround by mirrors. 168 6.2 Vacuum polarisation for the scalar field in the exterior region. 174 xii Preface The research included in this thesis was carried out in the School of Mathematical Sciences of the University of Nottingham. Chapters 3 and 6 are the outcomes of collaboration with Jorma Louko. Chapter 5 is the outcome of the author's own research. The results of this research have lead to the following publications: • H. R. C. Ferreira, Stability of warped AdS3 black holes in Topologically Massive Gravity under scalar perturbations, Phys.Rev. D87 (2013), no. 12 124013 (Ref. [1]). • H. R. C. Ferreira and J. Louko, Renormalized vacuum polarization on rotating warped AdS3 black holes, Phys.Rev. D91 (2015), no. 2 024038 (Ref. [2]). • H. R. C. Ferreira, Renormalized vacuum polarization of rotating black holes, Int.J.Mod.Phys. (2015) 1542007 (Ref. [3]). xiii xiv Notation In this thesis, we use metric signature (−; +; ··· ; +).
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