UNIVERSITA` DEGLI STUDI DI CAGLIARI Facolt`adi Scienze Matematiche, Fisiche e Naturali Applications of perturbation theory in black hole physics Paolo Pani A thesis submitted for the degree of Doctor of Philosophy Supervisor: Prof. Mariano Cadoni January 2011 Abstract Black holes have many faces. Arguably, they are the most interesting objects in theoretical physics, revealing the elusive connection between gravity and quantum mechanics. Within the gauge/string duality they provide useful insights on strongly coupled quantum field theories and on quantum gravity. Furthermore, probing the strong curvature regime of any grav- ity theory, black holes carry the imprint of possible strong curvature corrections to General Relativity. Finally, beside their unique theoretical properties, several experimental evidences suggest that astrophysical black holes exist in nature and they are believed to be very com- mon objects in the universe. In this dissertation we discuss several applications of linear perturbation theory in black hole physics. As applications in theoretical physics, we study perturbations of dilatonic black holes in Einstein-Maxwell theory and the holographic prop- erties of the dual field theory via the Anti de Sitter/Condensed Matter duality. Furthermore we discuss a method to compute long-lived quasinormal modes of Schwarzschild-Anti de Sit- ter black holes and we study vortex black hole solutions in three dimensional Anti de Sitter gravity. As applications in astrophysics, we discuss how the characteristic oscillations of black holes in string-inspired theories of gravity can provide observable signatures of deviations from General Relativity. We study two well-motivated effective theories: Dynamical Chern-Simons gravity and Einstein-Dilatonic-Gauss-Bonnet gravity. We conclude by discussing the black hole paradigm. Motivated by the lacking of a definitive answer on the existence of astrophysi- cal black holes, we study some viable alternatives, generally called “black hole mimickers”. We focus on two representative cases: static thin-shell gravastars and superspinars. We discuss their stability, gravitational-wave signature and viability as astrophysical objects. KEY-WORDS: Black holes; Perturbation theory; AdS/CFT correspondence; Holography; Modified gravity; Effective theories; Gravitational waves; Quasinormal modes; Black hole mimickers. a Giu `equalche cosa che hai fatto, suppongo. Acknowledgements I wish to thank the University of Cagliari and the Italian National Institute of Nuclear Physics (INFN) for financial and logistical support and the Master and Back foundation programme of Regione Sardegna for a grant. This work has been also partially founded by FCT - Portugal through project PTDC/FIS/098032/2008. I warmly thank my supervisor, Mariano Cadoni, for his guidance, for all the physics he taught me and for the possibility he gave me to find my way in gravitational physics. I am also extremely grateful to Vitor Cardoso. It is a pleasure and an honor to work with him, being always conscious that I will have something important to learn. A special thanks goes to Emanuele Berti. Although we have not met (yet!), the useful discussions we had, and his irreplaceable contributions to our works, certainly made me a better physicist. During these years, I had the pleasure to work with Enrico Barausse, Yanbei Chen, Giuseppe D’Appollonio, Leonardo Gualtieri, Carlos Molina, Richard Norte and Matteo Serra. Many parts of this work are the outcome of various fruitful collaborations with them. I would like to thank my gravity-group mates in Cagliari: Maurizio Melis and Cristina Monni, and those I met in Lisbon: Mariam Bouhmadi, Jordi Casanellas, Magdalena Lemos, Andrea Nerozzi, Jorge Rocha and Helvi Witek. I hope we will have many other interesting discussions in future. I cannot express my gratitude enough to my parents Francesco and Rosanna and to my sister Carly, for their constant support, to the rest of my family, especially to my grandmothers and to my aunt Lella, who helped me in correcting this thesis, and to my family-in-law: Albi, Dianella, Fra and Gino. To my long, long-dated friends, Ale, Alessandra, Beccio, Fede, Giorgio, Marzia, Matte, Pierma, Ricca, Save, Valeria(s). They share the misfortune to be the people who know me better and, in spite of this, they will always support me with their love. Se’ bon a’ sutt! I would like to thank all the members of the Physics Department in Cagliari, who contribute to make it a wonderful place to work in. In particular my physics-class mates and friends, Francesco, Giamba, Nicola, Scrilly and Gary, the No-Trash Cougar. Although we took different paths and we are now spread around the world, we keep constantly nonsensing about physics and white weapons. In this difficult period, during which Italian Governments are degrading Public Education and Academic Research year by year, I cannot forget to express my gratitude to my high school Science professors: Maria Tonina Fancello, Guido Orr`u, Maria Clorinda Todde and Luigi Uras. If now I am physicist, this is also due to their early guidance. Last, as usual, comes Giu. Sharing the life with you, with your unconditional love, energy, peace of mind and happiness, makes me a better person. I love you. Contents Preface .......................................... xiii Introduction 1 I Black holes in Anti de Sitter space 7 1 Black holes and Anti de Sitter/Condensed Matter duality 9 1.1 A new approach to strongly coupled field theories . .......... 9 1.2 AdS/CFTcorrespondenceinanutshell. ....... 11 1.3 Holographic properties of AdS black holes . ......... 13 2 Charged dilatonic black holes in AdS 21 2.1 Introduction.................................... 21 2.2 Instability of AdS-RN black holes in Einstein-Maxwell-dilaton gravity . 23 2.3 Numerical solutions of the field equations . ......... 28 2.4 Thezerotemperaturelimit . 30 2.5 Holographic properties of the new black hole solution . ............. 33 2.6 Conclusions ..................................... 40 2.7 Appendix: Scalar hairs in the probe limit. Schwarzschild-AdS background . 42 3 Phase transitions and holography of dyonic black holes in AdS 45 3.1 Introduction.................................... 45 3.2 Instability of dyonic AdS-RN black holes . ......... 46 3.3 Dyonic black holes with scalar hairs . ....... 47 3.4 Holographic properties of DDBHs . ...... 52 3.5 Purelymagneticblackholes . ..... 55 3.6 Discussion...................................... 59 4 Breit-Wigner resonances and the quasinormal modes of AdS black holes 61 4.1 Introduction.................................... 61 4.2 Quasi-bound states in SAdS black holes . ....... 62 4.3 Long-lived modes in the eikonal limit . ........ 67 4.4 Conclusionsandoutlook . 68 Table of Contents 5 Scalar hairs and exact vortex solutions in 3D AdS gravity 69 5.1 Introduction.................................... 69 5.2 Dynamics of a complex scalar field in 3D AdS spacetime . ......... 71 5.3 Scalarhairs ..................................... 72 5.4 Linearformofthefieldequations . ...... 73 5.5 Black hole solutions with real scalar hairs . .......... 74 5.6 Vortexsolutions ................................. 76 5.7 Conclusions ..................................... 78 II Signatures of black holes beyond General Relativity 81 6 Black holes in alternative theories of gravity 83 6.1 Do we really need some alternative to General Relativity?............ 83 6.2 Thehairyblackholeszoo .. .. .. .. .. .. .. .. 85 6.3 No-hairtheoremtests ............................. 86 6.4 Alternative theories of gravity. Two cases studies. ............. 87 7 Gravitational signature of black holes in Dynamical Chern-Simons gravity 93 7.1 Introduction.................................... 93 7.2 Perturbation equations and dynamical stability . ............ 96 7.3 Numericalapproach ............................... 97 7.4 Numericalresults................................ 101 7.5 Discriminating the QNMs: no-hair tests . ........107 7.6 Conclusions ..................................... 109 7.7 Appendix: Ghost-like instabilities for β < 0 ....................110 8 Black holes in Einstein-Dilatonic-Gauss-Bonnet gravity 113 8.1 Introduction.................................... 114 8.2 Spherically symmetric BHs in Einstein-Dilaton-Gauss-Bonnet theory . 116 8.3 Linearstabilityanalysis . 118 8.4 Slowly rotating BHs in EDGB theory . 122 8.5 BHGeodesicsinEDGBtheory . 124 8.6 Discussion...................................... 134 8.7 Appendix: Linear stability analysis: axial perturbations .............135 8.8 Appendix: Slowly rotating Kerr black holes in the Hartle approximation . 137 III Black hole mimickers 139 9 Are really black holes out there? 141 10 Gravitational-wave signature of a thin-shell gravastar 145 10.1Introduction................................... 146 10.2 Equilibriummodel ............................... 147 10.3 Gravitational perturbations . ........149 10.4 NumericalResults ............................... 155 x Table of Contents 10.5 Gravitational perturbations by a point particle . .............161 10.6 Gravitational flux from gravastars and black holes . ............165 10.7 Conclusionsandoutlook . 171 10.8 Appendix: Perturbation equations and matching conditions ........... 172 10.9 Appendix: The continued fraction method . .........177 10.10Appendix: High-compactness limit . .........181 11 Gravitational instabilities of superspinars 183 11.1Introduction................................... 183 11.2 A simple model of superspinar in four dimensions . ...........185 11.3Perfectmirror