Lecture Notes in Physics

Volume 906

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The series Lecture Notes in Physics (LNP), founded in 1969, reports new developments in physics research and teaching-quickly and informally, but with a high quality and the explicit aim to summarize and communicate current knowledge in an accessible way. Books published in this series are conceived as bridging material between advanced graduate textbooks and the forefront of research and to serve three purposes: • to be a compact and modern up-to-date source of reference on a well-deﬁned topic • to serve as an accessible introduction to the ﬁeld to postgraduate students and nonspecialist researchers from related areas • to be a source of advanced teaching material for specialized seminars, courses and schools Both monographs and multi-author volumes will be considered for publication. Edited volumes should, however, consist of a very limited number of contributions only. Proceedings will not be considered for LNP. Volumes published in LNP are disseminated both in print and in electronic for- mats, the electronic archive being available at springerlink.com. The series content is indexed, abstracted and referenced by many abstracting and information services, bibliographic networks, subscription agencies, library networks, and consortia. Proposals should be sent to a member of the Editorial Board, or directly to the managing editor at Springer:

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More information about this series at http://www.springer.com/series/5304 Richard Brito • Vitor Cardoso • Paolo Pani

Superradiance Energy Extraction, Black-Hole Bombs and Implications for Astrophysics and Particle Physics

123 Richard Brito Vitor Cardoso CENTRA CENTRA Departamento de Física Departamento de Física Instituto Superior Técnico, Universidade Instituto Superior Técnico, Universidade de Lisboa de Lisboa Lisbon, Portugal Lisbon, Portugal

Paolo Pani Dipartimento di Fisica “Sapienza” Universita di Roma & Sezione INFN Roma 1 Rome, Italy

ISSN 0075-8450 ISSN 1616-6361 (electronic) Lecture Notes in Physics ISBN 978-3-319-18999-4 ISBN 978-3-319-19000-6 (eBook) DOI 10.1007/978-3-319-19000-6

Library of Congress Control Number: 2015943376

Springer Cham Heidelberg New York Dordrecht London © Springer International Publishing Switzerland 2015 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made.

Printed on acid-free paper

Springer International Publishing AG Switzerland is part of Springer Science+Business Media (www.springer.com) Preface

Macroscopic objects, as we see them all around us, are governed by a variety of forces, derived from a variety of approximations to a variety of physical theories. In contrast, the only elements in the construction of black holes are our basic concepts of space and time. They are, thus, almost by definition, the most perfect macroscopic objects there are in the universe. – Subrahmanyan Chandrasekhar Superradiance is a very generic process involving dissipative systems, whereby energy is transferred from one medium to another, typically stimulated by wave scattering. With a 60-year-old history, superradiance has played a prominent role in optics, quantum mechanics, and especially in relativity and astrophysics. Superradiance in curved spacetime was born by the hand of Yakov Borisovich Zel’dovich, who realized that rotation of any macroscopic body with internal degrees of freedom could amplify incident radiation. Quantization of this process leads rotating objects—including black holes—to spontaneously radiate. Soon after, and inspired by this discovery, Hawking realized that a similar mechanism would trigger black hole evaporation very generically, even in the absence of rotation. A unified framework for superradiant effects can yield new insights into previously disconnected phenomena, for instance superradiant instabilities can occur in systems as diverse as fluids, stars, and black-hole spacetimes. Furthermore, the very necessity of superradiance in a variety of different systems can be understood by simple thermodynamical arguments. Recent developments in theoretical physics—in particular the realization that superradiance leads to new constraints on fundamental bosonic fields and to new hairy black holes—have further highlighted the need for a unified description of superradiant phenomena. Unfortunately, with the exception of the outstanding—but focused—work by Bekenstein and Schiffer [1], a proper overview on superradiance, including various aspects of wave propagation in black-hole spacetimes, does not exist. We hope that the current work will fill this gap. This book is addressed to researchers embarking on the subject, who wish to find a concise overview of the state-of-the-art and on the relevant methods to attack these problems. This work is also addressed to more experienced researchers wishing to dive quickly into a certain topic, by browsing through the relevant references. In view of the multifaceted nature of this subject, we present a unified treatment v vi Preface where various aspects of superradiance in flat spacetime are connected to their counterparts in curved spacetime, with particular emphasis on the superradiant amplification by black holes. In addition, we wish to review various applications of black-hole superradiance which have been developed in the last decade. These developments embrace—at least—three different communities, and our scope is to present a concise treatment that can be fruitful for the reader who is not familiar with the specific area. As will become clear throughout this work, some of these topics are far from being fully explored. We hope this study will serve as a guide for the exciting developments lying ahead, including the experimental implementation and observation of rotational superradiance! This work would not have been possible without the patient and constant encouragement of Ana Sousa (who has also kindly prepared the illustrations for us) and of Giulia Serra. We are indebted to Asimina Arvanitaki, Jacob Bekenstein, Óscar Dias, Sam Dolan, Roberto Emparan, Sean Hartnoll, Shahar Hod, Luis Lehner, Carlos Palenzuela, Robert Penna, Silke Weinfurtner for useful comments on a preliminary draft of this manuscript, and especially to João Rosa for comments and for comparing our superradiant amplification factors with his code. We are also much indebted to Enrico Barausse, Emanuele Berti, Óscar Dias, Roberto Emparan, Leonardo Gualtieri, Carlos Herdeiro, Luis Lehner, Hideo Kodama, Akihiro Ishibashi, José Lemos, Avi Loeb, Hirotada Okawa, Frans Pretorius, Thomas Sotiriou, Ulrich Sperhake, Helvi Witek, Shijun Yoshida, and Hirotaka Yoshino for many and interesting discussions throughout the years. We benefited from the generous support of a number of institutions. R. B. is supported by FCT foundation through grant SFRH/BD/52047/2012, and from the Fundação Calouste Gulbenkian through the Programa Gulbenkian de Estímulo à Investigação Científica. V.C. acknowledges financial support provided under the European Union’s FP7 ERC Starting Grant “The dynamics of black holes: testing the limits of Einstein’s theory” grant agreement no. DyBHo–256667. P.P. was supported by the European Community through the Intra-European Marie Curie contracts aStronGR-2011-298297, AstroGRAphy-2013-623439 and by FCT- Portugal through the projects IF/00293/2013 and CERN/FP/123593/2011. This research was supported in part by the Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Economic Development and Innovation. This work was supported by the NRHEP 295189 FP7- PEOPLE-2011-IRSES Grant.

Lisbon, Portugal Richard Brito Lisbon, Portugal Vitor Cardoso Rome, Italy Paolo Pani

Reference

1. J.D. Bekenstein, M. Schiffer, The many faces of superradiance. Phys. Rev. D58, 064014 (1998). arXiv:gr-qc/9803033 [gr-qc] Contents

1 Introduction ...... 1 1.1 Milestones...... 2 References...... 6 2 Superradiance in Flat Spacetime...... 11 2.1 Klein Paradox: The First Example of Superradiance ...... 11 2.1.1 BosonicScattering...... 12 2.1.2 FermionicScattering...... 13 2.2 SuperradianceandPairCreation...... 14 2.3 Superradiance and Spontaneous Emission by a Moving Object .... 16 2.3.1 CherenkovEmissionandSuperradiance...... 18 2.3.2 CherenkovRadiationbyNeutralParticles...... 18 2.3.3 Superradiance in Superfluids and Superconductors ...... 20 2.4 Sound Amplification by Shock Waves...... 21 2.4.1 Sonic“Booms”...... 21 2.4.2 Superradiant Amplification at Discontinuities ...... 22 2.5 RotationalSuperradiance...... 24 2.5.1 Example1.ScalarWaves...... 25 2.5.2 Example 2. Sound and Surface Waves: A PracticalExperimentalSetup?...... 26 2.6 Tidal Acceleration ...... 29 References...... 32 3 Superradiance in Black Hole Physics ...... 35 3.1 Action,EquationsofMotionandBlackHoleSpacetimes...... 36 3.1.1 Static, Charged Backgrounds...... 36 3.1.2 Spinning, Neutral Backgrounds ...... 37 3.1.3 GeodesicsandFrameDraggingintheKerrGeometry..... 38 3.1.4 TheErgoregion...... 39 3.1.5 Intermezzo: Stationary and Axisymmetric BlackHolesHaveanErgoregion...... 41 3.2 AreaTheoremImpliesSuperradiance...... 42

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3.3 Energy Extraction from BlackHoles:ThePenroseProcess...... 43 3.3.1 TheOriginalPenroseProcess ...... 44 3.3.2 TheNewtonianCarouselAnalogy ...... 47 3.3.3 Penrose’sProcess:EnergyLimits...... 48 3.3.4 ThePenroseProcessinGenericSpacetimes...... 50 3.3.5 The Collisional Penrose Process: Ultra-High-EnergyDebris...... 52 3.4 TheABCofBlackHoleSuperradiance ...... 54 3.5 SuperradiancefromChargedStaticBlackHoles ...... 56 3.5.1 LinearizedAnalysis:AmplificationFactors ...... 56 3.5.2 Backreaction on the Geometry: Mass andChargeLoss ...... 57 3.6 SuperradiancefromRotatingBlackHoles ...... 60 3.6.1 BosonicandFermionicFieldsintheKerrGeometry ...... 60 3.6.2 Energy Fluxes of Bosonic Fields at Infinity andontheHorizon...... 62 3.6.3 AmplificationFactors...... 64 3.6.4 DiracFieldsontheKerrGeometry ...... 65 3.6.5 LinearizedAnalysis:AnalyticVersusNumerics ...... 67 3.6.6 ScatteringofPlaneWaves...... 70 3.6.7 NonlinearSuperradiantScattering ...... 73 3.7 Boosted Black Strings: Ergoregions Without Superradiance ...... 74 3.8 SuperradianceinHigherDimensionalSpacetimes...... 77 3.9 Superradiance in Analogue Black Hole Geometries ...... 78 3.10 SuperradianceinNonasymptoticallyFlatSpacetimes...... 80 3.11 SuperradiancefromStars...... 81 3.12 Superradiance Beyond General Relativity...... 83 3.12.1 Superradiance of Black Holes Surrounded byMatterinScalar-TensorTheories...... 84 3.13 Microscopic Description of Superradiance and the Kerr/CFT Duality ...... 85 3.14 OpenIssues ...... 87 References...... 89 4 Black Holes and Superradiant Instabilities ...... 97 4.1 NoBlackHoleFissionProcesses...... 97 4.2 SpinningBlackHolesinConfiningGeometriesareUnstable...... 99 4.3 Superradiant Instabilities: Time-Domain Evolutions VersusanEigenvalueSearch...... 101 4.4 BlackHolesEnclosedina Mirror ...... 102 4.4.1 RotatingBlack-HoleBombs ...... 102 4.4.2 ChargedBlack-HoleBombs...... 105 4.5 Black Holes in AdS Backgrounds ...... 105 4.5.1 Instability of Small Kerr-AdS Black Holes andNewBHSolutions ...... 106 Contents ix

4.5.2 Charged AdS Black Holes: Spontaneous Symmetry Breaking and Holographic Superconductors ... 110 4.6 MassiveBosonicFields ...... 113 4.6.1 The Zoo of Light Bosonic Fields in Extensions oftheStandardModel ...... 114 4.6.2 MassiveScalarFields...... 116 4.6.3 MassiveVectorFields...... 119 4.6.4 MassiveTensorFields ...... 122 4.6.5 A Uniﬁed Picture of Superradiant Instabilities ofMassiveBosonicFields...... 124 4.7 BlackHolesImmersedina MagneticField...... 125 4.8 Superradiant Instability of Black Holes Surrounded by Conducting Rings ...... 127 4.9 NonminimalInteractions...... 128 4.9.1 Plasma-Triggered Superradiant Instabilities ...... 129 4.9.2 Spontaneous Superradiant Instabilities inScalar-TensorTheories...... 129 4.10 Kaluza-Klein Mass: Superradiant Instabilities in Higher Dimensions ...... 131 4.11 Ergoregion Instability ...... 132 4.11.1 Ergoregion Instability of Rotating Objects: A ConsistentApproach...... 133 4.11.2 Ergoregion Instability and Long-Lived Modes ...... 138 4.11.3 Ergoregion Instability in Fluids ...... 140 4.11.4 Ergoregion Instability and Hawking Radiation ...... 142 4.12 Black-Hole Lasers and Superluminal Corrections toHawkingRadiation ...... 143 4.13 Black Holes in Lorentz-Violating Theories: Nonlinear Instabilities ...... 144 4.14 OpenIssues ...... 144 References...... 147 5 Black Hole Superradiance in Astrophysics ...... 157 5.1 SuperradianceandRelativisticJets ...... 157 5.1.1 Blandford-ZnajekProcess...... 158 5.1.2 Blandford-Znajek Process and the Membrane Paradigm ...... 161 5.2 Superradiance, CFS Instability, and r-Modes of Spinning Stars..... 165 5.3 Evolution of Superradiant Instabilities: Gravitational-WaveEmissionandAccretion...... 168 5.3.1 Scalar Clouds Around Spinning Black Holes ...... 169 5.3.2 Gravitational-Wave Emission from the Bosonic Condensate ...... 170 5.3.3 GasAccretion...... 170 5.3.4 Growth and Decay of Bosonic Condensates Around Spinning Black Holes...... 172 x Contents

5.3.5 Superradiant Instabilities Imply No Highly-SpinningBlackHoles...... 174 5.3.6 Summary of the Evolution of Superradiant Instabilities ... 176 5.4 Astrophysical Black Holes as Particle Detectors...... 177 5.4.1 Bounds on the Mass of Bosonic Fields from GapsintheReggePlane...... 178 5.4.2 Gravitational-WaveSignaturesandBosenova...... 181 5.4.3 FloatingOrbits ...... 185 5.5 AreBlackHolesintheUniverseoftheKerrFamily? ...... 187 5.5.1 Circumventing the No-Hair Theorem with ComplexScalars ...... 188 5.5.2 Other Hairy Solutions and the Role of Tidal Dissipation ...... 190 5.5.3 Formation of Hairy Solutions and Bounds on BosonicFields ...... 191 5.6 PlasmaInteractions...... 192 5.7 IntrinsicLimitsonMagneticFields ...... 194 5.8 Phenomenology of the Ergoregion Instability...... 196 5.8.1 Ergoregion Instability of Ultracompact Stars ...... 197 5.8.2 Supporting the Black-Hole Paradigm: Instabilities of Black-Hole Mimickers ...... 198 5.9 OpenIssues ...... 202 References...... 204 6 Conclusions and Outlook ...... 213

A List of Publicly Available Codes ...... 215

B Analytic Computation of the Ampliﬁcation Coefﬁcients ...... 217

C Angular Momentum and Energy...... 221 C.1 Energy and Angular Momentum Fluxes at the Horizon ...... 222

D Electromagnetic Fluctuations Around a Rotating Black Hole Enclosed in a Mirror ...... 223

E Hartle-Thorne Formalism for Slowly-Rotating Spacetimes and Perturbations ...... 229 E.1 Background ...... 229 E.2 Perturbationsofa Slowly-RotatingObject...... 230 E.2.1 ScalarPerturbationsofa Slowly-RotatingStar...... 231

F WKB Analysis of Long-Lived and Unstable Modes of Ultracompact Objects ...... 233 References...... 236 Notation and Conventions

Unless otherwise and explicitly stated, we use geometrized units where G D c D 1, so that energy and time have units of length. We also adopt the .CCC:::/ convention for the metric. For reference, the following is a list of symbols that are used often throughout the text.

' Azimuthal coordinate # Angular coordinate m Azimuthal number with respect to the axis of rotation, jmjÄl l Integer angular number, related to the eigenvalue Alm D l.l C 1/ in four spacetime dimensions s Spin of the field ! Fourier transform variable. The time dependence of any field is ei!t For stable spacetimes, Im.!/ < 0 !R;!I Real and imaginary part of the QNM frequencies R; I Amplitude of reflected and incident waves, which characterize a wave- function ˆ Zslm Amplification factor of fluxes for a wave with spin s and harmonic indices 2 2 .l; m/. For scalar fields, Z0lm DjRj =jIj 1 with the asymptotic expansion at spatial infinity, ˆ Rei!t C Iei!t Occasionally, when clear from the context, we will omit the indices s and l and simply write Zm n Overtone numbers of the eigenfrequencies We conventionally start counting from a “fundamental mode” with n D 0 D Total number of spacetime dimensions (we always consider one timelike and D 1 spacelike dimensions)

xi xii Notation and Conventions

L Curvature radius of (A)dS spacetime, related to the negative cosmological constant ƒ in the Einstein equations (G C ƒg D 0) L2 D.D 2/.D 1/=.2ƒ/ is the curvature radius of AdS ( sign) or dS M Mass of the BH spacetime rC Radius of the BH event horizon in the chosen coordinates H Angular velocity of a zero-angular momentum observer at the BH horizon, as measured by a static observer at infinity S; V; T Mass parameter of the (scalar, vector or tensor) field In geometric units, the field mass is S;V;T „, respectively a Kerr rotation parameter: a D J=M 2 Œ0; M g˛ˇ Spacetime metric; Greek indices run from 0 to D 1 Ylm Spherical harmonics, orthonormal with respect to the integral over the 2-sphere Sslm Spin-weighted spheroidal harmonics Acronyms

ADM Arnowitt-Deser-Misner AGN Active Galactic Nucleus AdS Anti-de Sitter BH Black hole CFT Conformal ﬁeld theory GR General Relativity GW Gravitational Wave LIGO Laser Interferometric Gravitational Wave Observatory ODE Ordinary differential equation NS Neutron star PDE Partial differential equation QCD Quantum Chromodynamics QNM Quasinormal mode RN Reissner-Nordström ZAMO Zero Angular Momentum Observer

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