LECTURES ON QUANTUM GRAVITY Series of the Centro de Estudios Cient´ıficos SeriesEditor: ClaudioTeitelboim Centro de Estudios Cient´ıficos Valdivia, Chile IONICCHANNELSINCELLSANDMODELSYSTEMS,1986 EditedbyRam´onLatorre PHYSICALPROPERTIESOFBIOLOGICALMEMBRANESAND THEIRFUNCTIONALIMPLICATIONS,1988 EditedbyCeciliaHidalgo PRINCIPLESOFSTRINGTHEORY,1988 LarsBrinkandMarcHenneaux QUANTUMMECHANICSOFFUNDAMENTALSYSTEMS1,1988 EditedbyClaudioTeitelboim QUANTUMMECHANICSOFFUNDAMENTALSYSTEMS2,1989 EditedbyClaudioTeitelboimandJorgeZanelli TRANSDUCTIONINBIOLOGICALSYSTEMS,1990 EditedbyCeciliaHidalgo,JuanBacigalupo,EnriqueJaimovich, andJulioVergara JOAQUINLUCO:DosHistoriasdeUnaVida,1991 AnaTironiandPedroLabarca,EdicionesPedag´ogicasChilenasS.A. QUANTIZATIONOFGAUGESYSTEMS,1992 MarcHenneauxandClaudioTeitelboim,PrincetonUniversityPress QUANTUMMECHANICSOFFUNDAMENTALSYSTEMS3,1992 EditedbyClaudioTeitelboimandJorgeZanelli FROMIONCHANNELSTOCELL-TO-CELLCONVERSATIONS,1997 EditedbyRam´onLatorre THEBLACKHOLE25YEARSAFTER,1998 EditedbyClaudioTeitelboimandJorgeZanelli,WorldScientific BLACKHOLESANDTHESTRUCTUREOFTHEUNIVERSE,2000 EditedbyClaudioTeitelboimandJorgeZanelli,WorldScientific THEPATAGONIANICEFIELDS:AUniqueNaturalLaboratoryfor EnvironmentalandClimateChangeStudies,2002 EditedbyGinoCasassa,FranciscoSepu´lvedaandRolfM.Sinclair LECTURESONQUANTUMGRAVITY,2004 EditedbyAndr´esGomberoffandDonaldMarolf PUMPS,TRANSPORTERSANDIONCHANNELS StudiesontheirStructure,FunctionandCellBiology,2004 EditedbyFranciscoV.Sepu´lvedaandFranciscoBezanilla LECTURES ON QUANTUM GRAVITY Editedby ANDRE´SGOMBEROFF CentrodeEstudiosCient´ıficos(CECS) DONALDMAROLF UniversityofCalifornia,SantaBarbara(UCSB) Andres Gomberoff Donald Marolf Centro de Estudios Cientificos Physics Department Avenida Arturo Prat 514 University of California Valdivia, Chile Santa Barbara, CA 93106 Series Editor: Claudio Teitelboim Centro de Estudios Cient´ıficos Valdivia, Chile Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. Cover image by Andres Gomberoff. Library of Congress Cataloging-in-Publication Data Lectures on quantum gravity / edited by Andrés Gomberoff, Donald Marolf. p. cm. – (Series of the Centro de Estudios Científicos) Includes bibliographical references and index. ISBN 0-387-23995-2 1. Quantum gravity. I. Gomberoff, Andrés. II. Marolf, Donald. III. Series. QC178.L38 2005 530.14'3–dc22 2004065095 ISBN-10: 0-387-23995-2 e-ISBN 0-387-24992-3 Printed on acid-free paper. ISBN-13: 978-0387-23995-2 ISSN: 1571-571X (cid:164) 2005 Springer Science+Business Media, Inc. All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, Inc., 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now know or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if the are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed in the United States of America. (SBA) 9 8 7 6 5 4 3 2 1 SPIN 11329367 springeronline.com Foreword TheCentrodeEstudiosCientificos(CECS)begananewphaseofitsexistence attheendof1999,whenitmovedtothecityofValdivia,800kilometersSouth of the capital of Chile, Santiago. The letter “S", which stood for Santiago in the original acronym has been maintained to provide a sense of historical continuity,anditisnow-whennecessary-explainedasarisingfromtheplural inthewordCientíficos. Valdivia used to be part of the “frontier" in the early days of the country andonestillbreathesfrontierairinit. ThisfrontierairhasinspiredtheCenter toundertakenewandbolderchallengesinscienceandexploration,suchasan unprecedented airborne exploration of the Amundsen Sea in West Antarctica andthedevelopmentofastateoftheartTransgenicFacility. However, in the midst of all this excitement and frenetic activity, we were distinctlyremindedbythephysicistswhocametoValdiviafrommanycountries to takepart in the School of Quantum Gravitythat, as Richard Feynman used to say: “there is nothing better in life than eating cookies and talking about Physics". ClaudioTeitelboim Director,CentrodeEstudiosCientíficos Valdivia,April2004. v Contents ContributingAuthors xi Preface xiii TheThermodynamicsofBlackHoles 1 RobertM.Wald 1 Introduction 2 2 ClassicalBlackHoleThermodynamics 4 3 HawkingRadiation 10 4 TheGeneralizedSecondLaw(GSL) 15 4.1 ArgumentsforthevalidityoftheGSL 15 4.2 Entropybounds 18 5 CalculationsofBlackHoleEntropy 23 6 OpenIssues 28 6.1 Does a pure quantum state evolve to a mixed state in the processofblackholeformationandevaporation? 28 6.2 What(andwhere)arethedegreesoffreedomresponsiblefor blackholeentropy? 30 IntroductiontoQuantumFieldsinCurvedSpacetimeandtheHawkingEffect 39 TedJacobson 1 Introduction 39 2 Plancklengthandblackholethermodynamics 40 2.1 Plancklength 41 2.2 Hawkingeffect 41 2.3 Blackholeentropy 42 3 Harmonicoscillator 43 4 Quantumscalarfieldincurvedspacetime 46 4.1 Conformalcoupling 47 4.2 Canonicalquantization 48 4.3 Hilbertspace 49 4.4 Flatspacetime 50 4.5 Curvedspacetime,“particles",andstresstensor 51 4.6 Remarks 53 4.6.1 Continuumnormalizationofmodes 53 vii viii LECTURESONQUANTUMGRAVITY 4.6.2 Masslessminimallycoupledzeromode 53 5 Particlecreation 54 5.1 Parametricexcitationofaharmonicoscillator 54 5.1.1 Adiabatictransitionsandgroundstate 55 5.1.2 Suddentransitions 56 5.1.3 Relation between in and out ground states & the squeeze operator 56 5.2 Cosmologicalparticlecreation 58 5.3 Remarks 61 5.3.1 Momentumcorrelationsinthesqueezedvacuum 61 5.3.2 Normalizationofthesqueezedvacuum 61 5.3.3 Energydensity 61 5.3.4 Adiabaticvacuum 61 5.4 deSitterspace 61 5.4.1 Primordialperturbationsfromzeropointfluctuations 62 6 Blackholeevaporation 63 6.1 Historicalsketch 64 6.2 TheHawkingeffect 65 6.2.1 Averagenumberofoutgoingparticles 65 6.2.2 Normofthenegativefrequencypart&thermalflux 69 6.2.3 Thequantumstate 72 6.3 Remarks 73 6.3.1 Localtemperature 73 6.3.2 Equilibriumstate: Hartle-Hawkingvacuum 73 6.3.3 Stimulatedemission 73 6.3.4 Unruheffect 74 6.3.5 Rotatingblackhole 74 6.3.6 deSitterspace 75 6.3.7 Higherspinfields 75 6.3.8 Interactingfields 75 6.3.9 Stress-energytensor 75 6.3.10 Back-reaction 76 6.3.11 Statisticalentropy 76 6.3.12 Informationloss 77 6.3.13 Roleoftheblackholecollapse 78 7 Thetrans-Planckianquestion 79 7.1 Stringtheoryviewpoint 80 7.2 Condensedmatteranalogy 80 7.3 Hawkingeffectonafallinglattice 81 7.4 Remarks 84 7.4.1 FiniteEntanglemententropy 84 7.4.2 StimulatedemissionofHawkingradiationatlatetimes 84 7.4.3 Latticetimedependenceandgeometryfluctuations 85 7.5 Trans-Planckianquestionincosmology 85 LargeN fieldtheoriesandgravity 91 JuanMaldacena 1 GeneralIntroduction 91 2 TheCorrespondence 96 Contents ix 3 TestsoftheAdS/CFTCorrespondence 106 3.1.1 TheFieldTheorySpectrum 107 3.1.2 TheStringTheorySpectrumandtheMatching 113 4 CorrelationFunctions 118 5 WilsonLoops 128 6 TheoriesatFiniteTemperature 135 LecturesonD-branes,tachyoncondensation,andstringfieldtheory 151 WashingtonTaylor 1 Introduction 151 2 D-branes 157 3 TachyonsandD-branes 164 4 OpenstringfieldtheoryandtheSenconjectures 168 5 BasicsofSFT 176 6 EvidencefortheSenconjectures 184 7 Furtherdevelopments 193 8 Conclusionsandopenproblems 196 Billiarddynamics 207 ThibaultDamour,MarcHenneaux,HermannNicolai 1 Introduction 207 2 Kasnersolution–Diagonalcase 211 2.2.1 SupermetricandHamiltonian 213 2.2.2 Hyperbolicspace 214 3 Kasnersolution–Non-diagonalcase 216 3.2.1 BKLlimit 218 3.2.2 Firstencounterwithbilliards 219 4 Asymptoticdynamicsinthegeneralcase-Gravitationalbilliards 222 4.3.1 Computationofcurvature 225 4.3.2 BKLlimit 227 4.3.3 Remarks 228 4.4.1 Electricwalls 229 4.4.2 Magneticwalls 230 5 Velocity-dominance-Strongcoupling/smalltensionlimit 234 6 MiscellanyandConclusions 235 Appendix: A.Iwasawadecompositionandasymptoticsofnon-diagonal3d Kasnermetric 237 Appendix: B.Freezingtheoff-diagonalvariables: atoymodel 240 Appendix: C.KasnerframeversusIwasawaframes 241 Appendix: D.Hamiltonianreduction 241 TalltalesfromdeSitterspace 249 RobertC.Myers 1 Prologue 249 2 DeSitterspacebasics 251 3 TheHuntforΛ-N 255 x LECTURESONQUANTUMGRAVITY 4 IntroductiontothedS/CFTcorrespondence 262 5 AgeneralizeddeSitterc-theorem 270 6 Theglobalperspective 274 7 CFTontwoboundaries 280 8 DiscussionofdS/CFT 288 Appendix: ScalarfieldmodesindSspace 292 CausalSets: DiscreteGravity 305 RafaelD.Sorkin 1 Introduction 305 2 Originsofthecausetidea 306 3 Whatisacausalset? 309 4 Causalsetkinematicsingeneral 311 5 “Howbig”isacausetelement? 311 6 ThereconstructionofM4 312 7 Sprinkling,coarse-graining,andthe“Hauptvermutung” 312 8 Dimensionandlength 314 9 Alengthestimator 314 10 Dynamics 315 11 Fluctuationsinthecosmologicalconstant 317 12 Linksacrossthehorizon 318 13 Whatarethe“observables”ofquantumgravity? 319 14 Howthelargenumbersofcosmologymightbeunderstood: a“Tolman- Boltzmann”cosmology 320 15 Fieldsonabackgroundcauset 320 16 Topologychange 320 ThermalDecayoftheCosmologicalConstantintoBlackHoles 329 Andre´sGomberoff,MarcHenneaux,ClaudioTeitelboim,FrankWilczek 1 Introduction 330 2 TheThermalon 332 3 Lorentziancontinuation 336 4 ActionandProbability 337 5 Nariaithreshold 340 6 DecayofdeSitterspacethroughthecosmologicalthermalon 341 7 Canthethermalonaccountforthesmallpresentvalueofthecosmo- logicalterm? 345 8 Conclusions 346 Appendix: A.Instabilityofthermalons 347 Appendix: B.GravitationEssentialforExistenceofCosmologicalThermalons.348 Contributing Authors ThibaultDamourInstitutdesHautesE´tudesScientifiques(IHE´S). Andre´sGomberoffCentrodeEstudiosCient´ıficos(CECS). MarcHenneauxUniversite´ LibredeBruxelles. TedJacobsonUniversityofMaryland. JuanMaldacenaInstituteforAdvancedStudy. Robert C. Myers Perimeter Institute for Theoretical Physics, University of WaterlooandMcGillUniversity. HermannNicolaiMax-Planck-Institutfu¨rGravitationsphysik,Albert-Einstein- Institut. RafaelD.SorkinSyracuseUniversity. WashingtonTaylorCenterforTheoreticalPhysics,MassachusettsInstituteof Technology(MIT). ClaudioTeitelboimCentrodeEstudiosCient´ıficos(CECS). RobertM.WaldUniversityofChicago. FrankWilczekCenterforTheoreticalPhysics,MassachusettsInstituteofTech- nology(MIT). xi