Lectures on Quantum Gravity

356 Pages·2005·1.497 MB·English

by  Andres Gomberoff, Donald Marolf

LECTURES ON QUANTUM GRAVITY
Series of the Centro de Estudios Cient´ıficos
SeriesEditor: ClaudioTeitelboim
Centro de Estudios Cient´ıficos
Valdivia, Chile
IONICCHANNELSINCELLSANDMODELSYSTEMS,1986
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EditedbyAndr´esGomberoffandDonaldMarolf
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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)
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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