Quantum aspects of black holes

331 Pages·2015·4.81 MB·English

by  Calmet, Xavier (eds)

Fundamental Theories of Physics 178
Xavier Calmet Editor
Quantum
Aspects of
Black Holes
Fundamental Theories of Physics
Volume 178
Series editors
Henk van Beijeren
Philippe Blanchard
Paul Busch
Bob Coecke
Dennis Dieks
Detlef Dürr
Roman Frigg
Christopher Fuchs
Giancarlo Ghirardi
Domenico J.W. Giulini
Gregg Jaeger
Claus Kiefer
Nicolaas P. Landsman
Christian Maes
Hermann Nicolai
Vesselin Petkov
Alwyn van der Merwe
Rainer Verch
R.F. Werner
Christian Wuthrich
More information about this series at http://www.springer.com/series/6001
Xavier Calmet
Editor
Quantum Aspects
of Black Holes
123
Editor
XavierCalmet
Department of Physicsand Astronomy
Universityof Sussex
Brighton
UK
ISBN 978-3-319-10851-3 ISBN 978-3-319-10852-0 (eBook)
DOI 10.1007/978-3-319-10852-0
LibraryofCongressControlNumber:2014951685
SpringerChamHeidelbergNewYorkDordrechtLondon
©SpringerInternationalPublishingSwitzerland2015
Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof
the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,
recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionor
informationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar
methodology now known or hereafter developed. Exempted from this legal reservation are brief
excerpts in connection with reviews or scholarly analysis or material supplied specifically for the
purposeofbeingenteredandexecutedonacomputersystem,forexclusiveusebythepurchaserofthe
work. Duplication of this publication or parts thereof is permitted only under the provisions of
theCopyrightLawofthePublisher’slocation,initscurrentversion,andpermissionforusemustalways
beobtainedfromSpringer.PermissionsforusemaybeobtainedthroughRightsLinkattheCopyright
ClearanceCenter.ViolationsareliabletoprosecutionundertherespectiveCopyrightLaw.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this
publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexempt
fromtherelevantprotectivelawsandregulationsandthereforefreeforgeneraluse.
While the advice and information in this book are believed to be true and accurate at the date of
publication,neithertheauthorsnortheeditorsnorthepublishercanacceptanylegalresponsibilityfor
anyerrorsoromissionsthatmaybemade.Thepublishermakesnowarranty,expressorimplied,with
respecttothematerialcontainedherein.
Printedonacid-freepaper
SpringerispartofSpringerScience+BusinessMedia(www.springer.com)
Preface
The decision to write this book arose in discussions among members of the
WorkingGroup1(WG1)oftheEuropeanCooperationinScienceandTechnology
(COST)actionMP0905“BlackHolesinaViolentUniverse,”whichstartedin2010
and ended in May 2014.
The four years of the action have been absolutely fantastic for the research
themes represented by WG1. The discovery of the Higgs boson which completes
thestandardmodel ofparticle physicswas crowned bythe2013 Nobelprize.This
discoveryhasimportantimplicationsfortheunificationofthestandardmodelwith
general relativity which is important for Planck size black holes. Understanding at
what energy scale these forces merge into a unified theory, will tell us what is the
lightest possible mass for a black hole. In other words, the Large Hadron Collider
(LHC) at CERN data allows us to set bounds on the Planck scale. We now know
thatthePlanck scale isabove5TeV.Thus,Planckian black holes areheavierthan
5TeV.ThefactthatnodarkmatterhasbeendiscoveredattheLHCintheformofa
newparticlestrengthenstheassumptionthatprimordialblackholescouldplaythat
role.
The data from the Planck satellite reinforce the need for inflation. Planckian
black holes can make an important contribution at the earliest moment of our
universe, namely during inflation if the scale at which inflation took place is close
enough to the Planck scale. There have been several interesting proposals relating
theHiggsbosonofthestandardmodelofparticlephysicswithinflation.Indeed,the
LHC data imply that the Higgs boson could be the inflation if the Higgs boson is
non-minimally coupled to space-time curvature.
In relation to the black hole information paradox, there has been much excite-
mentaboutfirewallsorwhathappenswhenanobserverfallsthroughthehorizonof
ablackhole.However,firewallsrelyonatheorembyBanks,SusskindandPeskin
[Nucl. Phys. B244 (1984) 125] for which there are known counter examples as
shownin1995byWaldandUnruh[Phys.Rev.D52(1995)2176–2182].Itwillbe
interesting to see how the situation evolves in the next few years.
v
vi Preface
These then are the reasons for writing this book, which reflects on the progress
madeinrecentyearsinafieldwhichisstilldevelopingrapidly.Aswellassomeof
themembers ofour working group, severalother international expertshave kindly
agreed to contribute to the book. The result is a collection of 10 chapters dealing
with different aspects of quantum effects in black holes. By quantum effects we
mean both quantum mechanical effects such as Hawking radiation and quantum
gravitational effects such as Planck size quantum black hole.
Chapter1ismeanttoprovideabroadintroductiontothefieldofquantumeffects
inblackholesbeforefocusingonPlanckianquantumblackholes.Chapter2covers
thethermodynamicsofblackholeswhileChap.3dealswiththefamousinformation
paradox.Chapter4discussesanothertypeofobject,so-calledmonsters,whichhave
more entropythan black holes ofequal mass. Primordialblack holesare discussed
in Chaps. 5 and 6 reviews self-gravitating Bose-Einstein condensates which open
up the exciting possibility that black holes are Bose-Einstein condensates. The
formation of black holes in supersymmetric theories is investigated in Chap. 7.
Chapter 8 covers Hawking radiation in higher dimensional black holes. Chapter 9
presentsthelatestboundsonthemassofsmallblackholeswhichcouldhavebeen
producedattheLHC.Lastbutnotleast,Chap.10coversnon-minimallengtheffects
inblack holes.Allchaptershave been througha strict reviewingprocess.
This book would not have been possible without the COST action MP0905. In
particularwewouldliketothankSilkeBritzen,thechairofouraction,themembers
of the core group (Antxon Alberdi, Andreas Eckart, Robert Ferdman, Karl-Heinz
Mack, Iossif Papadakis, Eduardo Ros, Anthony Rushton, Merja Tornikoski and
Ulrike Wyputta in addition to myself) and all the members of this action for
fascinating meetings and conferences. We are very grateful to Dr. Angela Lahee,
ourcontactatSpringer,forherconstantsupportduringthecompletionofthisbook.
Brighton, August 2014 Xavier Calmet
Contents
1 Fundamental Physics with Black Holes . . . . . . . . . . . . . . . . . . . . 1
Xavier Calmet
1.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Quantum Black Holes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Low Scale Quantum Gravity and Black Holes at Colliders . . . . 5
1.4 An Effective Theory for Quantum Gravity. . . . . . . . . . . . . . . . 11
1.5 Quantum Black Holes in Loops . . . . . . . . . . . . . . . . . . . . . . . 13
1.6 Quantum Black Holes and the Unification
of General Relativity and Quantum Mechanics. . . . . . . . . . . . . 16
1.7 Quantum Black Holes, Causality and Locality . . . . . . . . . . . . . 20
1.8 Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2 Black Holes and Thermodynamics: The First Half Century . . . . . 27
Daniel Grumiller, Robert McNees and Jakob Salzer
2.1 Introduction and Prehistory . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.2 1963–1973 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.3 1973–1983 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.4 1983–1993 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.5 1993–2003 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.6 2003–2013 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.7 Conclusions and Future. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3 The Firewall Phenomenon. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
R.B. Mann
3.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.2 Black Holes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.2.1 Gravitational Collapse. . . . . . . . . . . . . . . . . . . . . . . . 75
3.2.2 Anti de Sitter Black Holes. . . . . . . . . . . . . . . . . . . . . 77
3.3 Black Hole Thermodynamics. . . . . . . . . . . . . . . . . . . . . . . . . 78
vii
viii Contents
3.4 Black Hole Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
3.4.1 Quantum Field Theory in Curved Spacetime . . . . . . . . 80
3.4.2 Pair Creation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.5 The Information Paradox. . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
3.5.1 Implications of the Information Paradox . . . . . . . . . . . 94
3.5.2 Complementarity . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
3.6 Firewalls. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
3.6.1 The Firewall Argument. . . . . . . . . . . . . . . . . . . . . . . 98
3.6.2 Responses to the Firewall Argument. . . . . . . . . . . . . . 100
3.7 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
4 Monsters, Black Holes and Entropy. . . . . . . . . . . . . . . . . . . . . . . 115
Stephen D.H. Hsu
4.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
4.2 What is Entropy?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
4.3 Constructing Monsters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
4.3.1 Monsters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
4.3.2 Kruskal–FRW Gluing. . . . . . . . . . . . . . . . . . . . . . . . 120
4.4 Evolution and Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . 123
4.5 Quantum Foundations of Statistical Mechanics. . . . . . . . . . . . . 124
4.6 Statistical Mechanics of Gravity? . . . . . . . . . . . . . . . . . . . . . . 126
4.7 Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
5 Primordial Black Holes: Sirens of the Early Universe. . . . . . . . . . 129
Anne M. Green
5.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
5.2 PBH Formation Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . 130
5.2.1 Large Density Fluctuations . . . . . . . . . . . . . . . . . . . . 130
5.2.2 Cosmic String Loops . . . . . . . . . . . . . . . . . . . . . . . . 132
5.2.3 Bubble Collisions. . . . . . . . . . . . . . . . . . . . . . . . . . . 132
5.3 PBH Abundance Constraints . . . . . . . . . . . . . . . . . . . . . . . . . 133
5.3.1 Evaporation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
5.3.2 Lensing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
5.3.3 Dynamical Effects . . . . . . . . . . . . . . . . . . . . . . . . . . 136
5.3.4 Other Astrophysical Objects and Processes . . . . . . . . . 137
5.4 Constraints on the Primordial Power Spectrum and Inflation . . . 138
5.4.1 Translating Limits on the PBH Abundance
into Constraints on the Primordial Power Spectrum. . . . 139
5.4.2 Constraints on Inflation Models . . . . . . . . . . . . . . . . . 141
5.5 PBHs as Dark Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
5.6 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
Contents ix
6 Self-gravitating Bose-Einstein Condensates. . . . . . . . . . . . . . . . . . 151
Pierre-Henri Chavanis
6.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
6.2 Self-gravitating Bose-Einstein Condensates . . . . . . . . . . . . . . . 155
6.2.1 The Gross-Pitaevskii-Poisson System . . . . . . . . . . . . . 155
6.2.2 Madelung Transformation . . . . . . . . . . . . . . . . . . . . . 156
6.2.3 Time-Independent GP Equation . . . . . . . . . . . . . . . . . 158
6.2.4 Hydrostatic Equilibrium . . . . . . . . . . . . . . . . . . . . . . 158
6.2.5 The Non-interacting Case . . . . . . . . . . . . . . . . . . . . . 159
6.2.6 The Thomas-Fermi Approximation. . . . . . . . . . . . . . . 160
6.2.7 Validity of the Thomas-Fermi Approximation . . . . . . . 161
6.2.8 The Total Energy. . . . . . . . . . . . . . . . . . . . . . . . . . . 162
6.2.9 The Virial Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 163
6.3 The Gaussian Ansatz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
6.3.1 The Total Energy. . . . . . . . . . . . . . . . . . . . . . . . . . . 164
6.3.2 The Mass-Radius Relation. . . . . . . . . . . . . . . . . . . . . 164
6.3.3 The Virial Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 169
6.3.4 The Pulsation Equation. . . . . . . . . . . . . . . . . . . . . . . 169
6.4 Application of Newtonian Self-gravitating BECs
to Dark Matter Halos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
6.4.1 The Non-interacting Case . . . . . . . . . . . . . . . . . . . . . 170
6.4.2 The Thomas-Fermi Approximation. . . . . . . . . . . . . . . 170
6.4.3 Validity of the Thomas-Fermi Approximation . . . . . . . 172
6.4.4 The Case of Attractive Self-interactions. . . . . . . . . . . . 172
6.5 Application of General Relativistic BECs to Neutron Stars,
Dark Matter Stars, and Black Holes . . . . . . . . . . . . . . . . . . . . 173
6.5.1 Non-interacting Boson Stars. . . . . . . . . . . . . . . . . . . . 174
6.5.2 The Thomas-Fermi Approximation for Boson Stars . . . 175
6.5.3 Validity of the Thomas-Fermi Approximation . . . . . . . 177
6.5.4 An Interpolation Formula Between
the Non-interacting Case and the TF
Approximation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
6.5.5 Application to Supermassive Black Holes . . . . . . . . . . 178
6.5.6 Application to Neutron Stars and Dark Matter Stars . . . 179
6.5.7 Are Microscopic Quantum Black Holes
Bose-Einstein Condensates of Gravitons? . . . . . . . . . . 180
6.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
6.7 Self-interaction Constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
6.8 Conservation of Energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
6.9 Virial Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
6.10 Stress Tensor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
6.11 Lagrangian and Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . 189
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191