Gravitational Solitons

273 Pages·2001·1.287 MB·English

by  Belinski V., Verdaguer E.

GravitationalSolitons
This book gives a self-contained exposition of the theory of gravitational
solitons and provides a comprehensive review of exact soliton solutions to
Einstein’sequations.
The text begins with a detailed discussion of the extension of the inverse
scattering method to the theory of gravitation, starting with pure gravity and
thenextendingittothecouplingofgravitywiththeelectromagneticfield. There
followsasystematicreviewofthegravitationalsolitonsolutionsbasedontheir
symmetries. These solutions include some of the most interesting in gravita-
tional physics, such as those describing inhomogeneous cosmological models,
cylindrical waves, the collision of exact gravity waves, and the Schwarzschild
andKerrblackholes.
This work will equip the reader with the basic elements of the theory of
gravitational solitons as well as with a systematic collection of nontrivial
applicationsindifferentcontextsofgravitationalphysics. Itprovidesavaluable
referenceforresearchersandgraduatestudentsinthefieldsofgeneralrelativity,
string theory and cosmology, but will also be of interest to mathematical
physicistsingeneral.
VLADIMIR A. BELINSKI studied at the Landau Institute for Theoretical
Physics, where he completed his doctorate and worked until 1990. Currently
he is Research Supervisor by special appointment at the National Institute
for Nuclear Physics, Rome, specializing in general relativity, cosmology and
nonlinear physics. He is best known for two scientific results: firstly the
proof that there is an infinite curvature singularity in the general solution of
Einsteinequations,andthediscoveryofthechaoticoscillatorystructureofthis
singularity,knownastheBKLsingularity(1968–75withI.M.Khalatnikovand
E.M.Lifshitz),andsecondlytheformulationoftheinversescatteringmethodin
generalrelativityandthediscoveryofgravitationalsolitons(1977–82,withV.E.
Zakharov).
ENRIC VERDAGUER received his PhD in physics from the Autonomous
UniversityofBarcelonain1977,andhasheldaprofessorshipattheUniversity
ofBarclelonasince1993. Hespecializesingeneralrelativityandquantumfield
theory in curved spacetimes, and pioneered the use of the Belinski–Zakharov
inverse scattering method in different gravitational contexts, particularly in
cosmology,discoveringnewphysicalpropertiesingravitationalsolitons. Since
1991 his main research interest has been the interaction of quantum fields with
gravity. He has studied the consequences of this interaction in the collision
of exact gravity waves, in the evolution of cosmic strings and in cosmology.
More recently he has worked in the formulation and physical consequences of
stochasticsemi-classicalgravity.
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†Issuedasapaperback

Gravitational Solitons
V.Belinski
NationalInstituteforNuclearPhysics(INFN),Rome
E.Verdaguer
UniversityofBarcelona
         
The Pitt Building, Trumpington Street, Cambridge, United Kingdom
  
The Edinburgh Building, Cambridge CB2 2RU, UK
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http://www.cambridge.org
©V. Belinski and E. Verdaguer 2004
First published in printed format 2001
ISBN 0-511-04170-5 eBook (netLibrary)
ISBN 0-521-80586-4 hardback
Contents
Preface page x
1 Inversescatteringtechniqueingravity 1
1.1 OutlineoftheISM 1
1.1.1 Themethod 2
1.1.2 Generalizationandexamples 5
1.2 Theintegrableansatzingeneralrelativity 10
1.3 Theintegrationscheme 14
1.4 Constructionofthe n-solitonsolution 17
1.4.1 Thephysicalmetriccomponentsg 23
ab
1.4.2 Thephysicalmetriccomponent f 26
1.5 Multidimensionalspacetime 28
2 Generalpropertiesofgravitationalsolitons 37
2.1 Thesimpleanddoublesolitons 37
2.1.1 Polefusion 43
2.2 Diagonalbackgroundmetrics 45
2.3 Topologicalproperties 48
2.3.1 Gravisolitonsandantigravisolitons 53
2.3.2 Thegravibreathersolution 57
3 Einstein–Maxwellfields 60
3.1 TheEinstein–Maxwellfieldequations 60
3.2 ThespectralproblemforEinstein–Maxwellfields 65
3.3 Thecomponentsg andthepotentials A 69
ab a
3.3.1 The n-solitonsolutionofthespectralproblem 69
3.3.2 Thematrix X 74
3.3.3 Verificationoftheconstraints 76
3.3.4 Summaryofprescriptions 80
vii
viii Contents
3.4 Themetriccomponent f 82
3.5 Einstein–Maxwellbreathers 84
4 Cosmology: diagonalmetricsfromKasner 92
4.1 Anisotropicandinhomogeneouscosmologies 93
4.2 Kasnerbackground 95
4.3 Geometricalcharacterizationofdiagonalmetrics 96
4.3.1 RiemanntensorandPetrovclassification 97
4.3.2 Opticalscalars 99
4.3.3 Superenergytensor 100
4.4 Solitonsolutionsincanonicalcoordinates 101
4.4.1 Generalizedsolitonsolutions 103
4.5 Solutionswithrealpoles 106
4.5.1 GenerationofBianchimodelsfromKasner 109
4.5.2 Pulsewaves 110
4.5.3 Cosolitons 114
4.6 Solutionswithcomplexpoles 115
4.6.1 Compositeuniverses 118
4.6.2 Cosolitons 121
4.6.3 Solitoncollision 123
5 Cosmology: nondiagonalmetricsandperturbedFLRW 133
5.1 Nondiagonalmetrics 133
5.1.1 Solutionswithrealpoles 134
5.1.2 Solutionswithcomplexpoles 135
5.2 BianchiIIbackgrounds 140
5.3 Collisionofpulsewavesandsolitonwaves 142
5.4 SolitonsonFLRWbackgrounds 148
5.4.1 SolitonsonvacuumFLRWbackgrounds 149
5.4.2 SolitonswithastiffperfectfluidonFLRW 151
5.4.3 TheKaluza–Kleinansatzandtheorieswithscalarfields 158
5.4.4 Solitonsonradiative,andother,FLRWbackgrounds 161
6 Cylindricalsymmetry 169
6.1 Cylindricallysymmetricspacetimes 169
6.2 Einstein–Rosensolitonmetrics 172
6.2.1 Solutionswithonepole 174
6.2.2 Cosolitonswithonepole 174
6.2.3 Solitonswithtwooppositepoles 175
6.2.4 Cosolitonswithtwooppositepoles 177
6.3 TwopolarizationwavesandFaradayrotation 178
6.3.1 Onerealpole 180
6.3.2 Twocomplexpoles 181
Contents ix
6.3.3 Twodoublecomplexpoles 182
7 Planewavesandcollidingplanewaves 183
7.1 Overview 183
7.2 Planewaves 185
7.2.1 Theplane-wavespacetime 185
7.2.2 Focusingofgeodesics 187
7.2.3 Plane-wavesolitonsolutions 192
7.3 Collidingplanewaves 194
7.3.1 Thematchingconditions 194
7.3.2 Collinearpolarizationwaves: generalizedsolitonsolutions 197
7.3.3 Geometryofthecollidingwavesspacetime 202
7.3.4 Noncollinearpolarizationwaves: nondiagonalmetrics 208
8 Axialsymmetry 213
8.1 Theintegrationscheme 214
8.2 General n-solitonsolution 216
8.3 TheKerrandSchwarzschildmetrics 220
8.4 Asymptoticflatnessofthesolution 224
8.5 GeneralizedsolitonsolutionsoftheWeylclass 227
8.5.1 Generalizedone-solitonsolutions 230
8.5.2 Generalizedtwo-solitonsolutions 234
8.6 Tomimatsu–Satosolution 238
Bibliography 241
Index 253