THE MATHEMATICALTHEORY OFCOSMIC STRINGS COSMICSTRINGSINTHEWIREAPPROXIMATION Series inHighEnergy Physics,CosmologyandGravitation Otherbooksintheseries Electron–PositronPhysicsattheZ MGGreen,SLLloyd,PNRatoffandDRWard Non-acceleratorParticlePhysics Paperbackedition HVKlapdor-KleingrothausandAStaudt IdeasandMethodsofSupersymmetryandSupergravity orAWalkThroughSuperspace Revisededition ILBuchbinderandSMKuzenko PulsarsasAstrophysicalLaboratoriesforNuclearandParticlePhysics FWeber ClassicalandQuantumBlackHoles EditedbyPFre´,VGorini,GMagliandUMoschella ParticleAstrophysics Revisedpaperbackedition HVKlapdor-KleingrothausandKZuber TheWorldinElevenDimensions Supergravity,SupermembranesandM-Theory EditedbyMJDuff GravitationalWaves EditedbyICiufolini,VGorini,UMoschellaandPFre´ ModernCosmology EditedbySBonometto,VGoriniandUMoschella GeometryandPhysicsofBranes EditedbyUBruzzo,VGoriniandUMoschella TheGalacticBlackHole LecturesonGeneralRelativityandAstrophysics EditedbyHFalckeandFWHehl THE MATHEMATICAL THEORY OF COSMIC STRINGS COSMIC STRINGSIN THEWIREAPPROXIMATION Malcolm R Anderson Departmentof Mathematics, UniversitiBrunei,Darussalam INSTITUTE OF PHYSICS PUBLISHING BRISTOL AND PHILADELPHIA c IOPPublishingLtd2003 (cid:1) All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical,photocopying,recordingorotherwise,withoutthepriorpermission of the publisher. Multiple copying is permitted in accordance with the terms of licences issued by the Copyright Licensing Agency under the terms of its agreementwithUniversitiesUK(UUK). BritishLibraryCataloguing-in-PublicationData AcataloguerecordforthisbookisavailablefromtheBritishLibrary. ISBN0750301600 LibraryofCongressCataloging-in-PublicationDataareavailable CommissioningEditor:JamesRevill ProductionEditor:SimonLaurenson ProductionControl:SarahPlenty CoverDesign:VictoriaLeBillon Marketing:NicolaNeweyandVerityCooke Published by Institute of Physics Publishing, wholly owned by The Institute of Physics,London InstituteofPhysicsPublishing,DiracHouse,TempleBack,BristolBS16BE,UK US Office: Institute of Physics Publishing, The Public Ledger Building, Suite 929,150SouthIndependenceMallWest,Philadelphia,PA19106,USA TypesetinLATEX2 byText2Text,Torquay,Devon PrintedintheUK(cid:1)byMPGBooksLtd,Bodmin,Cornwall Contents Introduction ix 1 Cosmicstringsandbrokengaugesymmetries 1 1.1 Electromagnetismasalocalgaugetheory 3 1.2 Electroweakunification 8 1.3 TheNielsen–Olesenvortexstring 15 1.4 StringsasrelicsoftheBigBang 24 1.5 TheNambuaction 27 2 Theelementsofstringdynamics 35 2.1 Describingazero-thicknesscosmicstring 35 2.2 Theequationofmotion 38 2.3 Gaugeconditions,periodicityandcausalstructure 41 2.4 Conservationlawsinsymmetricspacetimes 44 2.5 Invariantlength 48 2.6 Cuspsandcurvaturesingularities 49 2.7 Intercommutingandkinks 54 3 Stringdynamicsinflatspace 59 3.1 Thealignedstandardgauge 59 3.2 TheGGRTgauge 61 3.3 Conservationlawsinflatspace 63 3.4 Initial-valueformulationforastringloop 68 3.5 Periodicsolutionsinthespinorrepresentation 70 3.6 TheKibble–Turoksphereandcuspsandkinksinflatspace 73 3.7 Fieldreconnectionatacusp 80 3.8 Self-intersectionofastringloop 85 3.9 Secularevolutionofastringloop 92 4 Abestiaryofexactsolutions 99 4.1 Infinitestrings 99 4.1.1 Theinfinitestraightstring 99 4.1.2 Travelling-wavesolutions 100 4.1.3 Stringswithpairedkinks 102 4.1.4 Helicalstrings 103 vi Contents 4.2 Somesimpleplanarloops 105 4.2.1 Thecollapsingcircularloop 105 4.2.2 Thedoubledrotatingrod 106 4.2.3 Thedegeneratekinkedcusplessloop 107 4.2.4 Cat’s-eyestrings 108 4.3 Balloonstrings 112 4.4 Harmonicloopsolutions 114 4.4.1 Loopswithoneharmonic 114 4.4.2 Loopswithtwounmixedharmonics 117 4.4.3 Loopswithtwomixedharmonics 122 4.4.4 Loopswiththreeormoreharmonics 127 4.5 Stationaryrotatingsolutions 130 4.6 Threetoysolutions 135 4.6.1 Theteardropstring 135 4.6.2 Thecardioidstring 137 4.6.3 Thefigure-of-eightstring 141 5 Stringdynamicsinnon-flatbackgrounds 144 5.1 StringsinRobertson–Walkerspacetimes 144 5.1.1 Straightstringsolutions 145 5.1.2 Ringsolutions 147 5.2 StringsnearaSchwarzschildblackhole 152 5.2.1 Ringsolutions 153 5.2.2 Staticequilibriumsolutions 157 5.3 ScatteringandcaptureofastraightstringbyaSchwarzschildhole 159 5.4 RingsolutionsintheKerrmetric 167 5.5 StaticequilibriumconfigurationsintheKerrmetric 170 5.6 Stringsinplane-fronted-wavespacetimes 177 6 Cosmicstringsintheweak-fieldapproximation 181 6.1 Theweak-fieldformalism 182 6.2 Cuspsintheweak-fieldapproximation 185 6.3 Kinksintheweak-fieldapproximation 189 6.4 Radiationofgravitationalenergyfromaloop 191 6.5 Calculationsofradiatedpower 196 6.5.1 Powerfromcusplessloops 197 6.5.2 PowerfromtheVachaspati–Vilenkinloops 199 6.5.3 Powerfromthe p/q harmonicsolutions 202 6.6 Powerradiatedbyahelicalstring 204 6.7 Radiationfromlongstrings 208 6.8 Radiationoflinearandangularmomentum 211 6.8.1 Linearmomentum 211 6.8.2 Angularmomentum 213 6.9 Radiativeefficienciesfrompiecewise-linearloops 219 6.9.1 Thepiecewise-linearapproximation 219 Contents vii 6.9.2 Aminimumradiativeefficiency? 223 6.10 Thefieldofacollapsingcircularloop 226 6.11 Theback-reactionproblem 231 6.11.1 Generalfeaturesoftheproblem 231 6.11.2 Self-accelerationofacosmicstring 234 6.11.3 Back-reactionandcuspdisplacement 240 6.11.4 Numericalresults 242 7 Thegravitationalfieldofaninfinitestraightstring 246 7.1 Themetricduetoaninfinitestraightstring 246 7.2 Propertiesofthestraight-stringmetric 250 7.3 TheGeroch–Traschencritique 252 7.4 Isthestraight-stringmetricunstabletochangesintheequationof state? 255 7.5 Adistributionaldescriptionofthestraight-stringmetric 259 7.6 Theself-forceonamassiveparticlenearastraightstring 263 7.7 Thestraight-stringmetricin‘asymptotically-flat’form 267 8 Multiplestraightstringsandclosedtimelikecurves 271 8.1 Straightstringsand2+1gravity 271 8.2 Boostsandrotationsofsystemsofstraightstrings 273 8.3 TheGottconstruction 274 8.4 Stringholonomyandclosedtimelikecurves 278 8.5 TheLetelier–Gal’tsovspacetime 282 9 Otherexactstringmetrics 286 9.1 Stringsandtravellingwaves 286 9.2 Stringsfromaxisymmetricspacetimes 291 9.2.1 StringsinaRobertson–Walkeruniverse 292 9.2.2 AstringthroughaSchwarzschildblackhole 297 9.2.3 Stringscoupledtoacosmologicalconstant 301 9.3 Stringsinradiatingcylindricalspacetimes 303 9.3.1 Thecylindricalformalism 303 9.3.2 Separablesolutions 305 9.3.3 Stringsincloseduniverses 307 9.3.4 Radiatingstringsfromaxisymmetricspacetimes 310 9.3.5 Einstein–Rosensolitonwaves 316 9.3.6 Two-modesolitonsolutions 321 9.4 Snappingcosmicstrings 324 9.4.1 Snappingstringsinflatspacetimes 324 9.4.2 Otherspacetimescontainingsnappingstrings 329 viii Contents 10 Strong-fieldeffectsofzero-thicknessstrings 332 10.1 Spatialgeometryoutsideastationaryloop 334 10.2 Black-holeformationfromacollapsingloop 340 10.3 Propertiesoftheneargravitationalfieldofacosmicstring 343 10.4 A3+1splitofthemetricnearacosmicstring 346 10.4.1 Generalformalism 346 10.4.2 Somesamplenear-fieldexpansions 349 10.4.3 Seriessolutionsofthenear-fieldvacuumEinstein equations 352 10.4.4 Distributionalstress–energyoftheworldsheet 355 Bibliography 359 Index 367 Introduction Theexistenceofcosmicstringswasfirstproposedin1976byTomKibble,who drew on the theory of line vortices in superconductorsto predict the formation of similar structures in the Universe at large as it expanded and cooled during the earlyphasesofthe Big Bang. The criticalassumptionis thatthestrongand electroweak forces were first isolated by a symmetry-breaking phase transition which converted the energy of the Higgs field into the masses of fermions and vector bosons. Under certain conditions, it is possible that some of the Higgs field energy remained in thin tubes which stretched across the early Universe. Thesearecosmicstrings. Themassesanddimensionsofcosmicstringsarelargelydeterminedbythe energyscaleatwhichtherelevantphasetransitionoccurred.Thegrandunification (GUT)energyscaleisatpresentestimatedtobeabout1015GeV,whichindicates thattheGUTphasetransitiontookplacesome10−37–10−35saftertheBigBang, whenthetemperatureoftheUniversewasoftheorderof1028K.Thethicknessof acosmicstringistypicallycomparabletotheComptonwavelengthofaparticle with GUT mass or about 10−29 cm. This distance is so much smaller than the length scales important to astrophysics and cosmology that cosmic strings are usuallyidealizedtohavezerothickness. The mass per unit length of such a string, conventionally denoted µ, is proportionalto the square of the energyscale, and in the GUT case hasa value of about 1021 g cm−1. There is no restriction on the length of a cosmic string, although in the simplest theories a string can have no free ends and so must either be infinite or form a closed loop. A GUT string long enough to cross theobservableUniversewouldhaveamasswithinthehorizonofabout1016 M(cid:1), whichisnogreaterthanthemassofalargeclusterofgalaxies. Interest in cosmic strings intensified in 1980–81, when Yakov Zel’dovich and Alexander Vilenkin independently showed that the density perturbations generated in the protogalactic medium by GUT strings would have been large enoughtoaccountfortheformationofgalaxies.Galaxyformationwasthen(and remainsnow)oneofthemostvexingunsolvedproblemsfacingcosmologists.The extremeisotropyofthemicrowavebackgroundindicatesthattheearlyUniverse was very smooth. Yet structure has somehow developed on all scales from the planets to clusters and superclusters of galaxies. Such structure cannot be ix