Unstable even-parity eigenmodes of the regular sta

arXiv:gr-qc/0408060v118Aug2004Unstableeven-parityeigenmodesoftheregularstaticSU(2)Yang-Mills-dilatonsolutionsO.I.Streltsova∗,E.E.Donets,E.A.Hayryan,D.A.Georgieva,andT.L.BoyadjievJointInstituteforNuclearResearch,141980Dubna,RussiaAbstractInthispaperweobtainunstableeven-parityeigenmodestothestaticregularsphericallysymmetricsolutionsoftheSU(2)Yang-Mills-dilatoncoupledsystemofequationsin3+1Minkowskispace-time.ThecorrespondingmatrixSturm-LiouvilleproblemissolvednumericallybymeansofthecontinuousanalogueofNewton’smethod.Themethod,beingthepowerfultoolforsolvingbothboundary-valueandSturm-Liouvilleproblems,isdescribedindetails.1IntroductionRegularstationary/staticsolutionsofnonlinearfieldequationsplayaveryimportantroleforunderstandingofnon-perturbativeaspectsoffielddynamicsinvariousmodernfieldmodels,includinggravity.DuringthelastdecadesinceBartnikandMcKinnon’sdiscovery[1]oftheregularparticle-likesolutionstothecoupledsystemoftheEinstein-Yang-Mills(EYM)equationstherewasagrowinginterestinrevealingtheirpossiblephysicalandmathematicalsignificance[2]–[5].Atthesametime,thequalitativelysimilar(essentiallynon-Abelianones)regularsolutionswerefoundinothervariousfieldmodels,i.e.inthecoupledsystemoftheYang-Mills-dilaton(YMD)equations[6],inthecoupledsystemoftheEinstein-Yang-Mills-dilaton(EYMD)equations[7]–[10],inmoresophisticatedstringyinspiredfieldmodels[11]etc(seeRef.[12]forareview).Themainphysicalandmathematicalpropertiesofthementionedsolutionsaredeter-minedbythenonlinearnatureoftheSU(2)Yang-Mills(YM)fieldwhichisasubstantialpartofallthesemodels.Allthementionedcoupledsystemsofequationsadmitsphericallysymmetricparticle-likesolutionsinwhichtheYang-Millsfieldconfigurationisboundedbygravity(EYM),orbythescalardilatonfield(YMD),orbybothofthem(EYMD)∗E-mail:[e-mail]1inunstableequilibrium.(Note,pureYMfieldinthe3+1Minkowskispace-timeisself-repulsiveandnoregularpureYMconfigurationsexist.)Intermsofthelinearper-turbationanalysisthesesolutionshaveeven-parityunstablemodes[13],[6],[9],whicharecalledgravitating(EYM)ordilatonic(YMD)unstablemodes.Ontheotherhand,inthecaseofasphericalsymmetrytheSU(2)Yang-Millsself-interactionpotentialhasadouble-wellshapedformanditstwodistinctminimacorre-spondtothetopologicallydistinctYMvacua.TherelevantYMmagneticfunctionmusttendtothevacuumvaluesattheoriginandatthespatialinfinityforallregularsolu-tions.Itwasshownsomethoseregularsolutionscouldplayaroleatultramicroscopicdistanceanalogoustothatofelectroweaksphalerons[2]duetoexistenceoftheodd-parityYang-Millsnegativemodes[3](whicharecalledsphaleronicunstablemodes)andthecorrespondingfermionzeromodes,whichareresponsiblefortheanomalousfermionproduction[5].However,inthispaperweconcentrateonstudyoftheeven-parityunstablemodesintheYMDsystem,sincetheirsignificanceisinspiredbyaninterestingstory,relatedtothesingularityformationinnonlinearevolutionequations.Indeed,ithasbeendiscoveredinthenumericalstudiesofthemasslessfieldscollapse[14],therearetwotypesofthecollapsebehavior.Type-IIbehaviorischaracterizedbythemassgapabsenceintheblackholespectrumhencetheblackholeswithanarbitrarysmallmasscanbeobtainedinthisway.Lateron,theType-Ibehaviorwasobservedaswell[15],[16].IttakesplaceiftheconsideredsystemofEinstein-matterequationsadmitsregularstaticfiniteenergyasymptoticallyflatsolutions.Inthiscasethesmallestblackholehasafinitemasswhichisequaltothemassoftheloweststaticsolution.IntheEYMsystemtheloweststaticsolutionistheN=1Bartnick-McKinnon(BK)solutionanditsmassdeterminesthesmallestmassoftheformedblackholeinthistypeofacollapse.N=1BKsolutionisoccurredtobeanintermediateattractor,whichthecollapsingsolutionshouldattaininordertoturntotheType-Iblackholeformationscenario[15],[17],[18].Moreover,N=1BKsolutionisoccurredtobeathresholdconfiguration,whichseparatestheblackholeformationanddispersionscenarios.Therevealingofthethresholdnatureremainsanunresolvedproblemandthestudiesofthestaticconfigurationsdecayviatheirproperunstableeigenmodesseemtobeaverynaturalsteptowardsunderstandingthethresholdnature.Thus,thegoalofourpresentpaperistoobtainunstableeigenmodesoftheregularstaticsolutionsintheYMDsystem.Ithasbeenrealizedrecentlythatthesingularity(blackhole)formationingravityandblow-upinnonlinearwaveequationssharemanycommonfeatures[19],[20],[21],[22],sinceonedealswiththesameclassofthesupercriticalevolutionPDE’s[23].ItwasshowninRef.[24],acoupledsystemofYang-Mills-dilaton2equationsin3+1MinkowskispaceisalsoasupercriticalsysteminPDEterminologyanditdoesexhibitblow-up(singularityformation)behavior,whichis,inprinciple,equivalenttotheblackholeformation.So,westudytheunstablemodesofthestaticsolutionsintheYMDsystem,whichisrepresentativeforawideclassofasupercriticalevolutionPDE’s,includingself-gravitatingsystems.OurmaintooltoattacktheproblemnumericallyisacontinuousanalogueoftheNewton’smethod(CANM),whichisareallypowerfultoolforsolvingbothboundary-valueandSturm-Liouvilleproblems.ThekeypointofthemethodisanintroductionofformalevolutionparametertandacorrespondingreformulationoftheproblemintermsofanewsystemofPDEevolutionequations.Ifnecessaryconditionsar