Analytic Evidence for Continuous Self Similarity o

arXiv:hep-th/0607129v119Jul2006PreprinttypesetinJHEPstyle-HYPERVERSIONhep-th/0607129AnalyticEvidenceforContinuousSelfSimilarityoftheCriticalMergerSolutionVadimAsnin∗,BarakKol†andMichaelSmolkin‡RacahInstituteofPhysicsHebrewUniversityJerusalem91904,Israel∗[e-mail]†[e-mail]‡[e-mail]:Thedoublecone,aconeoveraproductofapairofspheres,isknowntoplayaroleintheblack-holeblack-stringphasediagram,andlikeallconesitiscontinuouslyselfsimilar(CSS).Itszeromodesspectrum(inacertainsector)isdeterminedindetail,anditimpliesthatthedoubleconeisaco-dimension1attractorinthespaceofthoseperturbationswhicharesmoothatthetip.Thisisinterpretedasstrongevidenceforthedoubleconebeingthecriticalmergersolution.Forthenon-symmetry-breakingperturbationsweproceedtoperformafullynon-linearanalysisofthedynamicalsystem.Thescalingsymmetryisusedtoreducethedynamicalsystemfroma3dphasespaceto2d,andobtainthequalitativeformofthephasespace,includinganon-perturbativeconfirmationoftheexistenceofthe“smoothedcone”.Contents1.IntroductionandSummary12.Perturbationsofthedoublecone62.1Solvingtheequations103.Non-linearsphericalperturbations133.1Analysisofthereduced2dphasespace18A.AnumericalsearchforaDSSsolution211.IntroductionandSummaryThephasediagramoftheblack-holeblack-stringtransition(seethereviews[1,2])wasconjecturedin[3]toincludea“merger”point–astaticvacuummetric(seefigure1)whichliesontheboundarybetweentheblack-stringandblack-holebranches.Itcanbethoughtaseitherablackstringwhosewaisthasbecomesothinthatithasmarginallypinched,orasablack-holewhichhasbecomesolargethatitspoleshadmarginallyintersectedandmerged(andhencethename“merger”).Themetriccannotbecompletelysmoothasitinterpolatesbetweentwodifferentspace-timetopologies,butitmayhaveonlyonenakedsingularity.Theblack-holeblack-stringsystemhasbeenthesubjectofintensivenumericalresearch[4,5,6,7,8].Naturally,themergerspace-timeitselfisunattainablenumericallysinceitincludesasingularity,butitmaybeapproachedbyfollowingeitherofthetwobranchesfarenough.Indeed,alltheavailabledataindicatesthattheblack-stringandtheblack-holebranchesapproacheachother,inaccordwiththemergerprediction.Atmergerthecurvatureisunboundedaroundthepinchpoint.Onedefinesthe“criticalmergersolution”tobethelocalmetricaroundthepinchpoint(atmerger),namelytheoneachievedthroughazoominglimitaroundthepoint.Itisnaturaltopredict[3,9]thatthecriticalmergersolutionwillloseallmemoryofthemacroscopicalscalesoftheproblem(thesizeoftheextradimensionandthesizeoftheblackhole)andmoreoverbeself-similar,namelyinvariantunderascalingtransformation.Thecentralmotivationofthispaperistodeterminethecriticalmergersolution.Self-similarmetricsbelongtooneoftwoclasses:ContinuousSelf-Similarity(CSS)orDiscreteSelf-Similarity(DSS):whileaCSSmetricisinvariantunderanyscaletransforma-tionandcanbepicturedasacone,aDSSmetricisinvariantonlyunderaspecificscalingtransformation(anditspowers)andcanbepicturedasawigglyconewithitswigglesbeinglog-periodic(seefigure2).Akeyquestionis:IsthecriticalmergersolutionCSSorDSS?–1–rzFigure1:Themergermetric.ristheradialcoordinateintheextendeddirections,zisperiodicallycompactified,whiletimeandangularcoordinatesaresuppressed.Theheavylinesdenotethehorizonofastaticblackobjectwhichisatthresholdbetweenbeingablack-holeandbeingablack-string.Thenakedsingularityisatthe×-shapedpinching(horizoncrossing)point.Uponzoomingontotheencircledsingularityitisconvenienttoreplace(r,z)byradialcoordinates(ρ,χ)radialcoordinates.Weshallbemostlyinterestedinthe“criticalmergersolution”–thelocalmetricnearthesingularity,namelytheencircledportionofthemetric(inthelimitthatthecircle’ssizeisinfinitesimal).ThesignificanceofthecriticalsolutionDSSCSSFigure2:Anillustrationofacontinuouslyself-similargeometry(CSS)asacone,andadiscretelyself-similargeometry(DSS)asawig-glycone.Thesingularityisatthetip.isthatcriticalexponentsofthesystemnearthemergerpointaredeterminedbypropertiesofthissolution(thisisknowntobethecaseinthecloselyrelatedsystem[9]ofChoptuikcriticalcollapse[10,11]).Considerationsandplan.Thedirectwaytosettleourquestionwouldbethroughnu-mericalsimulationofthesystem:onewouldneedtoobtainsolutionswhichareeverclosertothemergerandwitheverhigherresolutionnearthehighcurvatureregion(wherethesingularityisabouttoform).ThiscomputationwouldbecomparableindifficultytoChoptuik’soriginaldiscoveryofcriticalcollapse[10]inthatitrequiressuccessivemeshrefinementsoverseveralordersofmagnitude.However,sinceweareaskingalocalquestion,wemayexpectorhopethatalocalanalysiswouldsuffice,namelytheanalysisofthemetricclosetothesingularity.Ontheonehandalocalanalysisisdisadvantagedrelativetoasolutionofthewholesysteminbeingindirectandthereforeitsresultsneedsomeinterpretationwhichreducescertainty.Ontheotherhand,alocalanalysissuppliesmoreinsightintothemechanismthatdeterminesthelocalmetric,anditiseasiertoperform.Theselatteradvantagesinducedustopreferthelocalanalysisforthecurrentstudy.Thedemandingfull(numerical)analysisisyettobeperformed(seehoweverthesuggestivebutinconclusiveresultsin[12]).–2–Ourfirstassumptionisthattheloca