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1、Oscillatory and Power-law Mass Inflation in Non-Abelian Blac Interior structure of non-Abelian black holes is shown to exhibit in a general case either an oscillating mass-inflationary behavior, or power-law behavior with a divergent mass function. In both cases no Cauchy horizon forms. 7 9 9 1 c e
2、D 4 1 v 42 2 1 7 9 / c q - r :gviXraOSCILLATORYANDPOWER-LAWMASSINFLATIONINNON-ABELIANBLACKHOLESD.V.GALTSOVDepartmentofTheoreticalPhysics,MoscowStateUniversity,119899Moscow,RussiaE.E.DONETSLaboratoryofHighEnergies,JINR,141980Dubna,RussiaM.Yu.ZOTOVSkobeltsynInstituteofNuclearPhysics,MoscowStateUnivers
3、ity,119899Moscow,RussiaInteriorstructureofnon-Abelianblackholesisshowntoexhibitinageneralcaseeitheranoscillatingmass-in ationarybehavior,orpower-lawbehaviorwithadivergentmassfunction.InbothcasesnoCauchyhorizonforms.Massin ationinsideblackholesemergesasbackreactionontheperturbationscausedbythecross-
4、owofradiationtailsinthevicinityoftheCauchyhorizon(CH).Aparticularlysimplepictureofthise ectarisesinthecaseofhomogeneous(i.e.,tindependent)perturbationsinsphericalblackholes.Suchasituationmaybealsotreatednon-linearlyasaninteriorproblemforastaticblackholeintheframeworkofasuitableEinsteinmatterAnintere
5、stingexampleisprovidedbytheEinsteinYangMills(EYM)system,oritsextensionsincludingscalar elds:dilaton4orHiggs.3Inthe rsttheoryaninternalCauchyhorizonmayexistonlyforadiscretesequenceofblackholemasses.ForagenericmassthetrueCHisnotformed,but,whensuchawouldbeCHisapproached,themassfunctionstarts togrowexpo
6、nentially.However,thesingularityisnotformedinsteadofCHtothecontrarytotheperturbativeprediction.Inthefullnon-lineartreatmentthislocalmassin ationstopsshortly,andthemetricrelaxestothenextwouldbeCH.Thentheprocessisrepeatedagainresultinginanoscillatorybehaviorofthemassfunctionwithanin nitelygrowingampli
7、tude.Itisremarkablethatmaximalvaluesofthemassfunctionattainedinsubsequentcyclesalsoincreaseexponentially,sogloballywedealwithakindofaquantizedmassin ation.Theultimatesingularityisspacelikeandisnotpower-law.SphericalEYMblackholesaredescribedbyasingleYMfunctionW(r),andbytwometricfunctions =r2/grrand2=
8、gttgrr,whererisatwo-sphereradius.When approacheszero(beingnegative)insidetheblackhole,atsomer=rkthederivativeW=dW/drbecomesapproximatelyalinearfunctionofr,W=rUk,withanalmostconstantUk.Thenthebehaviorof isgovernedwithagoodaccuracybytheequation ( /r)+2 U2=0, 1 Interior structure of non-Abelian black h
9、oles is shown to exhibit in a general case either an oscillating mass-inflationary behavior, or power-law behavior with a divergent mass function. In both cases no Cauchy horizon forms. whichgiveslocally (r)= (rk) 2|Uk isexponentiallyin atingwhen|).Theonemassmovesfunctionleftwardmfrom(r)( rk=tor2R k
10、.2mrThe)thereforefunctionWstabilizesnearthelimitingvaluealthoughWmaybeverylargeatsometinyintervals.ThecorrespondingbehavioroffollowsfromanapproximateintegralZ= U/r=const,whichisvalidthroughouttheoscillationregion: (r)=(rk)exp Uk2(r2 r2k) . AfterpassingRk,anexponentialin(1)becomesoftheorderofunity,he
11、nce growslinearly,andthemassfunctionm(r)stabilizesatthevalueMk=m(Rk).Suchabehaviorholdsuntilthepointofthelocalmaximumof /r2,whichtakesplacewhen V2(V=W2 1)atthepoint r V2 k /2 r=x,| (rkke xkk)|=x k1, sowedealwithanin nitesequenceof“almost”Cauchyhorizonsasr0.Atthesametimethevaluesof| |andmatRkgrowrapi
12、dly | (Rk)|=x 3/2kexk/2,Mk Interior structure of non-Abelian black holes is shown to exhibit in a general case either an oscillating mass-inflationary behavior, or power-law behavior with a divergent mass function. In both cases no Cauchy horizon forms. W=W0+br2(1 ), = 2r(1 ),2 =c+lnr ,(2) withconst
13、antW0,b,c,.Parameterissubjecttosomerestrictions,whichensurescalardominance.Theydi erintheEYMDandEYMHcases.4,3Itfollowsfrom(2)thatthemassfunctiondivergesasr0accordingtothepowerlaw22m(r)=r .Thecorrespondingtendstozeroas(r)=1r,where1=const. Thisbehaviorseemstoberathergeneralforthetheories,whichincludes
14、calar elds.Wetermitpower-lawmassin ation.Noexponentialmassin ationisob-servedinsuchtheoriessincethemetricdoesnotapproachtotheinternalCHatall.So,inasense,power-lawmassin ationprovidesanalternativetothestan-dardmassin ationscenario.BothtypesofthemassandmetricfunctionsbehaviorfortheEYMandEYMDcasesaresh
15、owninthe gure(coordinaterandallthefunctionsaregiveninpower1/4). D.V.G.isgratefultotheOrganizingCommitteeforsupportduringtheconfer-ence.TheworkwassupportedinpartbytheRFBRgrants96-02-18899,18126. 1.E.PoissonandW.Israel,Phys.Rev.Lett.63,1663(1989);Phys.Rev.D 41,1976(1990);A.Ori,Phys.Rev.Lett.67,789(1991). 2.E.E.Donets,D.V.Galtsov,andM.Yu.Zotov,InternalstructureofEinstein YangMillsblackholes,gr-qc/9612067,Phys.Rev.D56,3459(1997). 3.D.V.GaltsovandE.E.Donets,Power-lawmassin ationinEinsteinYang MillsHiggsblackholes,gr-qc/97
