Fractal escapes in Newtonian and relativistic mult

arXiv:chao-dyn/9910033v125Oct1999FractalescapesinNewtonianandrelativisticmultipolegravitationalfieldsAlessandroP.S.deMoura∗InstitutodeF´ısicaGlebWataghin,UNICAMP,13083-970CampinasSP,BrazilPatricioS.Letelier†InstitutodeMatem´atica,Estat´ısticaeCiˆenciadaComputa¸c˜ao,DepartamentodeMatem´aticaAplicada,UNICAMP,13083-9790CampinasSP,BrazilAbstractWestudytheplanarmotionoftestparticlesingravitationalfieldsproducedbyanexternalmaterialhalo,ofthetypefoundinmanyastrophysicalsystems,suchasellipticalgalaxiesandglobularclusters.BoththeNewtonianandthegeneral-relativisticdynamicsareexamined,andintherelativisticcasethedy-namicsofbothmassiveandmasslessparticlesareinvestigated.Thehalofieldisgiveningeneralbyamultipoleexpansion;werestrictourselvestomultipolefieldsofpureorder,whoseNewtonianpotentialsarehomogeneouspolynomi-alsincartesiancoordinates.Apuren-polefieldhasndifferentescapes,oneofwhichischosenbytheparticleaccordingtoitsinitialconditions.Wefindthattheescapehasafractaldependencyontheinitialconditionsforn2bothintheNewtonianandtherelativisticcasesformassivetestparticles,butwithimportantdifferencesbetweenthem.Therelativisticmotionofmasslessparticles,however,wasfoundtoberegularforallthefieldswecouldstudy.∗email:sandro@ifi.unicamp.br†email:[e-mail]1Thebox-countingdimensionwasusedineachcasetoquantifythesensitivitytoinitialconditionswhicharisesfromthefractalityoftheescaperoute.I.INTRODUCTIONItiscommoninastrophysicstofindsystemscomposedbyanexternalmaterialhalosurroundinganinnerregionwhichiseithervoidorhasamassiveblackholeinthecenter.Examplesofsystemsthatarerealisticallydescribedbysuchamodelareellipticalgalaxiesandglobularclusters[1].Thegravitationalfieldintheinnerregionduetothehalosatisfiesthevacuumfieldequations,andinNewtoniangravitationitcanbeexpandedinasumofmultipoletermsofvariousorders.Duetotheintrinsicnonlinearityofgeneralrelativity,ingeneralthisisnottrueforgeneralrelativisticfields.However,inmanysituationsthehalocanbesupposedtobeaxisymmetric;inthiscase,thegeneralsolutionofEinstein’sequationsisknown,andtheinnerrelativisticfieldcanalsobeexpandedinaseriesofmultipolarterms.Inthisarticle,westudythedynamicsoftestparticlesinpuremultipolarhalofields(withoutanymatterintheinnerregion),bothintheNewtonianandrelativistictheories,forseveralmultipoleorders,formaterialparticlesand(intherelativisticcases)forlight.Wefindthatineachcasethesystemshavedistinctwell-definedescapes.Differentescapesarechosenbythetestparticlesdependingonitsinitialconditions;iftheescaperouteisafractalfunctionoftheinitialconditions,theoutcomeofaparticlestartingfromagiveninitialconditionshowsasensitivedependenceontheseinitialconditions.Inthissense,wesaythatthesystemischaotic[2–4].Associatedwiththis,wehaveaconvenientlydefinedfractaldimension,whichquantifiesthissensitiveness.Fractalescapeshavebeenfoundinanumberofphysicalsystems,includingtheHenon-Heilessystem[5],astrophysicalpotentials[6],themotionoftestparticlesingravitationalwaves[7],classicalionization[8],motionofparticlesinafluid[9],inflationarycosmology[10],andothers.Westudythefractaldimensionfordifferentmultipoleordersandfordifferentenergies,inthecaseofmaterialtestparticles.Forsimplicity,weconfineourselvestothecaseofzeroangularmomentum,whenthemotion2isplanar.Thisarticleisorganizedasfollows:insection2,weinvestigatetheclassicalmultipolegravitationalfield,showingthefractalityoftheescapebasinsandcalculatingthecorre-spondingdimensionforseveralmultipoleorders.Insection3,weturntotherelativisticcase,andinvestigatethedynamicsofbothmaterialparticlesandmasslessparticles;wecomparetheresultswiththoseobtainedintheclassicalcase.Insection4,wesummarizeourresultsanddrawsomeconclusions.II.NEWTONIANMULTIPOLEFIELDSTheNewtoniangravitationalfieldintheinnerregionduetothemassinthehalosatisfiesthevacuum(Laplace)fieldequation∇2Ψ=0,andthereforecanbeuniquelyexpandedinasumofmultipoleterms.Ifthefieldisaxisymmetric,asweadmitfromnowon,itsmultipoleexpansioncanbewrittenas:Ψ=∞Xn=0nanRn+bnR−(n+1)oPn(cosθ),whereR2=z2+r2,cosθ=z/R,andrandzaretheusualcylindricalcoordinates;thePnareLegendrepolynomials.Ifweconsidertheinnerregiontobeemptyofmatter,Ψcannothavesingularitiesthere.Thismeansthatbn=0forallnintheinnerregion.Thus:Ψ=a0+∞Xn=1anRnPn(cosθ),(1)wherea0isaconstantwhichcanwithoutlackofgeneralitybechosenaszero.ThemultipoleexpansionofΨcontainsonlytermsthatbecomesingularasR→∞(withtheexceptionoftheconstantterma0).Hereweareinterestedinstudyingthedynamicsofthefieldduetoeachmultipoletermseparately.Wethereforedefinethen-thordermultipolefieldΨnas:Ψn=anRnPn(cosθ).(2)3WenowconsideratestparticlemovinginthefieldΨnwithnoangularmomentuminthedirectionofthesymmetryaxisz;inthiscase,theparticle’smotionisrestrictedtoaplanethatcontainsthezaxis.Aconvenientsetofcoordinatesinthisplaneisgivenbythezaxisandtheextendedraxis,whichreachesnegativevalues.Inordertoavoidconfusion,weshalldenotetheextendedraxisbyx,with−∞x+∞.Ψiswrittenintermsofxsimplybyreplacingitforr.TheHamiltonianofatestparticleofunitmassacteduponbythefieldΨnis:Hn=12˙x2+˙z2+Ψn(x,z).(3)Thecorrespondingequationsofmotionare¨x=−∂Ψn/∂x,¨z=−∂Ψn/∂z.Sin