Exact Solutions of the Klein-Gordon Equation in th

arXiv:hep-th/0603036v212Jul2006ExactSolutionsoftheKlein-GordonEquationinthePresenceofaDyon,MagneticFluxandScalarPotentialintheSpecetimeofGravitationalDefectsA.L.CavalcantideOliveira∗andE.R.BezerradeMello†Dept.deF´ısica-CCEN.UniversidadeFederaldaPara´ıba58.059-970,J.Pessoa,PB.C.Postal5.008.BrazilFebruary7,2008AbstractInthispaperweanalysetherelativisticquantummotionofachargedspin−0particleinthepresenceofadyon,Aharonov-Bohmmagneticfieldandscalarpotential,inthespacetimesproducedbyanidealizedcosmicstringandglobalmonopole.Inordertodevelopthisanalysis,weassumethatthedyonandtheAharonov-Bohmmagneticfieldaresuperposedtobothgravitationaldefects.Twodistinctconfigurationsforthescalarpotential,S(r),areconsidered:i)thepotentialproportionaltotheinverseoftheradialdistance,i.e.,S∝1/r,andii)thepotentialproportionaltothisdistance,i.e.,S∝r.Forbothcasesthecenterofthepotentialscoincidewiththedyon’sposition.InthecaseofthecosmicstringtheAharonov-Bohmmagneticfieldisconsideredalongthedefect,andfortheglobalmonopolethismagneticfieldpiercesthedefect.Theenergyspectraarecomputedforbothcasesandexplicitlyshowntheirdependenceontheelectrostaticandscalarcouplingconstants.AlsoweanalysescatteringstatesoftheKlein-Gordonequations,andshowhowthephaseshiftsdependonthegeometryofthespacetimeandonthecouplingconstantsparameter.PACSnumbers:03.65.Pm,03.65.Ge,14.80.Hv∗E-mail:alo@fisica.ufpb.br†E-mail:emello@fisica.ufpb.br11.IntroductionTheinfluenceofthegravitationalfieldonquantummechanicalsystemshasattractedattentioninparticlephysicsseveralyearsago.Inthisway,theanalysisofthehydrogenatomincurvedspacetimehasbeenconsideredin[1,2,3].In[2]wasshownthattheshiftsintheenergyspectrumcausedbylocalcurvatureisdifferentfromtheusualgravitationalDopplershift.Ontheotherhand,thisshiftisappreciableonlyintheregionofstronggravitationalfield.Recentlytheanalysisoftheinfluenceofthetopologyofthespacetimeontheenergyspectrumofthehydrogenatom,hasbeenconsideredinanonrelativistic[4]andarelativistic[5]pointoviews.Inthesepapersthehydrogenatomisplacedinthespacetimeproducedbyanidealizedlinearcosmicstring,andapoint-likeglobalmonopole.Differentfromthepreviousanalysis,intheserecentinvestigationstheenergyspectraassociatedwiththehydrogenatomcouldbeexactlycalculated.Thefermion-dyonsystemhasbeenanalyzedin[6,7]underarelativisticpointofviewinaflatspacetime.Inthesepaperstheenergyspectrumassociatedwiththissystemwasshowntobesimilartothehydrogenatomoneiftheproductbetweentheelectricchargeoftheparticle,e,withthemagneticchargeofthemonopole,g,isanintegernumber.Thenonrelativisticquantumanalysisofchargedparticleinthepresenceofamagneticmonopoleandintheglobalmonopolespacetime,hasbeendevelopedin[8]consideringtheeffectoftheelectrostaticself-interactiononthechargedparticlecausedbythenon-trivialtopologyofthespacetime[9].Moreover,thechargedparticle-dyonsystemhasbeenanalyzedonaconicalspacetimeinanonrelativisticapproach[10].Inthesepublications,themagneticmonopoleanddyonwereconsideredsuperposedtotherespectivetopologicaldefects.Accordingtothemodernconceptsoftheoreticalphysics,topologicaldefectsmayhavebeenformedbythevacuumphasetransitionintheearlyUniverse[11,12].Theseincludedomainwall,cosmicstringsandmonopoles.Amongthem,cosmicstringsandmonopolesseemtobethebestcandidatetobeobserved.Cosmicstrings[13]andglobalmonopoles[14]areexotictopologicalobjects,theydonotproducelocalgravitationalinteraction,howevertheymodifythegeometryofthespacetimeproducingplanarandsolidangledeficit,respectively.In[15],Linetshowedthatacosmicstringspacetimecanbeproducedbythevortexsystem,i.e.,asystemcomposedbychargedscalarfieldandAbeliangaugefieldwhichundergoestospontaneoussymmetrybreaking.Admittingthattheparameterλ,associatedwiththescalarself-interactionpotential,andelectricchargee,bothgotoinfinity,keeping,however,therelatione2=8λ,thegeometryproducedbythissystemisgivenbythefollowinglineelementbelowexpressedincylindricalcoordinatesds2=−dt2+dz2+dρ2+b2ρ2dφ2,(1)withρ≥0and0≤φ≤2π.Intheexpressionabove,bisaparametersmallerthanunitywhichdependsonthelinearenergydensityofthevortex.Intheaxisρ=0thereexistalinearmagneticflux.Theglobalmonopoleisasphericallysymmetrictopologicalobjectformedbysystemcomposedbyaself-couplingscalartripletwhoseoriginalglobalO(3)symmetryisspontaneouslybrokentoU(1).Theinfluenceofthisobjectonthegeometryofthespacetimecanbeevaluatedbycouplingtheenergy-momentumtensorassociatedwiththissystemwiththeEinsteinequation.Admittingthemostgeneralstaticspherically2symmetricmetrictensor,givenbythelineelementbelow,ds2=−B(r)dt2+A(r)dr2+r2(dθ2+sin2θdφ2),(2)BarriolaandVilenkin[14]foundregularsolutionsfortheradialfunctionsB(r)andA(r),thatforpointsfarfromthemonopole’scorereadB(r)=A−1(r)=1−8πGη2−2GM/r,(3)ηbeingthescaleenergywherethesymmetryisbrokenandGthegravitationalconstant.TheparameterMisapproximatelythemassofthemonopole.Neglectingthemasstermandrescalingthetimevariable,wecanrewritethemonopolemetrictensorasds2=−dt2+dr2α2+r2(dθ2+sin2θdφ2),(4)wheretheparameterα2=1−8πGη2issmallerthanunity.In1931,P.M.Dirac,withtheobjectivetoobtainaglobaldualsymmetrybetweenelectricandmagneticfieldsintheMaxw