A gauge theoretical view of the charge concept in

arXiv:gr-qc/9907024v217Mar2000AgaugetheoreticalviewofthechargeconceptinEinsteingravity∗MarcToussaintInstituteforTheoreticalPhysics,UniversityofCologne50923K¨oln,Germany~mt/AbstractWewilldiscusssomeanalogiesbetweeninternalgaugetheoriesandgravityinordertobetterunderstandthechargeconceptingravity.Adimensionalanalysisofgaugetheoriesingeneralandastrictdefinitionofelementary,monopole,andtopologicalchargesareappliedtoelectromagnetismandtoteleparallelism,agaugetheoreticalformulationofEinsteingravity.Asaresultweinevitablyfindthatthegravitationalcouplingconstanthasdimen-sion~/ℓ2,themassparameterofaparticledimension~/ℓ,andtheSchwarzschildmassparameterdimensionℓ(whereℓmeanslength).Thesedimensionsconfirmthemeaningofmassaselementaryandasmonopolechargeofthetranslationgroup,re-spectively.Indetail,wefindthattheSchwarzschildmassparameterisaquasi-electricmonopolechargeofthetimetranslationwhereastheNUTparameterisaquasi-magneticmonopolechargeofthetimetranslationaswellasatopologicalcharge.TheKerrparameterandtheelectricandmagneticchargesareinterpretedsimilarly.WeconcludethateachelementarychargeofaCasimiroperatorinthegaugegroupisthesourceofa(quasi-electric)monopolechargeoftherespectiveKillingvector.Keywords:gaugetheoryofgravity,Kaluza-Klein,charge,monopole,mass,Taub-NUT.∗PresentedattheannualmeetingoftheGermanPhysicalSociety–Heidelberg,March1999.1CONTENTS2Contents1Introduction22Dimensionalanalysisofgaugetheories43Chargedefinitions63.1Monopolecharges................................63.2Topologicalcharges...............................63.3Elementarycharges...............................94Translationalmonopolecharges104.1TheKerr-Newmansolution..........................114.2TheTaub-NUTsolution............................125Relatingtootherformalisms136Summaryanddiscussion141IntroductionInthefifties,YangandMills[14]forthefirsttimeformulatedtheSU(2)-gaugetheorybystrictlykeepingtotheelectromagneticparadigm.Ataboutthesametime,Utiyama[13]formulatedthegeneralgaugetheoryofasemi-simpleLiegroup.Thesetheories,astheyexplaintheelectro-weakandstrongforces,weresupplementedbythegreatsuccessofparticlephysicstoclassifyallleptonsasrepresentationsoftheelectro-weaksymme-tryandallhadronsasrepresentationsoftheflavorsymmetry.O’Raifeartaigh[4]givesamoredetailedinsightintothehistoryofgaugetheories.Thegreatsuccessofsuchtheorieshasalsoinfluencedmodernformulationsofgravity–oneofthefourfundamentalforceswhichshouldalsoberepresentableintheframeworkofgaugetheory.However,theobviousdifferencebetweentheexternalspacetimesymmetriesandinternalsymmetries(asconsid-eredbyYangandMills)causessomedifficultiesforauniformformulationofallforces.Somead-hocassumption(thesoldering)solvesbasicproblemsbutperhapsdiminishesthebeautyofthetheory.Wereferto[10](moreintroductory[9])asageneralformulationofgravityasagaugetheory(seetable1).Forthisworkitismostimportanttounder-standteleparallelismasagaugetheoryoftranslationswiththeanholonomiccoframeϑαasgaugepotentialandtorsionTαasfieldstrength.Withaspecificlagrangian,thistheoryisequivalenttoEinsteingravity.ThiswillenableustoreformulatestandardEinsteiniansolutionsintheframeworkofteleparallelismandthustointerpretthesolutionparametersastranslationalcharges.Now,whatisacharge?Ingeneralisseemsplausibletodefineachargetobeaspecificand1INTRODUCTION3theorygaugegroupconnectionfieldstrengthgeneralgaugetheorysemi-simpleLiegroupGA∈Λ1(M,G)F=DΓ∈Λ2(M,G)electrodynamicsU(1)AF=dA(non-physical)affinegroupeeΓeeRaffinegaugetheorysolderedaffinegroupΓαβ,ϑαRαβ,TαteleparallelismsolderedtranslationsϑαTα=dϑaTable1:Gravitymaybedescribedbyformulatingagaugetheoryoftheaffinegroup.However,onehastoensurethatthegroup,i.e.theLie-algebravaluedconnection,appliestospacetime–issolderedtospacetime.ThisisdonebysplittingtheconnectionintoalinearpartΓαβ(withmatrixindicesαβthatworkonthebasiseαofthelocaltangentspace)andaninhomogeneouspartϑα(thatreplacestheholonomiccoframedxαandtherebyrealizesatranslationalgauge).ThefieldstrengthsplitsintothecurvatureRαβandthetorsionTα.Discardingthelineargauge(Γαβ≡0),thetheoryreducestoteleparallelism.invariantpropertyofaparticle(usuallygivenbyonenumber,perhapsaninteger).Sinceingaugetheorieswetakeparticlestobeelementsinarepresentationofthesymmetry,wearedirectlyledtothemostbasicnotionofacharge,theelementarycharge,classifyingtherepresentationofthesymmetrytheparticleisanelementof.Butalsoaspecificpropertyofthegaugefieldwhichaparticlenecessarilyinducesmaybeconsideredasacharge.Suchis,e.g.,themonopolecharacteroftheelectromagneticfieldaroundanelectron.Thisfieldisinducedbythecouplingoftheelectron’selementarychargetothegaugefield.Suchcouldalsobethemagneticmonopolecharacterof,say,theelectromagneticfieldaroundaDiracmonopole.But,sincethereexistsnomagnetic-typeelementarycharge,thereisnoreasonaforparticleinducesuchafield–exceptfortopology.Wewillseethatinthebundleformalismtopologicaleffectsalsomotivatethisthird,topologicalkindofcharge,includingthequasi-magneticmonopolecharge.InthefollowingwedefinethesethreekindsofchargesandapplythedefinitionsonTaub-NUTandKerr-Newmantypesolutionsofteleparallelism.