Violation of the Energy Conservation Law in Lorent

arXiv:hep-ph/9412391v130Dec1994VIOLATIONOFTHEENERGYCONSERVATIONLAWINLORENTZ-DIRACEQUATIONSFORMORETHANONECHARGED.VillarroelandR.RiveraDepartamentodeF´ısica,UniversidadT´ecnicaFedericoSantaMar´ıaCasilla110-V,Valpara´ıso,ChileAnexactsolutionofLorentz-Diracequationswheretheenergyconserva-tionlawisviolated,isdescribedhereinforthecaseoftwocharges.PACSnumbers:03.50.De,41.60-m1Lorentz-Diracequationsarecurrentlyacceptedastheequationsofmo-tionforchargedparticlesinclassicalelectrodynamics.TheseequationswerederivedbyDiracinaclassicalpaperin1938[1];thesameequationswerealsoobtainedbyRohrlichonthebasisofanactionprincipleinhiswell-knownbook[2].Inthecaseoftwocharges,theequationsare:m1aμ1=(e1/c)Fμαextv1α+(e1/c)Fμα2retv1α+2e21/3c3˙aμ1−1/c2aλ1a1λvμ1(1)whereweuseRohrlich’snotation.Theequationforchargee2isthesameas(1),butwithindexes1and2interchanged.Thefirsttermontheright-handside(RHS)in(1),isLorentz’sforceduetotheexternalfield.ThesecondtermontheRHS,linkschargee2chargee2bymeansofapurelyretardedinteraction,andrepresentstheonlymutualinteractionofthechargedpar-ticlesinLorentz-Diracequations.Finally,thethirdtermisthewellknownradiationreactionterm,constitutedbytwoparts:“Schottterm”,involvingthethirdderivativeofthechargeposition,and“Larmorterm”,involvingthesquareoftheacceleration.Inthecaseofonlyonechargeinanexternalfield,thesecondtermontheRHSof(1)disappears.Equationsin(1)arethefinalachievementofDirac’sconception,accord-ingtowhichitispossibletoformulateaconsistentsetofclassicalequationsofmotionforpointcharges.DiracreachedtheseequationsstartingfromMaxwellequationsandtheprinciplesofrelativity,maskingthedivergencesassociatedwiththesingularitiesofthefieldatthepositionofthecharges,bymeansofamassrenormalizationprocedure.Extensivetheoreticalwork,carriedoutespeciallybyRohrlich[2],ledtoconsiderthatLorentz-Diracequa-2tionscoherentlydescribetheone-chargecaseinanexternalfield.However,thesituationdoesnotremainthesamewhenseveralchargesareinvolved.Infact,afewyearsafterDiracderivedhisequations,Eliezer[3]appliedthemtothecaseoftwoequal-massparticleswithchargesofequalmagnitudebutoppositesigns,movinginastraightlineandcollidinghead-on.Thisledtoacontroversythatstillholdstrue[4].AccordingtoEliezer,thechargesstopbeforecolliding,thenturnbackandmoveawayfromeachother,withanaccelerationthatisalwaysdifferentfromzero,andwithavelocitythattendstowardthespeedoflight.Thisbehavioriscontrarytowhatcommonunderstandingwouldsuggestfortwochargesofoppositesigns,andisalsocontradictorytothelawofenergyconservation.Eliezer’sconclusionswerequestionedbyClavier[5],Plass[6],andRohrlich[2].However,morerecentdetailednumericalcalculationsbyBaylisandHuschilt[7],andKasher[8],cametosupportEliezer’sconclusions.ThepathologicalbehaviorofLorentz-DiracequationsinEliezerhead-oncollisionhasreceived,ingeneral,littleattentionintextbooks(oneexceptionisParrott’sbook[4]).Severalreasonsmayhelptoexplainthislackofat-tention.Inspiteoftheone-dimensionalcharacterofEliezer’scollision,theretardednatureoftheinteractionbetweenthechargesgivesrisetomathe-maticalcomplications.Inparticular,thenumericaltreatmentof[7]&[8]areratherelaborated,andquestionsabouttheconvergenceoftheiterativepro-cesscannotbeeasilyanswered.Thisproblemisrelated,inturn,totheissueoftheexistenceandtheuniquenessofthesolutionofLorentz-Diracequa-tions,whichstillremainsanopenquestioninthecaseofseveralcharges.3AnothersourceoftroublescomesfromthefactthatinEliezer’scollision,thekineticenergyofthecharges,thetotalrateoftheradiationemittedbythem,aswellastheenergystoredinthefieldofthecharges,arealltime-dependent.Thismakesitverydifficulttoanalyzetheenergyconservationlawatanygivenmoment.Althoughthekineticenergyandthetotalrateoftheradiationarewelldefinedconcepts,thesamedoesnotoccurwiththeconceptof“energystoredinthefieldofthecharges”,whichmakesforevenfurthertroubles.Wealsowanttopointoutthatthecatastrophicbehav-iorofEliezer’ssolutionshasbeenobtainedmerelyasamathematicalresultofLorentz-Diracequations,butwithoutidentifyinganyphysicalcause-anidentificationthatwouldbeconvenientforadeeperunderstanding.Fromourpointofview,Lorentz-Diracequationscannotbecorrect,be-causetheydonottakeintoaccountalltheradiationemittedbythecharges.Infact,inthecaseoftwocharges,radiationmustbedescribedby:aLarmortermforeachcharge,termsthatareindeedpresentin(1);plusatermthatconsiderstheinterferenceeffectbetweenthefieldsofbothcharges,whichisnotpresentin(1).ThesecondtermontheRHSof(1)istheonlyonethatconsidersbothparticles,butthistermhasverylittletodowiththeinterferenceradiation,whichmustinvolvetheproductoftheaccelerationoftwocharges,andthisisclearlynotthecase.So,inordertoputthiscon-jectureinaquantitativeway,itwouldbeidealtofindoneexactsolutionoftheLorentz-Diracequationsfortwochargeswherethetotalrateofradiation(includingtheinterferenceterm)canbecalculatedinapreciseway.Itisthepurposeofthisnotetopresentonesuchsolution,whichisthefirstexact4solutionfortheLorentz-Diracequationsformorethanonechargepresentintheliteraturerelatedtothesubject[9].Asitwillbeshownbelow,Lorentz-Diracequations,withappropriateexternalfieldstob