Hawking-UnruhRadiationandRadiationofaUniformlyAcceleratedChargeKirkT.M - PDF document

Hawking-UnruhRadiationandRadiationofaUniformlyAcceleratedChargeKirkT.McDonaldJosephHenryLaboratories,PrincetonUniversity,Princeton,NJ08544(February3,1998;updatedMarch4,1998)AbstractHawking-Unruhradiationisameasureofthequantum”uctuationsintheradi- ck(1)whereisthelocalaccelerationduetogravity,isthespeedoflight,¯isPlancksconstantand ck(2)whereistheaccelerationasmeasuredintheobserversinstantaneousrestframe.TheHawking-Unruhtemperature“ndsapplicationinacceleratorphysicsasthereason Thistopichasbeenreviewed,forexample,in[3]. photonsintheapparentthermalbathwouldbeinterpretedbyalaboratoryobserverasanextraŽcontributiontotheradiationrateoftheacceleratedcharge[8].ThepoweroftheextraŽradiation,whichIcallUnruhradiation,ŽisgivenbyUnruh =(energy”uxofthermalradiation)(scatteringcrosssection)(3)Forthescatteringcrosssection,weusethewell-knownresultforThomsonscattering,Thomson3,where/mcistheclassicalelectronradiusandisthemassoftheelectron.TheenergydensityofthermalradiationisgivenbytheusualexpressionofPlanck: d8 c3 h/kT(4)whereisthefrequency.The”uxoftheisotropicradiationontheelectronisjusttimestheenergydensity.Notethattheserelationsholdintheinstantaneousrestframeoftheelectron.Then,Unruh dtd c2 h/kT (5)Onintegratingoverwe“nd,Unruh 83¯0 2 h4=¯0a4 (6)usingtheHawking-Unruhrelation(2).Thepresenceof¯ineq.(6)remindsusthatUnruhradiationisaquantumeect.ThisequalstheclassicalLarmorradiationrate,dU/dt,when EE (7)whereisthequantumelectrodynamiccritical“eldstrength, V/cm=3(8)thatwas“rstnotedinthecontextofKleinsparadox[9,10,11].Inthiscase,theaccelerationisabout10Earths.TheUnruhradiationŽdeducedaboveis,however,notreallyanewtypeofradiation.Sciama[12]hasemphasizedhowtheapparenttemperatureofanacceleratedobservedshouldbeinterpretedinviewofquantum”uctuations.Unruhradiationisaquantumcorrectiontotheclassicalradiationratethatgrowslargeonlyinsituationswherequantum”uctuationsintheradiationratebecomeverysigni“cant.Thisphenomenonshouldbecontainedinthestandardtheoryofquantumelectrodynamics,butadirectdemonstrationofthisisnotyetavailable. 2RadiationDuringUniformlyAcceleratedMotionTheexistenceofUnruhradiationprovidesaninterestingcommentontheperpetualprob-lemŽofwhetherauniformlyacceleratedchargeemitselectromagneticradiation[13].ThisfamousproblemarisesindiscussionsoftheradiationreactionthatbeganwithLorentz[14],andPlanck[15,16].The(nonrelativistic)equationofmotionincludingtheradiationreactionis(inGaussianunits)(9)whereisanexternalforceontheelectrondueeithertoanelectromagneticorgravita-tional“eld, )(10)istheradiationreactionforce,isthevelocityoftheelectronandthedotindicatesdier-entiationwithrespecttotime.TheperpetualproblemŽiswhetherachargeradiatesifitsaccelerationisuniform,i.e.,if=0.Inthiscasetheradiation-reactionforce(10)vanishes,andmanypeoplehavearguedthatthismeansthereisnoradiation[17,18,19,20].Anadditionalperspectiveonthisproblemcomeswiththeuseofcovariantnotation.Therelativisticversionof(9)in4-vectornotationis (11)withexternal4-force,andradiation-reaction4-forcegivenby 3d2uµ Š (12)where 3 du 2e26 3c3v2Š(v×v)2 0(13)istheinvariantrateofradiationofenergyofanacceleratedcharge,)isthe4-velocity, cdistheinvariantintervalandthemetricis1).Thetimecomponentofeq.(11)canbewrittendmc Fv+ (14)where (15)isanenergy“rstidenti“edbySchott[21,22]asbeingstoredintheelectroninvirtueofitsacceleration,ŽandwhichwasgiventhenameaccelerationenergyŽbyhim.Thespatialcomponentsofeq.(11)aredm F2e22 3c3¨v+32 c2(v·v)v+2 c2(v·¨v)v+34 (16) Equationsequivalentto(14)-(16)were“rstgivenbyAbraham[23].VonLaue[24]wasthe“rsttoshowthattheseequationscanbeobtainedbyaLorentztransformationofthenonrelativisticresults(9)-(10).Thecovariantnotationofeqs.(11)-(13)was“rstappliedtotheradiationreactionbyDirac[25].Aninterestingdiscussionofthedevelopmentofeqs.(14)-(16)hasbeengivenrecentlybyYaghjian[26].Inthecaseofuniformacceleration,Schott[22]arguedthattheenergyradiatedbytheelectronisderivedentirelyfromitsaccelerationenergy;thereisasitwereinternalcompensationamongstthedierentpartsofitsradiationpressure,whichcausesitsresultanteecttovanish.ŽThisviewissomewhateasiertofollowifaccelerationenergyŽmeansenergystoredinthenearandinductionzonesoftheelectromagnetic“eld,asarguedbyThirring[28]andbyFultonandRohrlich[31].OthercommentaryonthisproblemincludesthatofDrukey[27],BondiandHoyle[29],DeWitt[30,36],Rohrlich[32,37],Rosen[33],Bradbury[34],Leibovitz[35],Nikishov[38],Ginzburg[13],Herrera[39]andColeman[40],allofwhomconcurthatauniformlyacceleratedchargeradiates.3ImplicationsofHawking-UnruhRadiationAnimmediateconsequenceoftheexistenceofUnruhradiationisthatitsinterpretationasameasureofthequantum”uctuationsabouttherateofclassicalradiationimpliesthattheclassicalradiationexists.Atpresent,Unruhradiationforuniformlyacceleratedmotionexistsonlyasatheoreticalconcept,notyetcon“rmedinthelaboratory.ExperimentalevidenceforHawking-Unruheectsdoesexistforuniformcircularmotion,asmentionedintheIntroduction.Itisnoteworthythatwhilediscussionofradiationbyanacceleratedchargeisperhapsmostintricateclassicallyincaseofuniformacceleration,thediscussionofquantum”uctua-tionsisthemoststraightforwardforthiscase.Inaddition,Hawking-Unruhradiationhelpsclarifyaresidualpuzzleinthediscussionoftheequivalencebetweenacceleratedchargesandchargesinagravitational“eld.Becauseofthedicultyinidentifyinganunambiguouswavezoneforuniformlyacceleratedmotionofacharge(inagravity-freeregion)andalsointhecaseofachargeinauniformgravitational“eld,thereremainssomedoubtastowhethertheradiationŽdeducedbyclassicalargu-mentscontainsphotons.Thus,onp.573ofthearticlebyGinzburg[13]weread:neitherahomogeneousgravitational“eldnorauniformlyacceleratedreferenceframecanactuallygenerateŽfreeparticles,expeciallyphotonsŽ.Wenowseethatthequantumviewisricherthananticipated,andthatHawking-Unruhradiationprovidesatleastapartialunderstand-ingofparticleemissioninuniformaccelerationorgravitation.Hence,wecanregardtheconcernsofBondiandGold[29],FultonandRohrlich[31],theDeWitts[36]andGinzburg[13]onradiationandtheequivalenceprincipleasprecursorstotheconceptofHawkingradiation.AcknowledgmentsIwishtothankBillUnruhfordiscussionsonHawking-Unruheects. 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