Understanding Hawking radiation in the framework o

arXiv:0806.3602v1[gr-qc]23Jun2008UnderstandingHawkingradiationintheframeworkofopenquantumsystemsHongweiYuandJialinZhangDepartmentofPhysicsandInstituteofPhysics,HunanNormalUniversity,Changsha,Hunan410081,China(Dated:June23,2008)AbstractWestudytheHawkingradiationintheframeworkofopenquantumsystemsbyexaminingthetimeevolutionofadetector(modelledbyatwo-levelatom)interactingwithvacuummasslessscalarfields.Thedynamicsofthedetectorisgovernedbyamasterequationobtainedbytracingoverthefielddegreesoffreedomfromthecompletesystem.Thenonunitaryeffectsarestudiedbyanalyzingthetimebehaviorofaparticularobservableofthedetector,i.e.,itsadmissiblestate,intheUnruh,Hartle-Hawking,aswellasBoulwarevacuaoutsideaSchwarzschildblackhole.WefindthatthedetectorinboththeUnruhandHartle-HawkingvacuawouldspontaneouslyexcitewithanonvanishingprobabilitythesameaswhatonewouldobtainifthereisthermalradiationattheHawkingtemperaturefromtheblackhole,thusreproducingthebasicresultsconcerningtheHawkingeffectintheframeworkofopenquantumsystems.PACSnumbers:04.70.Dy,03.65.Yz,04.62.+v1I.INTRODUCTIONBlackholesareintriguingobjectsandareworthstudyinginallpossiblevarieties.Theideaofblackholeshasbeenproventobehighlyfruitful.Inparticular,Hawking’sdiscoverythatblackholesarenot,afterall,completelyblack,butquantummechanically,emitradiationwithathermalspectrum[1],hasprovideduswiththeunderstandingthatblackholesmayplaytheroleof“Rosettastone”torelategravity,quantumtheoryandthermodynamics.Therefore,Hawkingradiation,asoneofthemoststrikingeffectsthatarisefromthecombi-nationofquantumtheoryandgeneralrelativity,hasattractedwidespreadinterestinphysicscommunityandithasbeenextensivelyexaminedfromdifferentprospectives,yieldingdif-ferentderivationsofit.Thesederivationsinclude(butnotlimitedto)Hawking’soriginalonewhichcalculatestheBogoliubovcoefficientsbetweenthequantumscalarfieldmodesoftheinvacuumstatesandthoseoftheoutvacuum[1,2],anEuclideanquantumgravityderivation[3]whichhasbeeninterpretedasacalculationoftunnellingthroughclassicallyforbiddentrajectory[4],anapproachbaseduponstringtheory[5,6],aninterestingproposalwhichtiesitsexistencetothecancellationofgravitationalanomaliesatthehorizon[7],andarecentstudywhichrevealsaninterestingrelationshipbetweentheexistenceofHawkingradiationandthespontaneousexcitationofatomsusingtheDDCformalism[8]thatsepa-ratesthecontributionsofvacuumfluctuationsandradiationreactiontotherateofchangeofthemeanatomicenergy[9].Inthecurrentpaper,weshalltrytounderstandtheHawkingradiationbyexaminingthetimebehaviorofastaticdetector(modelledbyatwo-levelatom)outsideaSchwarzschildblackholeimmersedinvacuummasslessscalarfieldsusingthewell-knowntechniquesdevel-opedinthestudyofopenquantumsystems.Asforanyopensystem,thefulldynamicsofthedetectorcanbeobtainedfromthecompletetimeevolutiondescribingthetotalsystem(detectorplusexternalfields)byintegratingoverthefielddegreesoffreedom,whichareinfactnotobserved.Itisworthnotingherethatanexaminationofasimilarissue,i.e.,theUnruheffectassociatedwithuniformlyacceleratedatomsintheparadigmofopenquantumsystemhasbeenalreadybeencarriedout[10].2Thepaperisorganizedasfollows.InnextSection,weshallreviewthebasicformalism,thederivationofthemasterequationdescribingsystemofthedetectorplusexternalvacuumscalarfieldsinweakcouplinglimitandthereduceddynamicsitgeneratesforthefinitetimeevolutionofthedetector.InSectionIII,weapplythemethodandresultsoftheprecedingSectiontodiscusstheprobabilityofspontaneoustransitionofdetectorfromthegroundstatetotheexcitedstatesoutsideaSchwarzschildblackhole.Finally,weconcludewithsomediscussionsinSectionIVII.THEMASTEREQUATIONWeshallconsidertheevolutioninthepropertimeofastaticdetector(two-levelatom)interactingwithvacuummasslessscalarfieldsoutsideaSchwarzschildblackholeandassumethecombinedsystem(detector+externalvacuumfields)tobeinitiallypreparedinafactor-izedstate,withthedetectorheldstaticintheexteriorregionoftheblackholeandthefieldsintheirvacuumstates.OurderivationofthemasterequationinthisSectionfollowscloselytothatinRef.[10].Theatomisassumedtobefullydescribedintermsofatwo-dimensionalHilbertspace,sothatitsstatescanberepresentedbya2×2densitymatrixρ,whichisHermitianρ+=ρ,andnormalizedTr(ρ)=1withdet(ρ)≥0.Inordertoachievearigorous,mathematicallysound,derivationofthereduceddynamicsofthedetector,weshallassumethattheinteractionbetweenthedetectorandthescalarfieldsareweaksothatthefinite-timeevolutiondescribingthedynamicsofthedetectortakestheformofaone-parametersemigroupofcompletelypositivemaps[11,12].Withoutlossofgenerality,wetakethetotalHamiltonianforthecompletesystemtohavetheformH=Hs+Hφ+λH′.(1)HereHsistheHamiltonianoftheatom,whichinthemostgenericcasetakestheformHs=ω023Xi=1niσi,(2)whereσi(i=1,2,3)arethePaulimatrices,ω0theenergylevelspacingandn=(n1,n2,n3)aunitvector.Inthepresentpaper,wewilltake,forsimplicity,ntobealongthethirdaxis3suchthatHssimplifiestoHs=ω02σ3.(3)HφisthestandardHamiltonianofmassless,freescalarfields,detailsofwhichneednotbespecifiedhereandH′istheinteractionHamiltonianoftheatomwiththeexternalscalarfieldsandisassumedtobegivenbyH′=σ3Φ(x).(4)Itshouldbepointedoutt