关闭

关闭

关闭

封号提示

内容

首页 String Physics and Black Holes

String Physics and Black Holes.pdf

String Physics and Black Holes

oztron
2012-07-25 0人阅读 0 0 0 暂无简介 举报

简介:本文档为《String Physics and Black Holespdf》,可适用于工程科技领域

arXiv:hepthvNovStringPhysicsandBlackHolesLeonardSusskindandJohnUglumaaDepartmentofPhysics,StanfordUniversityStanford,CAUSAIntheselectureswereviewthequantumphysicsoflargeSchwarzschildblackholesHawking’sinformationparadox,thetheoryofthestretchedhorizonandtheprincipleofblackholecomplementarityarecoveredWethendiscusshowtheideasofblackholecomplementaritymayberealizedinstringtheoryFinally,argumentsaregiventhattheworldmaybeahologramIntroductionAnoutsiderlisteningtothisconferencemightgettheideathatthereissuchathingasstringtheory,thatstringtheoryisarelativelycompletetheoryoftheworld,includinggravitationWedonotbelievethisisso,atleastnotyetThereisawideclassofphenomena,perhapsthemostinterestingphenomenaforfuturestudy,whichstringtheoryinitspresentformulationcannotaddressatthistime,andperhapscannotaddressatall–PlanckscalephysicsThedistinctionbetweenPlanckscalephysicsandstringscalephysicsisoftenignoredinstringtheory,butitisanimporantoneIfgisthestringcouplingandthenumberoflarge(uncompactified)spacetimedimensionsisD,thenthePlanckscaleℓPisrelatedtothestringscaleℓSbytherelationℓD−P=gℓD−SItisusuallyassumedthatthestringcouplingisverysmall,sothereisalargedifferencebetweenthetwoscalesMostofthephenomenathatweareabletodiscussinstringtheory–thespectrum,scatteringamplitudes,etcareallphenomenathathavetodowiththestringsscale,notthePlanckscaleThereisahostofproblemsatthePlanckscalewhichthepresentformulationofstringtheoryissimplyincapableofhandlingTheseinclude•ThethermodynamicsofstringsandtheirbehaviorattemperaturesabovetheHagedorntemperature•Veryhighenergyscatteringprocessesatverysmallimpactparameter•Blackholeevaporationandthepuzzlesassociatedwithit•ThecosmologicalconstantproblemInthecaseofahighenergycentralcollisionbetweentwostrings,itiseasytoguesstheanswer–ablackholeforms,andthenevaporatesButwecan’tstudythisinstringtheoryIfwedidunderstandquantumgravity,thenwecouldanswerallofthesequestionsTheplanoftheselecturesisasfollowsFirst,wewillreviewthephysicsofhorizons,includingtheirthermalbehavior,andallofthefundamentalphysicsoflargemassSchwarzschildblackholesWewillseethatalloftheparadoxesassociatedwithblackholeevaporationcanbeaddressedinthissimplecontext,andattemptstoresolvetheseparadoxeswillleadustotheideaofblackholecomplementarityAfterreviewingthelightfrontgaugeformulationofstrings,wewillbeabletoaskhowstringtheorymightbeabletoresolvetheparadoxesofblackholeevaporationInparticular,wewillbeinterestedinhowstringtheorystoresinformation,andhowthePlanckscaleisgeneratedfromstringtheoryWewilldiscusstheentropyofhorizonsinstringtheoryFinally,wewilldiscusssomeideasabouttheworldasahologram,dueto’tHooftandSusskindIfoneweretoplotthedistancescalesthatareprobedasoneincreasestheenergyofaprocess,weknowthatforordinaryrelativisticfieldtheory,thelengthscaledecreasesastheenergyincreasesWhatwearenowfindinginstringtheory,however,isthatthisbehaviordoesnotcontinueforeverIncreasingtheenergybeyondthePlanckenergy,onestartstoprobelargerdistancesinsteadofsmalleronesThisistheenergyregionwhichwemustunderstandtosolvetheaforementionedproblemsSchwarzschildBlackHolesThelineelementfortheeternalSchwarzschildblackholegeometry,inScwarzschildcoordinates(t,r,θ,ϕ),isgivenbyds=−(−GMr)dt(−GMr)−drrdΩ()whereGisNewton’sconstant,Mistheblackholemass,anddΩisthelineelementoftheunittwosphereThesurfacer=GM,t=∞isthefutureeventhorizonThereisalsoapasteventhorizonatr=GM,t=−∞,butwewillnotconcernourselvesmuchwiththis,sinceforblackholesformedbythegravitationalcollapseofmatteritisabsentThesingularityisatr=Lightsignalsfrompointsoutsidethehorizoncanreachinfinity,whereaslightsignalsfrompointsinsidethehorizonnecessarilyterminatewhentheyreachthesingularityWecanthereforethinkofthehorizonasconsistingofthosephotonswhichwerejustbarelytrappedbytheblackholePhysicsiscomplicatednearthesingularity,sowewillrestrictourattentiontophysicsstrictlyoutsidetheblackholeItisourbeliefthatmostoftheinterestingphysicsisatthehorizon,anywayToafreelyfallingobserver,thereisnothingspecialaboutthehorizonAllofthelocalgeometricalinvariantsremainsmallatthehorizon,sothereisnolocalsignalthathehascrossedintotheregionoftheblackhole,andhecancrossthehorizonintotheblackholeinafiniteamountofpropertimeNote,however,thatt=∞onthehorizon,soanexternalobserver,whosepropertimeisproportionaltot,willneverseeanythingcrossthehorizonThisisthefirstofaseriesofpeculiarsituationsinwhichobserversinsideandoutsidewilldisagreeSupposeweareinterestedinaregionofspaceveryclosetothehorizon,smallcomparedtothesizeoftheblackhole,butlargecomparedtoanymicroscopicscales(Equivalently,wemightbeinterestedinablackholewithverylargeM)Inthiscase,theSchwarzschildlineelementcanbesimplifiedIfwedefineρtobetheproperdistancefromtheeventhorizon,ρ=√r(r−GM)GMln(√r−GM√r√GM)()anddefinearescaledtimecoordinateω=tGM,thenforr−GM≪GM,Eq()canbeapproximatedasds=−ρdωdρdydz,()whereyandzdenotethedirectionstangenttothehorizonEq()isknownastheRindlermetric,andisnothingmorethanflatMinkowskispaceinhyperbolicpolarcoordinatesIf(T,X,Y,Z)denotethecoordinatesofMinkowskispace,thenT=ρsinh(ω),X=ρcosh(ω),Y=y,Z=z()AfreelyfallingobserversimplycorrespondstoanintertialMinkowskiobserver,andthereisclearlynothingspecialaboutthehorizon,whichisjustanordinarylightlikesurfaceThereis,however,somethinginterestingaboutourparametrizationofflatspaceThetimecoordinateωdoesnotbehavelikeanordinaryMinkowskitimevariable–infact,itcorrespondstoaLorentzboostparameter,andgoestoinfinityonthelightlikesurfaceofthehorizonSpacelikesurfacesofconstantωaccumulatenearthehorizonObserversatfixedρdescribehyperbolictrajectories,whichmeansthattheycorrespondtouniformlyacceleratedobserversinMinkowskispaceThismakessense,ofcourse,sinceanobserverwhowantstoremainoutsidetheblackholemusthavearocketorsomeothermeansofpropulsiontokeepfromfallinginAdiagramofRindlerspaceisshowninFigTheclassicalphysicsofobserverswhoarerestrictedtoremainonlyinthefirstquadrantofRindlerspace,correspondingtotheregionoutsidetheblackhole,iscompletelyconsistentAlthoughsignalswhichoriginateinquadrantIVcanIIIIIIIVRindlerhorizonxxω=∞ω=−∞FigureAdiagramofRindlerspaceshowingthefourquadrantsandthehorizoninfluenceeventsinquadrantI,theymustcrossthepasthorizonatω=−∞,andcanthereforebetreatedasinitialdataSimilarly,signalswhichpropagateoutfromquadrantIintoquadrantIImustcrossthesurfaceω=∞QuadrantIIIissimplyoutofcausalcontactwithquadrantI,andsignalsoriginatingtherehavenoeffectoneventsinquadrantIWhenitcomestodoingquantummechanicsinRindlerspace,however,thestorywillgetmorecomplicatedNowthatwehavesomenewintuitionaboutthenatureofspacetimenearthehorizonofablackhole,let’sreturntothefullSchwarzschildmetricandseeifwecanmakesomeoftheideasobtainedusingtheRindlerspaceapproximationmorepreciseLetusfirstdefinetheReggeWheelertortoisecoordinater∗=rGMln(rGM−)()andtheKruskalSzekerescoordinatesU=−exp((r∗−t)GM),V=exp((r∗t)GM)()eventhorizonsingularityFigureTheSchwarzschildgeometryinKruskalcoordinatesThenthelineelementcanbewrittends=−GMe−rGMrdUdVrdΩ()Themetricinthe(U,V)planeisconformaltoflatspace,andnothingspecialhappensatthehorizonr=GMNote,however,thatthesingularityatr=hasreappearedFigshowswhatthegeometrylookslikeinthe(U,V)planeUsingthisdiagramitiseasytounderstandthedisagreementbetweenfreelyfallingandexternalobserversAfreelyfallingobserversimplyfollowsageodesic,crossingthehorizoninafiniteamountofpropertimeandeventuallycrashingintothesingularityWhenthefreelyfallingobservercrossesthehorizon,hewillnolongerbeabletosendsignalstotheoutsideworldTheexternalobserver,inordertoavoidtheblackhole,mustbeconstantlyaccelerating,andthereforefollowsahyperbolictrajectoryThehorizoncorrespondstoasurfacewhichtheexternalobservercannotintersectinafinitetime,andthusmustbeatt=∞Anexternalobservercanonlyreceivesignalsfrompointsoutsidethehorizon,andcanthusnever“see”anythingcrossthehorizonInaddition,aswiththeRindlercase,thesurfacesofconstanttimeaccumulatenearthehorizon,andthetimecoordinatetactuallycorrespondstoakindofLorentzboostparameterAstincreases,therelativeboostbetweentheexternalobserverandaninfallingparticleincreasesItiseasytoshowthatastheparticlefallstowardthehorizon,itsmomentumasseenbytheexternalobserverincreaseslikeexp(tGM)Theblackholeistheultimateparticleaccelerator–themomentumofanyparticleasseenbytheexternalobserverwilleventuallybecomemuchlargerthanthePlanckmassQuantumPhysicsinRindlerSpaceNowletusconsiderthequestionofhowtodoquantummechanicsinRindlerspaceConsiderasinglerealfreemasslessscalarfieldφpropagatinginRindlerspaceItwillproveconvenienttoconsiderthewaveequationforφusingthetortoisecoordinates(t,r∗,θ,ϕ)Thereasonforthisisthatasr∗runsfrom−∞to∞,itcoverstheregionfromthehorizonouttoinfinityThusitcoversonlytheregionoutsidetheblackholeInthesecoordinates,thelineelementisds=(−GMr)−dtdr∗rdΩ,()andweseethatthe(t,r∗)partofthemetricisconformaltoflatspaceThus,foramasslessscalarfield,weexpectthewaveequationtotakeonasimpleformIfwedefineψ=rφandexpandψinsphericalharmonicsYmℓ,thentheactionforψmℓisS=∑ℓ∫dr∗dt(∂tψℓ)−(∂r∗ψℓ)−Vℓ(r∗)(ψℓ)()whereVℓ(r∗)=GMrℓ(ℓ)r−GMr()ThenewfeatureisVℓ,whichhastheformofapositiondependentmasstermforψVℓistherelativisticgeneralizationofacentrifugalbarrier,butitbehavesdifferentlythananordinarycentrifugalbarrier,becausewhileitisrepulsivefarfromtheblackhole,itisattractiveforr<GMThismeansthatparticlesofhighangularmomentumcanbetrappedintheregionbetweenthehorizonandr=GM,andcanrattlearoundinthisregionItisenlighteningtoexaminethewaveequationusingtheRindlerapproximationIfwedefineavariableu=ln(ρ),thenubehavesliker∗,inthatitgoesto∞atasymptoticinfinityand−∞atthehorizonWecanFourierexpandthefieldφtoobtaintransversemomentummodesφ~kInthecoordinates(ω,u),theLagrangianforφ~ktakesaparticularlysimpleform,L=(∂ωφ~k)−(∂uφ~k)−~keuφ~k()ThepotentialbarrierisnowgivenbyV~k=~keu,andweseethatonlythemodewith~k=canescapetoinfinityThenextimportantpieceofinformationwewillneedisthethermalnatureofRindlerspaceSpecifically,wewillseethataRindlerobserverdescribestheordinaryMinkowskivacuumbyathermaldensityoperatorThisisageneralresult,notrestrictedtotheabovecaseofamasslessscalarfieldConsiderdividingthehypersurfaceT=ofMinkowskispaceintotwohalves,onewithX<,andonewithX>Wewillcallthesehalvestheleftandrighthalves,respectivelyAssumethattheHilbertspaceHonthehypersurfaceT=factorizesintoaproductspaceHL⊗HRIf{|b〉L}isanorthonormalbasisforHLand{|a〉R}isanorthonormalbasisforHR,thenageneralket|ψ〉inHcanbewritten|ψ〉=∑b,aψ(b,a)|b〉L⊗|a〉R()IfwenowtraceoverthedegreesoffreedominHL,theresultingdensitymatrixfortherighthalfofthehypersurfaceisgivenbyρ(a,a′)=∑bψ(b,a)ψ∗(b,a′)()SincewearguedthatnocausalsignalfromthehypersurfaceT=,X<canenterquadrantI,thecompletesetofstatesonthehypersurfaceT=,X>isinfactthecompletesetofstatesneededtodescribephysicsinRindlerspaceforalltimeThus,theaboveconstructionispreciselythedensitymatrixusedbyaRindlerobserverinquadrantINowconsiderperformingthisdecompositionfortheMinkowskigroundstate|〉Givensomearbitrarysetoffields,whichwewilldenotebyφ,wecanrepresentthegroundstatewavefunctionalΨ(φ)byusingtheFeynmanKacformula,Ψ(φ)=∫FdAe−IA,()whereIistheEuclideanactionforthefieldandtheintegralisoverthesetFoffunctionsdefinedforT≥andwhichmatchφatT=LetHRdenotetheRindlerspaceHamiltonianforthefieldφ,whichgeneratestranslationsinωSinceωcorrespondstohyperbolicboostangleinthe(T,X)planeofMinkowskispace,whenweWickrotatetoEuclideanspace,HRbecomesthegeneratorofrotationsinthe(T,X)planeThuswecanwriteΨ(φ)=L〈φL|exp(−HRπ)|φR〉R()ThusthedensitymatrixfortheMinkowskivacuum,ρ(φR,φ′R),isgivenbyρ=∑φLR〈φR|e−πHR|φL〉LL〈φL|e−πHR|φ′R〉R=R〈φR|e−πHR|φ′R〉R,()andsothedensityoperatorfortheMinkowskispacevacuumisρ=exp(−πHR),whichisindeedthermal,withinversetemperatureβ=πThisphenomenonisknownastheUnruheffect:acceleratingobserversexperiencethermalradiationButisthereanysenseinwhichtheRindlerobserverisactuallyexperiencingabathofthermalizedparticlesWouldarealthermometermeasureatemperatureTheanswerisyesConsiderthefactthattherearealwaysfluctuationsofthevacuumThesefluctuationscanbedescribedasloopsinspacetimeSomeoftheseloopswillencircletheorigin,lyingpartiallyinsideandpartiallyoutsidequadrantIButsincetheseloopsintersectthesurfacesω=−∞andω=∞,asfarastheRindlerobserverisconcerned,theyareparticleswhicharepresentforalltimeTheRindlerobserverseesthesefluctuationsasabathofthermalparticleswhichareejectedfromthehorizoninfinitelyfarinthepastandwhichwilleventuallyfallbackontothehorizonintheinfinitefutureNotehoweverthatthisisaninterpretationofaparticularphenomenonbyaparticularobserverAfreelyfallinginertialobserverwouldnotdescribethesevacuumfluctuationsinthesamewayInfact,thefreelyfallingobserverwouldnotbeabletodistinguishthesefluctuationsfromanyothervacuumfluctuations,andwouldnoticenothingoutoftheordinaryItisonlytheRindlerobserverwhocandistinguishthisthermalradiation,andonlytheRindlerobserverwhomustdescribethevacuumusingathermaldensitymatrixThepropertemperaturemeasuredbyaRindlerobserveratdistanceρfromthehorizoncanbeobtainedfromtheRindlertemperatureTR=πbyusingthetransformationbetweenRindlertimeandpropertimeThepropertemperatureisthusgivenbyTproper=πρARindlerobserverwillthereforedescribetheregionclosetothehorizonasaveryhotplace,andinordertodescribephysicsinthisregion,hewillhavetounderstandphysicsatextremelyhightemperaturesIftheRindlerobserverusesaneffectivetheorywithsomecutoffmassΛ,thenitisnaturaltoimposeacutoffatadistanceoforderΛfromthehorizon,beyondwhichtheobservercannotpenetrateForconsistency,however,thisboundarysurfacemustbeendowedwithsomesetofdegreesoffreedomwhichrepresentthedegreesoffreedomintegratedouttoobtaintheeffectivetheoryatscaleΛ,andshouldbehavelikeahotmembraneThissurfaceeffectivelyaugmentsthehorizon,andisknownasastretchedhorizonWewillreturnlatertothisveryimportantideaNowthatwehavedeterminedthataRindlerobserverexperiencesatemperaturewhichdecreasesasonemovesawayfromthehorizon,letusreturntothequantumfieldsForeachtransversemomentummode~k=,thefieldφ~kisexcitedtoathermalspectrumEachmodeispopulatedaccordingtotheBoltzmanndistributionwithtemperatureπ,sothemodeswithenergiesgreaterthanπwillbeexponentiallysuppressedButforenergieslessthanπ,therewillexistabathofthermalparticleswhichcreateathermalatmospherearoundtheblackholeOnlyforthemodewith~k=cantheseparticlesescapetoinfinityAswesawearlier,RindlerspaceisagooddescriptionofanyregionofthehorizonsmallcomparedtotheentireblackholeLetusthenconsidertheinterpretationoftheseresultsforthefinitemassblackholeHighangularmomentumparticleswhichareejectedfromtheregionclosetothehorizonaredeflectedbythecentrifugalbarrierandbecomepartofthethermalatmosphereoftheblackholeOnlythelowestangularmomentummodescanescapethecentrifugalbarrierThisslowleakageofparticlesoutofthecentrifugalbarrierisknownasHawkingradiation,andleadstotheeventualevaporationoftheblackholeThetemperatureasobservedbyanobserveratasymptoticinfinityisT∞=TRdωdt=πGM,whichisknownastheHawkingtemperatureThisimpliesanevaporationtimefortheblackholeoforderGMThefactthatonlythelowestangularmomentummodescanescapeisthereasonforthelongevaporationtimeoftheblackholeGedankenExperimentsInvolvingBlackHolesIntheprevioussections,wehavearguedthatanexternalobservercandescribetheblackholeasahotmembranewhichcanabsorbandemitparticlesLetusnowconsideragedankenexperimentdesignedtotesttheexistenceofthestretchedhorizonSupposethatphysicsbelowsomeenergyscaleΛcanbedescribedbyamoreorlessstandardgrandunifiedtheoryIftheclaimthatthestretchedhorizonbehaveslikeahotmembraneiscorrect,thenanobservernearenoughtothehorizonoughttobeabletodetectbaryonnumberviolationNow,wecanmaketheblackholeaslargeaswelike–galacticsize,forexample–soallthetidalforcesatthehorizonareexceedinglysmall,anditishardtoimaginehowwearegoingtoseeanybaryonnumberviolationNevertheless,letuspressonandseewhatwefindImagineconstructingwhatwecalla“GUTbucket”TheGUTbucketissealed,sothatnoGUTparticlescanenterorleavethecontainer,andhasthepropertythatitcanwithstandGUTscaletemperatures,butnotPlanckscaletemperaturesAlthoughweknowofnowaytoconstructsuchaGUTbucket,thereisnoreasontothinkthatsuchacontainerinanywayviolatesthelawsofphysics,soitshouldbeperfectly

用户评价(0)

关闭

新课改视野下建构高中语文教学实验成果报告(32KB)

抱歉,积分不足下载失败,请稍后再试!

提示

试读已结束,如需要继续阅读或者下载,敬请购买!

评分:

/20

意见
反馈

立即扫码关注

爱问共享资料微信公众号

返回
顶部

举报
资料