关闭

关闭

关闭

封号提示

内容

首页 String Physics and Black Holes.pdf

String Physics and Black Holes.pdf

String Physics and Black Holes.…

上传者: oztron 2012-07-25 评分 0 0 0 0 0 0 暂无简介 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,主题内容包含arXiv:hepthvNovStringPhysicsandBlackHolesLeonardSusskindandJohnUglumaaDepa符等。

arXiv:hepthvNovStringPhysicsandBlackHolesLeonardSusskindandJohnUglumaaDepartmentofPhysics,StanfordUniversityStanford,CAUSAIntheselectureswereviewthequantumphysicsoflargeSchwarzschildblackholesHawking’sinformationparadox,thetheoryofthestretchedhorizonandtheprincipleofblackholecomplementarityarecoveredWethendiscusshowtheideasofblackholecomplementaritymayberealizedinstringtheoryFinally,argumentsaregiventhattheworldmaybeahologramIntroductionAnoutsiderlisteningtothisconferencemightgettheideathatthereissuchathingasstringtheory,thatstringtheoryisarelativelycompletetheoryoftheworld,includinggravitationWedonotbelievethisisso,atleastnotyetThereisawideclassofphenomena,perhapsthemostinterestingphenomenaforfuturestudy,whichstringtheoryinitspresentformulationcannotaddressatthistime,andperhapscannotaddressatall–PlanckscalephysicsThedistinctionbetweenPlanckscalephysicsandstringscalephysicsisoftenignoredinstringtheory,butitisanimporantoneIfgisthestringcouplingandthenumberoflarge(uncompactified)spacetimedimensionsisD,thenthePlanckscaleℓPisrelatedtothestringscaleℓSbytherelationℓDP=gℓDSItisusuallyassumedthatthestringcouplingisverysmall,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(rGM)GMln(rGMrGM)()anddefinearescaledtimecoordinateω=tGM,thenforrGMGM,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((rt)GM),V=exp((rt)GM)()eventhorizonsingularityFigureTheSchwarzschildgeometryinKruskalcoordinatesThenthelineelementcanbewrittends=GMerGMrdUdVrdΩ()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)Thereasonforthisisthatasrrunsfromto,itcoverstheregionfromthehorizonouttoinfinityThusitcoversonlytheregionoutsidetheblackholeInthesecoordinates,thelineelementisds=(GMr)dtdrrdΩ,()andweseethatthe(t,r)partofthemetricisconformaltoflatspaceThus,foramasslessscalarfield,weexpectthewaveequationtotakeonasimpleformIfwedefineψ=rφandexpandψinsphericalharmonicsYmℓ,thentheactionforψmℓisS=ℓdrdt(tψℓ)(rψℓ)Vℓ(r)(ψℓ)()whereVℓ(r)=GMrℓ(ℓ)rGMr()ThenewfeatureisVℓ,whichhastheformofapositiondependentmasstermforψVℓistherelativisticgeneralizationofacentrifugalbarrier,butitbehavesdifferentlythananordinarycentrifugalbarrier,becausewhileitisrepulsivefarfromtheblackhole,itisattractiveforr<GMThismeansthatparticlesofhighangularmomentumcanbetrappedintheregionbetweenthehorizonandr=GM,andcanrattlearoundinthisregionItisenlighteningtoexaminethewaveequationusingtheRindlerapproximationIfwedefineavariableu=ln(ρ),thenubehavesliker,inthatitgoestoatasymptoticinfinityandatthehorizonWecanFourierexpandthefieldφtoobtaintransversemomentummodesφ~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=factorizesintoaproductspaceHLHRIf{|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)=FdAeIA,()whereIistheEuclideanactionforthefieldandtheintegralisoverthesetFoffunctionsdefinedforTandwhichmatchφatT=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|eπHR|φL〉LL〈φL|eπHR|φ′R〉R=R〈φR|eπHR|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

用户评论(0)

0/200

精彩专题

上传我的资料

每篇奖励 +2积分

资料评价:

/20
0下载券 下载 加入VIP, 送下载券

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