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Hawking Radiation Large Distance (Much Bigger Than Planck Length) Resolving the information paradox Samir D. Mathur CERN 2009 Lecture 1: What is the information paradox ? Lecture 2: Making black holes in string theory : fuzzballs Lecture 3: Dynamical questions about black holes Lecture 4: Applying lessons to Cosmology, open questions Two basic points: The Hawking ‘theorem’ : If (a) All quantum gravity effects are confined to within a given distance like planck length or string length dn dn (b)√ TheN nvacuum√n + 1is unique√N√Nn √√nn++11 (n√+N1) √n + 1ndn (175)(n + 1) n (175) −√N n √n + ≈1 − √Ndt ∝√≈n + 1 dt(n∝+ 1) n (175) − ≈ 1 dt ∝ 1 gravity gravity ωR = [ l 2 mψm + mφn] = ω (176) Then there WILLωR be= inf[ormationl 2 m ψlossm + mφn] = ωR (17R6) R − − − 1 R − − − gravity ωR = [ l 2 mψm + mφn] = ω (176) m = n + n + 1, n = nm =nnL + nR + 1, n = nLR (1n7R7) (177) String theory: Bound statesL ofR quantaR − in− stringL−− Rtheory have a ‘large’− size λ m n + m m = 0, N = 0 (178) λ mψn + mφm= =n 0,+ nN =+0 ψ1, nφ= n n (178) (177) | − | L | R− | L − R This size grows withλ = the0, nmumber= l, ofn branes= 0λ, = 0inN, =them0ψ bound= l, staten = 0,, N(17=9)0 (179) ψ − − making the state a ‘horizon sizedλ quantumgramvitψy n + fuzzball’mφm = 0, gravNity = 0 (178) ωI |= ω−I |ωI = ωI (180) (180) 0 λψ= 0, <m0ψψ=0 0l, nψ = 0, < N0 ψ=(10801) (1(8117)9) | % | % | |% ≈%− | % | % ≈ gravity ωI = ωI (180) 0 ψ 0 ψ 0 (181) | % | % & | % ≈ 10 10 10 Lecture 1 What exactly is the black hole information paradox? (arXiv 0803.2030) The information problem: a first pass Hawking radiation Large distance (much bigger than planck length) How can the Hawking radiation carry the information of the initial matter ? If the radiation does not carry the information, then the final state cannot be determined from the initial state, and there is no Schrodinger type evolution equation for the whole system. So we lose quantum theory ... We would like the information to come out of the hole in the details of the Hawking radiation ... This would save us from having to contemplate more complicated solutions of the puzzle (remnants, wormholes, baby Universes, modifications of quantum mechanics ...) Is it possible for this to happen ? The central issue When a piece of coal burns away, a large number of photons are produced. The information of the coal is hidden in ‘delicate correlations’ between these photons’ Can this happen in a black hole ? Hawking’s computation: semiclassical (gravity classical, matter quantum) Get no correlations among radiation quanta What if we use quantum gravity ? Quantum gravity effects will be small But we need ‘subtle correlations’, so maybe the delicate effects of quantum gravity will generate just the needed correlations ? This is incorrect !! The effects of quantum gravity have to change the state of the radiation by order unity .... Basic point: SUBTLE correlations can carry information, as in the case of the burning coal But this CANNOT be achieved by SMALL corrections to the Hawking emission process We need an ORDER UNITY change in the evolution of field modes at the horizon Plan of the talk (A) Particle creation in curved space, black holes (B) Entangled nature of the produced pairs (C) The essential problem created by this entanglement (D) Formulating Hawking’s argument as a ‘theorem’ If (a) All quantum gravity effects are confined to within a given distance like planck length or string length (b) The vacuum is unique Then there WILL be information loss (A) Particle creation in curved space and black holes S1 y y : (0, 2πR) (175) → ClVˆ [l] Vˆ (176) N = n1n5 (177) dn √N n √n + 1 √N √n + 1 (n + 1) n (178) − ≈ dt ∝ 1 ω = [ l 2 m m + m n] = ωgravity (179) R R − − − ψ φ R m = n + n + 1, n = n n (180) L R L − R λ m n + m m = 0, N = 0 (181) | − ψ φ | λ = 0, mψ = l, n = 0, N = 0 (182) − gravity ωI = ωI (183) 0 ψ 0 ψ 0 (184) | & | & ' | & ≈ What helps is that we will usually detecnt1,pna2r,tnic3les inns4ome region which is far away (fr1o8m5) the region where spacetime is curved, for example1/ant1ans2ynm3 ptotic infinity in a black hole g(e1o8m6)etry. There is a natural choice of coordinates at infinityα, in which the metric looks like η . We (n1n5) lp (1µν87) can still make boosts that keep the metric in this form, but the change of time coordinate What helps is that we will usually detect particles in some region which is far away from the under these boosts does not channg1en5the vacukummk. =Wnh1na5t happne5ns is that positive fr(e1q8u8e)ncy region where spacetime is curved, for example at asymptotic infinity in a black hole geometry. modes change to other positivne !fr=eqnuencynm!! =odnes, giving tkhme ex=pnec! tne!d change of the e(n1e8r9g)y of There is a natural choice opf coo1rdinate1s at i5nfinity, in whkich thpe 1metric looks like ηµν. We a quantum when it is viewed from a moving frame. can still make boosts that keep th√e metric in this!form, but the change of time coordinate But even though this maSymibcreo a=n2aπtur2a√lnc1hno5ice of coSormdicirnoa=tes4,πg√ivni1nng5 a natural defi(n1i9ti0o)n of under these boosts does not change the vacuu2m. What happens is that positive frequency particlmeso,dwesecmhaanygesttioll oatshkerwphoysitwiveecfarenqnuoetncuysemsooRdmese, ogtivhienrg otthheeerxcpuercvteildincehaarngceooorfdtihneatee(n1es9ry1gs)yteomf and itas qcuoarrnetsupmonwdhienng itpaisrtviicelwese.d Tfrohme paominotviinsgtfhAraatmew.e have to know the following physics at Sbek = = Smicro (192) some poinBtu: twehvaent itshothueghentheirsgymacyarbrieeda nbaytuthraelsech2poGaicretiocflecso?orTdhinisatiensf,ogrimviantgioanniastnuroatl gdievfiennitbioyntohfe definitpioanrtiocfletsh,ewpeamrtaicylestmilloadseks;wrhaythweer,cwanennoetekudsmteoksok=mnonew1onttphheereontehregryc-umrvoimlineenatrumcootrednisnoarte(fo1s9ry3ts)hteemse particlaensdtaittsesc.orFroersptohnedpinhgyspiacratlicfileesl.dsTthheatpwoientcoisnstihdaetr,wwe ehavsesutmo eknthowat tthhee fpoallrotwicinlegs pdheyfisnicesdaitn ! 1 the flastocmoeorpdoiinnat:tewshyastteismthweietnhemrgeytrciacrrηieµdν bayreththesee!opnaerstiwclhesi?chTghiivseinthfoermexaptieocnteisdnpohtygsiivcea(nl19eb4ny)etrhgey definition of the particle modes; rather, we need to know the energy-momentum tensor for these of the state, an energy which shows up for examp4leπ in the gravitational attraction between these pparttiicclleess.tates. For the physical fields tha∆tEwe=consider, we assume that the particles d(e1fi9n5e)d in n1n5L Sotthheeflraetisconoordainmabteigsuyisttyeminwhitohwmpeatrrticicηleµsν aree tdheefionneeds awthicnhfigniivtey,thbeutexipf etchteerdepihsyssoicmaleerneegrigoyn of the state, an energy which shows up forgexam0 ple in the gravitational attraction (b1e9t6w)een of spacetime which is curved, then wavemodes t→hat travel through that region and back out these particles. g nonzero (197) to spatial infinity can have a nontrivial number of particles at the end, even though they may So there is no ambiguity in how particles are defined at infinity, but if there is some region ∆TCF T = ∆Tgravity (198) have sotfarstpeadcewtiimthe nwohipcharitsiccluesrveexdc,ittehdeninwatvheemmodaetsththeatsttarratv.elWthhraotugwhe tnheaetdrengoiwonisantdo bgeatcksoomute S GM 2 physictaol sfpeealtiinagl ifnofirntithyeclaennghtahveanadnotinmtreivsiaclalneu−sminbveorel−ovfedpairntictlheissaptrtohceesesndo,f epvaerntitchleoucgrheatt(hi1oe9yn9.m) ay Particle creation in cu∼rved sp2ace have started with no particles excited inetShbekm aetGtMhe start. What we need now is to g(e2t00s)ome ∼ 3.1 pPhaysritcaicl lfeelcinrgeafotriothne:lepnghtyhsaincdaltipmiecstcuarlees involvieωdt in this process of particle creation. Take a scalar field withΨ = Lagrangianψ(x)e− (201) Let us first get a simpler picture of why partic1les canµ get created when spacetime is curved. 3.1 Particle creation: physical pLic=tur∂eµφ∂ φ (202) We know that each fourier mode of a quantum 2field behaves like a harmonic oscillator, and if we areLient tuhsefiersxtcigteetdastsaimteplner pfoicrtutrheisoof swcihlylatpoarrttichleens cwaen hgaevtecrneaptaedrtiwclheesninsptahciestifmoue riisercumrvoedde.. | Expand! the field in fourier modes, and look at a particular mode. Thus tWhe kanmopwlithuadteeoafchthfiosufroiLeteurr imtheero damplitudemeoodfea, qw ofuha ithisncthu modemw10efice albedll bae,hhaavessaliLkeagarahnargmiaonniocf otshceillfaotromr, and if we are in the excited state n for this oscillator then we have n particles in this fourier mode. | ! 1 1 Thus the amplitude of thisThefo uLagrangianrier Lmo=d feor, aw˙ 2 h isich wωe2aca2ll a, has a Lagrangian of the form (3.30) 2 − 2 1 1 L = a˙ 2 ω2a2 (3.30) But as we move to later times, the spacetime 2can−di2stort, and the frequency of the mode can change, so that we will get But as we move to later times, the spacetime can distort, and the frequency of the mode can ! 1 1 ! change, so that we will get So we haveL a= harmonica˙ 2 oscillatorω 2a2. Wavefunction in ground (3.31) state gives vacuum,2 excited− 2 states mean particles in that ! 1 2 1 !2 2 mode L = a˙ ω a (3.31) We picture this situation in fig.1. Fig.1(a) show2s th−e 2potential where the frequency is ω. Let us requireWtehpaitcthuerreethairsesnitouaptiaorntiicnlefisgp.1r.esFeingt.1(ina)tshhioswfosuthrieerpomteondteia.l Twhherne wtheewfrielql uheanvcey tihs eω.vaLceutuums wavefurneqcutiiorenth0atfhoerrtehairsehnaormpaorntiicleosscpirlleasteonrt.inNtohwissfuopurpioersemwoedec.hTanhgene twheewpiolltehnavtiealthteo vtahceuuomne | ! for frewquaveenfcuyncωti!o;nth0is fpoorttehnitsiahlarinmosnhiocwonscinllafitogr..1(Nbo)w. Fsuoprptohsies wneewchpanogtentthiealp, ottheentviaalcutoumthestoantee | ! is a difffoerrfernetquweanvceyfuωn!;cthioisnpforotemnttiahleinonsehofworn
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