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1、Soft Hair on Black HolesStephen W. Hawking, Malcolm J. Perryand Andrew StromingerDAMTP, Centre for Mathematical Sciences,University of Cambridge, Cambridge, CB3 0WA UKCenter for the Fundamental Laws of Nature,Harvard University, Cambridge, MA 02138, USAAbstractIt has recently been shown that BMS sup

2、ertranslation symmetries imply an infinitenumber of conservation laws for all gravitational theories in asymptotically Minkowskianspacetimes. These laws require black holes to carry a large amount of soft (i.e. zero-energy) supertranslation hair. The presence of a Maxwell field similarly implies sof

3、telectric hair.This paper gives an explicit description of soft hair in terms of softgravitons or photons on the black hole horizon, and shows that complete informationabout their quantum state is stored on a holographic plate at the future boundary ofthe horizon. Charge conservation is used to give

4、 an infinite number of exact relationsbetween the evaporation products of black holes which have different soft hair but areotherwise identical. It is further argued that soft hair which is spatially localized tomuch less than a Planck length cannot be excited in a physically realizable process,givi

5、ng an effective number of soft degrees of freedom proportional to the horizon areain Planck units.arXiv:1601.00921v1 hep-th 5 Jan 2016Contents1Introduction12Electromagnetic conservation laws and soft symmetries43Conservation laws in the presence of black holes74Black hole evaporation95Quantum hair i

6、mplants126Gauge invariance147Supertranslations158Conclusion171IntroductionForty years ago, one of the authors argued 1 that information is destroyed when a black holeis formed and subsequently evaporates 2, 3. This conclusion seems to follow inescapablyfrom an unquestionable set of general assumptio

7、ns such as causality, the uncertainty prin-ciple and the equivalence principle.However it leaves us bereft of deterministic laws todescribe the universe. This is the infamous information paradox.Over the intervening years, for a variety of reasons, the initial conclusion that informationis destroyed

8、 has become widely regarded as implausible. Despite this general sentiment, inall this time there has been neither a universally accepted flaw discovered in the originalargument of 1 nor an a priori reason to doubt any of the unquestionable assumptions onwhich it is based.Recently such an a priori r

9、eason for doubt has emerged from new discoveries about theinfrared structure of quantum gravity in asymptotically flat spacetimes. The starting pointgoes back to the 1962 demonstration by Bondi, van der Burg, Metzner and Sachs 4 (BMS)that physical data at future or past null infinity transform non-t

10、rivially under, in addi-tion to the expected Poincare transformations, an infinite set of diffeomorphisms known assupertranslations. These supertranslations separately shift forward or backward in retarded(advanced) time the individual light rays comprising future (past) null infinity. Recently itwa

11、s shown 5, using new mathematical results 6 on the structure of null infinity, that a1certain antipodal combination of past and future supertranslations is an exact symmetryof gravitational scattering. The concomitant infinite number of supertranslation chargeconservation laws equate the net incomin

12、g energy at any angle to the net outgoing energyat the opposing angle. In the quantum theory, matrix elements of the conservation lawsgive an infinite number of exact relations between scattering amplitudes in quantum gravity.These relations turned out 7 to have been previously discovered by Weinber

13、g in 1965 8using Feynman diagrammatics and are known as the soft graviton theorem. The argumentmay also be run backwards: starting from the soft graviton theorem one may derive boththe infinity of conservation laws and supertranslation symmetry of gravitational scattering.This exact equivalence has

14、provided fundamentally new perspectives on both BMS sym-metry and the soft graviton theorem, as well as more generally the infrared behavior ofgravitational theories 5,7,9-39. Supertranslations transform the Minkowski vacuum to aphysically inequivalent zero-energy vacuum. Since the vacuum is not inv

15、ariant, supertrans-lation symmetry is spontaneously broken. The soft (i.e. zero-energy) gravitons are the asso-ciated Goldstone bosons. The infinity of inequivalent vacua differ from one another by thecreation or annihilation of soft gravitons. They all have zero energy but different angularmomenta.

16、1Although originating in a different context these observations do provide, as discussed in9, 10, a priori reasons to doubt the unquestionable assumptions underlying the informationparadox:(i) The vacuum in quantum gravity is not unique. The information loss argument assumesthat after the evaporatio

17、n process is completed, the quantum state settles down to a uniquevacuum. In fact, the process of black hole formation/evaporation will generically induce atransition among the infinitely degenerate vacua. In principle, the final vacuum state couldbe correlated with the thermal Hawking radiation in

18、such a way as to maintain quantumpurity.(ii) Black holes have a lush head of soft hair. The information loss argument assumes thatstatic black holes are nearly bald: i.e they are characterized solely by their mass M, chargeQ and angular momentum J. The no-hair theorem 40 indeed shows that static bla

19、ck holesare characterized by M, Q and J up to diffeomorphisms. However BMS transformationsare diffeomorphisms which change the physical state. A Lorentz boost for example mapsa stationary black hole to an obviously physically inequivalent black hole with differentenergy and non-zero momentum. Supert

20、ranslations similarly map a stationary black hole1None of these vacua are preferred, and each is annihilated by a different Poincare subgroup of BMS.This is related to the lack of a canonical definition of angular momentum in general relativity.2to a physically inequivalent one. In the process of Ha

21、wking evaporation, supertranslationcharge will be radiated through null infinity. Since this charge is conserved, the sum of theblack hole and radiated supertranslation charge is fixed at all times.2This requires thatblack holes carry what we call soft hair arising from supertranslations. Moreover,

22、when theblack hole has fully evaporated, the net supertranslation charge in the outgoing radiationmust be conserved. This will force correlations between the early and late time Hawkingradiation, generalizing the correlations enforced by overall energy-momentum conservation.Such correlations are not

23、 seen in the usual semiclassical computation. Put another way, theprocess of black hole formation/evaporation, viewed as a scattering amplitude from ItoI+, must be constrained by the soft graviton theorem.Of course, finding a flawed assumption underlying the information loss argument is afar cry fro

24、m resolving the information paradox.That would require, at a minimum, adetailed understanding of the information flow out of black holes as well as a derivation ofthe Hawking-Bekenstein area-entropy law 2, 3, 47. In this paper we take some steps in thatdirection.In the same 1965 paper cited above 8,

25、 Weinberg also proved the soft photon theo-rem. This theorem implies 41, 42, 43, 44, 45 an infinite number of previously unrecognizedconserved quantities in all abelian gauge theories - electromagnetic analogs of the super-translation charges. By a direct analog of the preceding argument black holes

26、 must carry acorresponding soft electric hair. The structure in the electromagnetic case is very similar,but technically simpler, than the gravitational one. In this paper we mainly consider theelectromagnetic case, outlining the gravitational case in the penultimate section. Details ofsoft supertra

27、nslation hair will appear elsewhere.The problem of black hole information has been fruitfully informed by developments instring theory. In particular it was shown 46 that certain string-theoretic black holes storecomplete information about their quantum state in a holographic plate that lives at the

28、horizon. Moreover the storage capacity was found to be precisely the amount predicted bythe Hawking-Bekenstein area-entropy law. Whether or not string theory in some form is acorrect theory of nature, the holographic method it has presented to us of storing informationon the black hole horizon is an

29、 appealing one, which might be employed by real-world blackholes independently of the ultimate status of string theory.Indeed in this paper we show that soft hair has a natural description as quantum pixelsin a holographic plate. The plate lives on the two sphere at the future boundary of the2In the

30、 quantum theory the state will typically not be an eigenstate of the supertranslation chargeoperator, and the conservation law becomes a statement about matrix elements.3horizon. Exciting a pixel corresponds to creating a spatially localized soft graviton or pho-ton on the horizon, and may be implem

31、ented by a horizon supertranslation or large gaugetransformation. In a physical setting, the quantum state of the pixel is transformed when-ever a particle crosses the horizon. The combination of the uncertainty principle and cosmiccensorship requires all physical particles to be larger than the Pla

32、nck length, effectively set-ting a minimum spatial size for excitable pixels. This gives an effective number of soft hairsproportional to the area of the horizon in Planck units and hints at a connection to thearea-entropy law.It is natural to ask whether or not the supertranslation pixels could con

33、ceivably store allof the information that crosses the horizon into the black hole. We expect the supertransla-tion hair is too thin to fully reproduce the area-entropy law. However there are other softsymmetries such as superrotations 48, 49, 50, 11, 12 which lead to thicker kinds of hairas discusse

34、d in 10. Superrotations have not yet been fully studied or understood. It is anopen question whether or not the current line of investigation, perhaps with additional newingredients, can characterize all the pixels on the holographic plate.This paper is organized as follows. Section 2 reviews the an

35、alog of BMS symmetriesin Maxwell theory, which we refer to as large gauge symmetries. The associated conservedcharges and their relation to the soft photon theorem are presented. In section 3 we constructthe extra terms in the conserved charges needed in the presence of a black hole, and showthat th

36、ey create a soft photon, i.e. excite a quantum of soft electric hair on the horizon. Insection 4 we consider evaporating black holes, and present a deterministic formula for theeffect of soft hair on the outgoing quantum state at future null infinity. In section 5 weconsider physical processes which

37、 implant soft hair on a black hole, and argue that a hairmuch thinner than a planck length cannot be implanted. In section 6 we discuss the gaugedependence of our conclusions. Section 7 presents a few formulae from the generalizationfrom large gauge symmetries and soft photons to BMS supertranslatio

38、ns and soft gravitons.We briefly conclude in Section 8.2Electromagnetic conservation laws and soft symmetriesIn this section we set conventions and review the conservation laws and symmetries of abeliangauge theories in Minkowski space.The Minkowski metric in retarded coordinates (u,r,z, z) near fut

39、ure null infinity (I+)readsds2= dt2+ (dxi)2= du2 2dudr + 2r2z zdzd z,(2.1)4where u is retarded time and z zis the round metric on the unit radius S2. These are relatedto standard Cartesian coordinates byr2= xixi,u = t r,xi= r xi(z, z).(2.2)Advanced coordinates (v,r,z, z) near past null infinity (I)

40、areds2= dv2+ 2dvdr + 2r2z zdzd z;r2= xixi,v = t + r,xi= r xi(z, z).(2.3)I+(I) is the null hypersurface r = in retarded (advanced) coordinates. Due to the lastminus sign in (2.3) the angular coordinates on I+are antipodally related3to those on Isothat a light ray passing through the interior of Minko

41、wski space reaches the same value ofz, z at both I+and I. We denote the future (past) boundary of I+by I+(I+), and thefuture (past) boundary of Iby I+(I). We use conventions for the Maxwell field strengthF = dA and charge current one-form j in which d F = e2 j (or aFab= e2jb) with e theelectric char

42、ge and the Hodge dual.Conserved charges can be constructed as surface integrals of F near spatial infinityi0. However care must be exercised as F is discontinuous near i0and its value dependson the direction from which it is approached. For example, approaching I+from I+, theradial Lienard-Wiechert

43、electric field for a collection of inertial particles with charges ekandvelocities vkis, to leading order at large r,Frt=e4r2Xkek(1 v2k)(1 vk x)2,(2.4)while going to I+from IgivesFrt=e4r2Xkek(1 v2k)(1 + vk x)2.(2.5)All fields may be expanded in powers of1rnear I. We here and hereafter denote the coe

44、fficientof1rnby a superscript (n). In coordinates (2.2),(2.3), the electric field in general obeys theantipodal matching conditionF(2)ru(z, z)|I+= F(2)rv(z, z)|I+.(2.6)3In coordinates with z z= 2/(1 + z z)2, the antipodal map is z 1/ z.5(2.6) is invariant under both CPT and Poincare transformations.

45、The matching conditions (2.6) immediately imply an infinite number of conservationlaws. For any function (z, z) on S2the outgoing and incoming charges defined by4Q+=1e2ZI+ FQ=1e2ZI+ F(2.7)are conserved:Q+= Q.(2.8)Q+(Q) can be written as a volume integral over any Cauchy surface ending at I+(I+).In t

46、he absence of stable massive particles or black holes, I+(I) is a Cauchy surface. Hencein this caseQ+=1e2ZI+d F +ZI+ j,Q=1e2ZId F +ZI j(2.9)Here we define on all of I by the conditions u = 0 = v. The first integrals on the righthand sides are zero modes of the field strength F and hence correspond t

47、o soft photons withpolarizations d. In quantum field theory, (2.8) is a strong operator equality whose matrixelements are the soft photon theorem 41, 42. The special case = constant corresponds toglobal charge conservation.The charges generates a symmetry under which the gauge field Azon I+transform

48、s as41, 425?Q+,Az(u,z, z)?I+= iz(z, z).(2.10)We refer to this as large gauge symmetry. It is the electromagnetic analog of BMS super-translations in gravity. This symmetry is spontaneously broken and the zero modes of Az the soft photons are its Goldstone bosons. An infinite family of degenerate vac

49、ua areobtained from one another by the creation/annihilation of soft photons.4Here is antipodally continuous from I+to I+. A second infinity of conserved charges involving thereplacement of F with F 45 leads to soft magnetic hair on black holes. This is similar to soft electric hairbut will not be d

50、iscussed herein.5The commutator follows from the standard symplectic two-form = 1e2RA dA, where A is aone-form on phase space and is a Cauchy surface.63Conservation laws in the presence of black holesIn this section we construct the extra term which must be added to the volume integralexpression (2.