Classical Limit of Black Hole Quantum N-Portrait and BMS Symmetry
aa r X i v : . [ h e p - t h ] S e p MPP–2015–215 , LMU–ASC 59/15
Classical Limit of Black Hole Quantum N -Portrait and BMS Symmetry Gia Dvali a,b,c , Cesar Gomez b,e and Dieter L¨ust a,b a Arnold Sommerfeld Center for Theoretical PhysicsDepartment f¨ur Physik, Ludwig-Maximilians-Universit¨at M¨unchenTheresienstr. 37, 80333 M¨unchen, Germany b Max-Planck-Institut f¨ur PhysikF¨ohringer Ring 6, 80805 M¨unchen, Germany c Center for Cosmology and Particle PhysicsDepartment of Physics, New York University,4 Washington Place, New York, NY 10003, USA e Instituto de F´ısica Te´orica UAM-CSICUniversidad Aut´onoma de Madrid, Cantoblanco, 28049 Madrid, Spain
Abstract
Black hole entropy, denoted by N , in (semi)classical limit is infinite.This scaling reveals a very important information about the qubitdegrees of freedom that carry black hole entropy. Namely, the mul-tiplicity of qubits scales as N , whereas their energy gap and theircoupling as 1 /N . Such a behavior is indeed exhibited by Bogoliubov-Goldstone degrees of freedom of a quantum-critical state of N softgravitons (a condensate or a coherent state) describing the black holequantum portrait. They can be viewed as the Goldstone modes of abroken symmetry acting on the graviton condensate. In this pictureMinkowski space naturally emerges as a coherent state of N = ∞ gravitons of infinite wavelength and it carries an infinite entropy.In this paper we ask what is the geometric meaning (if any) of theclassical limit of this symmetry. We argue that the infinite- N limitof Bogoliubov-Goldstone modes of critical graviton condensate is de-scribed by recently-discussed classical BMS super-translations brokenby the black hole geometry. However, the full black hole informationcan only be recovered for finite N , since the recovery time becomesinfinite in classical limit in which N is infinite. [email protected] [email protected] Entropy Scaling and Information Qubits
Black hole entropy scales as area in Planck units [1], N = R L P . (1)Here R = G N M is the gravitational radius of black hole of mass M and G N is the Newton’s constant. L P is Planck length, which in terms of G N andPlanck’s constant ~ is defined as L P ≡ ~ G N . (2)We have set the speed of light and all the other numerical factors to one.The Bekenstein’s formula tells us that the black hole is a quantum statewith degeneracy of micro-states scaling exponentially with N . The crucialquestion is: What are the microscopic qubit degrees of freedom describingthe above degeneracy?In answering this question, first notice that some important informa-tion about these qubits can be gained from analyzing the classical andsemi-classical limits of the Bekenstein’s formula and matching it with well-established known properties of classical black holes [2–5].The limits of interest are the ones in which the quantum back-reaction onclassical geometry vanishes. We shall distinguish the two limits of this sort.The first we shall refer to as the semi-classical limit . In this limit, we keepboth ~ and R finite, since we want to keep notions of geometry as well as ofquantum mechanics well-defined. The only consistent way of achieving thisis to take M → ∞ , G N →
0, but in such a way that we keep their product( R ) finite. Thus, with finite ~ , in order to keep geometry exact, we musttake the black hole to be infinitely massive , and simultaneously, gravity tobe infinitely weak .The second limit, is the classical one, ~ →
0. In this limit, we can keepthe geometry exact, even for finite values of M and G N .We now observe that no matter which of the two limits we choose, wehave L P → ) Number of qubits scales as N ;
2) Their energy gap and their coupling scales as /N .Putting it simply, although in classical limit the number of qubits becomesinfinite, they decouple and resolving the degeneracy of micro-states becomesinfinitely-hard. Correspondingly, the no-hair property is valid for any finiteobservation time of a classical black hole.The composite multi-graviton portrait of a black hole [2, 3] provides anatural candidate for information-carrier qubit degrees of freedom in form ofthe Bogoliubov-Goldstone degrees of freedom of the soft-graviton condensate.According to this picture, the black hole is well-described as a bound-state of N soft gravitons. Examining the coupling of these gravitons, it is evident thatthe condensate is at a quantum critical point. The qubit degrees of freedomare then identified with N Bogoliubov degrees of freedom that populate thespectrum within 1 /N energy gap. For large N , their coupling scales as 1 /N .These degrees of freedom can be described as Goldstone modes of cer-tain rank- N symmetry group acting on the graviton condensate [4–6] . Ithas been appreciated that Goldstone interpretation of these modes, withthe number of broken generators as well as the spontaneous breaking orderparameter scaling as N , automatically accounts for the right 1 /N scaling be-havior both of the energy gap as well as of the coupling of Goldstone-qubits.Correspondingly, this picture naturally accounts for various time-scales ofinformation-processing, such as, generating chaos and scrambling informa-tion within the time t scr ∼ √ N L P ln ( N ) [8], as well as, generating largeentanglement at the quantum critical point [4–7]. The information-recoverytime scales as t inf ∼ N / L P and becomes infinite both in classical and semi-classical limits.Thus, in this picture, a classical black hole carries infinite amount of hair,but it is undetectable, because the resolution time is infinite. For finite N , thehair carries all the information about the black hole quantum state, includ-ing some global charges that it may swallow [10]. The same hair manifestsitself in 1 /N -corrections to Hawking radiation [2, 3], which give deviations Here we use the word rank to refer to the number of generators of the algebra thatcreates Goldstone modes. At this point we do not need to commit ourselves to anyparticular realization of this symmetry. /N -corrections to the probe particle scattering at the critical gravi-ton condensate [9]. These corrections measure the 1 /N quantum deviationfrom the classical metric motion. The analogous 1 /N -corrections correctionsmeasure deviation from entropy suppression, e − N , in black hole formation inscattering of two energetic gravitons into the N soft ones [11].Notice, that for finite N the symmetry need not be exact, and Goldstonesare not exactly gapless, but the energy gap must vanish for infinite N . Below,in order to keep our discussion maximally general, we shall simply refer tothis symmetry as rank- N group.The question we would like to ask in the present paper, is: What isthe geometric interpretation, if any, of this symmetry, with spontaneously-broken N generators, in the classical limit? We shall try to argue that theanswer is the classical BMS symmetry [12], recently studied by Stromingeret al [14–16].In order to answer this question, we shall first make the connection tothe limiting case of Minkowski space, describing it as a coherent state ofgravitons. We shall start with the toy model of pure scalar gravity first. Consider amassless scalar field φ ( x µ ) in 3-space dimensions. In order to regularize theproblem, let us introduce a small mass m and then take the zero mass limit.The Lagrangian is, L = 12 (cid:0) ( ∂ µ φ ) − m φ (cid:1) . (3)In m = 0 zero limit, we have a continuous symmetry φ → φ + v , where v isan arbitrary constant. The Noether current of this symmetry is J µ ≡ ∂ µ φ ,which for m = 0 is conserved by the equation of motion. For nonzero mass,the divergence of the current is proportional to the mass term, which breaksthe shift symmetry explicitly ∂ µ J µ = − m φ . We shall work in the finitevolume V = (2 πR ) and then take the infinite volume limit. The standard4ode-expansion of the field φ is, φ ( x ) = Z d ( kR ) √ V ω k ( e − i ( ω k t − ~k~x ) a ~k + e i ( ω k t − ~k~x ) a + ~k ) . (4)with ω k = p m + | k | . The charge operator is, Q ≡ Z d xJ = Z d x∂ t φ ( x ) = − iq ( e − imt a − e imt a +0 ) (5)where q ≡ √ V m . Notice, in this parameterization Q has a dimensionalityof φ − . This is because the shift parameter v has the same dimensionalityas φ . Now acting with the shift operator on the vacuum state | i , we createanother state | v i , | v i = e − ivQ | i . (6)Notice that from the form of the shift generator (5), it is clear that thestate | v i is a coherent state of zero momentum quanta, characterized bythe complex parameter √ N e iθ , where the modulus measures the occupationnumber N = ( qv ) and the phase is θ = mt . It can be written in the form, | v i = e − N ∞ X n =0 ( N e i mt ) n √ n ! | n i , (7)where | n i are Fock states of zero momentum quanta of occupation number n . Notice, that from the scalar product of two coherent states |h v | v ′ i| = e − q ( v − v ′ ) , (8)it is clear that, for finite v, v ′ , they become orthogonal only for q → ∞ limit.If in this limit we keep m finite, the resulting state will describe a coherentlyoscillating scalar field of amplitude v and frequency m . The expectationvalue of the scalar field over such a coherent state behaves as a classical field, h v | φ | v i = ve − imt + h . c . . (9)However, if simultaneously, we take the limit, m →
0, the states withdifferent values of v become degenerate vacua. The expectation value of thefield φ in such a vacuum is h v | φ | v i = v . (10)Thus, we see that the vacua with constant values of the scalar field canbe understood as coherent states with infinite occupation number of infinitewave-length quanta. 5 .2 Gravitational Minkowski Vacua as Coherent States We are now ready to apply the coherent state picture to Minkowski space intheory with gravity. Since for Minkowski space all the curvature invariantsvanish and we are dealing with infinitely soft gravitons, we shall work withinthe weak field expansion. Consider a linearized sourceless equation for spin-2field, E h µν − m ( h µν − η µν h ) = 0 , (11)where E h µν ≡ (cid:3) h µν − η µν (cid:3) h − ∂ µ ∂ α h αν − ∂ ν ∂ α h αµ − ∂ µ ∂ ν h − η µν ∂ α ∂ β h αβ is the linearized Einstein tensor and h ≡ h αα . As we did in the scalar case,we have regularized the system by adding a small Pauli-Fierz mass term,which we shall later take to zero. As it is well known, such a regularizationintroduces the 3 additional propagating degrees of freedom in form of twohelicity-1 and one helicity-0 polarizations.In the basis in which kinetik terms are diagonal the decomposition ofmassive graviton h µν in terms of Einsteinian helicity-2 tensor component e h µν , helicity-1 vector A µ and a helicity-0 scalar φ has the following form, h µν = e h µν + ∂ µ A ν + ∂ µ A ν + 16 η µν φ + 13 ∂ µ ∂ ν m φ. (12)The massive graviton h µν is invariant under the gauge shift A µ → A µ − ξ µ , ˜ h µν → ˜ h µν + ∂ µ ξ ν + ∂ ν ξ µ , which acts on the Einsteinian component˜ h µν as the usual linear coordinate transformation. Below, we shall set thehelicity-1 component to zero.For m = 0 there exists a conserved current: J γ ( µν ) = ∂ γ h µν − η µν ∂ γ h − ( ∂ µ h γν + ∂ ν h γµ )+ ( η γµ ∂ ν + η γν ∂ µ ) h + η µν ∂ β h γβ . The divergence of this currentis simply the linearized Einstein tensor, ∂ γ J γ ( µν ) = E h µν , which vanishes bythe equation of motion of the massless theory.Moreover, notice that the trace of the current over µ, ν indexes, J γ ≡ η µν J γ ( µν ) = 2( ∂ ν h γν − ∂ γ h ) , (13)vanishes by the Pauli-Fierz constraint, which can be obtained by taking thedivergence of the equation of motion (11). We shall use this fact to makea shortcut for interpreting Minkowski as a coherent state. The current (13)vanishes on the Pauli-Fierz massive graviton h µν , but it is non-zero on theEinsteinian ˜ h µν and helicity-zero φ components separately. We are not inter-ested in the total Pauli-Fierz graviton, but only in its Einsteinian component6 h µν , which decouples from the rest in the massless limit. Hence, knowing thatthe action of the current on ˜ h µν and φ are exactly opposite, we can read offthe shift of massless graviton from the shift of φ .From the form of the charge, Q ≡ − Z d xJ = Z d x (cid:16) ∂ t ˜ h − ∂ ν ˜ h tν (cid:17) + Z d x∂ t φ , (14)it is clear that its action on φ is identical to (5). Thus the action on ˜ h µν isexactly the opposite, because Q must annihilate on any physical state of h µν .We can now proceed exactly as in the case of a pure scalar field. We acton the vacuum with the shift operator, | v i = e − ivQ | i , (15)and take the limit m →
0. The difference from the pure scalar case is thatwe now create the coherent states containing the infinite occupation numbersof both φ and ˜ h µν quanta of infinite wave-length. The expectation values ofthe quantum operators ˆ˜ h µν and ˆ φ on the state | v i satisfies, h v | ˆ˜ h µν | v i = − η µν h v | ˆ φ | v i = − η µν v . (16)In this way, the expectation value of the full would-be massive graviton van-ishes. However, in the massless limit, the independent physical degrees offreedom are φ and ˜ h µν and we can focus exclusively on ˜ h µν . For the probesources coupled to ˜ h µν the coherent state | v i is the exact Minkowski state.As it is clear from (16), the expectation value of Einsteinian graviton overthis state is nothing but a classical Minkowski space metric.Thus, we resolve Minkowski space as a coherent state of infinite occupa-tion number of infinite wave-length gravitons. Clearly, this state exhibits aninfinite degeneracy with respect to the change of this number. There exist a nice holographic interpretation of the previous exercise. Indeedfor finite volume V = R and finite value of the mass m the energy of thecoherent state | v i is E = mN = m R v . Since this energy is localized ina volume of size R we can use Bekenstein bound [13] on the entropy as ER ~ units. If now we identify this entropy with N and we try to saturatethe bound for the coherent state | v i we get the following relation between m and R , mR = 1 . (17)In other words, in this case the coherent state | v i saturates the Bekensteinbound. In the limit m = 0 we recover the infinite-volume limit and thecorresponding coherent state with infinite N . However, this double-scalinglimit m = 0 and R = ∞ with mR = 1 leads to an infinite total energy O ( Rv ) that vanishes only for v = 0. This is very natural, since the double-scaling limit corresponds to the interpretation of the Minkowski space as ofan infinitely-massive black hole. Although the total mass is infinite, all thecurvature invariants locally vanish. In this case, the holographic nature ofMinkowski space is simply a memory about the holographic nature of theinfinitely-massive black hole that it represents.In summary, we can think about the vacuum | v i as the double-scalinglimit of the finite-volume coherent state saturating Bekenstein bound. Thismakes the Minkowski vacuum essentially holographic, which matches thefact that it is precisely in this limit that Minkowski can be identified with aninfinite mass black hole.The above picture of Minkowski space also emerges as a limit from therepresentation of de Sitter space as of coherent state of gravitons of fre-quencies given by the Hubble parameter [9]. There it is shown that thisdescription fixes H = m , where H is the Hubble parameter (i.e., cosmolog-ical constant in Planck units). This implies that the Bekenstein bound issaturated for the Hubble volume for arbitrarily-small value of H . The zerocosmological constant limit is m = 0, leading to Minkowski as a coherentstate of infinitely-soft gravitons. In any case, the key point to be stressed isthat the coherent state representation of Minkowski space naturally capturesthe basic ingredient of holography. N limit of Bogoliubov-Goldstone Modesand BMS Symmetry. The fact that we can understand Miknowski space as infinite occupationnumber coherent state of zero momentum gravitons allows us to partially ad-dress the question about the classical limit of rank- N algebra. Indeed, since8inkowski can be viewed as a metric seen by an observer of an infinitely-massive black hole, it is natural to identify the large- N limit of rank- N symmetry of the quantum portrait as a Bondi, Metzner and Sachs (BMS)symmetry [12] of Minkowski vacuum in the spirit of [14–16]. With this identi-fication, the Bogoliubov-Goldstone modes that store black hole information,in N → ∞ limit are mapped on infinitely-soft graviton modes created byBMS supertranslations.However, the story is different for the classical limit of a finite massblack hole. As discussed above, despite the fact that N becomes infinite,the radius R stays finite. Therefore, the corresponding Goldstone modesmust carry the wave-length ∼ R . Thus, they cannot come purely from BMSdegeneracy of the asymptotic Minkowski vacuum. Thus, classical limit of thesymmetry group, must be spontaneously broken by finite R effects and not bythe asymptotic space [18]. The fact that these Goldstones have wavelength R , automatically fixes their number for any finite ~ to the value N , as itis predicted by the quantum portrait. Thus, it is natural to identify theGoldstone modes of the BMS group with the ~ → I ± . Its topology is R ⊗ S with R parametrized by advanced andretarded time coordinate u and v respectively. The S can be visualized asthe section of a light cone with the null infinity. We can parametrize this S with spherical coordinates θ and φ .The BMS supertranslations on I + are simply the transformations( u, θ, φ ) → ( u + f ( θ, φ ) , θ, φ ) , (18)with f an arbitrary real function. The super translation is simply a transla-tion in the retarded time dependent on the angular direction in S . The pre-vious definition implies that we have as many generators of supertranslationsas spherical harmonics, so we can formally characterize the supertranslations9s T l,m . One of the most interesting outputs of the recent research on theseasymptotic symmetries has been to realize that the Minkowski vacuum | i breaks spontaneously the invariance under supertranslations. Or in otherwords, the states obtained by acting on the vacuum T l,m | i = | l, m i , (19)which are soft gravitons of infinite wave length, can be interpreted as Gold-stone bosons [14, 16].Interestingly, this picture resonates with the former discussion on Minkowskias a coherent state of infinite wave-length gravitons. In fact, as alreadypointed out, this coherent state model saturates the Bekenstein bound for aformal entropy defined as the average number of quanta N of the correspond-ing coherent state. On the other hand, this is precisely the scaling regime inwhich Minkowski coherent state represents the infinite mass limit of a blackhole. Then, it becomes clear, why the entropy of the corresponding coherentstate must be infinite in this regime. We could think of this infinite numberas being associated with the infinite number of BMS Goldstone bosons.The canonical example of holography has been always the black hole itself.So, if the previous picture captures the essence of the holographic degrees offreedom, we must look for an analog of the BMS super translations actingon the black hole horizon. This is indeed what was announced in [18]. Inthese conditions we could think of the analog of (19), namely T ( l, m ) | BH i = | l, m ; BH i (20)for | l, m ; BH i the black hole Goldstone bosons. Since in this case we aredealing with the horizon, it is natural to think of these Goldstone bosonsas having wave length equal to the black hole radius, R . Moreover, sincefor finite ~ the black hole entropy is finite, we need somehow to reducethe effective number of these Goldstone bosons to be exactly the black holeentropy, N .At first sight the most interesting physics question could be to unveilwhy these finite wave-length graviton modes are Goldstone bosons relativeto the black hole quantum state. If we assume that they are, then they canprovide a family of inequivalent black hole quantum states nearly-degeneratein energy, and therefore, the right candidates to define the black hole entropy . The connection of BMS and the information paradox was already pointed out in [15].
10n what follows, we shall like to make contact between this geometricapproach to the black hole entropy and some of the main features of theportrait model of the black hole [2, 3]. In the black hole portrait the keyingredient is to identify the black hole with a self sustained condensate ofgravitons at a quantum critical point. In other words, we model the blackhole as a many body system of gravitons, but at a very special point, wherea quantum phase transition takes place. What makes the black hole spacial,is this quantum criticality that manifest itself precisely as the appearanceof nearly-gapless N Bogoliubov-Goldstone modes. These Goldstone bosonsare not there when we are away from criticality, as it could be the case fora star or some other massive object. They only appear when we reach thecriticality associated with the black hole formation .In essence this microscopic picture is extremely simple and points out thatonce we turn on gravity and we track the gravitational self energy in quantummechanical terms, the quantum state of the system breaks spontaneouslysome rank- N symmetry when it becomes a black hole, i.e., when the gravitoncondensate is at the critical point.This spontaneous symmetry breakdown comes with a set of Goldstonebosons that likely satisfy all the conditions to be the microscopic quantumversion of the black hole super translation Goldstone bosons. In this sense,the black hole horizon as a geometric notion is simply reflecting the criticalityof the many body system of gravitons, once we create a correspondencebetween the many-body Goldstone modes and the special BMS symmetriesof the horizon.In summary, once we decide to promote Goldstone to be the key wordunderlying black hole entropy we must look for the microscopic meaning ofthe underlying symmetry breakdown. To our view, the criticality of the blackhole portrait naturally fits the goal.We would like to stress that there are effects, captured by the quantum N -portrait, that are expected not to be visible in the classical BMS pictureand their account requires the finite N resolution. For example, for finite N the symmetry need not be exact and can (and in general will) be brokenexplicitly by 1 /N -effects. This explicit breaking gives a finite energy gap It is worth to notice that already in some simple toy models of critical Bose-Einsteincondensates we can assign l, m standard quantum numbers to the Goldstone modes andtrack how these many-body states become gapless at the critical point in N → ∞ limit [4].This illustrates the crucial role of quantum criticality in generating gapless Goldstonemodes.
11o Goldstone modes promoting them into pseudo-Goldstones. This is fullycompatible with the entropy scaling, since the (pseudo)Goldstone modes donot have to be exactly gapless, but rather populate the energy gap ∼ /N .Moreover, precisely these finite N effects allow the recovery of black holeinformation within the finite time. The information processing time-scales,such as the scrambling time for the critical condensate ( ∼ ln ( N )) [8], scalewith N . Therefore, understanding the finite N portrait is crucial for theinformation-recovery process.In order to make the closed contact between the large- N portrait [2, 3]and [14–16] one natural way would be to consider scattering of an externalsoft graviton at the critical N -graviton condensate and then take the semi-classical limit. Moreover, the BMS group can manifest itself also in blackhole formation in scattering of two energetic gravitons into the N soft ones,for which the right entropy suppression appears exactly when the N gravi-tons are at the quantum critical point [11]. At this point, the momentaof these gravitons are R − , with R being equal to the gravitational radiuscorresponding to the center of mass energy √ s of the process.Since, as we have argued, the Minkowski space corresponds to an in-finitely heavy black hole, there is at least one well defined limit, namely N = ∞ , R = ∞ , for which the scattering amplitude of [11] can be mappedto the infinitely-soft graviton processes of the type [14–16] controlled by BMSsupertranslations of asymptotic Minkowski space.The interesting regimes to be explored are with finite R , and with N either finite or infinite. In both cases, new types of corrections appear, bothdue to finite softness of gravitons and due to finite N . One possible way totake them into the account, could be via new kind of soft theorems, whichas shown recently [17], can arise from sub-leading contributions due to finitesoftness.It is also tempting to conjecture that for finite R , at least to leading orderin 1 /N , the scattering process can be understood in terms of BMS-typeGoldstone generators that are broken by finite- R geometry, as opposed tothe asymptotic flat space. Then these would be the natural candidates to beidentified with Bogoliubov-Goldstone modes of large- N graviton condensate.We leave this for the future work.The important point however is that finite N effects are crucial for theinformation recovery and must be taken into account.12 cknowledgements We like to thank I. Bakas and R. Isermann for useful discussions. The workof G.D. was supported by Humboldt Foundation under Alexander von Hum-boldt Professorship, by European Commission under ERC Advanced Grant339169 “Selfcompletion” and by TRR 33 ”The Dark Universe”. The workof C.G. was supported in part by Humboldt Foundation and by Grants:FPA 2009-07908, CPAN (CSD2007-00042) and by the ERC Advanced Grant339169 “Selfcompletion” . The work of D.L. was supported by the ERCAdvanced Grant 32004 “Strings and Gravity” and also by TRR 33.