Black Hole Evaporation, Quantum Hair and Supertranslations
BBlack Hole Evaporation, Quantum Hair andSupertranslations
César Gómez ∗ and Sebastian Zell † Instituto de Física Teórica UAM-CSIC, Universidad Autónoma de Madrid,Cantoblanco, 28049 Madrid, Spain Arnold Sommerfeld Center, Ludwig-Maximilians-Universität,Theresienstraße 37, 80333 München, Germany Max-Planck-Institut für Physik, Föhringer Ring 6, 80805 München, Germany
May 8, 2018
Abstract
In a black hole, hair and quantum information retrieval are interrelatedphenomena. The existence of any new form of hair necessarily implies the ex-istence of features in the quantum-mechanically evaporated radiation. There-fore, classical supertranslation hair can be only distinguished from globaldiffeomorphisms if we have access to the interior of the black hole. Indirectinformation on the interior can only be obtained from the features of thequantum evaporation. We demonstrate that supertranslations ( T − , T + ) ∈ BM S − ⊗ BM S + can be used as bookkeepers of the probability distributionsof the emitted quanta where the first element describes the classical injectionof energy and the second one is associated to quantum-mechanical emission.However, the connection between T − and T + is determined by the interiorquantum dynamics of the black hole. We argue that restricting to the di-agonal subgroup is only possible for decoupled modes, which do not bringany non-trivial information about the black hole interior and therefore do notconstitute physical hair. It is shown that this is also true for gravitational sys-tems without horizon, for which both injection and emission can be describedclassically. Moreover, we discuss and clarify the role of infrared physics inpurification. ∗ [email protected] † [email protected] a r X i v : . [ h e p - t h ] M a y ontents The Puzzle of Black Holes
A black hole is an extraordinary physical system. While in a classical theory,it is extremely simple for an outside observer, as a consequence of the no-hair-theorem (see e.g. [1]), its internal quantum complexity measured by the Bekenstein-Hawking-entropy [2, 3] N = M /M p is enormous. Both properties are obviouslyinterrelated. The black hole entropy appears because many different matter con-figurations can collapse into the same black hole geometry. The no-hair-theoremprevents an outside observer from resolving these differences which remain hid-den behind the horizon. Quantum-mechanically, the black hole evaporates [3] and2nitarity requires that along the evaporation process the black hole should deliverback the information which was classically hidden in its interior [4]. This meansthat although the classical metric has no hair, the evaporation products shouldhave features which compensate for this lack of information. In other words, thequantum radiation emitted during the black hole evaporation should carry thesame information which the classical no-hair-theorem prevents us to extract fromthe geometry.In the last twenty years it has become popular to use the AdS/CFT correspondenceas strong indication of the unitarity of black hole evaporation. However, this hopewill not be fulfilled until counting with the CFT dual of a small evaporating blackhole has been achieved. More generally, we shall argue in this note that the solutionto the evaporation problem requires to have a microscopic model of the black holeas a quantum system – whether obtained from AdS/CFT or differently. In [5,6], wehave developed such a model that, among other things, indicates the existence offorms of quantum hair effects of order 1 /N . Moreover, and in a model independentway, it is easy to see that taking into account the change of the black hole mass dueto Hawking evaporation leads to deviations from featureless emission on preciselythis order of magnitude [7]. Classical BMS-Hair
Recently, a new way to attack the problem has been suggested [8, 9] based onasymptotic BMS-symmetries [10]. This approach has received widespread atten-tion (see e.g. [11–22]). In particular, a potential new form of classical hair for ablack hole has been proposed [23, 24]. The idea is simply the following. One startswith a black hole of mass M and injects an energy µ in the form of incoming radia-tion with some angular features. This incoming radiation can be associated witha supertranslation in
BM S − which we denote by T − . Classically, the resultingsystem is a black hole with total mass M + µ but supertranslated by T − . One cando the same construction with identical µ but with different angular features, i.e.different supertranslations T − i , to obtain a family of different metrics all of themwith the same ADM-charges. Thus, it seems that one can indeed define classicalhair if all these metrics sharing the same ADM-charges are physically inequiva-lent. At the classical level, this means that those metrics are not just the samemetric written in different coordinate systems, i.e. that they are not related bya globally defined diffeomorphism. As we shall elucidate, the problem with thisform of classical hair is that for an observer outside, there is no way to decide if allthese metrics are different or simply the same metric in different coordinates. Inorder to decide that, the observer needs to have information about the interior ofthe black hole. In summary, defining hair by means of the classical gravitationalmemory associated to some incoming radiation is only operative if somehow wecan have extra information about the memory effects in the interior of the blackhole which is, in a different guise, the essence of the no-hair-theorem. All quantities will be properly defined at the beginning of section 2. a) In the classical limit, the change of the black hole only depends on the total absorbedenergy µ , in line with the no-hair-theorem. The black hole cannot emit.(b) Also in the semi-classical limit, the change of the black hole only depends on thetotal absorbed energy µ . The black hole can evaporate, but the evaporation products arefeatureless. In particular, they are distributed isotropically.(c) In the fully quantum treatment, the incoming radiation F in interacts with the micro-scopic description of the black hole. In doing so, it changes the microstate of the blackhole so that it can emit radiation F out with non-trivial angular profile. Figure 1: Absorption of a wave with energy profile F in by a black hole of mass M and possible subsequent evaporation in the classical, semi-classical and fullyquantum treatment.Fortunately, there is an indirect way to decide from the outside whether two blackhole metrics defined by injecting the same amount of energy but with differentangular features are physically different or not. We can just wait until the blackholes emit some radiation and compare the radiation produced by the two blackholes. For simplicity, we restrict ourselves in our discussion to a pure theory ofgravity in which only gravitational radiation can be emitted. The correspondingprocess is depicted in figure 1, where we distinguish the classical, semi-classicaland purely quantum contributions. The first thing to be noticed is that this testis purely quantum in the sense that only quantum-mechanically, the black holecan emit radiation. The second thing is that the information we can get on theemitted quantum radiation by actual measurements is necessarily encoded in theform of probability distributions. Thus, if those black holes defined by different T i − are indeed different, we should expect that the corresponding quantum probabilitydistributions are also different. 4 nsufficiency of Calculation in Classical Background Metric It is natural to expect that this difference has a non-trivial projection on devi-ations from isotropy, i.e. that the emitted quanta carry angular features. Thenwe obtain quantum probability distributions P i ( θ, φ ) from the measurement of theradiated quanta. We can use those to define a classical supertranslation T + i . Bythat we simply mean a classical supertranslation with the flow of emitted radia-tion determined by the measured quantum probability functions P i ( θ, φ ). In thissense, the former experiment produces a set of couples ( T − i , T + i ) where the firstsupertranslation in BM S − is classical and the second one in BM S + is determinedby the quantum probability distribution. From this point of view, if the classical T − i really implants hair, then the quantum T + i should be non-trivial, i.e. containspherical harmonics with l ≥
2. The crucial point is that this behavior cannot beachieved by the standard Hawking computation performed in a supertranslatedSchwarzschild metric, i.e. as pair creation in the background vacua defined by thenear horizon geometry. The reason is that the supertranslation acts as a diffeo-morphism near the horizon and does not change the local geometry. Therefore, itdoes not suffice if the P i only depend on the injected radiation and the geometryof the black hole. Instead, they must also depend on its internal dynamics.We can make the argument a bit more quantitative and assume that from the wholeenergy µ injected a fraction ˜ µ is associated to angular features. This means thatthe part of the incoming classical flow F in with angular labels l ≥ R d Ω F in with a value equal to ˜ µ . Clearly, ˜ µ = 0 would correspond to the injectionof featureless radiation. In order to parametrize how the P i depend on ˜ µ and theinternal structure of the black hole, we shall use the typical number of quantumconstituents of the black hole. In this sense, we expect P i ( θ, φ ; ˜ µ, N ) where thelabel i refers to the dependence on the incoming T − i and where we identify thenumber of quantum constituents of the black hole with the entropy N .Then the natural dimensionless parameter measuring the dependence of P i on theinternal structure is ˜ µ/ √ N where √ N is the black hole mass in Planck units. Inthis setup, angular features in the evaporation, i.e. finite N effects in P i ( θ, φ ; ˜ µ, N ),depend on the black hole microscopic model. Those will define a couple ( T − i , T + i )generically not in the diagonal subgroup BM S of BM S − ⊗ BM S + . Thus, thefirst important message of our note will be that the information about T − i cannotdetermine the quantum probability distribution T + i , i.e. we cannot predict thequantum probability distribution solely from the incoming radiation implantingthe hair. In this case, the associated supertranslation will not support angular features and will onlyproject on the l = 0 , ubleading Soft Modes We can investigate how this situation changes in the semi-classical limit M → ∞ ,i.e. N → ∞ , in which the Hawking computation becomes exact. In this case, theenergy associated to features becomes zero so that angular features can only beencoded in zero-energy modes. The effective decoupling of these modes will leadto a P i identical to the incoming radiation. This produces couples ( T − i , T + i ) in thediagonal subgroup BM S . In more concrete terms, the lim N →∞ P i ( θ, φ ; ˜ µ, N ) willonly capture local horizon physics or zero-energy modes. This brings us to our second point, namely how the actual features of the quantumprobability distribution P i depend on infrared physics. We know that in gaplesstheories such as gravity, evaporation interpreted as a S -matrix process has a zeroprobability amplitude without any accompanying soft gravitons. In order to obtaina finite answer, one has to include the emission of a certain class of soft radiation,namely IR-modes. However, this fact by no means implies that this companionradiation should carry the angular features that we need to purify the evaporation.On the contrary, we know from infrared physics that IR-radiation is only sensitiveto the initial and final scattering states. It is independent of the details of theprocess or in our case of the microscopic details of the black hole, i.e. cannotresolve the microstate.Independently of the question to what extent IR-radiation and hard quanta arecorrelated, we can quantitatively estimate the amount of information we couldlose when we integrate over unresolved IR-modes. From well-known results ofinfrared physics it follows that their number only grows logarithmically with theresolution scale (cid:15) , i.e. n soft ∼ − ln (cid:15) . However, what we have discussed impliesthat the natural resolution scale of features should be (cid:15) ∼ /N . Thus, the secondimportant message of our note is that unresolved IR-modes cannot account forthe bulk of information in black hole evaporation, but could only contribute as asubleading logarithmic correction. The part which carries features is the part ofthe radiation that can be resolved and that depends not on the infrared divergencesbut on the inner structure of the black hole, or in scattering language, on the detailsof the scattering process. A possible candidate is soft non-IR radiation, which isindependent of infrared divergences. As it should be, non-IR radiation dependson the details of the scattering process so that it cannot be predicted without amicroscopic theory of the black hole. It is important to stress that the pseudo Goldstone-Bogoliubov modes identified in [25] arenot equivalent to near horizon diffeomorphisms and consequently are good microscopic candidatesto describe the low energy effective changes of the microstate of the black hole during the processof absorption and evaporation. See [26] for a recent suggestion for purification by infrared modes. ummary and Outline In summary, non-trivial hair can be only defined by couples ( T − i , T + i ) ∈ BM S − ⊗ BM S + where the element in BM S − is classical and carries some finite energyand the element in BM S + is defined as a bookkeeper of the quantum probabilitydistribution of the radiated quanta. What concrete element T + i is associated witha given T − i cannot be derived solely from the classical geometry, but depends onthe internal quantum structure of the black hole. This non-trivial mapping isprecisely what makes the quantum hair informative. For a system with non-trivialdynamics, it is therefore impossible to restrict to a subgroup of BM S − ⊗ BM S + .Predictivity on this quantum output can be only achieved in the zero-energy (orequivalently N = ∞ limit) where we only get elements in the diagonal subgroup BM S . But since the soft modes are decoupled once the infrared divergences ofthe theory are properly taken into account [27–31], they cannot lead to observablefeatures. The outline of the paper is as follows. In section 2, we first recap some propertiesof BMS-gauge. In particular, we show how angular features of radiation define asupertranslation, which can be measured as a memory effect. Moreover, we discussthe role of soft modes. Then we use a combination of injected and emitted radi-ation of the same total energy to define Goldstone supertranslations as element( T − , T + ) ∈ BM S − ⊗ BM S + . In section 3, we first concentrate on a gravitationalsystem without horizon, which we shall call planet for concreteness, and show howwe can use Goldstone supertranslations to change its angular distribution of mass.In doing so, the key point is that it is impossible to infer T + from T − unless oneknows the internal dynamics of the planet. Moreover, we highlight the importanceof angular features by showing that it is impossible to determine the angular massdistribution of the planet without access to its interior. Subsequently, we applyGoldstone supertranslations to a black hole. We demonstrate how supertransla-tions can be used as bookkeeping tool for the emitted quanta. However, withoutknowledge of the microscopic dynamics of the black hole, they have no predictivepower. We also point out how we can use Page’s time to estimate the magni-tude of deviations from featureless evaporation. After concluding in section 4, weprovide a more detailed discussion of IR-physics in appendix A. In appendix B,we discuss the matching of the supertranslation field in advanced and retardedcoordinates and finally we explicitly calculate a Goldstone supertranslation of aplanet in appendix C. In [23] and [24] it is suggested to constraint the potential values of T + i using an infiniteset of conserved charges. Imposing these conservation laws makes the corresponding S -matrixcompletely insensitive to the internal structure of the black hole and consequently, in the languagewe are using here, can only capture unobservable zero-modes. This decoupling of soft modes is a quantum effect that should not be confused with theexistence, for instance in asymptotically Minkowski space time, of a non-trivial family of asymp-totically flat connections defining a representation of the
BMS -group (see [32] and referencestherein). This multiplicity of classical inequivalent vacua is quantum-mechanically reabsorbed inthe cancellation of infrared divergences. Quantum Hair
Retarded Coordinates
We first recap some properties of BMS-gauge, which is defined by the four gaugeconditions [10] g = g A = 0 , det g AB = r sin θ , (1)where A, B, . . . = 2 ,
3. Typically, BMS-gauge is used to study a spacetime asymp-totically, i.e. for r → ∞ , but it is possible to extend the metric to the bulk byimposing the conditions (1) to all orders in 1 /r . In a typical situation, however, ametric in BMS-gauge does not cover the whole spacetime.A metric in BMS-gauge exists both in retarded time u , which is suited to describeoutgoing radiation, and in advanced time v , which is suited to describe incom-ing radiation. The matching between these two metrics will be crucial for ourtreatment. Explicitly, an asymptotically flat metric in retarded time takes theform [10]:d s = − m + B r + O ( r − ) ! d u − (cid:16) O ( r − ) (cid:17) d u d r (2)+ r (cid:16) γ AB + C + AB r − + O ( r − ) (cid:17) d x A d x B + O ( r − )d x A d u , (3)where the metric on the sphere has to fulfill the requirement det g AB = r sin θ .Here m + B is the Bondi mass, γ AB the standard metric on the sphere and C + AB = (cid:16) D A D B − γ AB D (cid:17) C + (4)is determined by the supertranslation field C + , where D A is the covariant deriva-tive on the sphere. It is helpful to expand the supertranslation field in sphericalharmonics. Then the mode l = 0 represents a time shift and the mode l = 1corresponds to spatial translations. Therefore, all modes with l ≥ C + are connected via asymp-totic diffeomorphisms, i.e. the choice of the supertranslation field constitutes aresidual gauge freedom of BMS-gauge. These diffeomorphisms are the famous su-pertranslations. Therefore, we can define a supertranslation T + by the change itinduces in the supertranslation field: T + := ∆ C + . (5)In order to analyze the effect of supertranslations, we will need the constraintequation G = 8 πGT , whose leading order reads in BMS-gauge: ∂ u m + B = 14 G D ( D + 2) ∂ u C + − F out , (6)8here F out = 18 ( ∂ u C + AB )( ∂ u C + AB ) + 4 π lim r →∞ ( r T uu ) (7)is the total incoming null energy, composed of gravitational waves (first summand)and other forms of gravitating energy (second summand). Advanced Coordinates
The situation in advanced coordinates is very similar. The metric takes the formd s = − m − B r + O ( r − ) ! d v + (cid:16) O ( r − ) (cid:17) d v d r (8)+ r (cid:16) γ AB + C − AB r − + O ( r − ) (cid:17) d x A d x B + O ( r − )d x A d v , (9)where the supertranslation field and the supertranslations in advanced coordinatesare defined as in (4) and (5). The constraint equation becomes ∂ u m − B = 14 G D ( D + 2) ∂ u C − + F in , (10)where F in is the incoming energy, in analogy to (7). Measurement of the Supertranslation Field: Memory Effect
As already discussed, one can change the value of the supertranslation field by adiffeomorphism. Therefore, it follows by general covariance that the value of thesupertranslation field cannot have in general any experimental implication. How-ever, since it corresponds to physical outgoing or ingoing radiation, the differenceof the supertranslation field at different times does have experimental implica-tions: It describes the memory effect caused by the radiation, i.e. a permanentdisplacement of test masses after the radiation has passed [8, 33, 34].We will restrict ourselves to a simple situation in which we start with some sta-tionary metric g µν and we finish in a different stationary metric g µν . In between,there is a radiation epoch, i.e. F in/out only has support during this time span.Asymptotically on J ± , the process defines a non-stationary metric interpolatingbetween g µν and g µν which should be a solution to the Einstein equations.Since Birkhoff’s theorem implies that we can set ∂ A m ± B = 0 in a stationary metric,we can single out the zero-mode from (6) by integrating over the sphere: µ + = − R d u R d Ω F out π , (11)where we first consider retarded time and µ + = m + B, − m + B, is the total changeof Bondi mass due to the radiation epoch. This formula shows explicitly that9he Bondi mass m + B is monotonically decreasing, i.e. it measures the energy whichhas not yet left the bulk. Defining the emitted energy with non-trivial angulardistribution as ∆ ˜ F out := R d u F out − µ + , the constraint (6) becomes0 = 14 G D ( D + 2) T + − ∆ ˜ F out . (12)Thus, angular features in the outgoing radiation induces a supertranslation T + =∆ C + . Note that it is independent of the total emitted energy µ + .In advanced coordinates, we get from the constraint (10): µ − = R d v R d Ω F in π . (13)The advanced Bondi mass m − B is monotonically increasing, i.e. it measures theenergy which has already entered the bulk. Defining ∆ ˜ F in := R d v F in − µ − , theconstraint (10) becomes 0 = 14 G D ( D + 2) T − + ∆ ˜ F in . (14)This formula implies that an advanced supertranslation T − tracks angular featuresin the incoming radiation. As already pointed out, we shall define hair on the basis of scattering processeswhere some injected gravitational energy is radiated back by the system. The hairwill be encoded in the angular features of the injected radiation and the outgoingradiation. In this sense, we define hair as a typical response function . Throughthese formal scattering processes we define a map relating gravitational systems,black holes or planets, in different states sharing the same values for all the ADM-conserved quantities. We denote this induced map a Goldstone supertranslationsince it relates states which are degenerate in energy. Note that this scatteringdefinition of hair is tied to the mechanism of radiation whatever it could be.
Relationship to Antipodal Matching
As a first step, it is important to discuss whether there are general constraints onthis scattering process. Namely, it has been suggested in [35, 36] that any grav-itational S -matrix in an asymptotically flat spacetime must satisfy the followingrelation for an arbitrary initial quantum state | α i : ST − | α i = P ( T + ) S | α i , (15)where ( T − , T + ) ∈ BM S − ⊗ BM S + and P is the antipodal map on the sphere: P ( T + )( θ, ϕ ) = T + ( P ( θ, ϕ )) . (16)10mposing this invariance implies that if a matrix element h β | S | α i is non-vanishing,then h α | T − | α i = h β | P ( T + ) | β i . (17)This means that the memory effect of the outgoing wave, parameterized by T + ,must match the memory effect of the incoming wave, parameterized by T − , an-tipodally at each angle. Because of the constraints (12) and (14), this is equivalentto the statement that the outgoing energy ∆ ˜ F out matches the ingoing energy ∆ ˜ F in antipodally at each angle, in particular that ∆ ˜ F out is fully determined in terms of∆ ˜ F in .This criterion has a very interesting connection to IR-physics. As discussed inappendix A, we know that in a gapless theory such as gravity most process inwhich no soft modes are emitted have zero probability [37, 38]. In order to obtaina finite answer, one has to include the emission of a certain class of soft radiation,namely IR-modes. The sole exception are processes for which the kinematicalfactor B α, β defined in [38] is zero. The crucial point is that this happens if andonly if the ingoing energy matches the outgoing energy antipodally at each angle,as discussed in detail in [39]. Thus we conclude that h α | T − | α i = h β | P ( T + ) | β i ⇔ B α, β = 0 . (18)This means that restricting to processes which fulfill the condition (16) is equivalentto only considering processes that are IR-finite even without including IR-emission.A priori, there is nothing wrong with solely considering such processes. However,they only form a set of measure zero of those processes that occur in reality.Namely any realistic scattering is accompanied by the emission of soft IR-modes.Once we include soft IR-emission, we know that all processes – with an arbitrarynon-zero value of B α, β – are IR-finite. Thus in reality, any process can occur, i.e.also ones that do not fulfill the antipodal matching condition (16). For this reason,we will not restrict ourselves to processes that obey (16). Role of Soft IR-Gravitons
Since we consider processes that include the emission of soft IR-modes, it is naturalto ask if those modes could carry information about the black hole state and ifthey could even suffice to purify black hole evaporation. This is only possible iftwo conditions are fulfilled. First, IR-modes would have to be sensitive to themicrostate of the black hole. We expect this not to be the case since they onlydepend on the initial and final scattering state, but not on the details of theprocess. While we leave the above question for future work, we now focus on thesecond condition, namely that the number of resolvable IR-modes would have tobe big enough to be able to carry the whole black hole entropy. We will elaborate on this point in [40].
11n contrast to the proposal made in [26], we argue that generic properties of IR-physics imply that this is not the case. As follows from equation (37), the numberof unresolved soft modes scales logarithmically with the IR-resolution scale. Thus,when we lower the energy scale of resolution from (cid:15) to (cid:15) , the number of additionalIR-modes that we can resolve is: n ressoft ∼ B α, β ln (cid:15) (cid:15) , (19)where B α, β ∼ Gs is determined by the energy scale s of the process. We applythis formula to the single emission of a Hawking quantum of energy r − g . It will becrucial in this argument that Hawking radiation gets softer for bigger black holes.The worst resolution scale compatible with observing this process is (cid:15) = r − g . Thekey point is that the resolution scale in this process cannot be arbitrarily good.Namely, it is set by the time-scale of the process, (cid:15) ∼ t − . Since the life-time ofa black hole scales as t b-h ∼ N r g , we get n ressoft (cid:46) N ln N . (20)Thus, after the black hole has evaporated by emitting N Hawking quanta, themaximal entropy contained in the soft IR-modes is S soft (cid:46) ln N . (21)Independently of the question whether IR-modes are strongly correlated with theHawking quanta, this shows that they cannot account for the whole entropy ofthe black hole, but could only give a logarithmic correction. Of course, this leavesopen the possibility that non-IR soft modes could account for the bulk of blackhole information. However, since they are independent of IR-divergences and ac-companying dressing tools, the results of infrared physics do not constrain them.
Role of Zero-Energy Gravitons
Finally, we briefly discuss the role of zero-energy gravitons. To this end, we con-sider the process of a Goldstone supertranslation in the limit of zero energy injectedand zero energy radiated. This is equivalent to the scattering with a graviton ofzero energy. Since those carry no energy, they cannot emit IR-modes and there-fore obey the antipodal matching condition (16). This fact simply reflects thewell-known decoupling of soft modes [27–31]. The physical interpretation of thisphenomenon is that any bulk configuration is transparent for decoupled soft modesso that the energy profile of the outgoing wave is antipodally related to that ofincoming energy.But when the emitted/injected radiation does not carry energy, µ ± = 0, then theconstraint (11) (or respectively (13)) implies that R d Ω F in/out = 0. Since F in/out represents real gravitational radiation, it follows by the requirement of positiveenergy that F in/out = 0. Thus, only supertranslations with D ( D + 2) T ± = 012an occur in such a zero-energy process. This means that only the angular modes l = 0 , T + from the knowledge of T − is only possible forzero-energy modes. Those are, however, unphysical since they cannot be measuredin finite time. So we will only consider processes of non-zero energy in our paper.As explained, it is not possible for them to constrain or even predict T + from T − without detailed knowledge of the dynamics in the bulk. The response function,which determines T + in terms of T − , is trivial only for modes of zero energy. Physical Hair With Non-Zero Energy
So from here on, we consider the case where after we inject radiation F in of non-zero total energy µ , the system radiates back the same total amount of energy,but with a possibly different distribution F out . While such systems are of coursespecial, we will see that black holes can be one of them. This is a zero-energyprocess in the sense that the total energy of the system does not change. Thus,this process, which is depicted in figure 2, constitutes a transformation betweendegenerate systems and therefore defines hair.As far as we reduce ourselves to gravitational radiation, we can generically describethis process in terms of two supertranslations: At J − , T − is determined by theangular distribution ∆ ˜ F in of incoming energy according to the constraint (14) andat J + , T + follows from the angular distribution ∆ ˜ F out of outgoing energy via theconstraint (12). Thus, the whole process is associated to an element ( T − , T + ) ∈ BM S − ⊗ BM S + . It describes a zero-energy transition which interpolates betweentwo spacetimes of the same total energy, but contrary to the case of a zero-energymode, this transformation is non-trivial and it is not decoupled.It is crucial to note that for an asymptotic observer, T − and T + are independent.Whereas one is free to choose T − by preparing an appropriate incoming radiation, T + is sensitive to the properties of the system in the bulk. In other words, T + is a response of the system which does not only depend on the ingoing radiation,parameterized by T − , but also on the state of the system and its particular dy-namics, which are not entirely visible asymptotically. In particular, there is noreason why ( T − , T + ) should be in any subgroup of BM S − ⊗ BM S + . We recall that we restrict ourselves for now to a pure gravitational radiation, which propagatesalong null geodesics. Therefore, all emitted energy is bound to reach future null infinity J + . M Figure 2: A Goldstone supertranslation on a generic system of mass M . Radiationwith angular distribution F in scatters so that radiation with angular distribution F out is returned. Since R d v R d Ω F in = R d u d Ω F out , the total energy of the sys-tem remains unchanged. Here F in can be described in terms of the supertranslation T − and F out in terms of T + . Coordinate Matching
In order to compare ingoing and outgoing radiation, i.e. T − and T + , we need torelate the supertranslation field C − in advanced coordinates to the supertransla-tion field C + in retarded coordinates. Namely, we assume that we are given aclassical spacetime whose asymptotic behavior is fully known to us. Then it ispossible to describe this spacetime both in advanced and retarded BMS-gauge.Given an advanced coordinate system g vµν , we want to know if there is a uniqueretarded coordinate system g uµν we can associate to it. If we have such a mapping,it determines the relation of the advanced supertranslation field C − , defined asthe r part of g vAB , and the retarded supertranslation field C + , defined as the r part of g uAB .Given g vµν , we therefore have to find a diffeomorphism D such that g uµν := D ( g vµν )is in retarded BMS-gauge. Then we can read off from g uµν the C + associated to C − . However, we could have instead considered the diffeomorphism D = T + ◦D , where T + is a supertranslation diffeomorphism in retarded coordinates. Also D transforms the metric in advanced BMS-gauge to a metric in retarded BMS-coordinates. Clearly, if T + is a nontrivial supertranslation, the supertranslationfield in the resulting metric differs from the one in g u . From this consideration it isobvious that the matching between the advanced and the retarded supertranslationfield is in general not unique. 14he only hope we could have is that there is a natural way to identify C − and C + . In a static situation, a natural prescription is to require that the spatial partof the two metrics matches, i.e. g uAB = g vAB . (22)As is shown explicitly in appendix B for the example of the Schwarzschild metric,we can achieve this by identifying C + ( θ, ϕ ) = − C − ( θ, ϕ ), as also proposed in [18].Up to a sign, we match the supertranslation field angle-wise. Consequently, thesame matching holds for the supertranslations: T + ( θ, ϕ ) = − T − ( θ, ϕ ) . (23)There are several reason why the coordinate matching (23) is natural. First of all,the prescription (22) comes from a simple intuition. For an observer in a staticspacetime who lives on a sphere of fixed radius, the description of the sphere shouldbe the same independently of the choice of time coordinate. More generically, it ispossible to require that the action of advanced and retarded supertranslations isthe same in the bulk. This was done in [18, 19] for the cases of Schwarzschild andMinkowski.Moreover, we can consider a detector at big radius which is sensitive to gravita-tional memory. Then we investigate a process of back scattering, in which theangular distributions of incoming and outgoing energy are identical at each angle.This corresponds to a wall in the bulk which reflects the wave without furthermodifying it. In this case, the memory effect the ingoing wave causes, param-eterized by T − , is exactly canceled by the memory effect of the outgoing wave,parameterized by T + , so that there is no overall memory effect after the process.In that case, if we match T − and T + at each angle as in (23), it is possible tosimply describe the overall memory effect as T − + T + .However, it is crucial to stress that the coordinate matching (23) does not haveany constraining power on the physical process. It does not predict outgoing fromingoing radiation, but only shows how one and the same setup can be described indifferent coordinates. This is also evident from figure 2. The matching conditionat i only relates the absolute values of the supertranslation fields. In contrast,processes of non-zero energy solely determine a change of the supertranslationfield, as is clear from equations (12) and (14). Thus, radiation of non-zero energyis independent of the coordinate matching. In order to make the ideas presented above concrete, we discuss an explicit ex-ample, namely the application of a Goldstone supertranslation to a certain class15f planets. We start from a spherically symmetric nongravitational source T µν ,which sources a spherically symmetric spacetime g µν with ADM-mass M . In sucha spacetime, we want to realize a Goldstone supertranslation, i.e. we send in a wavewith total energy µ and angular distribution ∆ ˜ F in in such a way that after sometime, the planet emits a wave of the same energy µ but with a possibly differentangular distribution ∆ ˜ F out . Of course, only a special class of planets behaves inthat way.We explicitly construct such spacetimes in appendix C, to which we refer thereader for details of the calculation. First, we consider the incoming wave. Asdiscussed, the angular distribution ∆ ˜ F in of injected energy determines an advancedsupertranslation T − . As derived in equation (51), we can use it to describe thechange of the metric due to the injected radiation: δg vµν = τ v , v ( v ) s − ( r ) (cid:18) L ξ v ( T − ) g vµν + 2 µGr δ µ δ ν (cid:19) , (24)where L ξ v ( T − ) g vµν is an infinitesimal supertranslation which changes the super-translation field by a small amount T − . Whereas the asymptotic supertranslation T − only depends on the leading part of the incoming energy, it is crucial to notethat the transformation (24) also depends on a careful choice of the subleadingcomponents of the incoming wave. Only with a particular choice, the wave acts asa diffeomorphism not only asymptotically but also in the bulk outside the planet.We observe that the effect of the wave is twofold. First, it adds the total mass µ to the planet and secondly, it supertranslates the metric by T − . However, theseeffects are localized both in space and time. The function τ v , v ( v ) describes thesmooth interpolation between g vµν and g vµν + δg vµν , i.e. we have τ v , v ( v < v ) = 0and τ v , v ( v > v ) = 1. The function s − ( r ) describes the absorption of the wave,namely absorption takes place whenever s − ( r ) <
0. There is no absorption outsidethe planet, i.e. s − ( r > R ) = 1, where R is the radius of the planet, and the waveis fully absorbed before it reaches the center, s − ( r = 0) = 0. It will be crucial tonote that the transformation s − ( r ) L ξ v ( T − ) g vµν only acts as a diffeomorphism when s − ( r ) = 0.Moreover, the transformation (24) shows that we focus on planets which have asecond very special property aside from the fact that they emit as much energyas they receive: Namely there is no transport of energy between different angles.This means that the mass of the planet does not redistribute after absorption (thesame will be true after emission). The fact that this assumption is unnatural andnot true for generic systems will contribute to our conclusions.As a second step, we consider the emission of a wave by the planet. Of course, theproperties of the emitted wave depend on the internal dynamics of the source T µν .It is crucial to note we cannot resolve them in our purely gravitational treatment, Subleading terms are the 1 /r -term in T and the whole T A in (52). If one does not insistthat the wave acts as a supertranslation also in the bulk, one is free to choose the coefficient ofone of the two terms. The other one is determined by energy conservation: T ; µµν = 0. T + inducedby the angular distribution ∆ ˜ F out of outgoing energy: δg uµν = τ u , u ( u ) s + ( r ) (cid:18) L ξ u ( T + ) g uµν − µGr δ µ δ ν (cid:19) . (25)As for the case of absorption, the emission has two effects: It decreases the totalmass by µ and it supertranslates the metric by T + . Moreover, it is localized inspace and time in an analogous manner.We want to compare the planet before and after the Goldstone supertranslation,i.e. we are interested in the combined effect of the transformations (24) and (25).To this end, we have to specify a mapping between the advanced and retardedsupertranslations. As explained in section 2.2, we employ the angle-wise matching(23). Thus, we obtain the static final state of the planet: δg tot µν = θ ( r − R ) L ξ u ( T + − T − ) g µν + θ ( R − r ) (cid:16) s + ( r ) L ξ u ( T + ) g µν − s − ( r ) L ξ u ( T − ) g µν (cid:17) . (26)We get a planet which has the same ADM-mass but a different angular distribu-tion of mass. This is clear from the fact that the transformation (26) acts as adiffeomorphism only outside the planet.Since we used in our computation a planet with the special property that itsangular distribution of energy is frozen, we can read off the distribution fromdifference of energy distributions of the injected and emitted wave. In this case, T − − T + encodes all information about the angular energy distribution of theplanet in the bulk. However, this is no longer true for generic systems whichexhibit non-trivial dynamics after absorption and emission. In that case, T − and T + merely encode the initial state. Only with full knowledge of the theory whichgoverns the internal dynamics of the planet, we can infer the state of the planetat a later time from the asymptotic data T − and T + . The Role of Supertranslations
In summary, we obtain the following key properties of a Goldstone supertrans-lation in the case of a planet: Outside the planet, it acts as a diffeomorphism. For the planet with frozen energy distribution, there is also a very literal way in which one caninterpret the quantity T − − T + : One can imagine a gedankenexperiment where a source of light islocated in the interior of the planet after the Goldstone supertranslation and we collect the lightrays on the sky. The light sent from this common center point determines in this way a section atinfinity described by the supertranslation field T − − T + . Thus, the different redshift effects dueto the inhomogeneities of the planet matter distribution define a supertranslated section in thesky as the one for which light rays originate from a common spacetime point. This is reminiscentof Penrose’s concept of "good sections" [41].
17n particular, it does not change its ADM-mass. In contrast, it does not act as adiffeomorphism inside the planet where absorption takes place. Therefore, it is nota trivial global diffeomorphism but changes the spacetime physically. Thus, theGoldstone supertranslation encodes differences in the angular distribution amongmatter configurations degenerate with respect to the ADM-conserved quantities.It is crucial to discuss the role of supertranslations in this process: • For an asymptotic observer, ( T − , T + ) can be used as label for the angularfeatures of ingoing and outgoing radiation. • An asymptotic observer, however, cannot infer T + from T − . This is onlypossible with knowledge of internal dynamics of the planet. • Thus, ( T − , T + ) is a bookkeeping tool but without detailed information aboutthe interior, it does not have predictive power.As we shall discuss in a moment, the same conclusions hold in the black hole case.The only difference is that the internal dynamics leading to emission are fully quan-tum mechanical for a black hole. This will mean that in any classical description,supertranslation cannot constrain or even predict black hole evaporation.Using the example of the planet, it is easy to convince ourselves that antipodalmatching cannot play a role in processes of non-zero energy. Namely if it did,this would mean that the only planets which could exist would have the extremelyspecial property that they emit all energy they receive from one side exactly onthe other side. Hidden Angular Features
Finally, we discuss the transformation (26) when we do not have access to ( T − , T + ),i.e. when we do not record ingoing and outgoing radiation but only compare theinitial and final state of the planet. In that case, the planet possesses an inter-esting property, namely a special kind of no-hair-theorem. Concretely, we takethe perspective of an observer who has no access to the interior of the planet anddiscuss the difference between two planets which have the same mass but a dif-ferent angular mass distribution. As we have observed, the transformation (26)acts as a diffeomorphism outside the planet. Therefore, an outside observer can-not distinguish the two following cases when he is given a supertranslated outsidemetric. First, it could be the result of the transformation (26), where the planetwas physically changed due to a Goldstone supertranslation. Secondly, however,one can also obtain the supertranslated metric by acting on the initial planet witha global diffeomorphism. In this case, clearly, the planet does not change. Thus,also for a planet, an outside observer is not able to resolve angular features. Inorder to decide whether two asymptotic metrics differing by a supertranslation18escribe two different distributions of matter or the same distribution of matterin different coordinates, one needs access to the whole spacetime, i.e. the interiorof the source.We conclude that generic gravitational systems posses physical angular featureswhich are inaccessible for an outside observer. This strengthens our believe that themicrostates of a black hole have a non-trivial projection on angular features. Theonly difference is that while the restriction to outside measurements was artificial inthe case of the planet, an outside observer has in principle no access to the interiorof a black hole. As we will discuss in the next section, he can therefore never decidewhether a supertranslated metric corresponds to a physical change of the matterinside the black hole or to a global and therefore meaningless diffeomorphism. Thisis the reason for the classical no-hair theorem of a black hole and why we assignan entropy to the black hole and not to the planet. Supertranslations as Bookkeeping Device
Now we are ready to discuss the system of our interest, namely black holes. Sinceabsorption and emission are of different nature in that case, we will discuss themseparately. For absorption, we can proceed in full analogy to the planet and injecta wave with total energy µ and arbitrary angular distribution ∆ ˜ F in . By Birkhoff’stheorem, the spacetime outside the black hole is the same as for the planet so thatthe wave behaves identically. As in the case of the planet, the wave cannot beabsorbed outside the horizon and acts as a diffeomorphism everywhere outside theblack hole and also on the horizon.For the planet, we observed that the knowledge of injected energy alone does notsuffice to predict what radiation the planet emits. Instead, this can only be donewith knowledge of the interior dynamics of the planet. Those, however, can bedescribed classically in the case of the planet. For the black hole, the situation iseven worse. Not only do we not have access to any interior dynamics, but thesedynamics are also fully quantum. It is impossible to describe them even with fullclassical knowledge of the interior of the black hole.Before we elaborate on this point, we first show how it is possible to use super-translations as bookkeeping device for black hole evaporation. Unlike for the caseof the planet, this is a non-trivial question since the evaporation products aregeneric quantum states. In order to define an associated supertranslation, we shallproceed as follows. We consider an ensemble of quantum-mechanically identicalblack holes of mass M . For each black hole, we wait until it has emitted exactlyone Hawking quantum. We only record their angular features, i.e. the deviation Experimentally, we can realize this by preparing identical quantum states in such a way thatthey collapse and form black holes. l, m ). This defines a probability distribution for the angular features of theensemble: P ( l, m ) . (27)Obviously, the probability distribution (27) only contains a part of the quantum-mechanically available information. However, we will only focus on it since it canbe described in terms of classical supertranslations. At this point, it is crucial topoint out that the probability distribution (27) does not originate from a mixedstate but as a result of an ordinary quantum measurement. Thus, unlike in adescription in terms of a mixed state, it is not associated to any fundamental lossof information.Since we need to recover a featureless emission in the semi-classical limit, it followsthat P (0 ,
0) = 1 − (cid:15) , (28)where (cid:15) → (cid:15) of theemitted quanta carries features. For l ≥
2, we consequently get P ( l, m ) = (cid:15)A l,m , (29)where P ∞ l =2 P m = lm = − l A l,m = 1. The information contained in the P ( l, m ) is purelyquantum mechanical. At the semi-classical level, we have that P ( l, m ) = δ l andin the classical limit, we have no emission at all.Using the quantum probability distribution (27), we can associate to every Hawk-ing quantum an average energy flux: F out = (cid:126) r − g ∞ X l =0 m = l X m = − l P ( l, m ) Y l,m , (30)where Y l,m are the standard spherical harmonics. Just like for the case of theplanet, where we considered a classical process of emission, we can use the flux(30) to define a classical supertranslation T + . Of course, this is only possible aslong as (cid:126) = 0 since the energy flux is zero otherwise. When we record the quantum-mechanically emitted energy F out , we can proceed in analogy to the planet anduse the supertranslation fields T − and T + to track the evolution of the black hole.Concretely, in order to perform a Goldstone supertranslation, we first inject anenergy µ and then we wait until n H = µ/ ( (cid:126) r − g ) quanta have evaporated, as isdepicted in figure 3. Then we end up with a black hole of the same mass as beforethe process. Of course, the sensitivity of the final state on the initial state issuppressed by µ/M but unitarity dictates that the dependence is never trivial.20 Figure 3: A Goldstone supertranslation on a black hole of mass M . First, itabsorbs radiation with angular distribution F in and then it evaporates radiationwith angular distribution F out . Since R d v R d Ω F in = R d u d Ω F out , the totalenergy of the black hole remains unchanged. Here F in can be described in termsof the supertranslation T − and F out in terms of T + . Insufficiency of Supertranslation Hair
However, it is impossible to predict T + solely from the knowledge of T − . The rea-son is that the wave that we inject acts as a diffeomorphism outside the horizonand also on the horizon. Therefore, the geometry outside the black hole is un-changed after the wave has passed. Since the semi-classical Hawking calculationis only sensitive to the geometry on the horizon and outside the black hole, itsresult cannot change as a result of a supertranslation diffeomorphism. Therefore,additional knowledge about the interior is required to predict T + .We can make this argument more concrete by taking the perspective of an observerwho lives in a Schwarzschild metric supertranslated by T − . The observer has norecord of how the black hole was formed and is only allowed to make experimentoutside the horizon. Her goal is to determine the microstate of the black hole.More specifically, she wants to know if the black hole is in the bald microstate,whose evaporation products are featureless and in particular perfectly isotropic,or in a non-trivial microstate, whose evaporation products carry some angularfeatures. By our definition of microstate, one way to do so is to wait till the blackhole has evaporated and to determine the properties of the evaporation products.The question we are asking is if there is another way to determine the microstateof a black hole. The answer is negative, for the following reason: When an outsideobserver finds herself in a black hole metric with supertranslation field T − , thiscan happen because of two very distinct reason. Firstly, it could be the resultof injecting a wave with a non-trivial angular distribution of energy into a blackhole. In that case, the black hole is in a non-trivial microstate and T − indeed21haracterizes the microstate.However, there is a second way in which we can obtain a supertranslated Schwarz-schild metric. Namely, we can consider a featureless microstate, whose evaporationproducts are isotropic, and apply a supertranslation diffeomorphism to this setup.In this way, we do not change the physical state of the black hole but only describeit in a different metric. Thus, T − can also correspond to a featureless microstatedescribed in different coordinates.Without access to the evaporation products, the only way to distinguish thosetwo cases – injection of wave with angular features versus global diffeomorphism– is to enter the black hole. There, the wave acts non-trivially, i.e. not as adiffeomorphism, whereas the global diffeomorphism still does. Since the sameexterior metric can correspond to both a trivial and a non-trivial microstate, themetric alone cannot suffice to predict the evaporation products. From the outside,it is therefore impossible to distinguish classical supertranslation hair and globaldiffeomorphisms.In summary, as in the case of a planet, we can use ( T − , T + ) as a natural bookkeep-ing device for the black hole to track the angular features of ingoing and outgoingradiation. However, knowing T − does not suffice to predict T + , i.e. an observeroutside the black hole cannot infer T + from T − . This is only possible with amicroscopic model of the interior dynamics of the black hole. Generalization to Evaporation
Having discussed how we can implant hair on a black hole with a Goldstone su-pertranslation, it is trivially to consider the case of pure evaporation. We obtainit if we just leave out the first part of the Goldstone supertranslation, namely theinjection of a wave. Therefore, it suffices to consider J + as screen, where theconstraint (12) determines the retarded supertranslation field T + in terms of theangular distribution ∆ ˜ F out . In that case, the metric outside the black hole changesaccording to (25): δg uµν = τ u , u ( u ) (cid:18) L ξ u ( T + ) g uµν − µGr δ µ δ ν (cid:19) . (31)This equation shows that the back reaction splits in two parts. First, energy con-servation dictates that the mass of the black hole is reduced by the total emittedenergy µ = R d u R d Ω F out . This part of the back reaction is undebatable butdoes not suffice to ensure unitarity of the process. Fortunately, F out contains moreinformation than just the emitted energy, namely the supertranslation T + . Conse-quently, we obtain the back reacted black hole not only by reducing its mass, butby supertranslating it by T + . This approach is only valid if the supertranslationacts non-trivially in the interior of the black hole, e.g. because it is induced by aphysical wave. But in that case, the ability to associate hair to a black hole isequivalent to the ability to purify its evaporation.22 .3 A Comment on Page’s Time So far, we have not specified the magnitude of deviations from a thermal evapo-ration. We can estimate them by requiring that we reproduce Page’s time in ourapproach. In its most basic formulation, Page’s time is a direct consequence ofdescribing the black hole evaporation in a Hilbert space of fixed dimension. Inbrief, if we keep the dimension of the full Hilbert space, which describes at anytime both the black hole and the emitted radiation, fixed and equal to 2 N , thenat t = t P , which corresponds to the half life-time, i.e. the evaporation of ∼ N/ N and keeping it finite.Page’s time can be defined as the time-scale for the emission of the order of N quanta. Therefore, we first consider an ensemble of N identical quantum mechan-ical black holes and for each of them, we record the first emitted quantum. For ameasurement on a single black hole, the standard deviation is σ ∼ O (1) (32)since the quanta are distributed isotropically to leading order. However, when weaverage over N measurements, the standard deviation decreases as σ N ∼ √ N . (33)Features become visible as soon as their strength becomes bigger or equal thanthe uncertainty of the measurement. After Page’s time we can therefore resolvefeatures with the relative amplitude (cid:15) ∼ √ N . (34)In the formulation of the probability distribution (28), this means that after O ( N )measurement, those features becomes visible which are only carried by a fraction1 / √ N of the quanta.So far, we have only considered one emission for N identical black holes. If weconsider instead O ( N ) emissions of a single black hole, the difference is that theprobability distribution for each emission step is generically different. This istrue because of the back reaction of the previously emitted quanta. However,the argument in terms of the resolution stays the same, i.e. after Page’s time, wecan still resolve those features which are only carried by a fraction (cid:15) ∼ / √ N ofquanta. This argument provides evidence for the black hole N -portrait [5] wherefeatures are 1 /N -effects with a resolution scale O (1 / √ N ). In particular, in an An interesting question that we shall not discuss in this note but that can be worth to mentionis the possibility that a quantum computer designed using a Grover like algorithm [42] can reduce t P from O ( N ) to O ( √ N ). -matrix analysis along the lines of [43], where black hole formation was studiedas 2 → N -scattering process, angular features should appear as 1 /N -correction tothe leading amplitude. The main message of this note is easy to summarize. Any form of black hole hairshould imply the existence of features in the black hole evaporation products, i.e.in the emitted radiation. This obvious requirement immediately entails, given theintrinsically quantum nature of black hole radiation, that black hole hair should bedefined quantum-mechanically and that such a definition is inseparable from themechanism through which the black hole delivers, in the radiation, informationabout its internal structure.In this note, we have suggested to define hair on the basis of elementary processesof classical absorption followed by quantum emission. Moreover, we specialized toangular features in the radiation. This simplification has been done in order touse the asymptotic symmetry group and the corresponding supertranslations toparametrize both the incoming and the emitted radiation. Since we have completecontrol over the angular features of the injected radiation, we can define hair onthe basis of the angular features of the quantum-mechanically emitted radiation.These features encode information about the internal structure of the black holewhich can be measured by an external observer. In this sense, they provide anoperative and intrinsically quantum definition of hair.In principle, we can imagine two different sources of those features of the emittedradiation. The first one is a classical modification of the near horizon geometrythat will modify the corresponding semi-classical Bogoliubov transformations. Thesecond one is a real quantum interaction of the injected radiation with the quan-tum constituents of the black hole. The first possibility requires to define localchanges of the horizon geometry that preserve all the ADM-charges. Thus, locally,they can be always tuned to be equivalent to a diffeomorphism. Therefore, theycannot have observable consequences, i.e. classical supertranslations do not sufficeto define observable black hole hair. So the only real possibility of quantum emit-ted radiation with features is having a non-trivial scattering between the injectedradiation and the microscopic constituents of the black hole. This means that thefeatures that define hair in the way we are suggesting depend on the microscopicquantum structure, which we can parametrize as a dependence on the black holeentropy N . Thus, the hair that we are defining vanishes in the limit N = ∞ .As it is clear from the discussion, this way of addressing the definition of hair iswhat we can call an S -matrix approach, where by that S -matrix we simply meanthe dynamics involved in the complex process of actual absorption and quantumemission. If we focus on angular features, we can encode the properties of the hairin terms of the commutators, as operators, of this S -matrix and the generators24f the asymptotic symmetry group. Associating with the injected energy a super-translation T − in BM S − , a way to approach the existence of hair is by consideringthe commutator [ S, T − ]. Generically, the non-trivial hair will be associated withthe symmetry generators that are broken since those are the ones that will createnet differences between the angular features of the injected and emitted radia-tion. Although the infrared dynamics of gravity selects the zero-energy modes asnatural symmetries of S , they are not able to tell us anything about the internalstructure of the black hole since they are decoupled. Zero-modes are unable toencode observable features.What we have presented in this note is just the general framework to address theproblem of quantum hair. In order to go further, it is necessary to use a concretemodel of the black hole interior. The model in [5] provides, in principle, the toolsto address this questions in a quantitative way, something to which we hope tocome back in the future. A Recap of IR-Effects
In this appendix, we shall collect some well-known facts about infrared physicswhich could be useful to clarify some controversial aspects on the meaning of softmodes. Some of these issues have been revisited recently in a series of papers[27–29, 36, 44–48]. • In QED, asymptotic physical states associated with freely moving chargedparticles should be dressed in order to satisfy the Gauss law constraints.This dressing simply adds to the freely moving charge its companion electro-static field, i.e. the non radiative part of the retarded Lienard-Wiechert-fieldbehaving at large distances as 1 /r . In quantum field theory, this dressingcan be defined using a coherent state of off-shell photons [29] with dispersionrelation ω ( k ) = k v for v the velocity of the charged particle. This coherentstate dressing contains an infinite number of k = 0 modes and it is identicalto the dressing operator defined in [49]. In scattering theory, one can de-fine physical asymptotic states and an IR-safe S matrix using this dressingoperator. • Alternatively, one can use no dressing. Then, in perturbative QED aswell as in perturbative gravity, we find IR-divergences due to virtual pho-ton/graviton loops. These, after a careful analysis of overlapping divergences,can be resummed and exponentiated [37, 38]. When we consider the transi-tion from an initial state | α i to a final state | β i , we obtain S loop α, β = e B α, β ln λ Λ / S α, β , (35)where S α, β is the amplitude without taking into account soft loops whereas S loop α, β contains them. Here Λ is a UV-cutoff that defines what is soft, λ is25 IR-cutoff and B α, β is a non-negative number, which only depends on theinitial state | α i and the final state | β i . It is zero if and only if the ingoingcurrent in | α i matches the outgoing current in | β i antipodally at each angle.In the case of gravity, it scales as B α, β ∼ Gs , where s is the energy of theprocess. For B α, β = 0, soft loops clearly lead to a vanishing amplitude inthe limit λ → • In order to cancel the IR-divergences due to virtual photons/gravitons, theBloch-Nordsieck-recipe [51] requires to add a certain class of soft emissionprocesses. Again the effects of emitting theses soft IR-modes of energiesbelow (cid:15) can be resummed and exponentiated, yielding the rate [37, 38] | S full α, β | := X γ | S soft α, βγ | = e B α, β ln (cid:15)λ f ( B α, β ) | S soft α, β | , (36)where f ( B α, β ) is due to energy conservation and f ( B α, β ) ≈ B α, β .Combing the contribution from (35) and (36), one obtains a rate which isindependent of the IR-cutoff λ and in particular finite for λ →
0. Thiscancellation leads to the connection, highlighted in [49], between the softphoton theorem and the electrostatic coherent state dressing. In QED, wedo not have new symmetries besides the decoupling of zero-energy photons.The same is true in perturbative gravity. • In the correction factor e B α, β ln (cid:15)λ in (36), the n th summand of the exponentialseries comes from the emission of n IR-modes. Therefore, we can estimatethe number of soft modes from the term which gives the biggest contributionin the series. This gives n unressoft ∼ B α, β ln (cid:15)λ . (37)We conclude that the number of unresolved soft modes only scales logarith-mically with the infrared resolution scale (cid:15) . B Matching in Schwarzschild Coordinates
In this section, we demonstrate explicitly how we can transform a Schwarzschildmetric with non-trivial supertranslation field from advanced to retarded coordi-nates. In this way, we show how we can naturally identify the advanced super-translation field C − with the retarded one C + . We start from the Schwarzschildmetric g v , µν in advanced coordinates without supertranslation field:d s = − (1 − GMr )d v + 2d v d r + r dΩ . (38)The corresponding generators of supertranslations are ξ vv = f − , (39a) That this scaling also holds for graviton scattering at an ultra-Planckian center of mass energywas shown in [50]. rv = − D f − , (39b) ξ Av = f − , A r , (39c)which are characterized by an arbitrary function f − on the sphere. Thus, thesupertranslated metric is g vµν ( f − ) = g v , µν + L ξ v ( f − ) g v , µν . (40)In retarded coordinates, the Schwarzschild metric g u , µν without supertranslationfield is: d s = − (1 − GMr )d u − u d r + r dΩ . (41)The corresponding generators of supertranslations are ξ vu = f + , (42a) ξ ru = 12 D f + , (42b) ξ Au = − f + , A r , (42c)where it is important to note that the signs of ξ ru and ξ Au have changed with respectto (39). The supertranslated metric is: g uµν ( f + ) = g u , µν + L ξ u ( f + ) g u , µν . (43)The task now is to transform g vµν to retarded coordinates. As explained in section2.2, there can in general not be a unique way to match the advanced and retardedsupertranslation fields. However, a natural choice in a static metric is to requirethat the spherical metrics match: g vAB ( f − ) = g uAB ( f + ). Therefore, we use thediffeomorphism D m defined by v = u + 2 Z rr − GMr d r − D f − − GMr − f − . (44)Then it turns out that D m (cid:16) g vµν ( f − ) (cid:17) = g u , µν − L ξ u ( f − ) g u , µν = g uµν ( − f − ) . (45)Thus, we identify f + = − f − . (46)Up to a sign, the supertranslation field in advanced coordinates matches the re-tarded one angle-wise. With this choice, not only the spherical metrics match, butalso the g -components, i.e. the Newtonian potentials.27 Explicit Solution for Goldstone Supertranslation ofa Planet
Step 1: Absorption
The Goldstone supertranslation consists of two steps: First, an initially sphericallysymmetric planet absorbs as wave. As is well-known (see e.g. (9.3) in [52]), themetric of a static spherically symmetric spacetime can be cast in the general formd s = − A ( r )d t + B ( r )d r + r dΩ , (47)where all physical information is contained in the tt - and rr -components. Since wewant to describe a planet, there should neither be a surface of infinite redshift, i.e. A ( r ) > ∀ r , nor an event horizon, i.e. B ( r ) < ∞ ∀ r . Furthermore, asymptoticflatness implies that A ( r ) r →∞ −→ B ( r ) r →∞ −→ v = t + Z rr d r s BA , (48)we obtain the metric g vµν in advanced BMS-gauge:d s = − A d v + 2 √ AB d v d r + r dΩ , (49)which is suited to describe incoming radiation. Note that this metric describes thewhole spacetime and not only its asymptotic region, i.e. r → ∞ .We will restrict ourselves to infinitesimal supertranslations. In advanced time,these are generated by ξ vv = f − , (50a) ξ rv = − rD B ξ Bv , (50b) ξ Av = f − , A Z ∞ r d r ( √ ABr ) , (50c)where an arbitrary function f − on the sphere determines the change of the su-pertranslation field. We denote it by f − instead of T − in this appendix to avoidconfusion with the energy-momentum-tensor of the wave. The minus-superscriptindicates that we deal with a supertranslation in advanced coordinates. Our goalis to realize the infinitesimal diffeomorphism defined by (50) in a physical process,i.e. outside the planet, we want to have the stationary metric g vµν before some time v and after some point of time v , we want to end up in the stationary metric g vµν + L ξ v ( f − ) g vµν . For v < v < v , physical radiation interpolates between the twometrics. Inside the planet, the wave should be absorbed so that the transformationfades out and the metric around the origin remains unchanged. Adding as final28ngredient a change of the Bondi mass µ , which is necessary to ensure the positiveenergy condition, we obtain δg vµν = τ v , v ( v ) s − ( r ) (cid:18) L ξ v ( f − ) g vµν + 2 µGr δ µ δ ν (cid:19) , (51)where 0 ≤ τ v , v ( v ) ≤ τ v , v ( v < v ) = 0 and τ v , v ( v > v ) = 1. The function s − ( r ) describes the absorption of the wave bythe planet. It has the property that it is monotonically increasing with s − (0) = 0and s − ( ∞ ) = 1, where s − (0) = 0 ensures that the wave is fully absorbed beforethe origin and no black hole forms. Moreover, s − ( r ) = 0 is only permissiblewhenever the local energy density of the planet is non-zero. The magnitude of s − ( r ) determines how much absorption happens at r . It is crucial to note that thetransformation s − ( r ) L ξ v ( f − ) g µν is a diffeomorphism only where s − ( r ) is constant.Thus, the transformation (51) acts as a diffeomorphism only outside the planet,but not inside. This reflects the fact that we want to obtain a physically differentplanet. A transformation which acts as a diffeomorphism everywhere could notachieve this.Since we work with infinitesimal supertranslations, it is important that we staywithin the regime of validity of this first-order approximation, i.e. that terms lin-ear in f − dominate. As it will turn out in the calculation, this is the case ifmax ( θ,ϕ ) | f − | (cid:28) v − v . This means that the time-shift induced by the super-translation must be much smaller than the time-scale of the process, i.e. the su-pertranslation must be performed slowly. We will choose v − v such that this isthe case and so that we can neglect all higher orders in f − when we calculate theEinstein equations.We have to show that the transformation (51) leads to a valid solution of the Ein-stein equations. Thus, if we calculate the Einstein G µν and consequently the newenergy-momentum-tensor T µν , we have to demonstrate that this is a valid source.To this end, we have to perform two checks. First of all, it must be conserved, T ; µµν = 0. This is trivially true in our construction because of the Bianchi iden-tity, G ; µµν = 0. Secondly, we have to show that T µν fulfills an appropriate energycondition. For that purpose, we first note that this perturbation only depends onthe local geometry, except for ξ µv and s − ( r ), which also depend on spacetime pointsat bigger radii. Thus, outside the planet, we have the same solution as in [24],except for the fact the we perform our supertranslation slowly: T = 14 πr (cid:20) µ − D ( D + 2) f − + 3 M r D f − (cid:21) τ v , v ( v ) , (52a) T A = 3 M πr D A f − τ v , v ( v ) , (52b)where we used that there is no absorption outside the planet: s − ( r > R ) = 1.Obviously, the energy condition is fulfilled. At this point, we remark that leavingout all subleading parts, which are proportional to M , would also lead to a validwave in the metric (49), i.e., T ; µµν would also be true to all orders if one onlyconsidered the leading order of (52). This means that we add the subleading29arts to (52) not because of energy conservation but since we want to realize thetransition (51) not only to leading order in 1 /r , but to all orders. Fortunately, we do not either have to worry about the energy condition inside theplanet. For a small enough perturbation, this is true since the energy conditioninside a planet is not only marginally fulfilled. This means that s − ( r ) can benon-zero inside the planet: This corresponds to absorption of the wave by theplanet.Lastly, we have to show that the wave is still a valid solution after it has been partlyabsorbed. For the purpose of illustration, we model the planet as a sequence ofmassive shell with vacuum in between, T µν = 0. In that case, the only non-trivialquestion is whether (51) fulfills the energy condition after it passed some or allof the shells. Therefore, we calculate the energy-momentum-tensor in this region.By Birkhoff’s theorem, the local geometry corresponds to a Schwarzschild solutionwith diminished mass ˜ M (where ˜ M can be zero). It only depends on the matterwhich it has passed via ξ µv . We parameterize the difference of ξ µv and the vectorfield one would get in a pure Schwarzschild geometry of mass ˜ M by σ := Z R max R min d r (( √ AB − r ) , (53)where we have no matter for r > R max and between r and R min . Explicitly, thismeans that we can write ξ Av = f − , A (cid:18) r + σ (cid:19) , where it is important that σ does not depend on r in our region of interest. Ofcourse, σ = 0 corresponds to the case when there is no matter outside.With the help of Mathematica [53], we compute: T = 14 πr " ˜ µ − (1 + σr ) D ( D + 2) ˜ f − − M r D ˜ f − ! τ v , v ( v ) , (54a) T A = " M πr D A ˜ f − + σ π D A ( D + 2) ˜ f − τ v , v ( v ) , (54b) T AB = − σr π h(cid:16) D A D B − γ AB D (cid:17) ˜ f − i τ v , v ( v ) , (54c)where ˜ f − = s − ( r ) f − is the supertranslation which is attenuated because of ab-sorption in the outer shells. It is crucial to note that s − ( r ) = 0 in this calculationsince we are not inside one of the shells of the planet and likewise ˜ µ = s − ( r ) µ . Aswe can estimate σ very crudely as σ < /R , we see that for sufficiently large µ ,the energy condition is fulfilled. With a more accurate estimate, we expect that Of course, energy conservation relates the two subleading parts of T and T A . When wechoose one, it determines the other. We use that AB = 1 in a Schwarzschild geometry of arbitrary mass. µ is not restricted when the wave passes a massive shell.In summary, we have shown that the metric (51), which describes the dynamicaltransition from a spherically symmetric planet to a counterpart with nontrivialangular distribution of mass, is a valid solution. Step 2: Emission
The second step is to describe the emission of the wave by the planet. Thus, ourinitial metric is the one after absorption, as determined by equation (51): δg vµν = s − ( r ) (cid:18) L ξ v ( f − ) g vµν + 2 µGr δ µ δ ν (cid:19) . (55)As we want to consider emission, our first step is to transform it to retardedcoordinates. Intuitively, it is clear that it should be possible to describe a slightlyasymmetrical planet also in retarded coordinates. While it is generically hardto write down the corresponding diffeomorphism which connects the two metrics,we can use that the metric of a planet does not differ from Schwarzschild in theexterior region. Therefore, we can use the diffeomorphism (44) to obtain g uµν = g u, µν + s − ( r ) (cid:16) L ξ u ( − f − ) g u, µν + θ ( R − r )dev (cid:17) , (56)where g u, µν is the metric of the initial, spherically symmetric planet in retardedcoordinates. This means that we apply a supertranslation in retarded coordinateswhich is defined by the function f − used to defined the advanced supertranslation.The function dev accounts for the fact that we do not know the continuation of thematching diffeomorphism (44) to the interior of the planet. Therefore, g uµν mightdeviate slightly from BMS-gauge but only in the interior. We expect, however,that the matching diffeomorphism can be continued such that dev = 0. Finally,we want to point out that g uµν ( r = 0) = g u, µν ( r = 0) since s − ( r = 0) = 0, i.e.the wave does not reach the center and the mass distribution of the planet is stillspherically symmetric around r = 0.The case of the planet provides us with another justification why the matching (46)is natural. With this identification, both the metric (55) in advanced coordinatesand the metric (56) in retarded coordinates cover the whole manifold. Extrapo-lating the results of [18, 19], where finite supertranslations of Schwarzschild andMinkowski are discussed, we expect that for any other matching, i.e. for any othervalue of the supertranslation field, this is no longer the case. If this is true, therequirement that the BMS-coordinate system covers the whole manifold singlesout a unique value of the advanced supertranslation field as well as a unique valueof the retarded supertranslation field, and therefore a coordinate matching.Next, we want to describe how the metric (56) emits a wave. This wave shouldrealize a supertranslation described by f + , which is generically different from f − : δg uµν = τ u , u ( u ) s + ( r ) (cid:18) L ξ u ( f + ) g u, µν − µGr δ µ δ ν (cid:19) , (57)31here we used that L ξ u ( f + ) g uµν = L ξ u ( f + ) g u, µν to first order in f + and f − . Thus,working only to first order simplifies our calculations significantly since we cansimply use the calculations for the absorption. The wave (54) becomes: T = 14 πr " ˜ µ − (1 + σr ) D ( D + 2) ˜ f + − M r D ˜ f + ! τ u , u ( u ) , (58a) T A = − " M πr D A ˜ f + + σ π D A ( D + 2) ˜ f + τ u , u ( u ) , (58b) T AB = − σr π h(cid:16) D A D B − γ AB D (cid:17) ˜ f + i τ u , u ( u ) . (58c)As for the absorption, we have shown that we can realize the transformation (57)with a physical wave.Finally, we analyze the joint effect of absorption and emission. Combining thetransformations (51) and (57), we get total total change of the metric: δg tot µν = θ ( r − R ) L ξ u ( f + − f − ) g u, µν + θ ( R − r ) (cid:16) s + ( r ) L ξ u ( f + ) g u, µν − s − ( r ) L ξ u ( f − ) g u, µν + dev (cid:17) , (59)where we used retarded coordinates. As desired, the mass of the planet staysinvariant. Moreover, δg tot µν acts as a diffeomorphism outside the planet, namely it isthe difference of the advanced supertranslation, described by f − , and the retardedsupertranslation, described by f + . If we furthermore assume that the term dev,which reflects our incomplete knowledge of the matching between advanced andretarded coordinates in the planet, is zero, we see that the metric does not changefor f − = f + . We obtain a trivial transformation if the angular energy distributionof ingoing and outgoing radiation is angle-wise the same. Acknowledgements
We are happy to thank Gia Dvali for many stimulating discussions and comments.We also thank Artem Averin, Henk Bart and Raoul Letschka for discussions. Thework of C.G. was supported in part by Humboldt Foundation and by Grants: FPA2009-07908 and ERC Advanced Grant 339169 "Selfcompletion".
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