PHYSICAL REVIEW LETTERS 127, 041301 (2021)
Order-Unity Correction to Hawking Radiation
Eanna E. Flanagan * Department of Physics, Cornell University, Ithaca, New York 14853, USA and Cornell Laboratory for Accelerator-Based Sciences and Education (CLASSE), Cornell University, Ithaca, New York 14853, USA
(Received 14 February 2021; revised 3 June 2021; accepted 22 June 2021; published 22 July 2021)
When a black hole first forms, the properties of the emitted radiation as measured by observers near future null infinity are very close to the 1974 prediction of Hawking. However, deviations grow with time 7=3 and become of order unity after a time t ∼ Mi , where Mi is the initial mass in Planck units. After an evaporation time, the corrections are large: the angular distribution of the emitted radiation is no longer dominated by low multipoles, with an exponential falloff at high multipoles. Instead, the radiation is redistributed as a power-law spectrum over a broad range of angular scales, all the way down to the scale Δθ ∼ 1=Mi, beyond which there is exponential falloff. This effect is a quantum gravitational effect, whose origin is the spreading of the wave function of the black hole’s center-of-mass location caused by the kicks of the individual outgoing quanta, discovered by Page in 1980. The modified angular distribution of the Hawking radiation has an important consequence: the number of soft hair modes that can effectively interact with outgoing Hawking quanta increases from the handful of modes at low multipoles l to a large 2 number of modes, of order ∼Mi . We argue that this change unlocks the Hawking-Perry-Strominger mechanism for purifying the Hawking radiation.
DOI: 10.1103/PhysRevLett.127.041301
Introduction.—In the half century since its discovery, the with n ≤ N. Given a correction Δρ to the density matrix ρ Hawking evaporation of black holes and its associated on H, we define conundrums have proved to be a fertile source of insights ε ¼k 0 Δρk; ð Þ and progress in quantum gravity, from black hole thermo- Hn trHn 1 dynamics to holography to links between quantum infor- pffiffiffiffiffiffiffiffiffi mation and geometry [1–3]. At the same time, unresolved where kAk¼tr A†A, which gives a measure of the size of theoretical tensions have led to repeated scrutiny of the the correction to the state when restricted to Hn. There exist Δρ ε n ≪ N robustness of Hawking’s predictions. An evaporating black perturbations for which Hn is small whenever , ε hole is characterized by the small dimensionless parameter but for which Hn is nevertheless of order unity when 1=M, where M is the mass in Planck units, and there are n ∼ N. Such corrections have long been anticipated for small corrections that are perturbative in 1=M, as well as Hawking radiation, since an order-unity correction to an smaller corrections nonperturbative in 1=M. Large correc- entanglement entropy is required [4] for unitarity of the tions, however, have been elusive. evaporation process [2,3]. Indeed, recent calculations using There is a subtlety in classifying the size of corrections to Euclidean path integrals have explicitly shown that the time Hawking radiation, related to the fact that the number of evolution of the entanglement entropy of the Hawking relevant field modes N ∼ M2 is large, and fractional radiation and the black hole is consistent with unitarity [6– corrections to expected values may be small for certain 11]. Hence there are corrections to Hawking radiation that classes of operators but large for other operators. Suppose are of order unity, for operators that involve ∼N modes, we decompose the Hilbert space H of radiation states at although the new computational techniques do not yet þ 0 future null infinity I as the product H ¼ Hn ⊗ Hn, allow computation of the corrected state. where Hn is the Hilbert space of a certain set of n modes In this Letter, we confine attention to operators that act on n ≪ N modes, for which the general expectation has been that corrections to Hawking radiation are small. We show that there are corrections at the level of individual Published by the American Physical Society under the terms of modes that are of order unity, arising from quantum the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to gravitational effects in the deep infrared. The mechanism the author(s) and the published article’s title, journal citation, is straightforward: secularly increasing fluctuations in the and DOI. Funded by SCOAP3. center-of-mass location of the black hole cause a change in
0031-9007=21=127(4)=041301(6) 041301-1 Published by the American Physical Society PHYSICAL REVIEW LETTERS 127, 041301 (2021) the angular distribution of the radiation, with most of the one side of I þ and advanced on the other. On a cut of fixed power being redistributed to small angular scales. Although retarded time, the energy flux due to this quantum will be the modifications to the radiation do not directly impact the localized to a thin strip on the sphere of width Δθ ∼ issue of how unitarity of the evaporation is achieved, we M=M2 ∼ 1=M (see Fig. 1), and so the power spectrum of will argue that there is an important indirect effect. the radiation as a function of angular scale will be peaked at In the remainder of the Letter, we first give a heuristic angular scales ∼1=M. argument for the effect, then give a detailed derivation, Redistribution of power to small angular scales: and conclude with a discussion of some implications. Derivation.—Although the mechanism that modifies the Throughout we use Planck units with G ¼ ℏ ¼ c ¼ 1. Hawking radiation is universal, for simplicity, we will Redistribution of power to small angular scales: Brief specialize here to a four-dimensional Schwarzschild black heuristic argument.—As described by Page [12], the hole coupled to a massless free scalar field Φ. Near I þ we emission of Hawking radiation causes the uncertainty in use retarded Bondi coordinates ðu; r; θ; ϕÞ. We resolve the the black hole’s center of mass to grow with time. This Bondi-Metzner-Sachs (BMS) transformation freedom in growth is easy to understand: each outgoing quantum these coordinates by choosing the canonical coordinates −1 carries off a momentum ∼M in a random direction, [13] associated with the approximate stationary state of the and the resulting perturbation to the velocity of the black black hole shortly after it is formed at u ¼ 0, say, before it −2 3 hole is of order ∼M . Over an evaporation time ∼M , this has time to emit appreciable amounts of Hawking radiation single kick yields a displacement of the center-of-mass [15]. This choice also determines a particular Poincar´e position of the black hole of order ∼M. Over the course of subgroup of the BMS group. the evaporation process, we have N ∼ M2 such kicks that We define the field φ on I þ by accumulate as a random walk, giving a total netpffiffiffiffi uncertainty ∼ NM ∼ M2 φðu; θ; ϕÞ 1 in the position of the black hole of order . Φðu; r; θ; ϕÞ¼ þ O ; ð2Þ Now if a black hole displaced by ∼M2 emits a single r r2 quantum in a wave packet mode of duration ∼M, the þ energy flux at future null infinity I þ is delayed by ∼M2 on and we denote by H the Hilbert space of out-states on I parametrized by φ. We denote by Mi the initial mass of the black hole at u ¼ 0, and by M ¼ Mðu1Þ
Gðu; θ; u0; θ0Þ¼tr½ρφðu; θÞφðu0; θ0Þ − h0jφðu; θÞφðu0; θ0Þj0i ; ð Þ out out 3 j0i θ ¼ðθ; ϕÞ where out is the out vacuum and .For stationary, spherically symmetric states, we have G ¼ GðΔu; γÞ, where Δu ¼ u − u0 and γ is the angle FIG. 1. An illustration of the standard Unruh state of an 0 between θR and θ . We define the Fourier transform evaporating nonspinning black hole, at a particular instant of G˜ ðω; γÞ¼ dΔueiωΔuGðΔu; γÞ retarded time at future null infinity, in a reference frame that is , and decompose this in displaced from the black hole center of mass by several angular harmonics as Schwarzschild radii. The quantity plotted is a typical realization X∞ 2l þ 1 of the Gaussian random process on the sphere whose two-point G˜ ðω; γÞ¼ P ð γÞSðω;lÞ: ð Þ function is given by taking the two-point function of a scalar field 4π l cos 4 in the Unruh state at future null infinity and subtracting the two- l¼0 point function of the out vacuum. Fluctuations in individual wave _ packet modes give rise to fluctuations on the sphere that are The quantity Sðω;lÞ is related to the energy flux E to confined to concentric thin strips, giving rise to a characteristic infinity per unit frequency ω in field multipoles [16] of angular scale that is small compared to unity. order l by
041301-2 PHYSICAL REVIEW LETTERS 127, 041301 (2021) dE_ 2l þ 1 1 Δ2 1 σ 2 W˜ ðΔ; ξÞ¼N − − ð1 − ε2Þξ2σ2 − iε p Δ ξ ; ¼ ω Sðω;lÞ: ð5Þ exp 2 p · dω l 2π 2 σΔ 2 σΔ ð8Þ We denote by GU the regularized two-point function of the Unruh vacuum, for which the corresponding energy flux is −3=2 −3 2 2 where N ¼ð2πÞ σΔ , the quantities σΔ and σp are the ε jεj < 1 dE_ 2l þ 1 ωjt j2 variances in position and momentum, and with is ¼ lω : ð Þ a correlation coefficient. The evolution of these para- dω 2π eβω − 1 6 U;l meters is studied in Ref. [25], which shows that ε is of order unity and Here tlω is the transmission coefficient through the effective potential and β ¼ 8πM is the inverse temperature of the 4 3 3 3 c0M ð1 − M =M Þ Mi − M ≪ Mi; σ2 ¼ i i pffiffiffiffiffiffi radiation. As is well known, most of the power in the Δ 4 l ∼ Oð1Þ c1Mi Mi ≪ M ≪ Mi; spectrum (6) is concentrated at , with an expo- pffiffiffiffiffiffi l 2 nential falloff at large . σp ¼ c2 lnðMi=MÞ Mi ≪ M; ð9Þ We now want to derive how the energy spectrum (6) as a function of frequency and angular scale is modified. The where c0, c1, and c2 are dimensionless constants of order key idea is to supplement the standard computation by unity. including the evolution of a small number of relevant We now turn to describing how the fluctuations in the infrared gravitational degrees of freedom, specifically the center of mass of the black hole affect the Hawking þ BMS charges as computed on cuts u ¼ const of I . In the radiation. In Minkowski spacetime we can define a dis- classical theory, the values of these charges determine the placement operator UΔ, which displaces any state by an spacetime geometry when the black hole is stationary, and amount Δ, which acts on the field operator according to we assume that this is still true in the quantum theory when † UΔΦðt; rÞUΔ ¼ Φðt; r − ΔÞ. This operator extends natu- both the charges and geometry have quantum fluctuations. rally to the Hilbert space H of out-states on the black hole ’ We focus, in particular, on the black hole s center of mass spacetime, where its action is defined by Δ, encoded in the orbital angular momentum associ- ated with the Poincar´e subgroup of the BMS group † UΔφðu; θÞUΔ ¼ φðu þ n · Δ; θÞ; ð10Þ discussed above. I þ The framework we use is anchored at , where the with n the unit vector in the direction specified by θ. The ðu; θ; φÞ H coordinate system and out Hilbert space for the Unruh state for a black hole displaced from the origin by an scalar field are unaffected by the large quantum fluctuations amount Δ can be written as of the gravitational charges and of the geometry in the X interior. As described above, the fluctuations in Δ grow jΔi ⊗ jχjiUΔjψ ji; ð11Þ with time due to repeated kicks from outgoing Hawking j I þ I þ quanta. We divide into an early portion early with uu jχ i jψ i 1 and a late portion late with 1. The Hilbert where j is a set of states on the future horizon and j is space H can be correspondingly factored [17] into the a set of states in H. Taking the trace over the horizon states H ⊗ H I þ tensor product early late. The state of the center of gives for the corresponding Unruh state at mass at time u1 is strongly correlated with the Hawking I þ jΔihΔj ⊗ U ρ U† ; ð Þ radiation on early, by momentum conservation for each Δ U Δ 12 emission event, and if we trace over H we obtain a P early ρ ¼ c2jψ ihψ j c2 ¼hχ jχ i mixed state for the center of mass [23]. This state can be where U j j j j with j j j . described in terms of its Wigner function WðΔ; pÞ,a Suppose now that the state of the black hole’s center of function of the three-dimensional position Δ and momen- mass were fixed and not evolving with time, given by tum p of the black hole. Denoting a position eigenstate by Eq. (7) for the fixed values of the parameters σΔ, σp, and ε jΔi, the corresponding state is evaluated at u ¼ u1. Then by linearity from Eqs. (7), (11), Z Z and (12), the corresponding out-state would be Z Z d3Δ d3ξW˜ ðΔ; ξÞjΔ − ξ=2ihΔ þ ξ=2j; ð7Þ 3 3 ˜ † d Δ d ξWðΔ;ξÞjΔ−ξ=2ihΔþξ=2j ⊗ UΔ−ξ=2ρUUΔþξ=2: R ˜ 3 where WðΔ; ξÞ¼ d p exp½−ip · ξ WðΔ; pÞ. Since the ð13Þ kicks from the individual outgoing quanta are uncorrelated, the Wigner function W is very nearly Gaussian by the Tracing over the center-of-mass Hilbert space gives the multivariate central limit theorem. Hence W˜ has the form corrected version of the Unruh state
041301-3 PHYSICAL REVIEW LETTERS 127, 041301 (2021) Z ˜ 3 ˜ † [26], which is of order unity, for such frequencies GU varies ρU; ¼ d ΔWðΔ; 0ÞUΔρUU : ð14Þ corr Δ with γ only on angular scales of order unity; there are no other dimensionless parameters on which the function ˜ Of course the state of the center of mass is evolving with depends. It follows that GU has negligible variation over time and not fixed. Nevertheless, the corrected Unruh state 2 the range 0 ≤ γ ≲ 1=ðωσΔÞ ∼ M=M ≪ 1 that is not expo- n i (14) should give a good approximation to the -point nentially suppressed by the exponential factor in Eq. (17). I þ u functions on of the field at retarded times that satisfy Hence we can evaluate this function at γ ¼ 0 and pull it ju − u j ≪ M3 two conditions: (i) We have 1 , so the mass of outside the integral, and using Eqs. (4) and (5) we re- the black hole as well as the state of the center-of-mass have express it in terms of the total power per unit frequency u ¼ u P not evolved significantly from their values at 1. dE=d_ ω ¼ ðdE=d_ ωÞ 2 l l in the Unruh state.R We evaluate the (ii) We have u − u1 ≫ σΔ ∼ Mi . This ensures that the aμ remaining integral using the identity dμPlðμÞe ¼ displacements (10) in retarded time caused by the operators pffiffiffiffiffiffiffiffiffiffiffi 2π=aIlþ1=2ðaÞ, which expresses it terms of a modified UΔ in Eq. (14) do not generate a dependence on degrees of I þ Bessel function of the first kind [27]. The final result is freedom on early, which we have already traced over to rffiffiffi compute the state (7). −ω2σ2 dE_ π dE_ ð2l þ 1Þe Δ We now turn to showing that the modifications inherent ¼ I ðω2σ2 Þ: dω 2 dω ωσ lþ1=2 Δ in the corrected Unruh state (14) are of order unity, for U;corr;l U Δ individual outgoing wave packet modes at sufficiently late ð18Þ times. Combining Eqs. (3), (10), and (14) gives for the regularized two-point function of the corrected Unruh state −1=2 a Using the approximate formula Ilþ1=2ðaÞ¼ð2πaÞ e ½1þ Z Oðl2=aÞ , this simplifies to [28] G ðu; θ; u0; θ0Þ¼ d3ΔW˜ ðΔ; 0Þ U;corr _ _ 2 0 0 0 dE dE ð2l þ 1Þ l × GUðu þ n · Δ; θ; u þ n · Δ; θ Þ; ¼ 1 þ O : ð Þ dω dω 2ω2σ2 ω2σ2 19 U;corr;l U Δ Δ ð15Þ This corresponds to a power-law spectrum for angular using that the Wightman function in the second term in 0 ≤ l ≪ l l ¼ ωσ scales in the range crit with crit Δ, with most Eq. (3) is invariant under translations. The corresponding l ∼ l l ≥ l of the power in the vicinity of crit. At scales crit the functions of frequency ω and angle γ are related by spectrum falls off exponentially,p fromffiffiffi the upperpffiffiffi bound −1=2 a Z Ilþ1=2ðaÞ ≤ ð2πaÞ e exp½−l=ð4 aÞ for l ≥ a ≫ 1. G˜ ðω; γÞ¼ d3ΔW˜ ðΔ; 0Þe−iωðn−n0Þ·ΔG˜ ðω; γÞ l ¼ ωσ U;corr U We now consider the critical angular scale crit Δ. 3 2 At sufficiently late times u ≳ Mi , we have σΔ ∼ Mi from 2 2 2 ˜ ¼ exp ½−2ω σ sin ðγ=2Þ GUðω; γÞ; ð16Þ l ∼ M2=M ≫ Δ Eq. (9), and so the critical angular scale is crit i −1 1 using ω ∼ M , which reduces to ∼Mi if M ∼ Mi.At where we have used Eq. (8). Note that the transformation u=M3 ∼ 1 − ˜ early times, we have from Eq. (9) and using i (16) preserves Gðω; 0Þ, which is proportional to the total −7=2 M3=M3 that l ∼ u3=2M , so the modification effect energy flux per unit frequency, summed over all multipoles. i crit i Hence the transformation redistributes power over angular first becomes of order unity after an interval of retarded u ∼ M7=3 scales, but not from one frequency to another. time i . We next combine Eqs. (4), (5), and (16) to obtain for the Discussion and conclusions.—We close with a number spectrum of outgoing radiation of comments. First, the modification to the Hawking radiation does not alter the amount of entanglement Z dE_ 1 between modes inside the horizon and those outside, 2 −ω2σ2 ð1−μÞ ˜ ¼ð2l þ 1Þω dμPlðμÞe Δ GUðω;γÞ; and so does not directly impact the unitarity of the dω −1 U;corr;l evaporation process. The exterior modes that are relevant ð17Þ at late times depend, through the position of the black hole, on which early time exterior modes are occupied (the total μ ¼ γ 2 where cos . We now specialize to frequencies of the number of relevant exterior modes has increased from ∼Mi ω ∼ M−1 4 order , where most of the outgoing power is to ∼Mi ). This effect generates nontrivial mutual informa- −1 located. We thus exclude high frequencies ω ≫ M , tion [2] between early Hawking radiation and late Hawking where the power is exponentially suppressed, and low radiation, but does not alter the total entanglement between −1 frequencies ω ≪ M , where it is power-law suppressed, interior and exterior modes. ˜ from the spectrum (6). Since the function ωGUðω; γÞ Second, the corrected Unruh state (14) is not a Gaussian depends on ω and M only through the combination ωM state, unlike the original Unruh state, although it is
041301-4 PHYSICAL REVIEW LETTERS 127, 041301 (2021) stationary and spherically symmetric. Thus it is not The modified angular distribution of the Hawking determined by the two-point function (18), although it is radiation completely changes this picture, since the source completely determined by the formulas (8) and (14). term in Eq. (20) now extends effectively up to multipoles of 2 Third, the general mechanism discussed here involving order l ∼ Mi. This makes ∼Mi soft hair modes potentially spatial translations clearly also applies to other generators accessible, enough for each outgoing quantum to interact of the BMS group. The black hole at late times determines a with its own soft hair mode. Note, however, that this BMS frame that is related to the initial BMS frame by a scenario cannot be analyzed within a single semiclassical transformation which includes a rotation, boost, and super- spacetime. The details of the interaction of the Hawking translation, and secularly growing quantum fluctuations in radiation with the soft hair is an intriguing subject for those transformations modify the outgoing Hawking radi- further study. ation. However, in Ref. [25] we estimate that the typical I thank Abhay Ashtekar, Venkatesa Chandrasekaran, and boost [29] velocity scale is ∼1=M, and that the length scale Kartik Prabhu for helpful discussions and an anonymous involved in the supertranslation fluctuations is ∼1, so the referee for useful comments. This research was supported corresponding modifications to the Hawking radiation in part by NSF Grants No. PHY-1404105 and No. PHY- are small. 1707800. Fourth, consider the result of interpreting the corrected Unruh state (14) on I þ in terms of a single semiclassical spacetime with the black hole at the origin. An outgoing þ * mode with l ∼ Mi near I corresponds near the black hole [email protected] to a large amplitude standing wave in a thin shell of width [1] S. W. Hawking, Breakdown of predictability in gravitational ∼1 in the nonevanescent region between the horizon and collapse, Phys. Rev. D 14, 2460 (1976). [2] D. Harlow, Jerusalem lectures on black holes and quantum potential barrier, which varies over transverse length scales ∼1 information, Rev. Mod. Phys. 88, 015002 (2016). along the horizon of order . This Planckian behavior of [3] D. Marolf, The black hole information problem: Past, the extrapolated corrected Unruh state illustrates the poten- present, and future, Rep. Prog. Phys. 80, 092001 (2017). tial pitfalls of thinking in terms of a single semiclassical [4] The fact that kΔρk¼Oð1Þ is required follows from the spacetime and focusing on near-horizon physics. identity jSðρ þ ΔρÞ − SðρÞj ≤ kΔρk log d þ 1=e, where Fifth, we argue that the modification to the Hawking S is von Neumann entropy and d is the dimension of process removes one of the primary objections to the the Hilbert space [5], together with Sðρ þ ΔρÞ¼0 and 2 proposal that soft hair on black holes plays a key role in SðρÞ ∼ log d ∼ M . resolving the information loss paradox [21,22,30–32]. Soft [5] M. A. Nielsen, Continuity bounds for entanglement, Phys. hair consists of charges measurable at future null infinity Rev. A 61, 064301 (2000). associated with an extension of the BMS group [33–36], [6] A. Almheiri, N. Engelhardt, D. Marolf, and H. Maxfield, l The entropy of bulk quantum fields and the entanglement higher- analogs of the center of mass that are encoded in wedge of an evaporating black hole, J. High Energy Phys. the asymptotic metric. Just as for the center of mass, the 12 (2019) 063. expected value of soft hair charges can be set to zero by a [7] G. Penington, Entanglement wedge reconstruction and the gauge transformation, locally in time, but their variances information paradox, J. High Energy Phys. 09 (2020) 002. cannot and can contain nontrivial information. Outgoing [8] A. Almheiri, R. Mahajan, J. Maldacena, and Y. Zhao, The Hawking quanta excite soft hair via the gravitational wave Page curve of Hawking radiation from semiclassical geo- memory effect. It has been suggested that the Hawking metry, J. High Energy Phys. 03 (2020) 149. radiation is purified at late times by its entanglement with [9] A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian, and soft hair degrees of freedom [30]. A. Tajdini, Replica wormholes and the entropy of Hawking radiation, J. High Energy Phys. 05 (2020) 013. A difficulty with this proposal has been that only low l [10] G. Penington, S. H. Shenker, D. Stanford, and Z. Yang, modes of the soft hair can be excited by the outgoing Replica wormholes and the black hole interior, arXiv: quanta, because of the exponential falloff of the spectrum 1911.11977. (6) at high l. The soft hair field ΦðθÞ is given in terms of the [11] A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian, and þ scalar field φ on I by [see, e.g., Eqs. (2.19) and (4.4) of A. Tajdini, The entropy of Hawking radiation, arXiv: Ref. [14] ] 2006.06872. Z [12] D. N. Page, Is Black-Hole Evaporation Predictable?, Phys. 2 2 2 D ðD þ 2ÞΦðθÞ¼32πP duφ;uðu; θÞ ; ð20Þ Rev. Lett. 44, 301 (1980). [13] See, for example, Sec. II. C of Ref. [14]. 2 [14] E. E. Flanagan and D. A. Nichols, Conserved charges of the where D is the Laplacian on the two-sphere and P is a l ¼ 0 extended Bondi-Metzner-Sachs algebra, Phys. Rev. D 95, projection operator that sets to zero , 1 modes. Thus 044002 (2017). only a handful of soft hair modes can be excited, too few to [15] Note that imposing this requirement in the region of I þ 2 play a relevant role for purifying the ∼Mi outgoing near u ¼ 0 determines the coordinates everywhere on Hawking quanta. I þ because of the properties of the BMS group.
041301-5 PHYSICAL REVIEW LETTERS 127, 041301 (2021)
[16] Note that there are two different methods of defining the [26] F. Gray and M. Visser, Greybody factors for Schwarzschild angular spectrum of Hawking radiation. One can decom- black holes: Path-ordered exponentials and product inte- pose the field into spherical harmonic modes or decompose grals, Universe 4, 93 (2018). the stress energy tensor. We use the former definition. The [27] F. W. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, two definitions are not equivalent since the stress energy NIST Handbook of Mathematical Functions, 1st ed. tensor depends nonlinearly on the field. Nevertheless, the (Cambridge University Press, Cambridge, 2010), formula qualitative result that power is redistributed from large 18.17.19. angular scales to small angular scales will clearly be be [28] For l ≫ 1 Eq. (19) can be more simply derived by valid for both definitions. approximating the sphere as a plane and replacing the [17] This is not quite true, as there are also “edge modes” transform (17) with a two-dimensional Fourier transform. associated with the boundary u ¼ u1 [18–20], equivalent to [29] The effect of boosts is also suppressed by the fact that the soft hair [21,22]. We neglect these modes here as they are operator describing the evolution of the quantum field is not relevant to the present discussion, but will return to them approximately diagonal on a position basis for the black later in the Letter. hole center of mass, but not on a momentum basis. This is [18] W. Donnelly and L. Freidel, Local subsystems in gauge why boost fluctuations are not present in Eq. (14), although W theory and gravity, J. High Energy Phys. 09 (2016) 102. they are present in the Wigner function . [19] A. J. Speranza, Local phase space and edge modes for [30] A. Strominger, Black hole information revisited, arXiv: diffeomorphism-invariant theories, J. High Energy Phys. 02 1706.07143. (2018) 021. [31] S. Pasterski and H. Verlinde, HPS meets AMPS: How soft [20] M. Geiller, Edge modes and corner ambiguities in 3d Chern- hair dissolves the firewall, arXiv:2012.03850. Simons theory and gravity, Nucl. Phys. B924, 312 (2017). [32] P. Cheng and Y. An, Soft black hole information paradox: [21] S. W. Hawking, M. J. Perry, and A. Strominger, Soft Hair on Page curve from Maxwell soft hair of black hole, Phys. Rev. Black Holes, Phys. Rev. Lett. 116, 231301 (2016). D 103, 126020 (2021). [22] S. W. Hawking, M. J. Perry, and A. Strominger, Super- [33] A. Strominger, Lectures on the Infrared Structure of Gravity rotation charge and supertranslation hair on black holes, and Gauge Theory (Princeton University Press, Princeton, NJ, 2018). J. High Energy Phys. 05 (2017) 161. [34] M. Campiglia and A. Laddha, Asymptotic symmetries and [23] One can think of this as the early Hawking radiation subleading soft graviton theorem, Phys. Rev. D 90, 124028 decohering the black hole location [24]. (2014). [24] A. Arrasmith, A. Albrecht, and W. H. Zurek, Decoherence [35] M. Campiglia and A. Laddha, New symmetries for the of black hole superpositions by Hawking radiation, Nat. gravitational S-matrix, J. High Energy Phys. 04 (2015) 076. Commun. 10, 1024 (2019). [36] G. Comp`ere, A. Fiorucci, and R. Ruzziconi, Superboost [25] E. E. Flanagan, Infrared effects in the late stages of black transitions, refraction memory and super-Lorentz charge hole evaporation, arXiv:2102.13629. algebra, J. High Energy Phys. 11 (2018) 200.
041301-6