aa r X i v : . [ h e p - t h ] F e b An order-unity correction to Hawking radiation
Eanna E. Flanagan ∗ Department of Physics, Cornell University, Ithaca, NY 14853 Cornell Laboratory for Accelerator-based Sciences and Education (CLASSE), Cornell University, Ithaca, NY 14853
When a black hole first forms, the properties of the emitted radiation as measured by observers near futurenull infinity are very close to the 1974 prediction of Hawking. However, deviations grow with time, and becomeof order unity after a time t ∼ M / i , where M i is the initial mass in Planck units. After an evaporation time thecorrections are large: the angular distribution of the emitted radiation is no longer dominated by low multipoles,with an exponential fall o ff at high multipoles. Instead, the radiation is redistributed as a power law spectrumover a broad range of angular scales, all the way down to the scale ∆ θ ∼ / M i , beyond which there is exponentialfallo ff . This e ff ect is is a quantum gravitational e ff ect, whose origin is the spreading of the wavefunction of theblack hole’s center of mass location caused by the kicks of the individual outgoing quanta, discovered by Pagein 1980.The modified angular distribution of the Hawking radiation has an important consequence: the number of softhair modes that can e ff ectively interact with outgoing Hawking quanta increases from the handful of modes atlow multipoles l , to a large number of modes, of order ∼ M i . We argue that this change unlocks the Hawking-Perry-Strominger mechanism for purifying the Hawking radiation. Introduction – In the half century since its discovery, theHawking evaporation of black holes and its associated co-nundrums have proved to be a fertile source of insights andprogress in quantum gravity, from black hole thermodynam-ics to holography to links between quantum information andgeometry [1–4]. At the same time, unresolved theoretical ten-sions have led to repeated scrutiny of the robustness of Hawk-ing’s predictions. An evaporating black hole is characterizedby the small dimensionless parameter 1 / M , where M is theblack hole mass in Planck units, and there are small correc-tions that are perturbative in 1 / M , as well as smaller correc-tions nonperturbative in 1 / M . Large corrections however havebeen elusive.There is a subtlety in classifying the size of corrections toHawking radiation, related to the fact that the number of rel-evant field modes scales as N ∼ M and the dimension of theHilbert space is exponential in N . Given a correction ∆ ρ tothe density matrix ρ , and an algebra A of operators A on theHilbert space, we define ε A = max A ∈A tr[ ∆ ρ A ]tr[ ρ A ] , (1)which gives an upper bound to the fractional corrections to ex-pected values for that algebra. There exist perturbations ∆ ρ tothe density matrix for which ε A is small for operators whichact only on n modes with n ≪ N , but for which ε A is neverthe-less of order unity for operators that act on ∼ N modes. Suchcorrections have long been anticipated for Hawking radiation,since an order-unity correction to an entanglement entropy isrequired for unitarity of the evaporation process [3, 4]. Indeed,recent calculations using Euclidean path integrals have explic-itly shown that the time evolution of the entanglement entropyof the emitted Hawking radiation and the black hole is consis-tent with unitarity [5–10]. It follows that there are correctionsto Hawking radiation that are of order unity, for operators thatinvolve ∼ N modes, although the new computational tech-niques do not yet allow computation of the correction to thestate. In this Letter we confine attention to operators that act on n ≪ N modes, for which the general expectation has beenthat corrections to Hawking radiation are small. We show thatthere are corrections at the level of individual modes that areof order unity, arising from quantum gravitational e ff ects inthe deep infrared. The mechanism is straightforward: secu-larly increasing fluctuations in the center of mass location ofthe black hole cause a change in the angular distribution of theradiation, with most of the power being redistributed to smallangular scales. Although the modifications to the radiationdo not directly impact the issue of how unitarity of the evap-oration is achieved, we will argue that there is an importantindirect e ff ect.In the remainder of the Letter, we first give a heuristic ar-gument for the e ff ect, then give a detailed derivation, and con-clude with a discussion of some implications. Throughout weuse Planck units with G = ~ = c = Redistribution of power to small angular scales: brief heuris-tic argument – As described by Page [11], the emission ofHawking radiation causes the uncertainty in the black hole’scenter of mass to grow with time. This growth is easy tounderstand: each outgoing quantum carries o ff a momentum ∼ M − in a random direction, and the resulting perturbationto the velocity of the black hole is of order ∼ M − . Over anevaporation time ∼ M this single kick yields a displacementof the center-of-mass position of the black hole of order ∼ M .Over the course of the evaporation process we have N ∼ M such kicks that accumulate as a random walk, giving a to-tal net uncertainty in the position of the black hole of order ∼ √ N M ∼ M .Now if a black hole displaced by ∼ M emits a single quan-tum in a wavepacket mode of duration ∼ M , the energy flux atfuture null infinity I + is delayed by ∼ M on one side of I + and advanced on the other. On a cut of fixed retarded time,the energy flux due to this quantum will be localized to a thinstrip on the sphere of width ∆ θ ∼ M / M ∼ / M , and so thepower spectrum of the radiation as a function of angular scalewill be peaked at angular scales ∼ / M . Redistribution of power to small angular scales: derivation –Although the mechanism that modifies the Hawking radiationis universal, for simplicity we will specialize here to a fourdimensional Schwarzschild black hole coupled to a masslessfree scalar field Φ . Near I + we use retarded Bondi coordi-nates ( u , r , θ, φ ). We resolve the Bondi-Metzner-Sachs (BMS)transformation freedom in these coordinates by choosing thecanonical coordinates associated with the approximate sta-tionary state of the black hole shortly after it is formed at u = ϕ on I + by Φ ( u , r , θ, φ ) = ϕ ( r , θ, φ ) r + O r ! , (2)and we denote by H the Hilbert space of out states on I + parameterized by ϕ . We denote by M i the initial mass of theblack hole at u =
0, and by M = M ( u ) < M i the Bondi massat some later retarded time u with u ≫ M i . We denote by ρ U the Hawking radiation state on H for an eternal black holeof mass M , i.e. the Unruh vacuum. In the standard calculationit is argued that this state provides a good approximation tothe n -point functions on I + of the state for the gravitationalcollapse spacetime, for retarded times u with | u − u | smallcompared to the evaporation time M .For any density matrix ρ on H we define the regularizedtwo-point function G ( u , θ ; u ′ , θ ′ ) = tr (cid:2) ρϕ ( u , θ ) ϕ ( u ′ , θ ′ ) (cid:3) − out h | ϕ ( u , θ ) ϕ ( u ′ , θ ′ ) | i out , (3)where | i out is the out vacuum and θ = ( θ, φ ). For stationary,spherically symmetric states we have G = G ( ∆ u , γ ), where ∆ u = u − u ′ and γ is the angle between θ and θ ′ . We definethe Fourier transform ˜ G ( ω, γ ) = R d ∆ u e i ω ∆ u G ( ∆ u , γ ), and de-compose this in angular harmonics as˜ G ( ω, γ ) = ∞ X l = l + π P l (cos γ ) S ( ω, l ) . (4)The quantity S ( ω, l ) is related to the energy flux ˙ E to infinityper unit frequency ω in field multipoles of order l by d ˙ Ed ω ! l = l + π ω S ( ω, l ) . (5) See, for example, Sec. II.C of Ref. [12]. Note that there are two di ff erent methods of defining the angular spectrumof Hawking radiation. One can decompose the field into spherical har-monic modes, or decompose the stress energy tensor. We use the formerdefinition. The two definitions are not equivalent since the stress energytensor depends nonlinearly on the field. Nevertheless, the qualitative resultthat power is redistributed from large angular scales to small angular scaleswill clearly be be valid for both definitions. We denote by G U the regularized two point function of theUnruh vacuum, for which the corresponding energy flux is[13] d ˙ Ed ω ! U , l = l + π ω | t l ω | e βω − . (6)Here t l ω is the transmission coe ffi cient through the e ff ectivepotential and β = π M is the inverse temperature of the radi-ation. As is well known, most of the power in the spectrum(6) is concentrated at l ∼ O (1), with an exponential fall o ff atlarge l .We now want to derive how the energy spectrum (6) as afunction of frequency and angular scale is modified. The keyidea is to supplement the standard computation by includingthe evolution of a small number of relevant infrared gravita-tional degrees of freedom, specifically the BMS charges ascomputed on cuts u = const of I + . In the classical theory,the values of these charges determine the spacetime geometrywhen the black hole is stationary, and we assume that this isstill true in the quantum theory. We focus in particular on theblack hole’s center of mass ∆ , encoded in the orbital angu-lar momentum associated with the Poincar´e subgroup of theBMS group discussed above.As described above, the fluctuations in ∆ grow with timedue to repeated kicks from outgoing Hawking quanta. We di-vide I + into an early portion I + early with u < u , and a lateportion I + late with u > u . The Hilbert space H can be corre-spondingly factored into the tensor product H early ⊗H late . Thestate of the center-of-mass at time u is strongly correlatedwith the Hawking radiation on I + early , by momentum conser-vation for each emission event, and if we trace over H early weobtain a mixed state for the center-of-mass. This state can bedescribed in terms of its Wigner function W ( ∆ , p ), a functionof the three dimensional position ∆ and momentum p of theblack hole. Denoting a position eigenstate by | ∆ i , the corre-sponding state is Z d ∆ Z d ξ f W ( ∆ , ξ ) | ∆ − ξ / i h ∆ + ξ / | , (7)where f W ( ∆ , ξ ) = R d p exp[ − i p · ξ ] W ( ∆ , p ). Since thekicks from the individual outgoing quanta are uncorrelated,the Wigner function W is very nearly Gaussian by the centrallimit theorem. Hence f W has the form f W ( ∆ , ξ ) = N exp − ∆ σ ∆ −
12 (1 − ε ) ξ σ p − i ε σ p σ ∆ ∆ · ξ , (8)where N = (2 π ) − / σ − ∆ , the quantities σ ∆ and σ p are thevariances in position and momentum, and ε with | ε | < This is not quite true, as there are also “edge modes” associated with theboundary u = u [14–17], equivalent to soft hair [18, 19]. We neglect thesemodes here as they not relevant to the present discussion, but will return tothem later in the paper. correlation coe ffi cient. The evolution of these parameters isstudied in Ref. [20], which shows that ε = √ / σ ∆ = ( c M i (1 − M / M i ) M i − M ≪ M i , c M i √ M i ≪ M ≪ M i ,σ p = c ln( M i / M ) √ M i ≪ M , (9)where c , c and c are dimensionless constants of order unity.We now turn to describing how the fluctuations in the centerof mass of the black hole a ff ect the Hawking radiation. InMinkowski spacetime we can define a displacement operator U ∆ which displaces any state by an amount ∆ , which acts onthe field operator according to U † ∆ Φ ( t , r ) U ∆ = Φ ( t , r − ∆ ). Thisoperator extends naturally to the Hilbert space H of out stateson the black hole spacetime, where its action is defined by U † ∆ ϕ ( u , θ ) U ∆ = ϕ ( u + n · ∆ , θ ) (10)with n the unit vector in the direction specified by θ . TheUnruh state for a black hole displaced from the origin by anamount ∆ can be written as | ∆ i ⊗ X j (cid:12)(cid:12)(cid:12) χ j E U ∆ (cid:12)(cid:12)(cid:12) ψ j E , (11)where (cid:12)(cid:12)(cid:12) χ j E is a set of states on the future horizon and (cid:12)(cid:12)(cid:12) ψ j E a setof states in H . Taking the trace over the horizon states givesfor the corresponding Unruh state at I + | ∆ i h ∆ | ⊗ U ∆ ρ U U † ∆ , (12)where ρ U = P j c j | ψ j >< ψ j | with c j = < χ j | χ j > .Suppose now that the state of the black hole’s center ofmass were fixed and not evolving with time, given by Eq. (7)for the fixed values of the parameters σ ∆ , σ p and ε evaluatedat u = u . Then by linearity from Eqs. (7), (11) and (12) thecorresponding out state would be Z d ∆ Z d ξ f W ( ∆ , ξ ) | ∆ − ξ / i h ∆ + ξ / |⊗ U ∆ − ξ/ ρ U U † ∆+ ξ/ . (13)Tracing over the center of mass Hilbert space gives the cor-rected version of the Unruh state ρ U , corr = Z d ∆ f W ( ∆ , ) U ∆ ρ U U † ∆ . (14)Of course the state of the center of mass is evolving withtime and not fixed. Nevertheless, the corrected Unruh state(14) should give a good approximation to the n -point func-tions on I + of the field at retarded times u that satisfy twoconditions: • We have | u − u | ≪ M , so the mass of the black hole aswell as the state of the center-of-mass have not evolvedsignificantly from their values at u = u . • We have u − u ≫ σ ∆ ∼ M i . This ensures that thedisplacements (10) in retarded time caused by the oper-ators U ∆ in Eq. (14) do not generate a dependence ondegrees of freedom on I + early , which we have alreadytraced over to compute the state (7). We now turn to showing that the modifications inherent inthe corrected Unruh state (14) are of order unity, for individualoutgoing wavepacket modes at su ffi ciently late times. Com-bining Eqs. (3), (10) and (14) gives for the regularized twopoint function of the corrected Unruh state G U , corr ( u , θ ; u ′ , θ ′ ) = Z d ∆ f W ( ∆ , ) × G U ( u + n · ∆ , θ ; u ′ + n ′ · ∆ , θ ′ ) , (15)using that the Wightman function in the second term in Eq. (3)is invariant under translations. The corresponding functions offrequency ω and angle γ are related by˜ G U , corr ( ω, γ ) = Z d ∆ f W ( ∆ , ) e − i ω ( n − n ′ ) · ∆ ˜ G U ( ω, γ ) = exp h − ω σ ∆ sin ( γ/ i ˜ G U ( ω, γ ) , (16)where we have used Eq. (8). Note that the transformation (16)preserves ˜ G ( ω,
0) which is proportional to the total energy fluxper unit frequency, summed over all multipoles. Hence thetransformation redistributes power over angular scales, but notfrom one frequency to another.We next combine Eqs. (4), (5) and (16) to obtain for thespectrum of outgoing radiation d ˙ Ed ω ! U , corr , l = (2 l + ω Z − d µ P l ( µ ) e − ω σ ∆ (1 − µ ) ˜ G U ( ω, γ ) , (17)where µ = cos γ . We now specialize to frequencies of theorder ω ∼ M − , where most of the outgoing power is lo-cated. We thus exclude high frequencies ω ≫ M − wherethe power is exponentially suppressed, and low frequencies ω ≪ M − where it is power-law suppressed, from the spec-trum (6). Since the function ω ˜ G U ( ω, γ ) depends on ω and M only through the combination ω M [21], which is of orderunity, for such frequencies ˜ G U varies with γ only on angu-lar scales of order unity; there are no other dimensionless pa-rameters on which the function depends. It follows that ˜ G U has negligible variation over the range 0 ≤ γ . / ( ωσ ∆ ) ∼ M / M i ≪ γ = d ˙ E / d ω = P l ( d ˙ E / d ω ) l in the Unruhstate. We evaluate the remaining integral using the identity R d µ P l ( µ ) e a µ = √ π/ aI l + / ( a ) which expresses it terms of amodified Bessel function of the first kind [22]. The final resultis d ˙ Ed ω ! U , corr , l = r π d ˙ Ed ω ! U (2 l + e − ω σ ∆ ωσ ∆ I l + / ( ω σ ∆ ) . (18)Using the approximate formula I l + / ( a ) = (2 π a ) − / e a [1 + O ( l / a )] this simplifies to d ˙ Ed ω ! U , corr , l = d ˙ Ed ω ! U (2 l + ω σ ∆ + O l ω σ ∆ . (19)This corresponds to a power-law spectrum for angular scalesin the range 0 ≤ l ≪ l crit with l crit = ωσ ∆ , with most ofthe power in the vicinity of l ∼ l crit . At scales l ≥ l crit the spectrum falls o ff exponentially, from the upper bound I l + / ( a ) ≤ (2 π a ) − / e a exp[ − l / (4 √ a )] for l ≥ √ a ≫ l crit = ωσ ∆ . Atsu ffi ciently late times u & M i we have σ ∆ ∼ M i from Eq.(9), and so the critical angular scale is l crit ∼ M i / M ≫ ω ∼ M − , which reduces to ∼ M i if M ∼ M i . At earlytimes we have from Eq. (9) and using u / M i ∼ − M / M i that l crit ∼ u / M − / i , so the modification e ff ect first becomes oforder unity after an interval of retarded time u ∼ M / i . Discussion and conclusions: – We close with a number ofcomments. First, the general mechanism discussed here in-volving spatial translations clearly also applies to other gen-erators of the BMS group. The black hole at late times de-termines a BMS frame which is related to the initial BMSframe by a transformation which includes a rotation, boost andsupertranslation, and secularly growing quantum fluctuationsin those transformations modify the outgoing Hawking radia-tion. However, in Ref. [20] we estimate that the typical boostvelocity scale is ∼ / M , and that the lengthscale involved inthe supertranslation fluctuations is ∼
1, so the correspondingmodifications to the Hawking radiation are small.Second, one e ff ect of the interaction of Hawking quantawith the black hole center of mass is that whether or not anearly outgoing mode is occupied influences which outgoingmodes are relevant at late times. This e ff ect generates non-trivial mutual information [3] between early Hawking radia-tion and late Hawking radiation. However, it does not alterthe amount of entanglement between modes inside the hori-zon and those outside, and so does not impact the unitarity ofthe evaporation process.Third, consider the result of interpreting the corrected Un-ruh state (14) on I + in terms of a single semiclassical space-time with the black hole at the origin. An outgoing mode with l ∼ M i near I + corresponds near the black hole to a large am-plitude standing wave in a thin shell of width ∼ ∼
1. This Planckian behavior of the extrapolated cor-rected Unruh state illustrates the potential pitfalls of thinkingin terms of a single semiclassical spacetime and focusing onnear-horizon physics.Fourth, we argue that the modification to the Hawking pro-cess removes one of the primary objections to the proposal For l ≫ that soft hair on black holes plays a key role in resolving theinformation loss paradox [18, 19, 23–25]. Soft hair consistsof charges measurable at future null infinity associated withan extension of the BMS group [26–29], higher- l analogs ofthe center-of-mass that are encoded in the asymptotic metric.Just as for the center-of-mass, the expected value of soft haircharges can be set to zero by a gauge transformation, locallyin time, but their variances cannot and can contain nontrivialinformation. Outgoing Hawking quanta excite soft hair viathe gravitational wave memory e ff ect. It has been suggestedthat the Hawking radiation is purified at late times by its en-tanglement with soft hair degrees of freedom [23].A di ffi culty with this proposal has been that only low l modes of the soft hair can be excited by the outgoing quanta,because of the exponential fall o ff of the spectrum (6) at high l . The soft hair field Φ ( θ ) is given in terms of the scalar field ϕ on I + by [see, eg. Eqs. (2.19) and (4.4) of Ref. [12]] D ( D + Φ ( θ ) = π P Z du ϕ , u ( u , θ ) , (20)where D is the Laplacian on the two-sphere and P is a pro-jection operator that sets to zero l = , ∼ M i outgoing Hawking quanta.In addition it is possible for two successive outgoing quantato cancel one another and give e ff ectively zero contribution tothe integral (20).The modified angular distribution of the Hawking radia-tion completely changes this picture, since the source termin Eq. (20) now extends e ff ectively up to multipoles of order l ∼ M i . 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