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Black Hole Radiation in the Presence of

arXiv:hep-th/9303103v1 18 Mar 1993Black Hole Radiation in the Presence ofa Short Distance CutoffTed Jacobson∗Department of Physics, University of MarylandCollege Park, Maryland 20742andInstitute for Theoretical PhysicsUniversity of California, Santa Barbara, CA 93106March, 1993UMDGR93-32NSF-ITP-93-26hep-th/9303103AbstractA derivation of the Hawking effect is given which avoids referenceto field modes above some cutofffrequency ωc ≫M−1 in the free-fall frame of the black hole.To avoid reference to arbitrarily highfrequencies, it is necessary to impose a boundary condition on thequantum field in a timelike region near the horizon, rather than on a(spacelike) Cauchy surface either outside the horizon or at early timesbefore the horizon forms.Due to the nature of the horizon as aninfinite redshift surface, the correct boundary condition at late timesoutside the horizon cannot be deduced, within the confines of a theorythat applies only below the cutoff, from initial conditions prior to theformation of the hole. A boundary condition is formulated which leadsto the Hawking effect in a cutofftheory.

It is argued that it is possiblethe boundary condition is not satisfied, so that the spectrum of blackhole radiation may be significantly different from that predicted byHawking, even without the back-reaction near the horizon becomingof order unity relative to the curvature.∗jacobson@umdhep.umd.edu1

1IntroductionThe Hawking radiation from a black hole of mass M is most copious at awavelength of order M.1In this sense it is a long distance effect, whosescale is set by the mass of the hole.Thus it is odd that all derivationsof the Hawking effect refer in some manner to arbitrarily short distances.For instance, consider Hawking’s original derivation [1]: the annihilationoperator for an outgoing quantum field mode at late times is expressed, viathe free field equations, in terms of annihilation and creation operators foringoing modes at early times, before the matter has collapsed to form thehole. The thermal character of the state at late times is then deduced fromthe boundary condition specifying that the initial state is the vacuum (orvacuum plus some excitations of finite total energy.

)The fishy thing about this derivation is that the frequency of the ingoingmodes diverges as the time of the corresponding outgoing modes goes toinfinity. This is because all of the outgoing modes, for all eternity, originateas incoming modes that arrive at the hole before the formation of the eventhorizon.

An infinite number of oscillations of the incoming modes must thusbe packed into a finite time interval, so their frequency must diverge.Other derivations of the Hawking effect also make reference to arbitrarilyshort distances. A recent derivation by Fredenhagen and Haag [2] is basedon the form of the singularity in the two-point function ⟨φ(x)φ(y)⟩as xapproaches y just outside the horizon.

Similarly, arguments based on theproperties of the correlation functions on the Euclidean continuation of theblack hole metric [3] assume that the correlation functions have the requistiteanalytic behavior, which involves the form of the short distance singularities.Finally, arguments (for conformal fields in two dimensions) based on conser-vation of the stress-energy tensor [4, 5] assume the value of the trace anomaly,which is the result of regulating a short distance divergence of the theory.Since the scale of the process is set by the mass of the hole, it would seemthat it should be possible to avoid the role of ultra high freqencies muchhigher than M−1 in deriving its existence. In a previous paper [6] this issuewas discussed in detail, and two arguments were offered to support this pointof view, one involving the response of accelerated particle detectors and oneinvolving conservation of the stress-energy tensor.These arguments were1We use units with G = c = ¯h = 1.2

not conclusive but they did make it plausible that the Hawking effect wouldoccur even if there were a Planck frequency cutoffin the frame of free-fallobservers that fall from rest far from the hole.It now seems a mistake to focus on a Planck frequency cutoff, since thesame arguments would support the existence of Hawking radiation as long asthe high frequency cutoffωc is much larger than M−1. In the present paperit will be shown how Hawking’s original analysis can be modified to avoidreference to ultra high frequencies.

This will require the use of an alternateboundary condition, which states roughly that observers falling freely into theblack hole (starting from rest far away) see no particles at frequencies muchhigher than M−1 but less than some cutoffωc. That this condition impliesthe existence of black hole radiation was implicit in Hawking’s original paper[1], and was later stressed by Unruh [7].

One contribution of the presentpaper is to demonstrate in detail how the derivation can be structured so asto entirely avoid invoking the behavior of ultra high frequency modes. Thisanalysis involves several sticky technicalities, which we have attempted toaddress as thoroughly as possible.This alternate boundary condition is not an initial condition, since it isimposed for all times.

Moreover, for the reason explained above, it cannotbe derived from the early time vacuum inital condition. It is in the nature ofthe horizon as an infinite redshift surface that the state of the outgoing fieldmodes at low frequencies descends from presently unknown physics at veryhigh frequency (in the free-fall frame).

Thus the validity of the boundarycondition cannot be proved within a theory that is only valid below somehigh frequency cutoff. The possibility of justifying the boundary condition onenergetic grounds will be addressed in section 6.

Our conclusion will be that itis quite possible the boundary condition is not satisfied, so that the spectrumof black hole radiation may be significantly different from that predicted byHawking, even without the back-reaction near the horizon becoming of orderunity relative to the curvature. Violations of the boundary condition leadingto a large back-reaction also seem possible, however in such a situation thequasi-static, semiclassical framework of our calculations is unjustified.The rest of the paper is organized as follows.In section 2 Hawking’soriginal derivation is reviewed.

In section 3 the role of ultra high frequenciesin this derivation is discussed, and in section 4 our alternate boundary con-dition is formulated and discussed in detail. It is shown in section 5 that thisboundary condition implies the existence of the usual Hawking radiation.

In3

section 6 the physical basis of the boundary condition is discussed, and theimplications of a violation of the boundary condition are studied. Section7 contains some concluding remarks, and the appendices contain technicalmaterial needed in the rest of the paper.2Hawking’s reasoningIn this section Hawking’s original derivation [1] of black hole radiation froma non-rotating, uncharged black hole will be reviewed.

We use Wald’s formu-lation [9] in terms of individual wavepackets, rather than Bogoliubov trans-formations between orthonormal bases, because selection of a complete basisis distracting and unnecessary for our our purposes.Consider an outgoing positive frequency wavepacket P at late times farfrom the black hole, centered on freqency ¯ω and retarded time ¯u. (Theretarded time coordinate is defined in Appendix A.) Suppose P is normalizedin the Klein-Gordon norm, so the annihilation operator for this wavepacketis given bya(P) = ⟨P, Φ⟩,(1)where the bracket notation denotes the Klein-Gordon (KG) inner product.

(See Appendix B for the definition of the KG inner product, and AppendixC for a discussion of this characterization of annihilation and creation op-erators.) We are interested in the state of the quantum field “mode” corre-sponding to this wavepacket.

This is partly2 characterized by the expectationvalue of the number operator,⟨N(P)⟩= ⟨Ψ|a†(P)a(P)|Ψ⟩. (2)Using the field equation ∇2Φ = 0, this number operator can be expressed interms of operators whose expectation values are fixed by initial conditions orother assumptions on the properties of the state |Ψ⟩.Propagating the wavepacket P backwards in time, it breaks up into a“reflected piece” R that scatters offthe curvature outside the matter andout to past null infinity I−, and a “transmitted” piece T that propagates2For simplicity we focus on the expectation value of the number operator.

In fact,the form of the annihilation operator a(P) discussed below implies also the true thermalnature of the state. (See for example [9, 10, 11].

)4

back through the collapsing matter and then out to I−. (See Fig.

1.) Theoriginal wavepacket P can be exressed as the sum of these two solutions, asP = R + T,(3)and the annihilation operator for P (1) can thus be decomosed asa(P) = a(R) + a(T).

(4)Since both the wavepackets and the field operator satisfy the wave equation,the KG inner products in (4) are conserved, and can therefore be evaluatedon any Cauchy hypersurface.Because of time translation invariance in the part of the spacetime ex-terior to the matter, the reflected packet R consists of the same frequencieswith respect to the Schwarzschild time coordinate at I−as the packet P atfuture null infinity I+. Thus the operator a(R) = ⟨R, Φ⟩is an annihilationoperator for an incoming wavepacket centered on frequency ¯ω.

Assumingthat this mode of the quantum field started out in its ground state, we havea(R)|Ψ⟩= 0,(5)so that⟨N(P)⟩= ⟨Ψ|a†(T)a(T)|Ψ⟩. (6)At I−the transmitted packet T is composed of both positive and negativefrequency components with respect to the asymptotic Schwarzschild time,T = T(+) + T(−),(7)and we have the expansiona(T) = a(T(+)) −a†(T(−)∗).

(8)Thus a(T) is a combination of annihilation and creation operators for incom-ing wavepackets at I−. Assuming that both the positive frequency part andthe complex conjugate of the negative frequency part of the packet T startedout in their ground states, we havea(T(+))|Ψ⟩= 0a(T(−)∗)|Ψ⟩= 0.

(9)5

Thus, using (6), (8), (9) and the commutation relation between annihilationand creation operators (49,50) we have⟨N(P)⟩= −⟨T(−), T(−)⟩. (10)For a wavepacket with a spread of frequencies ∆ω ≪κ, the Klein-Gordonnorm of the negative frequency packet T (−) can be evaluated in terms of thenorm of T as described in Appendix D, and one finds, using (57),−⟨T(−), T(−)⟩= ⟨T, T⟩(exp(2π¯ω/κ) −1)−1.

(11)This is just what the emission would be from a body at temperature κ/2π =1/8πM, for a mode of energy ¯ω with absorption coefficient ⟨T, T⟩.3Ultra high frequenciesThe difficulty with this analysis is that at past null infinity, the incom-ing packet t consists of extremely high frequency components, whose fre-quency (with respect to the asymptotic rest frame of the hole) grows as∼exp(¯u/4M) ω as the retarded time ¯u of the outgoing wavepacket goes toinfinity. This exceeds Planck frequency ωP for ¯u > 4M ln(ωP/ω), that is, af-ter only several light crossing times for the hole.

That the frequency divergesin some such manner is immediately evident from inspection of Fig. 1.

Aninfinite amount of time at infinity corresponds to the interval between anyfinite u and the horizon at u = ∞. The correspondingly infinite number offield oscillations must all be packed into the finite range of advanced timesbetween some v and v0, the advanced time of formation of the horizon.It is unsatisfactory from a physical point of view to base the predictionof black hole evaporation on an assumption that involves the behavior ofarbitrarily high frequency modes.

We are ignorant of what physics mightlook like at those high frequencies or corresponding short distances. In orderto be confident of the prediction of Hawking radiation, one should formulate aderivation that avoids this ignorance while invoking only known physics—orat least only more reasonable extrapolations of known physics.It is not the unknown physics of high energy interactions that we areconcerned about here.

Although we are dealing with incoming wavepacketswith arbitrarily high frequency relative to the frame of the collapsing matter6

that forms the black hole, there is no interaction between these wavepacketsand the collapsing matter. The reason is that these incoming wavepacketmodes are in their ground state, so there is nothing for the collapsing matterto interact with.What we are concerned about is the need to assume that the physics isLorentz invariant under arbitrarily large boosts.

Assuming Lorentz invari-ance, one can of course argue that although the frequency of the transmittedwavepacket t grows as exp(¯u/4M) with respect to the asymptotic rest frameof the black hole, there is always a local Lorentz frame in which the fre-quency appears as low as one wishes. The velocity of this frame relative tothe black hole approaches the speed of light as ¯u →∞, with a boost factorγ = (1 −v2)−1/2 = exp(¯u/4M).We have no observations that confirm Lorentz invariance at the levelof such arbitrarily high velocity boosts [12, 13, 14].

Probably the highestboost factors at which Lorentz invariance might be checked anytime soonarise in cosmic ray proton collisions. We are basically at rest with respectto the cosmic microwave background (CMB) radiation.

Assuming Lorentzinvariance, one predicts that for proton energies greater than about 1020 eV(relative to the CMB frame), the head-on collision of a proton with a CMBphoton can produce a pion. This process would leave its mark on the cosmicray proton spectrum.

If this mark is eventually observed, it will lend supportto the assumption of Lorentz invariance that went into the calculation.3 Theboost factor here relating the CMB frame to the center of mass frame of thecollision is a “modest” γ ∼1012.In the black hole situation, after a retarded time interval ∆u ∼4M ln 1012 ≃102M, the boost factor required to transform an incoming wavepacket to lowfrequency would have increased by more than 1012. Thus the above deriva-tion of a steady flux of Hawking radiation depends on the assumption ofLorentz invariance arbitarily far beyond its observationally verified domainof validity.3According to Sokolsky [15], it should be possible to confirm this prediction in thecoming decade.7

4Cutoffboundary conditionTo avoid the need to make assumptions regarding arbitrarily high frequencybehavior we will have to give up the attempt to derive the properties ofthe state of the quantum field at late times from the initial condition thatit is the vacuum state before the hole forms. Instead, we will formulate adifferent “boundary” condition on the state that will still imply the existenceof Hawking radiation.The alternate boundary condition is expressed in terms of the particlestates defined by free-fall observers near the horizon that have fallen in fromrest far away from the hole.

For frequencies much higher than M−1, theseparticle states are well defined by field modes with positive frequency withrespect to the proper time of the free-fall observers. Our boundary conditionwill be that outgoing, high freqency field modes are in their ground states.How high is “high”?

Roughly, to predict Hawking radiation to an accuracyη ≪1, it will suffice to assume that the outgoing modes of free-fall frequency∼η−2M−1 are in their ground state. The statement of the boundary condi-tion just given is appropriate for a massless, free field.

We defer to subsection4.6 a brief discussion of the modifications required for a treatment of massiveand/or interacting fields.To derive this alternate boundary condition from the condition that theinitial state is vacuum requires appeal to arbitrarily high frequency modes,for the reason discussed earlier. Thus we make no attempt here to derivethis alternate boundary condition, but rather take it as given.

The questionof physical plausibility of the condition will be taken up in section 6.4.1Precise formulation of the boundary conditionActually imposing the alternate boundary condition in terms of the propertime of the family of free-fall observers is somewhat complicated. Instead,shall employ the affine parameter along radial ingoing null geodesics as therelevant “time” variable.

This turns out to amount to the same thing nearthe horizon, as will now be explained.First note that the usual radial coordinate r is an affine parameter alongthe radial null rays (see Appendix A). To find the rate of change of r withrespect to the proper time τ along the free-fall geodesic, note that the quan-tity pv = gvµdxµ/dτ = (1 −2Mr )dv/dτ −dr/dτ is conserved, since the metric8

is independent of v in Eddington-Finkelstein (EF) coordinates (41). If thegeodesic starts from rest at ∞, one has at infinity dv/dτ = 1 and dr/dτ = 0,so pv = 1.

It follows then that at the horizon r = 2M, one has dr/dτ = −1.That is, r is changing at the same rate as the proper time.An outgoing solution f to the wave equation near the horizon is nearlyindependent of v in EF coordinates, since the lines of constant r are nearlynull there. Along the free-fall world line near the horizon, we therefore havedf/dτ ∼= (∂f/∂r)dr/dτ ∼= −(∂f/∂r).

Thus, for outgoing modes near thehorizon, the frequency with respect to r on a constant v surface is effectivelythe negative of the frequency with respect to the free-fall observers.The particle states of our boundary condition will correspond to wavepack-ets f composed of field modes on a constant v null hypersurface Σ of the formfωlm(r, θ, φ) = r−1 exp(iωr)Ylm(θ, φ) . (12)In an effort to avoid confusion I will call these positive r-frequency modes,because they have positive frequency with respect to the proper time of thefree-fall observers.

We can regard the operator a(f) = ⟨f, Φ⟩as (proportionalto) an annihilation operator for a one particle state provided that the Klein-Gordon (KG) norm of f is positive. (This is discussed in Appendix C.)To evaluate the Klein-Gordon inner product (44) on Σ, we use the metriccomponents in EF coordinates (41) and the surface element (46) to find√−ggµνdΣν = −δrµ r2sinθdrdθdφ.

Thus the KG inner product takes theform⟨f, g⟩= −i2ZdΩZ ∞0dr r2 (f ∗∂rg −g∂rf ∗). (13)This shows that the modes fωlm (12) indeed have positive norm for ω > 0,as do localized wavepackets constructed by superposing them.44It is tempting to try to define a full Hilbert space of one-particle states on a constantv-surface using the positive r-frequency modes.

However, the fact that the r-integral runsonly over the interval [0, ∞) leads to a problem with this definition. Positive frequencymodes of the form fωlm and fω′lm (12) are not orthogonal for ω ̸= ω′, and linear combi-nations of positive frequency modes can have negative norm.

This is not a problem if onerestricts attention to wavepackets that have negligible support near r = 0, since for themit makes no difference whether the r-integration is over [0, ∞) or (−∞, ∞). (One cannottake wavepackets of compact support since that would be inconsistent with their beingcomposed of purely positive frequencies.) In any case, we will refer to only one wavepacketat a time, with no need to consider the full Hilbert space of one particle states.9

Our alternate boundary condition can thus be implemented as follows.We choose to calculate the expectation value of the number operator corre-sponding to wavepackets P with the property that on some constant v sur-face, v = vc, their transmitted piece T has only components with r-frequencyω(r) much higher than M−1 but less than some cutofffrequency ωc,ωc > ω(r) ≫M−1. (14)(If the frequency at infinity ω is much greater than M−1, we also requireω(r) ≫ω.) Then, instead of propagating the transmitted piece T of thewavepacket P all the way back through the collapsing matter and out to pastnull infinity, we stop when it reaches v = vc.

There we decompose it into itspositive and negative r-frequency parts and impose the boundary conditionthat the positive r-frequency part (and the complex conjugate of the negativefrequency part) are in their ground states.5 To carry out this program, itmust first be established that there exist positive u-frequency wavepacketswith the property than on some surface v = vc, their r-frequency componentssatisfy (14). This will be accomplished in subsection 4.3 below.4.2Self-consistency of the boundary conditionNote that for a wavepacket centered on frequency ¯ω and retarded time ¯u, thesurface v = vc must necessarily move to the future as ¯u grows with ¯ω fixed,in order to avoid the occurence of r-frequency components above the cutofffrequency.

Thus our boundary condition is not being imposed on a singleCauchy surface, so is not an “initial” condition. This raises the questionwhether our boundary condition is consistent with the field dynamics.For simplicity, let us think of the boundary condition as being imposedon a surface of fixed radius, r = rb.c., just outside the horizon.6 This surfaceis timelike, so the site of the part of the boundary condition imposed at5Although the wavepacket P is completely outside the horizon, its positive and negativer-frequency parts have support both inside and outside the horizon.

(See equations (30),(53), (54). )6Actually, the boundary condition refers to the region inside the horizon as well, sincethe positive and negative frequency parts have support inside the horizon.

It is thereforemore accurate to think of the boundary condition as being imposed on a pair of surfacesof constant r, one just outside the horizon and one just inside. Since the one inside isspacelike, no question of consistency arises for that part of the boundary condition.10

advanced time v includes, within its past, sites of parts of the conditionimposed at earlier advanced times. Is the condition imposed at v consistentwith the earlier ones?The boundary condition refers to the state of outgoing modes with r-frequency ω(r) in the range ωc > ω(r) ≫M−1.

The modes of frequency ωccome from two sources: modes that propagate out from yet closer to thehorizon with yet higher frequencies, and modes that have scattered offthegeometry. The state of the former modes can be freely specified, since theyare above the cutoffuntil they reach advanced time v and hence no conditionat all is imposed on them until then.

Thus there is enough freedom to con-sistently assign the state of the outgoing modes at ωc. But one may still askif the gound state boundary condition is the appropriate one, in view of thecontributions from the modes that have backscattered.

For instance, someHawking radiation can scatter back towards the hole and then scatter againout from the hole, apparently leading to some non-zero occuption number inan outgoing mode that the boundary condition assigns to its ground state.The scattering amplitude for these modes in this region of the spacetime isvery small however, so such processes should affect the state only very little.Now let us consider the modes with frequency less than the cutoff. Thestate of these modes can not really be independently specified, since they canbe traced back (primarily) to modes yet closer to the horizon with frequencyωc, on which a (ground state) boundary condition has already been imposed.Thus the state of the modes with frequency ω(r) < ωc must be calculated, notassigned.

In fact, it follows from the argument in section 5 that no modes areexcited while they are propagating close to the horizon; it is not until theyclimb away significantly (on the scale of M) that the presence of Hawkingradiation becomes apparent in the free-fall frame.Thus it appears not inconsistent to impose our ground state boundarycondition, at least to the order of precision of our calculations. Note thatwe can really only check self-consistency of the calculation: As shown in thenext two subsections, the unavoidable spread of the wavepackets makes itnecessary to imose a boundary condition on a wide range of frequencies fromthe beginning.

Then all we can do is verify that this boundary condition isself-consistent.11

4.3Existence of the required wavepacketsLet pωlm denote the solution to the massless scalar wave equation in Schwarzschildspacetime that is purely outgoing at future null infinity (and is therefore out-going at the horizon as well), and is of the formpωlm = (2πω)−1/2 exp(−iωt)r−1fωl(r)Ylm(θ, φ),(15)withfωl(r) =(eiωr∗+ Aωle−iωr∗as r∗→+∞Bωleiωr∗as r∗→−∞,(16)where r∗is the tortoise coordinate defined in eqn. (42).

These modes arenormalized according to ⟨pωlm, pω′l′m′⟩= δ(ω−ω′)δll′δmm′. Using these modes,we seek to construct wavepackets that satisfy the condition (14) restrictingthe r-frequency components on a constant v surface, v = vc.The wavepackets we will employ are of the following form:P¯ω¯ulm = NZ ¯ω+∆ω¯ωdω B−1ωl exp(iω¯u) pωlm.

(17)P¯ω¯ulm is a unit norm, positive t-frequency wavepacket centered on frequency¯ω + ∆ω2 . N is a normalization factor, and the factor B−1ωl (inverse of thetransmission amplitude) is included in the integrand so that we will havecontrol over the spread of the part of the packet near the horizon.Thewavepacket P¯ω¯ulm is defined by its (purely outgoing) behavior at I+ and thefact that it vanishes on the horizon.

Alternatively, propagating it backwardsin time from I+ as in section 2, one sees that it is generated by data on aCauchy hypersurface formed by a constant v surface v = vc together withthe part of I−that lies to the future of vc. The wavepacket generated bythe data at v = vc alone will be called the “transmitted packet” T¯ω¯ulm, andthat generated by the data at I−will be called the “reflected packet” R¯ω¯ulm.Thus we have P¯ω¯ulm = T¯ω¯ulm + R¯ω¯ulm.For each ¯ω and for ¯u sufficiently long after the collapse that formed theblack hole, one can always choose vc sufficiently far in the past so that T¯ω¯ulmis concentrated near the horizon.

In this case, the asymptotic form fωl ∼=Bωl exp(iωr∗) can be accurately substituted in the integrand (17) and oneobtainsT¯ω¯ulm = N (2π)−1/2r−1Ylm(θ, φ)Z ¯ω+∆ω¯ωdω ω−1/2 exp(iω(¯u −u)). (18)12

This transmitted wavepacket is localized in retarded time u, centeredroughly on ¯u, with a spread ∆u ≃8π/∆ω. More precisely, the spread of T¯ω¯ulmin u is of course infinite, but the packet is well localized in the following sense.7After carrying out the angular integrals the KG norm (13) of the packet (18)calculated at v = vc reduces to a numerical factor times an integral overx of the quantity (sin x/x)2, where x = ∆ω(u −¯u)/2.

One can show thatR y0 (sin x/x)2 dx = (π/2)[1−(1/πy)+O(y−2)]. Thus, defining η as the fractionof the full norm omitted in a range ∆u, one has η ≃1/πy = 4/π∆ω ∆u, orη ≃1/∆ω ∆u .

(19)In Hawking’s paper [1], wavepackets of the form (17) (without the factorof B−1ωl ) were also employed, however ∆ω was chosen very small comparedwith the surface gravity κ = 1/4M, so that the wavepackets relevant tothe black hole radiation would be very peaked in frequency, thus simplifyingthe analysis. From our point of view, the difficulty with this is that such apacket cannot be squeezed close enough to the horizon without containingr-frequencies above the cutoffωc.In fact, one must take ∆ω>∼κ, and tomaximize the precision of our derivation one should take ∆ω ∼√ωcκ, aswill now be shown.4.4Precision of the derivationThe precision of the derivation we will give is limited by the fact that thewavepackets will not be infinitely squeezed up against the horizon.Theresulting “error” is of order Cmax ≡(1 −2M/rmax), where rmax is the largestvalue of r to occur in the wavepacket.8 Of course, strictly speaking, rmax =∞, but a fraction (1 −η) if the wavepacket is contained within a smallerrange of r values, given by ∆u ≃1/η∆ω.

Thus to minimize the errors we7The wavepacket P¯ω¯ulm at I+ does not have the same width in u as does T¯ω¯ulm atvc. The wavepacket is somewhat dispersed, since the different frequency components haveunequal transmission amplitudes.

We included the factor B−1ωl in the definition (17) ofP¯ω¯ulm so that our packet would be well localized at vc; it will not bother us that P¯ω¯ulm isnot as well localized at I+.8Actually, since only a fraction of the wavepacket is located at r ∼rmax, with the restat smaller values of r, the error is somewhat smaller. To keep the crude analysis thatfollows from getting too complicated, we will simply make the conservative error estimateusing the largest value of r.13

should minimize the combined error due to the fraction η of the wavepacketbeyond rmax, and due to Cmax not vanishing. To carry out this minimizationcalculation, we must express Cmax as a function of η and ∆ω, and minimizethe error functionE2(η, ∆ω) ≡η2 + C2max(η, ∆ω) .

(20)The relation between u and r at constant v is given (cf. (42),(43)) by∂u/∂r|v = −2(1−2Mr )−1 = −2C−1, where C = (1−2Mr ).

It is this factor thatconverts between u-frequency and r-frequency at fixed v, ω(r) = −2C−1ω.We assume that on the constant v surface, the wavepacket is squeezed verynear to the horizon, since that is in any case required in order to deduce theexistence of Hawking radiation from our boundary condition. Then we have(with κ = 1/4M)Cmax/Cmin ≃exp(κ∆u) ∼exp(κ/η∆ω) .

(21)Now assuming the highest r-frequency present in the wavepacket is the cutofffrequency, we have ωc = ω(r)max = C−1minωmax, so that Cmin = ωmax/ωc. Togetherwith (21) this yieldsCmax ∼exp(κ/η∆ω) (¯ω + ∆ω)/ωc ,(22)where we have returned to the notation ¯ω ≡ωmin.

For the purposes of mini-mizing the error, we will consider ¯ω as fixed, since this is really determinedby which frequencies we want to learn about.Already (22) shows us that it is not acceptable to choose δω ≪κ asHawking did. For instance, suppose that ¯ω ∼O(κ), so the frequencies mostcopious in the Hawking radiation will be included, and suppose that ∆ω =0.01κ and η = 0.01.

Then we have Cmax = exp(10, 000) κ/ωc, which will besmaller than unity only if κ/ωc is much smaller than we want to assume!To minimize the error (20), we use (22) and set ∂E/∂η = 0 and ∂E/∂∆ω =0. Up to factors of O(1), this yields at the minimum:η ∼(∆ω)3/κω2candCmax ∼(∆ω)5/κ2ω3c ,(23)where ∆ω satisfies¯ω + ∆ω ≃(∆ω)5/κ2ω2c .

(24)14

As long as ¯ω ≪√κωc, the solution is given by∆ω ∼√κωc,η ∼qκ/ωc,Cmax ∼qκ/ωc. (25)Note that for such a “minimum error” wavepacket with ω(r)max = ωc, we haveω(r)min = C−1max¯ω ∼(¯ω2ωc/κ3)1/2 κ , which will satisfy the condition ω(r) ≫κ aslong as ¯ω ≫(κ/ωc)1/2 κ.We conclude that one can work with wavepackets with r-frequencies inthe required range, with a built-in imprecision of the calculation9 limited toan error of orderqκ/ωc.4.5Horizon fluctuationsAnother point that should be checked is how close to the horizon is ourboundary condition being imposed?

If this is within the expected range ofquantum fluctuations of the horizon itself, then we will not have succeeded informulating a derivation free of short distance uncertainties. To estimate theradius rb.c.

at which the boundary condition is being imposed, note that fora mode of frequency M−1 coming from a hole of mass M, we have ω(r) ∼ωcwhen (1 −2Mr )−1M−1 ∼ωc, or rb.c. ∼2M + lc, where lc = ω−1c .

The scaleof quantum fluctuations of the horizon δr can be estimated by using theBeckenstein-Hawking entropy S = 14A/l2P and setting δS ∼1, which is char-acteristic of thermal fluctuations about equilibrium.10 Assuming the horizonshould be treated as N ≡A/l2P independent fluctuating area elements, eachof area a and radius r, we have δA ∼√Nδa ∼lPδr, so δA ∼l2P gives δr ∼lP.Thus for a Planck scale cutoff, we are perhaps not justified in ignoring thequantum fluctuations of the horizon in our derivation. The simple way outis to take the cutofflength much longer than the Planck length.

This isfine until we come to discussing the physical justification for the boundary9It may be that the derivation can be improved, reducing the imprecision.Thewavepacket analysis employed here seems a rather clumsy approach to the problem. Theproblem can also be formulated using the approach of Fredenhagen and Haag[2], whichfocuses on the behavior of the two-point function.

That approach may turn out to bemore suitable for maximizing the precision of the derivation.10This gives the same scale as the one obtained by York using the uncertainty pricipleand the spectrum of quasinormal modes [16], or using the Euclidean partition functionapproach [17].15

condition, or violations of it. It should be kept in mind that if the modes arefollowed all the way back to where they are squeezed up within one Plancklength of the horizon, several grains of salt should be added to the wholeanalysis.4.6Massive or interacting fieldsIn order to apply the arguments of section 5, it is necessary that the prop-agation be governed by the massless wave equation for a sufficiently longinterval of advanced time v. Thus for a free field of mass m one must imposethe boundary condition on wavepackets satisfying ω(r) ≫m, in addition tothe condition ω(r) ≫M−1 already discussed in section 4.1.

Then one findsthat particles corresponding to these wavepackets are created near the blackhole just as are massless ones, and they then propagate away from the hole asmassive particles. As long as the mass is much less than the cutofffrequency,m ≪ωc, there is no obstruction to extending our argument to cover the caseof massive particles.It is generally believed that the Hawking effect occurs for interacting fieldsas well as for free fields, although this has never been demonstrated explicitly.For the purposes of determining what would be emitted by a real black hole,some researchers [18] have assumed that the process can be divided into twostages, much as for the massive free field just discussed.

In the first stage,which takes place very near the horizon, the dynamics of the field is governedby the asymptotically free regime. In QCD for example, free quarks andgluons are assumed to be radiated with a thermal spectrum.

In the secondstage, as the particles climb away from the horizon, the self-interactions ofthe field become important, and the free particle states hadronize into jets.A direct demonstration of the validity of this picture has never beengiven, although there are various arguments that support it. Gibbons andPerry[3] argued that the periodicity of the Euclidean section of Schwarzschildspacetime implies the thermal character of Hawking radiation for interactingfields.

This argument applies only to the thermal equilibrium state on theeternal black hole spacetime. Moreover, it rests heavily on the assumptionthat a state that is regular on the horizon must arise by analytic continu-ation from a state that is regular on the (periodically identified) Euclideansection.

While this condition seems natural in some sense, it has not beendemonstrated to be necessary.16

Another argument advanced in favor of thermality is that of Unruh andWeiss[19], who demonstrated that the Minkowski vacuum of an interactingfield theory is a thermal state when viewed by a uniformly accelerating familyof observers. More precisely, correlation functions in the Rindler wedge aregiven by the thermal density matrix relative to the Hamiltonian that gener-ates translations along the boost Killing field.

This is a purely kinematicalresult. It is, in a sense, a local version of the Euclidean section argumentthat avoids the need for assumptions about regularity of the analyticallycontinued correlation functions on the Euclidean section.

To turn it into aderivation of Hawking radiation for interacting fields, one presumably mustassume the field is in a state that “looks like” the Minkowski vacuum verynear the horizon, use the Unruh-Weiss result to describe it from the point ofview of the static observers as a thermal state, and then propagate this ther-mal state out away from the hole. The result will depend on the interactionsand on what state is incoming from infinity, since this would interact withthe outgoing Hawking radiation.For weakly coupled fields one can study this process using perturbationtheory.

Massless λφ4 theory in a 2-dimensional black hole spacetime wasstudied by Leahy and Unruh[20], who showed that for an ingoing thermalstate at the Hawking temperature, the interaction preserves the thermalnature of the outgoing state.For an ingoing vacuum state however, theoutgoing state is not thermal.It does not appear to be entirely straightforward to extend the argu-ments of our paper to the case of interacting fields, since we use the linearityof the field equation to express the annihilation operator corresponding toa wavepacket at one time in terms of annihilation and creation operatorsassociated with wavepacket at another time. In order to extend our argu-ment, one can presumably use the fact that for the first part of the process,as the excitations are created, only the propagation of the field “near thelight cone” is relevant.

That is, one can presumably show that only smallspacetime intervals are involved, and thus use the fact that the correlationfunctions behave like free field ones in this region, due to asymptotic free-dom. This picture of the process was outlined by Fredenhagen and Haag inthe discussion section of [2], but to my knowledge it has never been workedout in any detail.17

5Hawking radiation in the presence of a cut-offHaving formulated in the previous section a boundary condition on the quan-tum state near the horizon that refers only to modes below the cutoff, it isnow our task to determine the properties of the state far from the hole.5.1Evaluating the occupation numbersSuppose now that P¯ω¯ulm = R¯ω¯ulm + T¯ω¯ulm is an outgoing wavepacket of theform (17), and propagate T¯ω¯ulm back to a constant v surface v = vc onwhich its r-frequency components satisfy ωc > ω(r) ≫M−1. (More precisely,it will be composed of both positive and negative r-frequency modes withfrequencies in this range.

)Now we would like to evaluate the expectation value of the number oper-ator N(P¯ω¯ulm), subject to our “boundary conditions” on the quantum state.These are that1. the reflected piece R¯ω¯ulm is in its ground state at I−, and2.

the positive r-frequency part and the complex conjugate of the negativer-frequency part of the transmitted piece T¯ω¯ulm are in their groundstates on the surface v = vc on which the r-frequency componentssatisfy ωc > ω(r) ≫M−1.Subject to these boundary conditions, the evaluation of ⟨N⟩goes through asin section 2 and we find⟨N(P¯ω¯ulm)⟩= −⟨T(−,r)¯ω¯ulm, T(−,r)¯ω¯ulm⟩(26)where T(−,r)¯ω¯ulm denotes the negative r-frequency part of the wavepacket T¯ω¯ulm,evaluated on the surface v = vc.Now the KG norm in (26) cannot have the form of (11) because, asexplained in sections 4.3 and 4.4, the spread of frequencies ∆ω in the packetmust be taken to be at least of order κ (or even much larger in order tomaximize the precision). In order to exploit the simple formula (54) that isapplicable to the negative r-frequency part of a wavepacket of the form (17)18

with ∆ω ≪κ, we break up the packet P¯ω¯ulm into a large number of pieces,definingP¯ω¯ulm =N−1Xj=0pj(27)pj = NZ ¯ω+(j+1)∆ω/N¯ω+j∆ω/Ndω B−1ωl exp(iω¯u) pωlm(28)and the corresponding transmitted packets tj. The {pj} (and the {tj}) arean orthogonal (but non-normalized) set of wavepackets, of the type usedin Hawking’s original derivation when N is chosen large enough so that∆ω/N ≪κ.

(Note that for such large N, B−1ωl does not vary much overthe range of integration in (28) and can be pulled out of the integral. )Each packet tj has a width of order ∆u ∼N/ηκ, and therefore containsr-frequency components in the ratio ω(r)max/ω(r)min ∼eN/η (see equation (21)).Nevertheless, the full wavepacket T¯ω¯ulm contains only r-frequencies in therange (14); the other r-frequency components in the tj’s must cancel in thesum (27), since the sum gives a much more localized wavepacket (whichsuffers much less differential redshift).

It is important to stress that althoughwe work with the packets tj as a technique to evaluate the r.h.s. of (26), wedo not attribute any direct physical significance or quantum state to them.Since extracting the negative frequency part is a linear operation, we have⟨T(−,r)¯ω¯ulm, T(−,r)¯ω¯ulm⟩=Xj,k⟨t(−,r)j, t(−,r)k⟩.

(29)To evaluate the KG inner products ⟨t(−,r)j, t(−,r)k⟩we would like to make useof the expression (54) for t(−,r)jas a linear combination of tj and the “timereflected” packet etj. That is, we would like to use the formulat(−,r)j= c−(e−πωj/κtj + etj),(30)wherec−= e−πωj/κ(e−2πωj/κ −1)−1(31)andωj = ¯ω + j∆ω/N.

(32)Now this expression for t(−,r)jwas derived in Appendix D assuming that thewavepacket tj is squeezed close to the horizon. However, although T¯ω¯ulm is19

squeezed close to the horizon, the individual wavepackets tj may not be, sincetheir width ∆u is much larger than that of T¯ω¯ulm.Fortunately this is not a problem, for the following reason. Since theKG norm is conserved, we can choose to evaluate (26) on an earlier surfacev < vc, on which not only T¯ω¯ulm but also all the tj are squeezed close to thehorizon.

Moreover, the negative r-frequency part of T¯ω¯ulm at v = vc evolvesto the negative r-frequency part at v < vc. This is because T¯ω¯ulm is a functionof r only through u in this region.

Since u = v −2r −4M ln( r2M −1), a shiftin v is equivalent to a scaling of r near the horizon (where the logarithm isdominant) by a linear transformation r →ar + b, which leaves the negativer-frequency part unchanged. This means we can evaluate the r.h.s.

of (26)at a surface upon which the tj are sufficiently sqeezed to justify use of theformula (30).The cross-terms in the sum (29) vanish, since ⟨tj, tk⟩= ⟨etj, etk⟩= ⟨tj, etk⟩=0 for j ̸= k. The diagonal terms are given by the result (57), so we have finally⟨N(P¯ω¯ulm)⟩=Xj⟨tj, tj⟩(exp(2πωj/κ) −1)−1. (33)This is just what the expected occupation number would be for a wavepacketmode of the form (17) (equivalently (27)) emitted from a body at temperatureκ/2π with absorption coefficients ⟨tj, tj⟩for the component wavepackets pj.6Physics of the boundary conditionIn this section we take up the question of whether there is any way to arguethat the boundary condition is in fact satisfied.

Recall that because of thegravitational redshift there is no way, within a cutofftheory, to derive thequantum state of the high frequency outgoing modes just outside the horizon.The natural expectation would be that they will be in their “free-fall” groundstate, because from their point of view, there is nothing special about thehorizon and they are merely propagating along just as they would in flatspacetime. The problem with this line of reasoning is that it ignores the veryquestion we are trying to address: does the fact that these modes have beenredshifted down from physics above any cutoffscale leave an imprint on theirquantum state?20

6.1Is this a one-scale problem?Together with the presence of the horizon, the absence of any scale otherthan the size of the black hole is really the essence of the Hawking effect.One can almost deduce the Hawking result from the fact that the boundarycondition introduces no length scale other than the Schwarzschild radiusinto the problem.In our form, the boundary condition states that fieldmodes near the horizon with r-frequencies satisfying ωc > ω(r) ≫M−1 arein their ground state. (Since we impose this boundary condition for alltimes, no condition need be imposed on modes with ω(r) > ωc.) This groundstate is a pure state, however the state of every mode outside the horizon iscorrelated to that of another mode inside the horizon.

When only the fieldoutside is accessible, there is missing correlation information. An observer farfrom the hole can never determine the state of the modes inside the horizon,so the relative phases of the states of all those outgoing modes at infinitythat emerged from the region of the horizon are completely unknown.

Thestate thus cannot be a pure state, but is rather one in which the missinginformation must be maximized in some sense. A maximum entropy stateis a thermal one, so the state of the outgoing modes should appear thermal(modulo absorption coefficients) far from the hole.

Since the cutoffωc playsno quantitative role in the problem as formulated, the only scale is M, sothe temperature must be proportional to 1/M. Calculation shows it to beTH = 1/8πM.In the formulation where the boundary condition is imposed in the asymp-totic past, the insensitivity of the black hole radiation to the details of theinital state before the hole forms follows from the nature of the horizon as aninfinite redshift surface: the more time passes, the higher the frequency ofthe relevant ingoing modes.

In the limit of infinite time, all that matters isthe fact that the infinitely high frequency modes are assumed to be initiallyin their ground state.But what if one does not assume that physics is invariant under infiniteblueshifting of scale? If there is new physics at some short distance scale,whether it be the Planck scale or something longer, then the gravitationalredshift may lead to a communication from short to long distance scalesoutside the horizon.

That is, the redshift effect leads to a breakdown of theusual separation of scales.Thus it seems perfectly possible that the quantum state of the outgoing21

field modes near the horizon might not be the ground state. The precise stateof these modes could reflect details of physics at much shorter distances.

Forinstance, there may be amplitudes for the excited states that could only becalculated from a knowledge of the short-distance theory. If this is the case,then the spectrum of black hole radiation may be quite different from thatdeduced by Hawking.11 For example, if one of these modes were to emergeat the cutoffin an excited state, then the emission in that mode would be acombination of the spontaneous Hawking radiation, the stimulated emission,and the original excitation.12 Thus the flux of energy at infinity would begreater than the Hawking flux.6.2Constraints on the stress-energy tensorIn this subsection we will analyze the implications for the stress-energy tensorof a violation of the ground state boundary condition near the horizon.

Thegoal is to determine what restrictions energy considerations may place on theform of the quantum state of the outgoing modes near the horizon. If thecomponents of ⟨Tµν⟩in the free-fall frame become too large, then neglect ofthe back-reaction is unjustified.

I see no reason in principle why this maynot happen in actuality. It may be that, in fact, the problem of quantumfields interacting with gravity in a black hole spacetime defies treatmentwhich neglects the back-reaction or which treats it as a small perturbationthat produces only slow evaporation of the black hole mass.However, ifthis is the case, then the (static) method of analysis used in this paper isinapplicable.Under what conditions can the back-reaction be treated as a small pertur-bation?

From the semi-classical Einstein equation Gµν = 8πl2P⟨Tµν⟩, we inferthat the back-reaction will be small provided the stress tensor componentsin the free-fall frame near the horizon are small compared with l−2Ptimes thetypical curvature components there, i.e.,⟨Tµν⟩≪1/l2PM2. (34)11This has nothing to do with the fact that for interacting fields, the spectrum of blackhole radiation will reflect the dressing and decay of the interacting particle states.

Rather,we are referring to a difference in the state of the high frequency modes, before the inter-actions have had their effect.12Stimulated emission by black holes is analyzed in Ref. [23].22

In the Unruh or Hartle-Hawking states, one has ⟨Tµν⟩= O(M−4) in thefree-fall frame near the horizon, hence in that state the back-reaction is verysmall indeed as long as the hole is much larger than Planck size. In fact, onemust increase the stress tensor by a factor of order (M/MP)2 before the back-reaction becomes more than a small perturbation.

This leaves alot of leewayin the form of the state near the horizon, and demonstrates that even withinthe approximation that treats the back-reaction as a small perturbation,there is no particular reason why the ground state boundary condition at thehorizon should hold.This leeway in the state at the horizon does not necessarily mean thatthe black hole flux would differ significantly from the Hawking flux however.The reason is that the energy carried by outgoing modes near the horizon isvastly redshifted by the time they make it out far from the hole. In order tomake a significant difference in the flux at infinity, an excited outgoing modenear the horizon must have a very high energy with respect to the free-fallframe.To obtain a very crude estimate of the energy density associated with suchan excited mode, consider a wavepacket that far from the hole is centeredon a frequency ω with a width ∆ω ∼ω and a spread in retarded time∆u ∼ω−1.

Suppose this mode is occupied in a one particle state near thehorizon at some r. As discussed in section 4.3, its energy relative to thefree-fall frame will be roughly (1 −2Mr )−1 ω, and the proper volume of thethin spherical shell containing it will be roughly (1 −2Mr ) ω−1M2 (since ithas a thickness ∆u ∼ω−1 at infinity).Thus the energy density will beroughly (1−2Mr )−2 ω2M−2.13 If this mode is followed all the way back to thehorizon, the energy density diverges, and the neglect of the back-reaction istotally unjustified. If on the other hand the mode is follwed only back to thevalue of r for which the r-frequency is equal to the cutoffωc, then one has(1 −2Mr )−1 ∼ωc/ω, and the energy density is 1/l2cM2.

Note that this resultis independent of ω, even though the extra power emitted ω/∆u ∼ω2 is not.Now if the cutoffrepresents not just an arbitrary scale beyond which weare pleading ignorance, but is rather a physical scale at which the nature ofpropagation might fundamentally change, then it might make sense to halt13As discussed in section 4.3, the finite width of the wavepacket leads to a differentialredshift across the packet, so this simple analysis is too crude to produce reliable numericalcoefficients.23

the backward-in-time propagation when the r-frequency reaches ωc. Let usentertain this possibility.Suppose then that the energy density near the horizon due to the pres-ence of an extra particle in the black hole radiation is given by 1/l2cM2 assuggested by the above computation.

More extra particles would just multi-ply this by the number of particles, irrespective of their frequency.14 (Notehowever that in order not to over-count the degrees of freedom the inde-pendent modes should be spaced in frequency by the spread adopted above,∆ω ∼ω.) Similarly, for each particle missing from the Hawking flux, oneexpects a negative contribution to the energy density of the same magnitude.Now if the back-reaction is a large effect, then our analysis on the staticblack hole background is actually not correct.

We see no reasoning by whichthis scenario can be ruled out, but we can say nothing more about it. If onthe other hand the back-reaction is small, then at least one of the followingmust be true:1.

The outgoing modes have only small amplitudes to be not in theirground state.2. There is near-perfect cancellation between the energy densities due to“over-occupied” and “under-occupied” modes.3.

The cutofflength lc is much longer than the Planck length.It is not even entirely clear that (1) is consistent with a small back-reaction, since it only implies a small expectation value for the energy density,but still allows fluctuations of order 1/l2M2. It would seem to require a fullquantum theory of gravity to determine whether or not the back-reactioncould really be neglected in such circumstances.

While (2) cannot be ruledout, it seems somewhat implausible, since there is no apparent reason forsuch cancellation to occur. Also (3) does not seem very likely, since there iscurrently no evidence of any fundamental length scale other than the Plancklength.

Nevertheless, let us just accept these as the logical possibilities thatthey are. Is there any further difficulty with such a scenario of deviationfrom the Hawking spectrum maintaining small back-reaction?14The estimated energy density breaks down however if the frequency is too low, because∆u ∼ω−1 will become so broad that the differential redshift across the wavepacket totallyinvalidates the assignment of a particular r-frequency to the packet near the horizon.24

If the spectrum of radiation is different, but the luminosity is the sameas the Hawking luminosity, then there must be cancellations of positive andnegative energy contributions, as mentioned in item (2) above. Althoughthis scenario does not seem likely, there seems to be no way to rule it out.The possibility that the net luminosity differs from the Hawking luminosityappears to be somewhat constrained however by general properties of thestress energy tensor if the back-reaction is to remain small.As first shown in the 70’s [4, 5], given some relatively “theory-independent”constraints on the behavior of the stress-energy tensor one can derive a for-mula for the net radiation flux far from a (quasi)static black hole.

Theseconstraints are:• ⟨Tµν⟩ν; = 0;• ⟨Tµν⟩is nonsingular on and outside the horizon (in regular coordinates);• ⟨Tµν⟩is static and spherically symmetric (in four dimensions);• no radiation is incoming from infinity at late times.If all these properties hold then it can be shown[5] that the luminosity L ofthe black hole is given in two spacetime dimensions byL = 12 MZ ∞2M dr r−2⟨T αα ⟩(D = 2)(35)and in four dimensions byL = 2π MZ ∞2M dr ⟨T αα ⟩+ 4πZ ∞2M dr (r −3M) ⟨T θθ ⟩(D = 4). (36)Let us consider first the two-dimensional case.

Then the luminosity isdetermined entirely by the trace of the stress-energy tensor. If we consider aconformally invariant massless scalar field, the trace is determined in a state-independent manner by the trace anomaly to be ⟨T αα ⟩= R/24π where R is theRicci scalar.

Putting this in (35) yields the Hawking flux LH = 1/768πM2.Any deviation from the Hawking flux for a conformally invariant field intwo dimensions thus implies that at least one of the properties of the stresstensor assumed above must fail to hold. It seems that the most questionableassumption is that of the value of the trace.

But what would be the physicalbasis for a deviation from the usual trace anomaly formula?25

It was argued in [6] that the presence of a high frequency cutoffωc is onlylikely to affect the value of the trace by terms of order O(R/ω2c). This argu-ment was based on the assumption that the origin of the quantum violationof conformal invariance can be located entirely in the regulated functionalmeasure in the manner of Fujikawa [24].

If correct this implies that the cor-rections to the trace (and to the Hawking flux) are very small indeed for holesmuch larger than the cutofflength. However, if there is fundamentally newphysics at the cutoffscale, then the violation of conformal invariance willnot be due simply to the non-invariance of the regulated functional measure.This opens up the possibility of a more significant deviation from the usualtrace anomaly.

Nevertheless, the fact that the usual trace anomaly is state-independent (assuming the state has the usual short distance form down tosome cutoffmuch smaller than the radius of curvature of the spacetime) sug-gests strongly that no significant deviation from the usual trace would occur.Thus, at least in this two-dimensional model, it is hard to see how the fluxcould differ from the Hawking flux and still have a small back-reaction.In the four dimensional case (36) the situation is perhaps different. For aconformally invariant field the trace is still determined by the trace anomaly,and is given by⟨T αα ⟩= βC2/48 = βM2/r6,(37)where 60π2β = 1,74, 33/60π2 for fields of spin-0, 1/2 and 1 respectively, andC2 is the square of the Weyl tensor.

Now however, the trace does not suffice todetermine the luminosity. One free function of r remains undetermined.

Thereason is that, unlike in two dimensions where all metrics are conformally flat,the Schwarzschild spacetime is not conformally flat, so even a conformallycoupled field scatters in a non-trivial way. Both the spin of the field andthe detailed radial dependence of the metric affect the radial dependence of⟨Tµν⟩and the net flux at infinity.

Numerical computations [25] show, forexample, that for a massless, minimally coupled scalar field in the Unruhvacuum in Schwarzschild spacetime, the contribution of the second integralto (36) is relatively small, and the Hawking luminosity is of order LH ∼(4800πM2)−1 ∼10−4M−2.A deviation from the Hawking luminosity could be produced, as in thetwo dimensional case, by a deviation from the usual trace anomaly, howeverthe same arguments as given in that case make this seem unlikely. But infour dimensions there is another possibility: Any change in the tangential26

stress ⟨T θθ ⟩will entail a change in the luminosity L, without violating theabove assumptions on the behavior of ⟨Tµν⟩. Can this be exploited to allowfor a deviation from the Hawking luminosity?

While it is not clear why thereshould be any fundamental difference between the two and four dimensionalcases with regard to the possibility of deviating from the Hawking radiation,let us just take the result (36) and see what can be done with it.Note first that the r-dependence of ⟨T θθ ⟩has alot to do with the scat-tering behavior of fields propagating in the Schwarzschild geometry. Thusat most, we should think of the possibility of freely modifying ⟨T θθ ⟩at onepoint, letting the behavior everywhere else be determined by the scatteringoffthe background geometry.

A change in the luminosity of order δL couldbe produced, consistent with (36), in two qualitatively different ways: (i) achange δT θθ ∼O(δL/M−2) over a range δr ∼M, or (ii) a very large changeδT θθ over a very small range of r near the horizon. The second way seemsinconsistent with the scattering behavior of the field, since the effective po-tential that governs the scattering is well behaved near the horizon.

Thefirst way requires only a relative change δT θθ /T θθ of order unity to change theluminosity by order unity. Thus there seems to be no obstacle to the physicsat the cutoffscale leading to a deviation from the Hawking luminosity, evenif the back-reaction is to remain small.7ConclusionWhat has been accomplished in this paper?

We have succeeded in formulat-ing a derivation of the Hawking effect (for massless free fields) that avoidsreference to field modes above some cutofffrequency in the frame of thefree-fall observers that are asymptotically at rest. To stay below the cutoffit is necessary to impose a boundary condition on the field near the hori-zon for all times.

The boundary condition states roughly that the outgoinghigh frequency field modes are in their “ground state” as viewed by free-fallobservers. This boundary condition is not derivable from the initial statewithin the cutofftheory.The precision of our derivation is controlled by the ratio of the cutofflength to the Schwarzschild radius of the black hole, and is limited byqlc/M.For a black hole large compared with the cutofflength, the largest source ofimprecision is the unavoidable spread of the wavepackets employed, and the27

associated large differential in the redshift suffered across the packet when itis near the horizon.The boundary condition we impose may or may not be physically thecorrect one. If it fails to hold, then there will be a deviation from the Hawkingspectrum.

It seems this could occur either with or without a large back-reaction. If the back-reaction is to remain small, then either the deviationmust be small, or there must be cancellation between positive and negativeenergy contributions, or there must be a physical cutoffmuch longer than thePlanck length.

In four dimensions, the generally expected behavior of thestress-tensor cannot be used to definitively rule out any of these scenarios.Even a very small deviation from the thermal nature of the Hawkingradiation would seem to entail a breakdown in the generalized second law ofthermodynamics [21, 22, 26]. Thus one has reason to suspect that the physicsat the cutoffscale somehow conspires to produce precisely the “thermal”state.

However, that is not to say that the ordinary effects of quantum fieldpropagation in the black hole background should not leave their mark on theradiation. The scattering of wavepackets by the geometry is one well-knownaspect of this mark, but it is conceivable that the redshifting of the physicsat the cutoffis another one.

If that is the case, then the thermodynamicbehavior of physics in a black hole spacetime may turn out to be much moresubtle than was previously thought.Given a candidate theory with a short distance cutoff, it will certainlybe interesting to study its behavior in a black hole spacetime, in which theredshift effect acts as a microscope to reveal consequences of short-distancephysics at larger scales.AcknowledgmentsI am grateful to John Dell, Kay Pirk and Jonathan Simon for several helpfuldiscussions. This research was supported by the National Science Foundationunder Grant Nos.

PHY91-12240 and PHY89-04035.28

Appendix A: Black hole line elementThe static, spherically symmetric black hole line element in Schwarzschild,tortoise, double-null, and ingoing Eddington-Finkelstein coordinates takesthe following forms:ds2=(1 −2Mr )dt2 −(1 −2Mr )−1dr2 −r2dΩ2(38)=(1 −2Mr )(dt2 −dr∗2) −r2dΩ2(39)=(1 −2Mr )dudv −r2dΩ2(40)=(1 −2Mr )dv2 −2dvdr −r2dΩ2(41)withr∗= r + 2M ln( r2M −1)(42)u = t −r∗,v = t + r∗. (43)The coordinates u and v are called the retarded and advanced time coordinatesrespectively.If xµ(λ) is an affinely parametrized geodesic, then it is a stationary pointof the integralR gµν ˙xµ ˙xνdλ, where the dot · = d/dλ.

To see that r is anaffine parameter along ingoing radial null geodesics, it is convenient to usethe Eddington-Finkelstein coordinates, so that ˙v = ˙θ = ˙φ = 0. Upon varyingv(λ), one immediately finds ¨r = 0, so r = aλ + b for some constants a and b.Appendix B: Klein-Gordon inner productThe Klein-Gordon inner product ⟨f, g⟩between two initial data sets f and gon a Cauchy surface Σ is defined by⟨f, g⟩=Zjµ dΣµ(44)jµ = i2√−ggµν(f ∗∂νg −g∂νf ∗).

(45)The surface element dΣµ is given bydΣµ = 16ǫµijk dσidσjdσk,(46)29

where σi (i = 1, 2, 3) are coordinates on the surface Σ. For solutions of theKG equation of compact support, (44) is independent of the Cauchy surfaceon which the integral is evaluated, since the current vector density jµ isdivergence free, ∂µjµ = 0.

We shall have occasion to evaluate the KG innerproduct on a surface that is null, which can be thought of as a limiting caseof Cauchy surfaces.Appendix C: Quantum field theoryThe field operator Φ for a real, free scalar field is a Hermitian operator thatsatisfies the wave equation ∇2Φ = 0. We define an annihilation operatorcorresponding to an initial data set f on a surface Σ bya(f) = ⟨f, Φ⟩Σ.

(47)If the data f is extended to a solution of the wave equation then we canevaluate the KG product in (47) on whichever surface we wish. The hermitianadjoint of a(f) is called the creation operator for f and it is given bya†(f) = −⟨f ∗, Φ⟩Σ.

(48)The commutation relations between these operators follow from the canon-ical commutation relations satisfied by the field operator.The latter areequivalent to[a(f), a†(g)] = ⟨f, g⟩,(49)provided this holds for all choices of f and g. Now it is clear that only if fhas positive, unit KG norm are the appelations “annihilation” and “creation”appropriate for these operators. From (49) and the definition of the KG innerproduct it follows identically that we also have the commutation relations[a(f), a(g)] = −⟨f, g∗⟩,[a†(f), a†(g)] = −⟨f ∗, g⟩.

(50)A Hilbert space of “one-particle states” can be defined by choosing adecomposition of the space S of complex initial data sets (or solutions tothe wave equation) into a direct sum of the form S = Sp ⊕Sp∗, where allthe data sets in Sp have positive KG norm and the space Sp is orthogonalto its conjugate Sp∗. Then all of the annihilation operators for elements of30

Sp commute with each other, as do the creation operators.A “vacuum”state |Ψ⟩corresponding to Sp is defined by the condition a(f)|Ψ⟩= 0 forall f in Sp, and a Fock space of multiparticle states is built up by repeatedapplication of the creation operators to |Ψ⟩. (Instead of thinking of the Hilbert space as the Fock space correspondingto some decomposition Sp⊕Sp∗as above, it is perhaps conceptually preferableto take the point of view of the algebraic approach to quantum field theory[27], according to which a “state” is simply a positive linear functional ρon the ⋆-algebra of field operators.

Thus for example, to express the ideathat a given field mode f is in its ground state, one says that the stateρ satisfies ρ(Oa(f)) = 0 for all operators O. This language is preferableif, as is often the case for quantum fields in curved space, one wishes tosimultaneously consider a state as an element of two completely differentlyconstructed (for example “in” and “out”) Fock spaces.In the algebraicapproach, no mysterious “identification” of the two Fock spaces is required.Another advantage is that whereas the statement that the field operatoris “hermitian” is meaningless until the Hilbert space on which it acts hasbeen specified, the statement that Φ goes into itself under the abstract ⋆operation is always well defined.

The algebraic approach is clearly preferablein contexts (e.g. [27, 28]) in which one wishes to obtain results valid for aclass of quantum states that is as wide as possible.

)Appendix D: Negative frequency partof the transmitted wavepacketConsider the transmitted part t¯ω¯u of a wavepacket p¯ω¯u propagating in theSchwarzschild black hole spacetime, narrowly peaked in u-frequency about ¯ω(at large r) and about some late retarded time ¯u. For the original Hawkingargument one needs to determine the KG norm of the negative frequencypart of t¯ω¯u at I−in terms of the norm of t¯ω¯u itself.

For our argument in thispaper, it is the negative r-frequency part on a constant v surface that is ofinterest. It was Hawking’s original argument that these two are related, usingthe geometrical optics approximation to propagate the very high frequencymodes in question back out to I−.Consider a collection of null surfaces, wavefronts for such a mode.

In Fig.31

2 one surface is shown that is outgoing at retarded time u and ingoing atadvanced time v(u). The key fact is that for late retarded times, the valueof the affine parameter r(u, vc) where this wavefront intersects the surfacev = vc is linearly related to the advanced time v(u).15 Therefore the negativer-frequency part of t¯ω¯u at v = vc propagates back to the negative v-frequencypart at I−.

Thus the corresponding KG norms are identical, so in both caseswe can carry out the calculation at v = vc.Now there is an observation [8, 9] that makes the extraction of the nega-tive frequency part simple: let U be defined by κu = −ln(−κU), and considerthe functions q and eq, defined byq(U) =(e−iωufor U < 00for U > 0(51)andeq(U) = q(−U). (52)That is, eq is just the function q reflected over the line U = 0 (u = ∞).

Thenone can easily show that the functionsq(+)=c+(q + e−πω/κ eq)and(53)q(−)=c−(e−πω/κq + eq)(54)are pure positive and negative U-frequency packets respectively. One cansolve for the normalization factors c+ and c−by setting q = q(+) + q(−).

Thisyieldsc−=−e−πω/κ c+,(55)c+=(1 −e−2πω/κ)−1. (56)Finally, the KG norm of q(−) is calculated from (54) and (55,56) using ⟨eq, eq⟩=−⟨q, q⟩and ⟨q, eq⟩= 0, yielding⟨q(−), q(−)⟩= −⟨q, q⟩(e2πω/κ −1)−1.

(57)15Hawking argued that this is because as one goes from (u, vc) back along the wavefromtand out to I−, the “vector” that connects the wavefront to the horizon (and earlier tothe null ray that becomes the generator of the horizon) is parallel transported into itself.This is not actually correct, since the connecting vector satisfies not the parallel transportequation but the geodesic deviation equation. Nevertheless, one still obtains a finite linearscaling of the connecting vector, which is all that is required for the argument [29].32

The preceeding calculation is directly applicable to the wavepacket t¯ω¯u,squeezed near the horizon on the surface v = vc. The relation between r andu along v = vc is given by (42,43), u = vc −2r∗= vc −2r −4M ln( r2M −1).For our wavepacket near the horizon, the spread in r is very small comparedwith 2M, so the wavepacket only has support where one has κu ≃−ln( r2M −1) + const..Thus U and r are linearly related (via −κU =r2M −1), sothe negative r-frequency part q(−,r) is equal to the negative U-frequency partq(−) (54) with ω = ¯ω, provided the packet is sufficiently peaked in frequencyabout ¯ω (∆ω ≪κ) so that the expressions (53,54) for the positive andnegative frequency parts still hold.

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FIGURE CAPTIONSFigure 1. Conformal diagram depicting wavefronts of the wavepacket P =R + T propagating in the spacetime of a spherically symmetric collapsingbody.Figure 2.

Conformal diagram depicting the propagation of a wavefront. Thepoint (vc, r(u, vc)) is connected by a radial null geodesic to a point on I−atadvanced time v(u).

The affine parameter r along the line v = vc is linearlyrelated to v(u) for late retarded times u.35

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