Utrecht-THU-95/26 hep-th/9601073 Hawking Spectrum and High Frequency Dispersion 6 9 Steven Corley and Ted Jacobson 9 1 Institute for Theoretical Physics, University of Utrecht n P.O. Box 80.006, 3508 TA Utrecht, The Netherlands a J and 5 Department of Physics, University of Maryland 1 College Park, MD 20742-4111, USA 1 corley, [email protected] v 3 7 0 1 Abstract 0 6 We study the spectrum of created particles in two-dimensionalblack hole geometries for 9 a linear, hermitian scalar field satisfying a Lorentz non-invariant field equation with higher / h spatialderivativetermsthataresuppressedbypowersofafundamentalmomentumscalek0. t The preferred frame is the “free-fall frame” of the black hole. This model is a variation of - p Unruh’s sonic black hole analogy. We find that there are two qualitatively different types of e particle production in this model: a thermal Hawking flux generated by “mode conversion” h at the black hole horizon, and a non-thermal spectrum generated via scattering off the : v background into negative free-fall frequency modes. This second process has nothing to do Xi with black holes and does not occur for the ordinary wave equation because such modes do not propagate outside the horizon with positive Killing frequency. The horizon component r a of the radiation is astonishingly close to a perfect thermal spectrum: for the smoothest metric studied, with Hawking temperature TH 0.0008k0, agreement is of order (TH/k0)3 ≃ at frequency ω = TH, and agreement to order TH/k0 persists out to ω/TH 45 where the thermal number flux is O(10−20). The flux from scattering dominates at ≃large ω and becomes many orders of magnitude larger than the horizon component for metrics with a “kink”, i.e. a region of high curvature localized on a static worldline outside the horizon. Thisnon-thermalfluxamountstoroughly10%ofthetotalluminosityforthekinkiermetrics considered. The flux exhibits oscillations as a function of frequency which can be explained by interference between the various contributions to the flux. 1 Introduction Black holes are boost machines. They process high frequency input and deliver it as low fre- quency output, owing to the gravitational redshift. Thus they provide a glimpse of the world at very short distance scales. This short distance world consists of nothing but vacuum fluc- tuations. A black hole acts like a microscope, allowing us to peer into the vacuum and see something of the nature of these short distance fluctuations. When looking directly at a black hole, we see the vacuum fluctutations as processed by quantum field dynamics. In ordinary continuum quantum field theory, this processing results in Hawking radiation[1], with a perfect thermal spectrum. The older the black hole, the higher is the boost interpolating between the input and output. In fact this boost grows exponentially as exp(t/4M) with the age t of the hole. Therefore, according to ordinary quantum field theory, the phenomenon of Hawking radiation involves physics at arbitrarily high frequencies and short distances. If there is new physics at some length scale, then the output of the black hole will be the result of processing at least down to that scale. Perhaps,therefore,theexistenceandpropertiesofHawkingradiationcanteach ussomething about physics at very short distances. Note that the term “short” here refers to measurements in the asymptotic rest frame, or the free-fall frame, of the black hole. If one assumes exact Lorentz invariance and locality, the large boosts provided by the black hole are just symmetry transformations, and one can learn nothing new. However, the assumption of unlimited boost invariance is beyond the range of observational support, so we shall not make it. Instead, we consider in this paper the effect on black hole radiation of (local and non-local) Lorentz non- invariant modifications to quantum field theory. It is worth pointing out that even if one does assume exact Lorentz invariance, there is still room for short distances to play a crucial role in black hole physics. One way this might happen is via nonlocality, such as in string theory[2]. Another possibility is that the infalling matter or vacuum fluctuations might have intense gravitational interactions with the outgoing trans-Planckian degrees of freedom[3]. Thus, even if exact Lorentz invariance is assumed, the use of ordinary field theory in analyzing physics around black holes might be unjustified. Tosomeextent,onecansidesteptheshort-distanceregimebyimposingaboundarycondition onthequantumfieldinatimelikeregionoutsidetheeventhorizon[4]. Assumingthatfieldmodes propagate in the ordinary way below some cutoff frequency ω , and assuming that the outgoing c modes with frequencies below the cutoff but well above the Hawking temperature T are in H their ground state, then the usual Hawking effect can be deduced in an approximation that gets better as T /ω gets smaller. This calculation shows that a conservative upper bound on the H c deviations from the thermal spectrum is of order O((T /ω )1/2). Other estimates[5], based on H c the behavior of accelerated detectors near the horizon or on the trace anomaly, suggest that the deviations will be much smaller, of order O(T /ω ). Such arguments leave some room for H c interesting dependence on short distance physics however, due to cumulative effects. In view of the gentle curvature of spacetime outside the black hole one might expect no excitation of high energy degrees of freedom. On the other hand, even if there is only a small amplitude for excitation in a time of order M, it is conceivable that the amplitude has a secular part which grows with time as the trans-Planckian degrees of freedom creep, while redshifting over extremely long times (e.g. M3), away from the horizon. (For the purposesof this paperthe term “Planck scale” will refer to the scale at which hypothetical Lorentz non-invariant physics occurs.) Furthermore, even small deviations in the spectrum might have a large effect when 1 integrated all the way up to trans-Planckian wavevectors. Given a particular model of short distance physics, we would thus like to ask the following questions: 1. Where do the outgoing modes come from? 2. Does the above mentioned out vacuum boundary condition hold? 3. Exactly how large are the deviations from the thermal Hawking flux? 4. Are the deviations from the thermal Hawking flux small even at very short wavelengths? 5. Do the deviations for short wavelengths accumulate to make a large difference in any physical quantity, such as the energy flux or energy density? The simple model we shall consider in this paper is a quantum field in two spacetime di- mensions satisfying a linear wave equation with higher spatial derivative terms. The dispersion relation ω = ω(k) thus differs at high wavevectors from that of the ordinary wave equation. The particular dispersion relation we shall study in detail is ω2 =k2 k4/k2. A modified dispersion − 0 relation occurs ubiquitously in all sorts of physical situations. Whenever there is new structure at some scale, for example as in a plasma or a crystal, wave propagation senses this, and the structureisreflectedinthedispersionrelation. Unruh[6]recently studiedamodellikethiswhich was motivated by a sonic analog of a black hole. Although he describes the model in terms of sound propagation in an inhomogeneous background fluid flow, the model is in fact identical to that of a scalar field in a black hole spacetime, with the co-moving frame of the background flow replaced by the free-fall frame of the black hole. By numerical integration of the altered partial differential wave equation (PDE), Unruh studied the propagation of wavepackets in this model and established that, to the numerical accuracy of his calculation, Hawking radiation still occurs and is unaffected by the altered dispersionrelation. Thenumerical accuracy was not quitegood enough to ruleoutdeviations at the upper bound referred to above. Perhaps the most interesting thing about the model is the peculiarbehaviorofwavepackets sentbackwardsintimetoward thehorizon: ratherthangetting squeezed in an unlimited way against the horizon and ceaselessly blueshifting, the wavepackets reach a minimum distance of approach, then reverse direction and propagate back away from the horizon. The blueshift at the closest approach to the horizon is independent of the retarded time about which the outgoing wavepacket was centered, and the packet continues to blueshift on the way out going backwards in time. Subsequently, Brout, Massar, Parentani and Spindel (BMPS) [8] made an analytical study of the Unruh model, and came to similar conclusions in a leading order approximation in 1/M. In addition, BMPS introduced another model, differing from the Unruh model in that the altered dispersion relation is defined with respect to Eddington-Finkelstein coordinates. In the BMPS model, an outgoing wavepacket propagated backward in time does not reverse direction but rather hugs the horizon at a distance of one “Planck length”, with exponentially growing wavevectors. For this model the usual Hawking effect at leading order in 1/M was established by analytical methods. Theprimarypurposeof the presentpaper is to determineprecisely the spectrumof Hawking radiation for a model with a nonlinear dispersion relation as in the Unruh and BMPS models. In our model the field equation has fourth order spatial derivatives in the free-fall frame. We 2 wish to evaluate quantitatively the deviations from the thermal spectrum, including the high wavevector region. Toachieve thisaim, numericalintegration ofthePDE isimpractical (atleast for us), and leading order approximations are insufficient. Instead, we employ a two-pronged attack. First, we exploit the stationarity of the background metric to simplify the problem. Thus, instead of solving a PDE, we numerically solve alot of ordinary differential equations (ODE’s) for the mode functions. Second, as a check on the accuracy of our numerical solutions, we develop the exact solution for a subclass of the models. To our surprise we have found in the models studied here that, in addition to the Hawking radiation, radiation is produced via scattering from the static curvature. A second purpose of our paper is to give a physical picture of the Hawking effect in the context of these models with altered dispersion relation. What we describe has alot in common with the picture explained by BMPS in [8] (although we developed our picture independently before becoming aware of their paper). The picture has two essential features, reversal of group velocity without reflection and “mode conversion” from one branch of the dispersion relation to another. Interestingly, both these phenomena can occur for linear waves in inhomogenous plasmas[9, 10, 11], and undoubtedly occur in many other settings as well. The propagation of a wavepacket and the direction-reversal phenomenon can be understood using the WKB approximation. At the turn-around point partial mode conversion from a positive free-fall frequency to a negative free-fall frequency wave takes place. This mode conversion gives rise to the Hawking effect. A third purpose of our paper is to discuss the “stationarity puzzle” in these models: If the wavepackets go from infinity to infinity, without ever passing through the collapsing matter, then how can there be any particle production? The remainder of our paper is organized as follows. Section 2 defines the model to be studied and section 3 describes the wavepacket propagation, mode conversion, and scattering in this modelusinga WKB analysis. Section 4 lays out the computational techniques we employed to obtain the precise quantitative results that are reported and interpreted in section 5. In section 6 the stationarity puzzle is discussed, and section 7 contains a summary of our results. Throughout the paper we use units in which h¯ = c= G = 1, unless otherwise specified. 2 The model and its quantization The model we shall consider consists of a free, hermitian scalar field propagating in a two dimensional black hole spacetime. The dispersion relation for the field lacks Lorentz invariance, and is specified in the free-fall frame of the black hole, that is, the frame carried in from the rest frame at infinity by freely falling trajectories. This is the same frame as the one used in the Unruhmodel[6],butthedispersionrelationweadoptisdifferent. TheBMPSmodelontheother hand adopts the same dispersion relation as Unruh, but applys it in the Eddington-Finkelstein coordinate frame. 2.1 Field equation Let uα denote the unit vector field tangent to the infalling worldlines, and let sα denote the orthogonal, outward pointing, unit vector, so that gαβ = uαuβ sαsβ. (See Fig. 1.) − 3 x x t 1 2 1 t 2 u δ t δx=s free-fall worldline Figure1: Apatchofspacetimeshowingafree-falltrajectoryandsometandx(Lemaˆıtre-like)coordinate lines. u and s are orthonormalvectors, and the derivative along s is modified, while that along u is just the partial derivative. The notations δ and δ denote ∂/∂t and ∂/∂x respectively, and δ is the Killing t x t vector. The action is assumed to have the form: 1 S = d2x√ ggαβ φ∗ φ, (1) α β 2 − D D Z where the modified differential operator is defined by α D uα = uα∂ (2) α α D sα = Fˆ(sα∂ ). (3) α α D The time derivatives in the local free-fall frame are thus left unchanged, but the orthogonal spatial derivatives are replaced by Fˆ(sα∂ ). The function Fˆ determines the dispersion relation. α For the moment it will be left unspecified. Invariance of the action (1) under constant phase transformations of φ guarantees that there is a conserved current for solutions and a conserved “inner product” for pairs of solutions to the equations of motion. However, since is not α D in general a derivation, simple integration by parts is not allowed in obtaining the equations of motion or the form of the current. We shall obtain these below after further specifying the model. The black hole line elements we shall consider are static and have the form ds2 = dt2 (dx v(x)dt)2. (4) − − This is a generalization of the Lemaˆıtre line element for the Schwarschild spacetime, which is given by v(x) = 2M/x (together with the usual angular part). We shall assume v < 0, − dv/dx > 0, and v v as x . ∂ is a Killing vector, of squared norm 1 v2, and the event →po → ∞ t − horizon is located at v = 1. The curves given by dx vdt = 0 are timelike free-fall worldlines − − which are at rest (tangent to the Killing vector) where v = 0. Since we assume v < 0 these are ingoing trajectories. v is their coordinate velocity, t measures proper time along them, and they are everywhere orthogonal to the constant t surfaces (see Fig. 1). We shall refer to the function v(x) as the free-fall velocity. The asymptotically flat region corresponds to x . → ∞ In terms of the notation above, the orthonormal basis vectors adapted to the free-fall frame are given by u = ∂ +v∂ and s = ∂ , and and in these coordinates g = 1. Thus the action t x x − (1) becomes 1 S = dtdx (∂ +v∂ )φ2 Fˆ(∂ )φ2 . (5) t x x 2 | | −| | Z (cid:16) (cid:17) 4 If we further specify that Fˆ(∂ ) is an odd function of ∂ , then becomes a derivation “up to x x α D total derivatives”, and integration by parts yields the field equation (∂ +∂ v)(∂ +v∂ )φ= Fˆ2(∂ )φ. (6) t x t x x The conserved inner product in this case is given by (φ,ψ) = i dx φ∗(∂ +v∂ )ψ ψ(∂ +v∂ )φ∗ , (7) t x t x − Z (cid:16) (cid:17) where the integral is over a constant t slice and is independent of t if φ and ψ satisfy the field equation (6). The inner product can of course be evaluated on other slices as well, but it does not take the same simple form on other slices1. The dispersion relation for this model in flat spacetime, or in the local free-fall frame (as- suming v(x) constant), is given by ≈ ω2 = F2(k), (8) where F(k) iFˆ(ik). Unruh’s choice for the function F(k) has the property that F2(k) = k2 ≡ − for k k and F2(k) = k2 for k k , where k is a wavevector characterizing the scale of the ≪ 0 0 ≫ 0 0 newphysics. Weusuallythinkofk asbeingaroundthePlanckmass. Specifically, heconsidered 0 the functions F(k) = k0 tanh[(k/k0)n] n1. Of course there are many other modifications one { } could consider. Perhaps the simplest is given by F2(k) = k2 k4/k2. (9) − 0 Thisdispersionrelationhasthesamesmallk behaviourasUnruh’s,butbehavesquitedifferently for large k. It has the technical advantage that the field equation (6) has no derivatives higher than fourth order, and for this reason it is the one on which all the calculations in this paper are based, although we shall briefly discuss the behavior for alternate choices in the final section. These two dispersion relations are plotted in Figure 2 along with the dispersion relation for the ordinary wave equation. 2.2 Quantization Toquantizethefieldweassumethefieldoperatorφˆ(x)isself-adjointandsatisfiestheequationof motion (6) and thecanonical commutation relations. In setting up thecanonical formalism, it is simplest to use the time function and evolution vector for which only firstorder time derivatives appear in the action. (Otherwiseone mustintroduce extra momenta which are constrained, and then pass to the reduced phase space.) This just means that we define the momenta by π = δL/δ(∂ φ) = (∂ +v∂ )φ, t t x i.e.,πisthetimederivativealongthefree-fallworldlines. Theequaltimecanonicalcommutation relations are then [φ(x),π(y)] = iδ(x,y), as usual. 1In fact the inner product is non-local when evaluated on other slices if Fˆ(sα∂ ) is nonlocal. The conserved α current density jα is determined by theequation ∂ jα = √ ggαβ(φ∗ φ φ φ∗) . α Da − 1Dβ 2− 2Dβ 1 (cid:16) (cid:17) 5 a 1.0 ω b 0.5 c 0.0 0.0 0.5 1.0 k Figure 2: Curve a is the standarddispersionrelationfor the masslesswaveequation, curveb is the type used by Unruh, and curve c is the one used in this paper (9). We define an annihilation operator corresponding to an initial data set f on a surface Σ by a(f)= (f,φˆ), (10) where the inner product is evaluated on Σ. If the data f is extended to a solution of the field equation then we can evaluate the inner product (10) on whichever surface we wish. The hermitian adjoint of a(f) is called the creation operator for f and it is given by a†(f)= (f∗,φˆ). (11) − The commutation relations between these operators follow from the canonical commutation relations satisfied by the field operator. The latter are equivalent to [a(f),a†(g)] = (f,g), (12) providedthisholdsforallchoicesoff andg. Nowitisclearthatonlyiff haspositive, unitnorm are the appelations “annihilation” and “creation” appropriate for these operators. From (12) and the definition of the inner product it follows identically that we also have the commutation relations [a(f),a(g)] = (f,g∗), [a†(f),a†(g)] = (f∗,g). (13) − − A Hilbert space of “one-particle states” can be defined by choosing a decomposition of the space S of complex initial data sets (or solutions to the field equation) into a direct sum of the form S = S S ∗, where all the data sets in S have positive norm and the space S p p p p ⊕ is orthogonal to its conjugate S ∗. Then all of the annihilation operators for elements of S p p commute with each other, as do the creation operators. A “vacuum” state Ψ corresponding | i to S is defined by the condition a(f)Ψ = 0 for all f in S , and a Fock space of multiparticle p p | i states is built up by repeated application of the creation operators to Ψ . | i It is not necessary to construct a specific Fock space in order to study the physics of this system. In fact, any individualpositive norm solution p defines annihilation and creation opera- tors and a number operator N(p)= a†(p)a(p). The physical significance of the number operator depends of course on the nature of p. 6 There are two types of positive norm wavepackets in which we are interested. The first are those corresponding to the quanta of Hawking radiation. These have positive Killing frequency, that is, they are sums of solutions satisfying ∂ φ = iωφ with ω > 0. It is not obvious that t − such solutions have positive norm in the inner product (7), and in fact they do not in general. However, usingthefactthattheKillingfrequencyisconserved,weknowthatifapositiveKilling frequency wavepacket were to propagate out to infinity (or any other region where v = 0), the integral for its norm would be manifestly positive. Since the norm is conserved, this suffices. The other type of positive norm wavepackets we shall employ are those which correspond to particles as defined by the free-fall observers. These have positive free-fall frequency, that is, they are sums of solutions satisfying (∂ +v∂ )φ = iω′φ, with ω′ > 0, on some time slice. t x − These have manifestly positive norm (if the solutions summed are orthogonal to each other like, for example, harmonic modes in a constant v region), although the free-fall frequency is not conserved. Finally, we conclude this section on quantization with a cautionary remark. One sees from the dispersion relation (9) that, for k2 > k2, the field has imaginary frequency modes which are 0 well-behaved in space. In principle these modes must be included in a complete quantization of the model. Although imaginary frequency modes can be quantized[12, 13], the resulting model is unstable in that the energy spectrum is unboundedbelow. However, these modes play no role in our analysis of the Hawking effect, so we shall simply ignore them as an irrelevant unphysical feature of the model. 3 Wavepacket propagation and mode conversion In this section we describe, by way of pictures, the production of Hawking radiation from an initial vacuum state by means of a process known as “modeconversion”. We also describea new process of particle production via scatering in a static geometry that happens in the dispersive models studied here. We assume that all ingoing positive free-fall frequency wavepackets are unoccupied, at some given time, far (but not infinitely far—see section 6) from the hole where v(x) is approximately (or exactly) constant. Given this initial state, we wish to calculate the numberof particles, in a given outgoing packet, detected by an observer far from the hole whois at rest with respectto the hole. Following thestandard technology (see section 4.2), the number of particles in this packet is obtained by propagating the packet back in time to where the initial ground state boundary condition is imposed and taking the norm of its negative free-fall frequency piece. The behavior of a wavepacket propagated back in time can be understood qualitatively as follows. Assume a solution to the field equation (6) of the form φ = e−iωtf(x) and solve the resulting ODE (17) for f(x) by the WKB approximation. That is, write f(x)= exp(i k(x)dx) and assume the quantities ∂ v and ∂ k/k are negligible compared to k. The resulting equation x x R is the position dependent dispersion relation (ω v(x)k)2 = F2(k). (14) − This is just the dispersion relation in the local free-fall frame, since the free-fall frequency ω′ is related to the Killing frequency ω by ω′ = ω v(x)k. (15) − 7 The position-dependent dispersion relation is useful for understanding the motion of wavepack- ets that are somewhat peaked in both position and wavevector. A graphical method we have employed is described below. The same method was used by BMPS[8], who also found a Hamil- tonian formulation for the wavepacket propagation using Hamilton-Jacobi theory. 1.0 0.5 v o ω’ 0.0 -0.5 -1.0 -0.75 -0.25 0.25 0.75 k o Figure 3: Graphicalsolution of the position-dependent dispersionrelation (14), with F(k) givenby (9), in units where k0 = 1. The line labeled v0 corresponds to a position far from the hole. The other line corresponds to the classical turning point. The k values of the intersections of the straight and curved lines are the solutions to the dispersion relation for fixed ω and v. For the v0 line these are denoted from left to right by k−, k−s, k+s, and k+ in the text. The filled arrowheads indicate the direction of propagationof wavepackets,in momentum space, as discussed in the text. Graphs of the squareroot of both sides of equation (14) are shownin Figure 3 for F(k) given by (9) and for two different values of v. As x varies, the slope v(x) (= v(x)) of the straight − | | line representing the left hand side of (14) changes, butfor a given wavepacket the intercept ω is fixedsincethe Killingfrequencyis conserved. For a given x, theintersection points on thegraph correspond to the possible wavevectors in this approximation. These solutions to the dispersion relation for fixed ω and v will be denoted, in increasing order, as k(ω) = k , k , k , k . (16) − −s +s + (Thesubscript“s”isintendedtosuggest“smaller”inabsolutevalue.) Notethatfortheordinary wave equation one would have only the two roots with ω > 0 corresponding to k and k at −s +s the velocity v . 0 The coordinate velocity dx/dt of a wavepacket is the group velocity v = dω/dk. This may g also be expressed, using (15), as v = v′ +v(x), where v′ dω′/dk is the group velocity in the g g g ≡ free-fall frame which corresponds to the slope of the curved line in Figure 3. The group velocity is positive at k and negative at k , k and k . Thus while there is one outgoing mode at +s − −s + fixed positive ω, there are three ingoing modes. Of crucial importance is the fact that the k − mode outside the horizon (v > 1) has negative free-fall frequency when the Killing frequency − is positive. Now consider a wavepacket located far from the hole, centered about frequency ω, and containingonlykvaluesaroundk . Thisisanoutgoingwavepacketso,goingbackwardsintime, +s the packet moves towards the hole. Two qualitatively different effects govern the wavepacket 8 propagation, namely, mode conversion at the horizon and scattering off the geometry. These will now be discussed in turn. 3.1 Mode conversion at the horizon As the wavepacket propagates backwards in time towards the black hole v(x) increases, so the | | slope of the straight line in Figure 3 increases, until eventually the straight line becomes tangent to the dispersion curve. At this point v drops to zero. If ω is very small compared to k , then g 0 this stopping point x occurs when v(x) is very close to 1, that is, just barely outside the t − horizon. What happens at the stopping point? It was incorrectly suggested in Ref. [5] that the wavepackets just asymptotically approach limiting position x and wavevector k . However, t t near the stopping point the point particle picture of the wavepacket motion is inadequate, and the spread in both k and x must be considered. One can determine qualitatively what happens byconsideringthebehaviorof nearbysolutions tothedispersionrelation as follows.2 As pointed out by Unruh[14], it is an unstable situation for the wavepacket to just sit at the stopping point: for k slightly above k the group velocity drops below zero (i.e. the comoving group velocity t drops below the magnitude of the free-fall velocity) so, backwards in time, the wavepacket tends to move back away from the horizon and therefore to the right (to higher wavevectors) on the dispersion curve. Once this begins to happen, k continues to increase as the wavepacket moves further away. Exactly this behavior was found in Unruh’s numerical solution[6] to the PDE. In brief, a long wavelength k -packet went in, and a short wavelength k -packet came out! This +s + is an example of the phenomenon of “mode conversion”[10, 11], but it is only half the story. There is another short wavelength solution to the dispersion relation as x approaches x , at t k on the negative wavevector, negative free-fall frequency branch of the dispersion curve, that − mixesin. Wewilldiscussinamomentaquantitativemeasureoftherelativeamplitudesofthek − andk packetsarisingfromthismodeconversionprocess,bylookingathowitworksfortheusual + wave equation. Suffice it to say here that the negative wavevector mode mixes in strongly for suffientlysmallωforbothUnruh’sdispersionrelationand(9),asshownbothbyUnruh’ssolution of the PDE and by the ODE methods applied by BMPS[8] and ourselves. The “converted”, negativewavevector, wavepacketalsohasanegativegroupvelocity, andsoalsomoves,backwards in time, away from the hole. The end result thus consists of two wavepackets, one constructed of large positive k wavevectors and the other of large negative k wavevectors, both propagating away from the hole (at different group velocities) and reaching the asymptotically flat (constant free-fall velocity) region. The number of created particles in the final, late time, wavepacket is given by (minus) the norm of the negative wavevector (and negative free-fall frequency) part of the initial, early time wavepacket. Let us see how the conversion amplitude is determined in the case of the ordinary wave equationwiththelineardispersionrelation. Thiswillalsoindicatehowitworksforthenonlinear dispersion relations. First note that the wavepacket vanishes inside the horizon (from the causal behavior of the ordinary wave equation), so it must have some negative wavevector component, since a purely positive wavevector wavepacket cannot vanish on the half line (or any open interval). But how large is this negative wavevector piece? The WKB form of a single frequency mode is φ exp(i kdx), and the dispersion relation ω vk = k yields k = ω/(1 + v). ∼ − 2In fact, the WKB aRpproximation breaks down as the stopping point is approached, however this does not preventus from obtaining qualitative information about themotion of the wavepacket as described here. 9