Pre-Hawking Radiation from a Collapsing Shell
aa r X i v : . [ g r- q c ] N ov Pre-Hawking Radiation from a Collapsing Shell
Eric Greenwood, Dmitry Podolsky, and Glenn Starkman
CERCA, Department of Physics, Case Western Reserve University, Cleveland, OH 44106-7079
We investigate the effect of induced massive radiation given off during the time of collapse of amassive spherically symmetric domain wall in the context of the functional Schr¨odinger formalism.Here we find that the introduction of mass suppresses the occupation number in the infrared regimeof the induced radiation during the collapse. The suppression factor is found to be given by e − βm ,which is in agreement with the expected Planckian distribution of induced radiation. Thus a massivecollapsing domain wall will radiate mostly (if not exclusively) massless scalar fields, making itdifficult for the domain wall to shed any global quantum numbers and evaporate before the horizonis formed. I. INTRODUCTION
Black holes embody one of the greatest frontiers intheoretical physics – the intersection of quantum the-ory with general relativity. So far, attempts to confrontthis intersection and produce a single theory incorporat-ing both quantum mechanics and general relativity, havebrought along with them several interesting properties(such as Hawking radiation, see [1]), solidified existingarguments (such as black hole entropy, see [2]), and in-troduced unanswered paradoxes associated with the for-mation and evaporation of a black hole. These includethe information loss due to the thermal nature of Hawk-ing evaporation (see [3–6]).One radical resolution of the information loss paradoxwas recently proposed in [7], where the collapse of a thinspherically symmetric domain wall (shell) was studied bymeans of the functional Schr¨odinger formalism (see also[8, 9]). Solving the Wheeler-de Witt equation for thecollapsing shell, the authors of [7] argued that (a) suchcollapse takes an infinite amount of time from the pointof view of a distant observer, even when quantum fluctua-tions of the shell are taken into account (but ignoring anyback-reaction due to the emission of radiation), and (b)the shell emits pre-Hawking massless radiation prior tothe formation of a trapped surface. This pre-Hawking ra-diation (PHR) could, they found, cause the shell to evap-orate completely in time, t evap ∼ R S (cid:16) R S l P (cid:17) (measuredfrom the moment when the radiation begins to effectivelycarry away the mass of the shell). If (a) and (b) are bothtrue, then no actual black hole can be formed in such acollapse, and, although the quantum state of PHR fromthe collapsing shell is entangled with the quantum stateof the shell, information about the initial quantum stateof the shell is continuously released during the process ofevaporation. They found that the spectrum of PHR isapproximately thermal [21] only at sufficiently large fre-quencies ωR s ≫
1, where R s is the Schwarzschild radiusof the collapsing shell. The infrared part ωR s ≪ M ≫ M P .First, we take into account quantum fluctuations of theSchwarzschild radius R S (“quantum trembling” of theevent horizon, see [10]) of the shell, present the correctionterms to the functional Schr¨odinger equation describingthe quantum collapse of the spherical shell and show thatsuch corrections make the collapse time finite and in factrather short: t coll ∼ R S , counted from the moment whenpre-Hawking evaporation starts being effective. Second,we show that the actual evaporation time t evap is signif-icantly longer than the time scale t evap ∼ R S (cid:16) R S l P (cid:17) forpre-Hawking evaporation through emission of masslessquanta, once one takes into account that the main con-tribution to the mass of the shell is given by constituentprotons and neutrons. The latter implies that a typicalcollapsing shell has very large baryon/lepton quantumnumbers, and in order for the shell to evaporate com-pletely prior to collapse, these quantum numbers musteither be erased by physical processes, or radiated awayto infinity. Since global quantum numbers, such as B and L , are only carried by quanta of massive quantum fields,such as leptons and baryons, the relevant time scale tobe compared to the total collapse time is the time ofevaporation due to emission of massive PHR.We extend the analysis of [7] to the case of PHR ofmassive particles and find the corresponding evapora-tion time. For masses of known leptons and baryons(including likely values of the neutrino masses) and as-trophysically interesting masses of the collapsing shell MM P ≫ M P m ≫ t coll .We note how non-perturbative Standard Model baryon-number violation softens, but does not eliminate, thisconclusion.This paper is organized as follows. In the Sec. II weconstruct the functional Schr¨odinger equation describingcollapse of a quantum spherically symmetric shell andfind corrections to this equation induced by fluctuationsof the metric. We then compare classical and quantumcollapse times for the shell. In Sec. III we analyze mas-sive PHR emitted by the collapsing shell by means offunction Schr¨odinger formalism, derive its spectrum andestimate its backreaction on the process of collapse. InSec. IV we discuss the effect of baryon and lepton num-ber violation on the process of collapse. Finally, the Sec.V is devoted to the discussion and conclusions. II. ESTIMATES OF THE COLLAPSETIMESCALEA. Collapse Time in Classical Physics
Following [7] let us consider a perfectly spherical do-main wall representing a spherical shell of collapsing mat-ter, so that the position of the shell is completely de-scribed by its radius, R ( t ). Our first goal is to checkthe well-known statement that from the point of view ofa distant observer, the collapse takes infinite amount oftime. Since we neglect any quantum effects, such as vac-uum fluctuations, the Schwarzschild radius of the shell, R s = 2 GM , remains unchanged during the course to col-lapse, see [11].Due to the spherical symmetry of the configurationand Birkhoff’s Theorem, the metric can be naturallyseparated into two different regions. The metric of thespace-time exterior to the collapsing shell is (see [11])Schwarzschild: ds = − (cid:18) − R s r (cid:19) dt + dr − R s /r + r d Ω , r > R ( t )(1)where as usual d Ω = dθ + sin θdφ . (2)In the interior of the shell, the metric is flat ds = − dT + dr + r dθ + r sin θdφ , r < R ( t ) (3)The interior time coordinate, T , is related to the asymp-totic observer time coordinate, t , via the proper time ofan observer moving with the shell, τ . The relations are dTdτ = s (cid:18) dRdτ (cid:19) (4)and dtdτ = 1 B s B + (cid:18) dRdτ (cid:19) (5)where B ( t ) ≡ − R s R ( t ) . (6)By taking the ratio of (4) and (5), the relationshipbetween the interior time T and the asymptotic time t isfound to be dTdt = p R τ B p B + R τ = r B − (1 − B ) B ˙ R , (7) where R τ = dR/dτ and ˙ R = dR/dt .Integrating the equations of motion for the shell givesan expression for the total mass of the shell [11], whichis the integral of motion, M = 12 [ p R τ + p B + R τ ]4 πσR . (8)Here σ is the surface tension of the shell. This expressionfor M is implicit since R s = 2 GM occurs in B . Solvingfor M explicitly in terms of R τ gives M = 4 πσR [ p R τ − πGσR ] . (9)When | R − R s | /R s ≪
1, i.e. when the shell is closeto its Schwarzschild radius, the velocity of the shell asobserved by an asymptotic observer, is given by˙ R ≈ − B . (10)Thus, to lowest order in ( R − R S ), R ( t ) ≈ R S + ( R − R S ) e − t/R S (11)where R = R ( t = 0) is the initial position of the shell[7].As we see, classical collapse of a spherically symmetricshell indeed takes infinite amount of time from the pointof view of an asymptotic observer, although the distancebetween the shell and its Schwarzschild radius quicklybecomes exponentially small. In what follows, we willcount time starting from the moment where the behaviorof the radius of the collapsing shell is approximately givenby (11). B. Collapse Time in Quantum Physics
We shall now consider how the classical estimate forthe collapse time is modified once quantum effects (quan-tum fluctuations of the collapsing shell and of the metric)are taken into account.As shown in [7], the collapse time will remain infinitein the presence of quantum fluctuations in the positionof the collapsing shell. Let us recall how exactly onecomes to this conclusion. First, one assumes that thequantum dynamics of the system (the collapsing shell,the background space-time and the distant observer) arecompletely described by the Wheeler-de Witt equation H Ψ = 0 (12)with the Hamiltonian H = H sys + H obs , (13)where H obs is the Hamiltonian of the distant observer.Since the proper distance between the collapsing shelland the observer is very large, the quantum state of theobserver is not entangled with the state of the shell andthe total wave function Ψ is separable:Ψ = X k c k Ψ sys k Ψ k obs . (14)One also expects that the state of observer Ψ obs satisfiesthe usual Schr¨odinger equation i ∂ Ψ obs ∂t = H obs Ψ obs , (15)where t is the observer’s time coinciding with the usualSchwarzschild time. Therefore, the quantum state of thecollapsing shell should also satisfy Schr¨odinger’s equation i ∂ Ψ sys ∂t = H sys Ψ sys . (16)The Hamiltonian of the shell is [7] H = B Π · B Π + B (4 πµR ) , (17)where Π = − i ∂∂R is the canonical momentum, B is givenby (6) and µ ≡ σ (1 − πGσR S ) is a constant. In the nearhorizon limit R → R S , only the “kinetic” term survivesin (17), and the “squared” functional Schr¨odinger equa-tion (16) becomes especially easy to solve using tortoisecoordinates t , u = R + R S log | R/R S − | : (cid:18) ∂ ∂t − ∂ ∂u (cid:19) Ψ sys ≈ . (18)Its solution describes a wave packet propagating withthe speed of light, with its width remaining fixed in thetortoise coordinates. Note however that the horizon islocated at u = −∞ , so that it takes infinite time for thewave packet to reach it.The considerations above do not, however, take allpossible physical effects into account. For example, inderiving (16), we assume a priori that neither the hori-zon scale R S , nor the mass of the shell M = R S G fluctu-ates. However, both should fluctuate, at least when wetry to measure their value with precision finer than thePlanck scale, simply because the metric of space-time it-self strongly fluctuates at scales of the order of Plancklength l P . Naively, it seems that these tiny Planckianfluctuations cannot be relevant for the physics discussedbecause the curvature of space-time at the scale R ≈ R S is significantly smaller than M P . Recall however thatthe size of the shell R approaches the Schwarzschild ra-dius R S exponentially quickly (see (11)), so that the dif-ference R − R S ∼ R e − t/R S soon becomes comparableto the Planckian length l P . Therefore, at time scales t ≫ R s log( R /l P ) fluctuations of the metric must betaken into account.How can one then account for these fluctuations at thelevel of the Wheeler-de Witt equation in (12)? By sepa-rating the metric fluctuations in the full Wheeler-de Wittequation from the fluctuations of the matter degrees of freedom associated with the collapsing shell and distantobserver, one can write H Ψ = (cid:18) − πG · G ab,cd δ δg ab δg cd − πG √ gR (3) ++ H sys + H obs ) Ψ = 0 , (19)where g ab is 3-metric of the space-time, R (3) is 3-curvature and G ab,cd = g − / ( g ac g bd + g ad g bc − g ab g cd )is the local de Witt supermetric. The functionalSchrodinger equation [12], as well as corrections to it [13],can be constructed from (19) by looking for solutions ofthe form Ψ = e iS (20)with the “effective action” S expanded in powers of G : S = G − S + S + GS + . . . . (21)The leading term S describes the classical backgroundspace-time and the classical collapsing shell, and equa-tions of motion derived by variating “the action” S areEinstein-Jacobi equations for the background. The term S takes into account (a) the vacuum fluctuations of ge-ometry and (b) fluctuations of the shell and their backre-action on the background geometry. Variation of S givesthe functional Schr¨odinger equation discussed above. Fi-nally, the term S and higher order terms describe thebackreaction of the vacuum fluctuations of geometry onthe background as well as interaction between fluctua-tions of the shell and fluctuations of geometry. Equationsof motion found by variation of S and higher order termsrepresent corrections to the functional Schr¨odinger equa-tion.The original Wheeler-de Witt equation (19) is notori-ously hard to solve for many reasons, one of them be-ing that it requires regularization in the same way asdoes the Schr¨odinger equation describing behavior of thewave functional of relativistic quantum fields — behaviorof a quantum field theory can be understood much moreeffectively by using second quantization and Feynman di-agrammatics than first quantization and the Schr¨odingerequation. In fact, it is natural to expect that the veryexpansion in (21) breaks down at the vicinity of eventhorizon R → R S , when the behavior of the radius of theshell is approximately given by the expression in (11) andwhere, as we shall see, the evaporation process seems tobe the most effective.Fortunately, the relevant physics of the perturbationtheory in powers of G can be understood using uncer-tainty relations. In particular, in order to estimate thecollapse time one can notice that the horizon is expectedto “tremble” at Planckian scales [14], since the fluctua-tions of the metric g can be estimated as δgg ∼ l P L , (22)where L is the length scale characterizing the curvatureof the space-time [22] [14]. Correspondingly, the charac-teristic size of fluctuation of the Schwarzschild radius canbe estimated as δR S ∼ l P R S . (23)Recall again that at t ≫ R S the radius of the shell R isexponentially close to the Schwarzschild radius R s , anddue to the uncertainty in (23) in determining the value of R S , the observer at infinity will be unable to determinewhether the actual event horizon is formed or not afterthe time t ∼ R s log (cid:18) R s R l P (cid:19) (24)which can therefore be understood as the collapse timeof the shell.Another way to reproduce the estimate of (24) is touse the fact that a distant observer can only determinethe position of the collapsing shell by sending quanta to-wards it, having them scattered off the shell and measur-ing the corresponding cross sections (scattering and/orgravitational capture). Apparently, the maximal energyof the quantum, which the observer can send to the shell,is of the order of M P . In the vicinity of the shell theblueshifted maximal energy of the quantum will there-fore be bounded by M P q − R s R . (25)This energy is related to the uncertainty in determiningthe value of the position of the shell R ( t ) by δR · M P q − R s R > , (26)showing that at t > R s log (cid:18) R l P (cid:19) (27)the observer will be unable to determine whether theshell has crossed its Schwarzschild radius. The estimateof (27) coincides with that of (24), up to the logarithmicprecision.As we see, once the fluctuations of metric are properlytaken into account, the collapse time is no longer infinite(and in fact rather short taking into account how small isthe value of the Schwarzschild radius R S for astrophysi-cally interesting scales). III. PRE-HAWKING RADIATION OF AMASSIVE SCALAR FIELD
In the process of collapsing toward its Schwarzschildradius, the shell emits PHR, losing mass. In principle, if this PHR has sufficient power, the shell could evapo-rate completely prior to formation of an event horizon, asargued in [7]. To determine whether collapse can success-fully conclude, the collapse time scale as given in (27) hasto be compared to the time scale of evaporation throughPHR emission. Since the spectrum of the massless PHRis nearly thermal, with the temperature T being of theorder of the Hawking temperature [7], the characteristictime scale for the evaporation of the collapsing shell ofinitial mass M through emission of massless quanta co-incides with the scale of Hawking evaporation for a blackhole with Schwarzschild radius R S = 2 GM t evap ∼ M P R S = R S (cid:18) R S l P (cid:19) . (28)As long as R S ≫ l P , this time scale is significantly longerthan the collapse time given in (27), and PHR cannotprevent the shell from collapse.Validity of this conclusion strongly depends on the es-timate in (27) for the collapse time. However, as weshall now argue, even if the estimate in (27) is ultimatelywrong and an exact solution of the Wheeler-de Wittequation (19) for a collapsing shell reveals a much longer(but finite) collapse time, the conclusion will hold un-der most (or perhaps all) astrophsically interesting con-ditions. The ultimate reason for its robustness is the factthat a collapsing shell made of protons, electrons andother constituents of the Standard Model carries globalquantum numbers. These global quantum numbers arenot known to be carried by massless particles. Sincequanta of PHR with non-zero global quantum numbershave to be massive, the rate of their emission is expectedto be exponentially suppressed by the usual Boltzmannfactor [1]. Unless the Hawking temperature is at or abovethe mass of the particles, this Boltzmann factor will bea sizable suppression.In this Section, we shall expand the results of [7] to thecase of massive radiation, derive its spectrum and analyzeits backreaction on the dynamics of a collapsing shell.In subsequent sections we shall examine the implicationof the exponential suppression of massive modes on thelikely evaporation pathways of astrophysical black holes. A. Functional Schr¨odinger Formalism for MassivePre-Hawking Radiation
Our goal in this section is to expand the results of [7]to include the case of massive quanta radiated during thetime of collapse. As a result, we will therefore carefullyfollow their development and notation.Let us consider a spherically symmetric configurationof a massive real scalar field Φ which propagates in thebackground of the collapsing shell. The action for thescalar field is S = Z d x √− g (cid:0) g µν ∂ µ Φ ∂ ν Φ + m Φ (cid:1) , (29)where g µν is the background metric given by (1) and (3).The field Φ can always be expanded into a complete setof eigenmodes Φ = X k a k ( t ) f k ( r ) (30)such that the Hamiltonian is a simple sum of terms. Thetotal wavefunction then factorizes and can be found bysolving a time-dependent Schr¨odinger equation of justone variable.From (1) and (3), one can split the action (29) intointerior and exterior parts S in = 2 π Z dt Z R ( t )0 drr ˙ T h − ( ∂ t Φ) ˙ T + ( ∂ r Φ) + m Φ i (31)and S out = 2 π Z dt Z ∞ R ( τ ) drr h − ( ∂ t Φ) − R s /r + (cid:18) − R s r (cid:19) ( ∂ r Φ) + m Φ i , (32)where we used (7). It was shown in [7] that (7) is givenby ˙ T = dTdt = B r − B ) R h (33)where h = M/ (4 πµ ) and µ = σ (1 − πσGR S ).The regime of most interest is the one when the radiusof the shell approaches the Schwarzschild radius R s .We see from (33) that in the near horizon limit, ˙ T ∼ B →
0. Therefore the kinetic term for S in diverges as( R − R S ) − in this limit. The kinetic term in S out divergeslogarithmically, so the S in kinetic term is dominant term.Similarly the potential term in S in vanishes while the po-tential term in S out becomes finite, so the potential termin S out dominates. Therefore we can write the action as S ≈ π Z dt h − B Z R s drr ( ∂ t Φ) + Z ∞ R s drr (cid:18) − R s r (cid:19) ( ∂ r Φ) (34)+ m Z ∞ R s drr ΦΦ i , where we have changed the limits of integration from R ( t )to R s since this is the region of interest.Using the expansion in the modes (30), we can rewritethe action as S ≈ Z dt h − B ˙ a k ( t ) M kk ′ ˙ a k ′ ( t ) + 12 a k ( t ) N kk ′ a k ′ ( t )+ m a k ( t ) P kk ′ a k ′ ( t ) i (35) where ˙ a = da/dt , and M , N and P are matrices that areindependent of R ( t ) and are given by M kk ′ = 4 π Z R s drr f k ( r ) f k ′ ( r ) (36) N kk ′ = 4 π Z ∞ R s drr (cid:18) − R s r (cid:19) f ′ k ( r ) f ′ k ′ ( r ) (37) P kk ′ = 4 π Z ∞ R s drr f k ( r ) f k ′ ( r ) . (38)To simplify (35), we can write S = Z dt (cid:20) − B ˙ a k ( t ) M kk ′ ˙ a k ′ ( t ) + a k ( t ) R kk ′ a k ′ ( t ) (cid:21) (39)where the matrix R is defined as R kk ′ = N kk ′ + m P kk ′ . (40)From the action (39), we can find the Hamiltonian and,according to the standard quantization condition,Π k = − i ∂∂a k ( t ) (41)is the momentum operator conjugate to a k ( t ). From (36)and (40), we see that the matrices are Hermitian, there-fore it is possible to use the principle axis transformationto simultaneously diagonalize them (see Sec. 6-2 of [15]for example). Then for a single eigenmode, the Hamilto-nian takes the form H = − m (cid:18) − R s R (cid:19) ∂ ∂b + 12 Rb (42)where m and R denote eigenvalues of M and R , while b isthe amplitude of the eigenmode. Hence the Hamiltonianfor a single eigenmode takes (42) the form of a harmonicoscillator with a time-dependent mass.To find the spectrum of the massive radiation, we mustdetermine the occupation number of the quanta inducedduring the collapse. From the functional Schr¨odinger for-malism, the wave function ψ ( b, t ) must satisfy i ∂ψ∂t = Hψ, or from (42) (cid:20) − m (cid:18) − R s R (cid:19) ∂ ∂b + 12 Rb (cid:21) = i ∂ψ∂t . (43)Re-writing (43) in the standard form we obtain (cid:20) − m ∂ ∂b + m ω ( η ) b (cid:21) ψ ( b, η ) = i ∂ψ ( b, η ) ∂η (44)where ω ( η ) = RmB ≡ ω B (45)and η = Z dt ′ B. (46)Hence, instead of considering a time-dependent mass weconsider the time dependent frequency. In (45) we de-fined ω ≡ R/m . Here we will take the angular frequency ω to be ω = p k + m , (47)which is the angular frequency for a relativistic particle.The exact solution to (44) is given in [16] ψ ( b, η ) = e iα ( η ) (cid:18) mπρ (cid:19) / exp (cid:20) im (cid:18) ρ η ρ + iρ (cid:19) b (cid:21) (48)where ρ η = dρ/dη and ρ is given by the real solution ofthe non-linear auxiliary equation ρ ηη + ω ( η ) ρ = 1 ρ (49)with initial conditions ρ (0) = 1 √ ω , ρ η (0) = 0 . (50)The phase α is given by α ( η ) = − Z η dη ′ ρ ( η ′ ) . (51)Complete information about the radiation in the back-ground of the collapsing shell is contained in the wave-function (48).If we consider that an observer at infinity will regis-ter quanta of the field Φ at different frequencies, (42)tells us that the observer will interpret the wavefunctionof a given mode at some later time in terms of simpleharmonic oscillator states, { ϕ n } , at the final frequency, ω f . From (11) we see that in the near horizon, i.e. latetime limit, B ∼ e − t/R S . Therefore we can write the finalfrequency as ω f = ω e t f / R S . (52)However, this is in terms of the conformal time, since thederivative in (44) is in respect to η not t . (46) tells usthat the frequency in t is B times the frequency in η ,and at time t f , this implies that the observed physicalfrequency is then ω ( t ) = Bω f ≈ e − t f /R s ω f = ω e − t f / R s (53)where the superscript ( t ) on ω refers to the fact that thisfrequency is with respect to t . The initial ( t = 0) vacuumstate for each modes is then simply the simple harmonicoscillator ground state ϕ ( b ) = (cid:16) mω π (cid:17) / e − mω b / (54) where ω = ω ( t = 0). As mentioned previously, thenumber of quanta in each mode can be evaluated by de-composing (48) in terms of the simple harmonic oscilla-tor states and computing the corresponding occupationnumber per mode. The wavefunction for a given modein terms of simple harmonic oscillator basis is given by ψ ( b, t ) = X n c n ( t ) ϕ ( b ) (55)where c n = Z dbϕ ∗ ( b ) ψ ( b, t ) (56)which is an overlap of the wavefunction at some latertime ψ ( b, t ) with the simple harmonic oscillator groundstate (54). The occupation number at the final frequency ω f is then given by N ( t, ω f ) = X n n | c n | . (57)The occupation number in the mode b is then given by(see Appendix B of [7]) N ( t, ω f ) = ω f ρ √ "(cid:18) − ω f ρ (cid:19) + (cid:18) ρ η ω f ρ (cid:19) . (58) B. Numerical Results
In the following plots, we have that the massless case(i.e. m = 0) is given by the blue line, m = 5 R s is givenby the purple line and m = 50 R S is given by the brownline.We have numerically evaluated the spectrum of modeoccupation numbers at finite time and show the resultsin Figure 1 for different values of t/R S . We see first thatthe occupation number of the massive radiation vanishesbelow ω ( t ) = me − t/ R S , as expected from (47). Abovethat, the spectra of the massless and massive cases areidentical, since at this point the spectrum depends on ω ( t ) R S only. It is also clear that the occupation numberis non-thermal for low frequencies. In particular, as in [7],there is no singularity in N at ω = 0 at finite time forthe massless case. We also observe oscillations in N ( ω )for both the massless and massive cases.We are interested in the effect that mass has on thespectrum of the pre-Hawking radiation given off duringthe time of collapse. We therefore plot log( N ) versus kR S in Figure 2 for different fixed values of t/R s . Herewe see that for small values of kR S the occupation num-ber is greater for smaller masses (largest for the masslesscase). However, for larger values of kR S the occupationof each of the masses begin to overlap, as expected from(53), since as kR S increases ω ( t ) becomes dominated by k and the effect of the mass is then unimportant. Theoccupation number of the massive case is obviously sup-pressed when compared to the massless case. We wish Ω H t L R S - - H N L t (cid:144) R S =
7, 8
FIG. 1: ln( N ) versus ω ( t ) R S for t/R S = 7 ,
8. Since ω ( t ) = √ k + m e − t/ R S , for the massive case ω ( t ) has a minimumvalue which is equal to me − t/ R S , as can be seen here. Notethat the occupation for the massless case and the massive caseoverlap each other, since we are plotting versus ω ( t ) , which isthe same for both cases after the minimum value is obtained.The occupation number grows as t/R S increases to determine by how much the massive radiation is sup-pressed.To determine the amount of suppression, we comparethe spectrum of mode numbers versus kR S with the oc-cupation numbers for the massless and massive Planckdistribution, which are given by N P ( ω ) = 1 e βω − ω = | k | for the massless case and ω = √ k + m for the massive case, and where β is the inverse temper-ature. In Figures 2 and 3 we plot ln( N ) versus kR S forfixed values of t/R S . Before we analyze the results weshould note that the value of β in Figure 2 is differentthan that in Figure 3. This is because here we are plot-ting versus kR s not ω ( t ) R S , hence we are ignoring thetime dependence of the frequency, however keeping thetime-dependence of the occupation number. Thereforeeach moment in time will have a different β associatedwith it.Figure 2 shows that for small values of kR S the numeri-cal distributions for the occupation number given in (58)are similar to the expected Planck distribution, shownhere as dotted lines. However, as kR S increases, the twodistributions differ from each other. This is due to thenon-thermal nature of the radiation.Figure 3 shows that for small values of kR S the numer-ical distributions for the occupation number given in (58)are in agreement with the expected Planck distribution,again shown here as dotted lines. This is due to the factthat as t/R S increases, the spectrum becomes more andmore thermal for low frequencies, since we are ignoringthe red-shift of the frequency.From (59) we see that for k << m the occupationnumber is exponentially suppressed by e − βm . Due to the
20 40 60 80 100 120 140 kR S - H N L t (cid:144) R S = FIG. 2: ln( N ) versus kR S for t/R S = 10. The dotted linescorrespond to the expected Planck distributions (59). Forsmall values of kR S the numerical values are similar, howeveras kR S increases the results differ. This is due to the non-thermal feature of the radiation for early times.
20 40 60 80 100 120 140 kR S H N L t (cid:144) R S = FIG. 3: ln( N ) versus kR S for t/R S = 13. The dotted linescorrespond to the expected Planck distributions. Here thenumerical results and expected Planck distributions are inagreement. This is due to the fact that as t/R S increases thespectrum becomes more and more thermal. agreement in Figure 3 between the theoretical expectedresult (59) and the functional Schr¨odinger equation, wecan also conclude that when considering massive radi-ation, the spectrum is exponentially suppressed by themass. This result is in agreement with Hawking’s predic-tion (see [1]). C. Backreaction
We are finally ready to estimate the rate of pre-Hawking evaporation of the collapsing shell through theemission of quanta with mass m . The behavior of themass M of the shell is given by dMdt = 12 G dR S dt = − πR (cid:18) m π R s (cid:19) / me − πmR S . (60)Taking into account (10), with solution (11) in thenear horizon regime (where the pre-Hawking emission ismostly effective), one can estimate t evap ≈ m p πmR S, exp (4 πmR S, ) (61)at mR S, ≫
1. Here R S, is the Schwarzschild radius atthe beginning of the collapse.As we explained in the Section II B, when the radius ofthe shell R becomes sufficiently close to R S , so that onecan use the expression in (11), one naturally expects theperturbation theory in powers of G for the Wheeler-deWitt equation in (19) to break down. This effect is ap-parently stronger than PHR, which is entirely containedin the order G of the expansion corresponding to the ap-proximation of the functional Schr¨odinger equation (notethough that the backreaction of the PHR is not containedin the order G ). This effect is very well known in physicsof black holes, where the effective Unruh temperature ofthe Hawking radiation increases when the observer ap-proaches the event horizon. We shall however assumethat the conservative lower bound for evaporation timein (61) holds in all orders in G , since one expects higherorders of G to describe gravitational interaction (capture)of PHR with the shell as well as gravitational scatteringof quanta of PHR on each other.Rewriting (61) in terms of the initial mass of the shell, M , t evap ∼ R S (cid:18) M P mM (cid:19) / exp (cid:18) πmM M P (cid:19) . (62)We see that for M ≫ M P / πm , t evap ≫ R S (cid:18) M M P (cid:19) ≫ R S (63)The condition M ≫ M P / πm , can be rewritten to placeit more clearly into an astrophysical context: M ≫ − M solar eV m (64)Note that among different global quantum numbers,baryon number B (and lepton number L ) is of the mostimportance. One can naturally expect that the maincontribution to the mass of the collapsing shell comesfrom masses of constituent protons and neutrons. Aslong as the pre-Hawking emission of particles carrying thebaryon number remains suppressed, the shell cannot loseits mass effectively through the pre-Hawking emission ofmassless quanta. For such shells with masses larger than10 kg, the pre-Hawking evaporation time is given by the time scale in (62) rather than that in (28). For shells ofsolar mass 10 kg the time scale involved will be of theorder of exp(10 ) years, apparently much longer thanany physically interesting time scale in the problem. IV. EFFECT OF BARYON AND LEPTONNUMBER VIOLATION
In the previous section, we have shown that the needto emit massive particles in order to radiate away globalquantum numbers, such as the baryon or lepton numbersof the shell (or equally, gauge quantum numbers, such asthe electro-magnetic charge of the shell), have the poten-tial to enormously extend the evaporation timescale ofthe shell. So far, however, we have not included the pos-sibility that physical processes which violate such globalquantum numbers could allow a significant fraction ofthe mass of the shell to be radiated in massless quanta,leaving behind a less massive shell with a higher effectiveHawking temperature.Indeed, the Standard Model contains a mechanism forchanging the baryon and lepton numbers of the shell –non-perturbative electroweak baryon and lepton numberviolation [17–19]. This process, which can, for example,convert 9 quarks into 3 anti-leptons [20], can allow thebaryon number of our shell to relax. However, the processconserves B − L , indeed, it conserves B/ − L e , B/ − L µ and B/ − L τ separately. (Here L i is the lepton numberof the i th generation.) Thus, the minimum mass to whicha shell of baryon number B (and lepton number L ≪ B)can evaporate by the emission of massless quanta is not ∼ Bm B , as one might have guessed, but rather M min ( B ) ≃ B m ν e + m ν µ + m ν τ ) ≃ B i m ν i . (65)In other words M min ≃ M i m ν i m B . (66)We should note, that the electroweak non-perturbativebaryon and lepton number violation is suppressed by theincredible e π/α ≃ e at temperatures below the elec-troweak symmetry breaking scale. Thus in order for itto affect the evaporation of the shell materially, the elec-troweak symmetry must be restored. One might expectthat this could happen in the region just above the sur-face of the shell as it reaches R s . We are going to addressthis issue in more details in the future work.If non-perturbative electroweak baryon number vio-lation does indeed convert the shell’s baryons to anti-neutrinos, and thus reduce the mass by many orders ofmagnitude, neutrino flavor mixing could reduce it stillfurther. Since ν e , ν µ and ν τ are not mass eigenstates, itcould in fact be that M min ≃ M i m ν i m B . (67)where now i enumerates the (three or more) mass eignes-tates. In this case we can rewrite t evap ∼ R S (cid:18) m B M P ( m minν ) M (cid:19) / exp (cid:18) π ( m minν ) M m B M P (cid:19) . (68)In this case, t evap ≫ R S only if M ≫ m B M P π ( m minν ) ≃ M solar (cid:18) eVm minν (cid:19) . (69)Interestingly, with m minν ≃ . V. CONCLUSIONS AND DISCUSSION
As was recently demonstrated in [7], a spherically sym-metric collapsing shell generally loses energy due to pre-Hawking radiation. The spectrum of this radiation isnearly thermal for modes with momentum k ≫ R − S ∼ M P M larger than the inverse Schwarzschild radius of theshell, while the occupation numbers of low momentummodes remain time-dependent until an event horizon isformed. If the time scale of this pre-Hawking evaporationis shorter than the collapse time, we expect this effect toprevent the formation of the event horizon and thereforelead to the resolution of the information loss paradox.The initial pure quantum state of the collapsing shell willevolve into a pure quantum state of the PHR, which con-tains the same amount information as the original state.The question whether such a scenario can be realizedin practice is ultimately related to the hierarchy betweentwo time scales in the problem: the collapse time for theshell and its evaporation time through emission of PHR.As we have argued, while classically the collapse time isinfinite, taking quantum fluctuations of the space-timemakes it finite and in fact rather small: t coll ≈ t + R S , (70)where t is the time necessary for the shell to collapse tothe near horizon regime given in (11), irrelevant for thephysics discussed, since pre-Hawking evaporation processis ineffective at t < t .The collapse time scale in (70) is to be compared tothe total evaporation time for the collapsing shell due toemission of PHR. If one has a “realistic” collapsing shellmade of baryons and leptons, its mass is proportional to (approximately) the total number of baryons composingthe shell and therefore to the total baryon number B ofthe shell. To evaporate the mass of the shell means tocarry this baryon number away to infinity, and the rele-vant evaporation time scale should be associated to theprocess of emitting quanta of massive radiation. As wedemonstrated, the spectrum of radiation is exponentiallysuppressed by the mass m of the emitted quanta, mean-ing that temperature of the black hole must be greaterthan that of the mass for that species to be radiated away.This would then imply that during most of the collapseof the massive domain wall, the radiation given off wouldbe that of massless particles. For example, if a protonwas to be radiated away during the collapse, the processwould be suppressed by a factor of e − m e /T H ∼ e − . In other words, it is difficult to have the collapsing matterevaporate before the trapped surface is created.Since massive radiation is only induced once the tem-perature of the black hole raises above that of the massof the radiation, the first massive particle that wouldbe radiated would be a neutrino, which carries leptonnumber, not baryon number. The lightest baryon, theproton, is at least ten orders of magnitude heavier, andthus its emission is horrendously suppressed. The shellcan therefore lose its baryon number (and its mass) onlyif baryon number is violated, say by conversion into lep-tons. Such processes exist within the standard model,but are suppressed by approximately e − , unless theelectroweak symmetry is restored. Moreover, such pro-cesses would leave a third of the B − L to be carried by theheaviest neutrino, which may be too heavy to experienceunsuppressed PHR. Thus, within the Standard Model,effective radiation of the shell’s mass may rely on allof electroweak symmetry restoration, non-perturbativebaryon violation, neutrino oscillations, and PHR of neu-trinos happening sufficiently rapidly to evade horizon for-mation. Grand unified theories contain B -violating op-erators, and B (and even B − L ) could be violated bygravitational operators above the Planck scale and shortcircuit some of this complication, but again, the ratesof such processes in the extremely thin range of radiusthat is at GUT or Planck temperatures would need tobe studied. Moreover, the radiation of the shell masswould proceed not so much by PHR as by the emissionof high energy particles as a byproduct of baryon numberviolating interactions at high energy/temperature.Finally, let us note that the region where the pair pro-duction happens is located within vicinity of the shell(where the Schwarzschild metric describing the exteriorof the shell is smoothly connected to the Minkowskispace-time in the interior of the shell), and while one par-ticle in the pair is escaped to infinity, the other particlefrom the pair enters the interior of the shell. Apparently,when the pre-Hawking evaporation process becomes ef-fective, the number density of particles in the interior ofthe shell grows, and eventually the original approxima-0tion of Minkowski space-time in (3) inside the shell breaksdown. At this point, as one might naturally expect, (a)the space-time inside the shell will be described by theclosed FRW metric and will collapse in finite time, and(b) density fluctuations in the gas of particles of PHRinside the shell will grow and collapse. This scenario ex-plains how exactly the formation of the trapped surfaceproceeds in the collapse of a spherically symmetric shell. Acknowledgments
The authors would like to thank A. Tolley for discus-sions. EG and DP were supported by NASA ATP grantto Case Western Reserve University. EG, DP and GDSwere supported by a grant from the US DOE to the the-ory group at CWRU. [1] S. W. Hawking, Commun. Math. Phys. , 199 (1975)[Erratum-ibid. , 206 (1976)].[2] J.D. Bekenstein, Black holes and the second law, Lett.Nuovo Cim. 4 (1972) 737; Black holes and entropy, Phys.Rev. D 7 (1973) 2333[3] S. W. Hawking, Phys. Rev. D14 (1976) 2460.[4] J. Preskill, arXiv:hep-th/9209058.[5] S. D. Mathur, Lect. Notes Phys. , 3 (2009)[arXiv:0803.2030 [hep-th]];S. D. Mathur, Class. Quant. Grav. , 224001 (2009)[arXiv:0909.1038 [hep-th]].[6] S. B. Giddings, arXiv:0911.3395 [hep-th].[7] T. Vachaspati, D. Stojkovic and L. M. Krauss, Phys. Rev.D , 024005 (2007); T. Vachaspati and D. Stojkovic,gr-qc/0701096 (2007).[8] E. Greenwood and D. Stojkovic, JHEP
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