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Toy Models for Black Hole Evaporation∗

arXiv:hep-th/9209113v2 11 Nov 1992UCSBTH-92-36hep-th/9209113Toy Models for Black Hole Evaporation∗Steven B. Giddings†Department of PhysicsUniversity of CaliforniaSanta Barbara, CA 93106-9530AbstractThese notes first present a brief summary of the puzzle of information loss to blackholes, of its proposed resolutions, and of the flaws in the proposed resolutions.Therefollows a review of recent attempts to attack this problem, and other issues in black holephysics, using two-dimensional dilaton gravity theories as toy models. These toy modelscontain collapsing black holes and have for the first time enabled an explicit semiclassicaltreatment of the backreaction of the Hawking radiation on the geometry of an evaporatingblack hole.

However, a complete answer to the information conundrum seems to requirephysics beyond the semiclassical approximation. Preliminary attempts to make progressin this direction, using connections to conformal field theory, are described.

*To appear in the proceedings of the International Workshop of Theoretical Physics, 6thSession, String Quantum Gravity and Physics at the Planck Energy Scale, 21 – 28 June 1992,Erice, Italy.† Email addresses: giddings@denali.physics.ucsb.edu, steve@voodoo.bitnet.

Since the discovery of black holes, physicists have been faced with the possibilitythat they engender a breakdown of predictability. At the classical level this breakdownarises at the singularity.

Classically we do not know how to evolve past it. Inclusion ofquantum effects may serve as a remedy, allowing predictable evolution, by smoothing outthe singularities of general relativity.

However, as suggested by Hawking [1,2], quantumeffects also present another sharp challenge to predictability through the mechanism ofHawking evaporation.To see this, consider a pure quantum state describing an infalling matter distribution.If this matter collapses to form a black hole, it will subsequently emit Hawking radiation. InHawking’s approximation where the backreaction of the emitted radiation on the geometryis neglected, the radiation is thermal and is described by a mixed quantum state.

Thissuggests that once the black hole evaporates the initial pure state has been converted toa final mixed state; information has been lost, and unitarity has been violated. Hawkingproposed that this represents a new and fundamental type of unpredictability inherent inquantum gravity.Beyond any prejudice that quantum gravity shouldn’t violate unitarity, there arepotential problems with this scenario.

In particular, as argued in [3-5], na¨ıve attempts toformulate unitarity violating dynamics typically run afoul of essential principles such asenergy conservation. We are therefore motivated to look for other possible resolutions tothe problem of what happens to information that falls into a black hole.There have been various proposals for resolving the black hole information problem,1but each of them appears to have flaws.

A list of these, in ascending order of speculativecontent, and together with objections, is as follows:1. Correcting Hawking’s calculation by including the backreaction renders the final statepure; the information escapes in the Hawking radiation.2Objection: this wouldappear to imply that either all of the information has been extracted from the infallingmatter by the time it crosses the horizon, or that information propagates acausallyfrom behind the horizon to outside.2.

The information is released in the last burst of radiation as the black hole evaporatesto the Planck scale and quantum-gravitational and backreaction effects dominate.Objection: Since the initial black hole could have been arbitrarily massive, it must be1 For more comprehensive reviews see [6-8].2 This alternative has for example been advocated in [9,10].1

possible for the remaining Planck scale energy to carry offan arbitrarily large amountof information corresponding to all of the possible black hole initial states. A largeamount of information can be transmitted using a small amount of energy only overa long period of time, e.g., through emission of many very soft photons.

This impliesthe next proposal[2,11,7].3. The black hole leaves behind a long-lived remnant with Planck-sized mass.Thisremnant must have infinitely many states to allow it to carry the unbounded amountof information that could have been present in the initial state.3Objection: aninfinite spectrum of states with Planck-size masses wreaks havoc with loop calculationsand with production probabilities both in thermodynamics and in background fields.In particular, such a spectrum implies infinite production of these particles in theHawking radiation from arbitrarily large black holes, and likewise appears to implyinfinite production from a thermal ensemble at any given temperature.

The resultinginstabilities are disastrous.44. Baby universes form and carry away the information that falls down the black hole,and thus unitarity, while apparently violated in our Universe, is restored for the systemincluding the baby universes[16].

Objection: in a different context, it has been argued[17,18] that wormholes simply shift coupling constants and don’t lead to such apparentviolations of unitarity.5. Information emerges from the black hole via a previously unsuspected mechanismrather than through small corrections to the Hawking radiation.

Such a mechanism issuggested both by the failure of other attempts to resolve the information problem andby arguments that there should be upper bounds on information content that arisefrom Planck scale physics[7].Objection: this proposal appears to require acausalbehavior behind the horizon.There are other variations on these basic possibilities. The objections are not iron-clad, and may have loopholes.

One way (and perhaps the only way) to actually solve theinformation problem is to gain control of backreaction and quantum gravity effects. Thisis a difficult task in four dimensions, and it behooves us to search for simple toy models togain more intuition.3 In one possible realization of this proposal[12-14] the infinite number of states arises fromexcitations of an infinite “internal” volume of the black hole.4 For an attempt to evade these see [15].2

One such toy model[12] is two-dimensional dilaton gravity, described by the actionS = 12πZd2x √−g"e−2φ R + 4(∇φ)2 + 4λ2−12NXi=1(∇fi)2#. (1)Here φ is the dilaton, 4λ2 the cosmological constant, and the fi are minimally coupledmatter fields.

Note for future reference that since the gravitational part of the action ismultiplied by e−2φ, the quantity eφ plays the role of the gravitational coupling constant.This toy model has several virtues. First, it is perturbatively renormalizable by powercounting; the only dimensionful coupling constant is λ. Secondly, it is completely solubleat the classical level.

Among these solutions are black holes, and these black holes Hawkingradiate f-particles. Finally, this model is the low energy effective theory for certain typesof four- and five-dimensional black holes, and this provides a direct application to higher-dimensional physics.Before pursuing the former points, let us recall the connection to higher-dimensions.5The low energy action for string theory is of the formS =Zd10x √−g e−2φR + 4(∇φ)2 −12F 2µν + · · · + O(α′)(2)where Fµν is the electromagnetic field strength and where terms involving other fieldsare neglected, as are higher-dimension operators.

This theory is known [19,20] to havemagnetically charged black hole solutions,ds2 = −1 −r+r1 −r−rdt2 +dr21 −r+r 1 −r−r + r2dΩ22 + ds26e−2φ = e−2φ01 −r−rFµν = Q ǫ(2)µν . (3)Here r+, r−, and φ0 are constants, ds26 denotes your favorite 6-dimensional string com-pactification, Q is the magnetic charge, and dΩ22, ǫ(2)µν are the line element and Levi-Civitatensor on the two-sphere.This solution has a causal structure identical to that of Schwarzschild (see Fig.

1),with a horizon at r = r+ and a singularity at r = r−. However, there is a crucial differencebetween their geometries.

Consider the slice S shown in Fig. 1; in Schwarzschild, this5 For a more complete explanation see [14].3

Fig. 1: The Penrose diagram for the magnetically charged dilatonic blackhole in four dimensions.

Also shown is a constant-time slice S through thegeometry.spatial slice gives the Einstein-Rosen bridge. In the present case there is also a throatconnecting two asymptotically flat regions (Fig.

2), but in the extremal limit M →Q thethroat becomes infinitely long. In this limit an observer in the asymptotic region sees thehorizon and singularity disappear to infinity; likewise an observer fixed near the horizonsees the asymptotic region recede to infinity.6 For the latter observer the universe is aninfinite tube terminating in a black hole; this solution takes the formds2 = ds22DBH(r, t) + Q2dΩ22e2φ = e2φ2DBH(r)Fµν = Q ǫ(2)µν .

(4)Here ds22DBH and φ2DBH are the metric and dilaton for the two dimensional black hole instring theory that was found by Mandal, Sengupta, and Wadia [21] and Witten [22]; wewill see their explicit forms shortly. The important point is that the solution (4) is a directproduct of two two-dimensional solutions.

The second of these is the round two-spherethreaded by a magnetic flux. A similar construction holds for five-dimensional black holes.6 At first sight one might think that the extremal Reissner-Nordstrom black hole has the sameproperty, since its spatial geometry also has an infinite throat.

However, the horizon does notbecome causally disconnected from the rest of spacetime as a result of the rapid falloffof g00 alongthe throat.4

Fig. 2: Pictured is the spatial geometry of the right half of the slice S ofFig.

1.The mass of the two-dimensional black hole in (4) depends on the asymptotic valueof the dilaton, φ0. If the latter is scaled to −∞as the extremal limit is taken, then themass can be arranged to be finite.

Conversely, if the asymptotic dilaton is fixed, then themass is zero and the resulting two-dimensional solution is the vacuum.With the dilaton (i.e. the coupling) fixed at infinity, and near the extremal limit,the solution far down the throat is closely approximated by (4) with a small mass.

Atextremality, and as seen by the asymptotic observer, the horizon is infinitely far down thethroat and the r, t solution far down the throat is the vacuum. Let us now consider low-energy scattering of particles from the extremal black hole.

Very low energy excitationsthat penetrate into the throat will not be able to excite the angular degrees of freedom,which have a threshold ∼1/Q. On the other hand, s-wave excitations may penetrate thethroat and raise the mass of the black hole above extremality.

This corresponds to raisingthe mass of the two-dimensional black hole above zero.The excess mass will later beemitted in Hawking radiation, corresponding to evaporation of the two-dimensional solu-tion back to zero mass. This provides a direct relationship between low-energy scatteringby near-extremal dilatonic black holes and formation and evaporation of two-dimensionalblack holes.

The low-energy effective theory describing such excitations is obtained bydropping the angular dependence in (2), and one obtains the action (1).77 To describe excitations that can actually penetrate the throat, one must add matter termsto the action (2). This is further described in [14,23].5

To investigate the two-dimensional problem we return to the action (1). Although thegeneral solution is easily found [12], we focus on some special cases; units are chosen sothat λ = 1.

First are the vacuum solutions, with fi ≡0:ds2 = −dx+dx−M −x+x−e−2φ = M −x+x−(5)where x± = x0 ± x1 are light-cone coordinates and M is an arbitrary parameter. Thechange of coordinates x+ = eσ+ , x−= −e−σ−(with σ± = τ ± σ) givesds2 = −dσ+dσ−1 + Meσ−−σ+φ = −12ℓnM + eσ+−σ−.

(6)For M = 0 the metric is flat, and this solution is identified as the ground state of thetheory. Note that the dilaton is thenφ = −σ;(7)the M = 0 solution is therefore called the linear dilaton vacuum.

For M > 0 one recoversthe two-dimensional black hole of [21,22]. The causal structure of this black hole is identicalto that of Schwarzschild; its Penrose diagram is given by Fig.

3. Solutions with M < 0have naked singularities.Fig.

3: The Penrose diagram for the two-dimensional eternal black hole.6

Black hole formation occurs when one allows matter to fall into the linear dilatonvacuum. For example, a general left-moving lump of classical matter is given byf = F(x+) ,(8)for an arbitrary function F(x+), and has stress tensorT++ = 12 (∂+F)2 .

(9)An F(x+) that vanishes outside x+f > x+ > x+i , or equivalently σ+f > σ+ > σ+i , cor-responding to a lump of finite width, yields a Penrose diagram as in Fig. 4: the matter“collapses” to form a black hole.

The metric for σ+ < σ+i isds2 = −dσ+dσ−(10)and for σ+ > σ+f isds2 = −dσ+dσ−1 + M eσ−−σ+ −∆eσ−. (11)Here the constants M and ∆are moments of the matter distribution,M =Z σ+fσ+idσ+T++(σ+) , ∆=Z σ+fσ+idσ+e−σ+T++(σ+) .

(12)The change of coordinatesξ+ = σ+ , ξ−= −ℓnhe−σ−−∆i(13)returns (11) to the asymptotically flat form,ds2 = −dξ+dξ−1 + M eξ−−ξ+ . (14)Hawking radiation arises from evolution of positive frequency incoming states to bothpositive and negative frequency outgoing states.8 In a fixed background the left and rightmoving f-quanta are decoupled, and in considering the Hawking radiation we focus on theright-movers.

The positive frequency modes in the respective regions areuω =1√2ω e−iωσ−(in)vω =1√2ω e−iωξ−(out)(15)8 For more details, see e.g., [24,25].7

Fig. 4: The Penrose diagram for an “infalling” lump of classical matter.

Alsoindicated are the “in” and “out” regions for right movers.and they are related by the Fourier transform,vωξ−(σ−)=∞Z0dω′ αωω′uω′(σ−) + βωω′u∗ω′(σ−). (16)The Fourier coefficients αωω′, βωω′ give the Bogoliubov transformation, and determine thespectrum of Hawking radiation.

For example, it is easily shown that the expectation valueof the out number operator in the in vacuum is given byin0N outω 0in =Z ∞0dω′ |βωω′|2 ;(17)nonvanishing βωω′ therefore implies Hawking radiation.In the present case the Bogoliubov coefficients can be evaluated exactly [25] (contrastfour-dimensional black holes! ); for example,βωω′ = 12πrω′ω ∆iωB (−iω −iω′, 1 + iω) .

(18)From this one can derive [25] the outgoing stress tensor,DT f−−E= 148"1 −11 + ∆eξ−2#(19)8

which, after an initial transitory period, describes a thermal flux of Hawking radiation.Next we wish to investigate the backreaction of the Hawking radiation on the geometry.It is most easily discussed by considering the vacuum functional integral,ZDgDφ eiS[φ,g]ZDgf e−i4πRd2x √−gNPi=1(∇fi)2. (20)Here we have split the action into the gravitational part and the matter part.

The subscripton the matter measure indicates that this functional measure is induced from the functionalmetric(δf1, δf2)g =Zd2x √−g δf1 δf2(21)in the usual fashion. The matter functional integral has been extensively studied in thestring literature and elsewhere, and givesZDgf e−i4πRd2x √−gNPi=1(∇fi)2= eiNSPL(22)whereSP L = −196πZd2x1√−gZd2x2√−g R(x1)−1(x1, x2)R(x2)(23)is the Polyakov-Liouville action.

(Here−1 denotes the Green function for thed’Alembertian.) Although this action is nonlocal, it appears local in conformal gauge,ds2 = −e2ρdx+dx−:SP L =124πZd2x (∇ρ)2 .

(24)Gravitational dynamics, including quantum effects of the matter, therefore follow fromthe functional integralZDgDφ eiS[g,φ]+iNSPL[g] . (25)One way to see the gravitational effects of the matter is to compute the quantum stresstensor,N 2π√−gδSP Lδgµν =T fµν.

(26)One finds either directly from (23), or equivalently from the well-known trace anomaly,DT f+−E= −N96R = −N12 ∂+∂−ρ . (27)9

The other components of the stress tensor can also be found by varying (23), although itis simpler to integrate the conservation law ∇µ⟨T fµν⟩= 0 using (27). This givesDT f−−E= −N12(∂−ρ)2 −∂2−ρ + t−(x−)(28)where t−(x−) is an integration function that must be fixed by boundary conditions.

(Equiv-alently it arises from the ambiguity in defining the Green function in (23)).Computation of ⟨T f−−⟩from this expression and (11)yields agreement with (19),confirming the relationship [26] between the conformal anomaly and Hawking radiation.We conclude that SP L incorporates both the Hawking radiation, and the effects of itsbackreaction on the geometry.What, then, are these effects? To begin, let’s determine when the backreaction due tothe f-fields has a substantial effect on the geometry.

This can be estimated by asking whenthe Hawking radiation (uncorrected by backreaction effects) has carried away a substantialfraction of the mass of the black hole,Zdξ−DT f−−E∼M . (29)It is straightforward to see that for large M this occurs at the retarded time x−evap ≃−∆, near the horizon, and where the dilaton at the trailing edge of the incoming matterdistribution satisfiese2φ ∼1M .

(30)Therefore if the mass of the black hole is taken to be large (and also x+i ∆≫1), the evap-oration process takes place at weak coupling. This helps us in separating the backreactionfrom quantum-gravitational effects.Indeed, the resulting weak-coupling can be used to justify a semi-classical analysis ofthe functional integral (25), via the classical equations arising from the actionSSC = 12πZd2x e−2φ −2ρ + 4(∇φ)2 + 4λ2+ N24πZd2x(∇ρ)2(31)where we use the conformal-gauge result R = −2ρ.

In order to do this we must takethe number of matter fields N to be large so that the second term dominates the otherquantum corrections to the dynamics, and can be treated on the same footing as the firstterm.10

Because the correction term in (31) is quadratic in ρ, and ρ = 0 for the linear dilatonvacuum, it remains a solution to the semiclassical theory. However, the theory is no longerexactly soluble and, in fact, other solutions are difficult to find.

We have, nonetheless,gained considerable insight into the structure of the solutions from numerical and generalarguments [27-30].As an example, from the resulting time-independent equations one can numericallyfind static solutions[28-30] that correspond to a black hole in equilibrium with an influx ofradiation that precisely balances the outward Hawking flux.In the dynamic case of a black hole formed from collapsing matter, one can arguethat an apparent horizon, determined by the condition (∇φ)2 = 0, forms and recedes asthe black hole evaporates. A surprise is that behind this apparent horizon is a singularityof the semiclassical equations[27,13].

The reason this is surprising is that it is distinct inbehavior from the original classical singularity; in particular, it occurs wheree2φ = e2φcr = 12N . (32)It is therefore (for large N) not at large eφ as was the classical singularity.

Mathematicallythis singularity arises as a result of the vanishing of an eigenvalue of the kinetic operator in(31); this signals a breakdown of the semiclassical approximation. This breakdown meansthat solutions of the equations following from (31) should really not be trusted for valuesof the dilaton φ ≥φcr −ǫ, for some small (N-dependent) ǫ.The resulting geometry is pictured in Fig.

5, Fig. 6.

Shown are both the apparenthorizon, and the effective horizon. The latter will in general be defined as the boundaryof the region from which future-directed causal curves can escape to null infinity with-out encountering Planck-scale physics; it is thus the boundary of the region where wecan make predictions using an effective field theory valid below the Planck scale.

In thepresent context “Planck scale” physics is identified with physics beyond the validity of thesemiclassical approximation. Eventually the line with φ = φcr −ǫ crosses the horizons.

Wecannot determine the physical behavior in the future light cone (or “shadow”) of this pointsince it depends on the presently ill-understood physics at φ ≥φcr −ǫ. For φ < φcr −ǫ,the solution asymptotes back to the linear dilaton vacuum.This semiclassical picture, although incomplete, may nonetheless give some cluesabout the information problem.

In particular, working order-by-order in 1/N, it is prob-able that one can construct an argument[25] that information does not come out in the11

Fig. 5: A Kruskal diagram for the evaporating two-dimensional black hole.Q denotes the line along which φ = φcr −ǫ; beyond this line a full quantumdescription of the collapse is presumably needed to make predictions.Hawking radiation before one reaches the shadow of φ = φcr −ǫ.

This is analogous to stat-ing that the information doesn’t come out of four-dimensional black holes until they reachthe Planck scale. It may therefore, in the present context, rule out the most conventionalproposed resolution, resolution 1), of the black hole information conundrum.On the other hand, semiclassical predictability has failed and we still can’t say whatdoes happen to information.

To proceed we must go beyond the large-N approximationand investigate the full quantum theory.Quantum dilaton gravity is most easily treated by gauge-fixing the metric to the formgµν = e2ρˆgµν(33)where ˆgµν is a fixed background metric. Although we have argued that dilaton gravityis renormalizable, the fields ρ and φ are dimensionless so in fact an infinite number ofcounterterms can occur.

The general parity invariant action is of the formS = −12πZd2xp−ˆgGMN(XP )ˆ∇XM · ∇XN + 12Φ(XP) ˆR + T(XP )(34)12

Fig. 6: A Penrose diagram for the evaporating two-dimensional black hole.Prediction of physics in the causal future of the line Q requires a quantumtreatment of dilaton gravity.

In particular, one cannot at present determinethe final fate of the black hole.where XP = (ρ, φ). Here GMN(XP), Φ(XP ), and T(XP ) are arbitrary functions, and weuse sigma-model notation.In quantizing dilaton gravity there are several physical restrictions on the generalaction (34).

The most obvious of these are:1) The theory should depend on ˆg, ρ only through the combination in (33); that is,the theory should be invariant under the background transformationˆgµν →e2ωˆgµνρ →ρ −ω . (35)This condition is on-shell equivalent to invariance under conformal rescaling ofthe background metric ˆg, which implies that the sigma-model β-functions mustvanish.

If we momentarily reinstate Planck’s constant, and work to leading order13

in ¯h, this implies∇MΦ∇MT −4T −¯h2T = 0∇M∇NΦ + ¯h2 RMN = 0(∇Φ)2 −¯h2Φ + (N −24)¯h3= 0 . (36)Off-shell these must be supplemented by the condition that the tangent vectorV M to the ρ direction satisfy VM = ∇MΦ/2.2) The theory must agree with dilaton gravity at weak coupling:GMN →eφ→0GcℓMN =−4e−2φ2e−2φ2e−2φ0Φ →Φcℓ= −2e−2φT →T cℓ= −4λ2e−2φ(37)Furthermore, outside the large-N approximation we must worry about the role of theghosts.

If the measures for the ghost and (ρ, φ) functional integrals are defined using themetric g, as in (21), then this results in the replacement N →N −24 in (31). This impliesthe nonsensical result that for N < 24, black holes accrete mass by Hawking radiatingnegative energy in ghosts!

The problem is easily resolved by instead using the metric e2φgto regulate the functional measures [31,32]. There follows a third condition:3) At weak coupling, subleading counterterms of the formGMN →eφ→0GcℓMN + ¯h2−2−224−N12+ · · ·Φ →Φcℓ+ ¯h24 −N6ρ −4¯hφ + · · ·(38)should appear in GMN, Φ.Finally there may be other physical constraints; one such restriction is4) The theory should have a sensible ground state.Writing down the full β-function equations (36), let alone solving them, is no smalltask.

One promising approach has been advocated in refs. [33-36,32].

As is easily seen, theleading order metric given in (38) is flat, and therefore trivially obeys the leading-orderβ-function equations. Furthermore, if T = 0 this theory is an exact CFT, that is, an exactsolution of the β-function equations.14

One can then identify the tachyon as the operator of conformal dimension (1,1) thatagrees with T cℓto leading order in eφ. The theory with T ̸= 0 is obtained by perturbingthe exact flat theory with this operator.

This is similar to steps used to define Liouvilletheory [37], and should yield an exact solution to the β-function equations.Although the resulting theory satisfies criteria 1–3, it does not satisfy criterion 4.It can be shown that there are regular solutions with mass unbounded from below [32].Hawking radiation in these theories does not shut off[36,35], and black holes appear toradiate to infinite negative mass. The necessary modifications for a stable theory are notobvious; one attempt to stabilize such models is by applying suitable boundary conditionsat the line where φ = φcr[38,39].Despite these facts, the general approach of attempting to identify exact conformalfield theories that represent evaporating black holes is worthy of pursuit; perhaps othermore realistic examples can be constructed.

One is still, however, left with the feelingthat uniqueness is lacking. Consideration of supersymmetric theories may provide suf-ficient uniqueness and solve the problem of negative mass.

A different tack is to viewthe problem of the non-uniqueness of quantum dilaton gravity as similar to that of four-dimensional gravity.In the latter case, we expect string theory to provide an escapefrom non-renormalizability. Perhaps two-dimensional dilaton gravity is best treated as thelow-energy limit of string theory as well.To conclude, we have succeeded in qualitatively understanding two-dimensional blackhole formation and evaporation until quantum effects become strong; this is analogousto understanding four-dimensional black holes up to the Planck regime.

Furthermore, wemay likely rule out the most conservative proposed resolution to the black hole informationproblem. This is potentially very interesting.

However, a solution to the information co-nundrum is still beyond the horizon. We haven’t seen quantum restoration of predictability,and probably won’t until we understand quantization of the family of theories of quantumdilaton gravity.

Although this is a challenge, it is a good toy problem to develop techniquesfor higher-dimensions: quantization of dilaton gravity should be an excellent warmup forunderstanding quantum gravity in four dimensions.AcknowledgementsI wish to thank my collaborators B. Birnir, C. Callan, J. Harvey, W. Nelson, and A.Strominger for a stimulating collaboration. I have also benefitted from discussions with S.Hawking, J. Preskill, L. Susskind, and L. Thorlacius.

This work was supported in part byDOE grant DE-FG03-91ER40168 and NSF PYI grant PHY-9157463.15

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+x +0oooo−−xx+x if−SingularityLinearHorizonInfalling matter0dilaton vacuumBlack hole"in""out"

x+ix f+Apparent horizonInfalling matter????? ?QEffective horizon??

?Q

r=r+r=r+r=r−r=r−S

+x+x ifLinearInfalling matterdilaton vacuumApparent horizon???? ?QEffective horizonLDV????

?Q

AF regionHorizonThroatMouth(r=r )+

x =0+x =0−x x =M−+x x =M−+

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